WEBVTT 00:00:00.000 --> 00:00:03.936 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:03.936 --> 00:00:12.030 align:middle line:90% 00:00:12.030 --> 00:00:14.400 align:middle line:84% GIAN PAOLO BERETTA: Good afternoon. 00:00:14.400 --> 00:00:26.110 align:middle line:84% OK, so the last time, we defined diffusive and convective fluxes. 00:00:26.110 --> 00:00:35.230 align:middle line:84% I want to review this definition with some graphics. 00:00:35.230 --> 00:00:42.260 align:middle line:84% The idea is that we have a continuum 00:00:42.260 --> 00:00:47.930 align:middle line:84% with all the assumptions that go under this description. 00:00:47.930 --> 00:00:53.910 align:middle line:84% And suppose that anywhere, somewhere inside your continuum, 00:00:53.910 --> 00:01:08.040 align:middle line:84% you set up a small surface area with an oriented normal, n. 00:01:08.040 --> 00:01:13.140 align:middle line:84% And then you look at what properties-- 00:01:13.140 --> 00:01:19.500 align:middle line:84% and here we are talking about the extensive properties, those 00:01:19.500 --> 00:01:23.040 align:middle line:84% that we said that can be described as integrals 00:01:23.040 --> 00:01:32.710 align:middle line:84% over volume of densities of that property like energy, entropy, 00:01:32.710 --> 00:01:36.640 align:middle line:90% mass, momentum. 00:01:36.640 --> 00:01:48.860 align:middle line:84% So this is how much of that property crosses that area 00:01:48.860 --> 00:01:51.450 align:middle line:90% and with which speed. 00:01:51.450 --> 00:01:55.670 align:middle line:84% So how much it transfers, it goes across, 00:01:55.670 --> 00:02:01.250 align:middle line:84% divided by the area of this surface that we call d omega 00:02:01.250 --> 00:02:10.780 align:middle line:84% and divided by the time interval during which we measure 00:02:10.780 --> 00:02:12.850 align:middle line:90% the amount that crosses. 00:02:12.850 --> 00:02:16.965 align:middle line:90% That defines the flux. 00:02:16.965 --> 00:02:22.520 align:middle line:90% 00:02:22.520 --> 00:02:28.220 align:middle line:84% So let's consider first the flux, the measurement, 00:02:28.220 --> 00:02:34.370 align:middle line:84% in terms of the reference frame of the laboratory. 00:02:34.370 --> 00:02:38.470 align:middle line:90% And that we call it J zero. 00:02:38.470 --> 00:02:42.280 align:middle line:84% I also call it Lagrangian, but maybe the term Lagrangian 00:02:42.280 --> 00:02:45.080 align:middle line:90% is not really appropriate. 00:02:45.080 --> 00:02:49.150 align:middle line:84% But in any case, it's the flux with respect to the laboratory 00:02:49.150 --> 00:02:51.580 align:middle line:90% reference frame. 00:02:51.580 --> 00:02:57.110 align:middle line:84% And once I have done that and I know the density, 00:02:57.110 --> 00:03:00.910 align:middle line:84% which means the property per unit volume 00:03:00.910 --> 00:03:08.030 align:middle line:84% that prevails in that position, that defines also 00:03:08.030 --> 00:03:12.680 align:middle line:84% a velocity, which is the velocity of transport 00:03:12.680 --> 00:03:19.690 align:middle line:84% of that property which is this vector here. 00:03:19.690 --> 00:03:23.070 align:middle line:84% So multiply-- I'm sorry-- the long vector is the flux. 00:03:23.070 --> 00:03:24.910 align:middle line:84% The short vector is the velocity. 00:03:24.910 --> 00:03:28.490 align:middle line:90% 00:03:28.490 --> 00:03:31.100 align:middle line:90% One of the properties is mass. 00:03:31.100 --> 00:03:33.780 align:middle line:84% So if I do that same measurement for mass, 00:03:33.780 --> 00:03:42.530 align:middle line:84% I get the flux of mass which is what we call momentum, momentum 00:03:42.530 --> 00:03:43.310 align:middle line:90% density. 00:03:43.310 --> 00:03:48.990 align:middle line:84% And the velocity associated with that flux, 00:03:48.990 --> 00:03:51.890 align:middle line:84% we call it barycentric velocity of the flow. 00:03:51.890 --> 00:03:56.920 align:middle line:84% It's the velocity of fluid mechanics. 00:03:56.920 --> 00:04:03.220 align:middle line:90% Then we define also a-- 00:04:03.220 --> 00:04:05.260 align:middle line:84% for property A, we go back to property A, 00:04:05.260 --> 00:04:09.430 align:middle line:84% and we define a relative velocity. 00:04:09.430 --> 00:04:14.770 align:middle line:84% So the difference between vA and the velocity 00:04:14.770 --> 00:04:22.480 align:middle line:84% with which the fluid parcels cross that surface, which 00:04:22.480 --> 00:04:25.660 align:middle line:90% is the barycentric velocity. 00:04:25.660 --> 00:04:28.930 align:middle line:84% So the difference you compute, since these 00:04:28.930 --> 00:04:32.810 align:middle line:84% are vectors this way, like the typical vector construction. 00:04:32.810 --> 00:04:36.050 align:middle line:84% So this blue vector is the difference between the two. 00:04:36.050 --> 00:04:39.200 align:middle line:90% It's the relative velocity. 00:04:39.200 --> 00:04:41.530 align:middle line:84% It's the velocity of transport of property A 00:04:41.530 --> 00:04:45.040 align:middle line:84% relative to the barycentric velocity. 00:04:45.040 --> 00:04:48.460 align:middle line:84% And then you multiply by the same factor 00:04:48.460 --> 00:04:52.060 align:middle line:84% that goes between the short and the long vector here. 00:04:52.060 --> 00:04:56.920 align:middle line:84% So the density of property A, and that gives you 00:04:56.920 --> 00:05:06.930 align:middle line:84% the diffusive flux of property A. The difference between J zero 00:05:06.930 --> 00:05:10.330 align:middle line:84% and the diffusive flux, is the convective flux, 00:05:10.330 --> 00:05:13.350 align:middle line:84% which is this vector, red vector, which, 00:05:13.350 --> 00:05:17.340 align:middle line:84% of course, by construction is directed in the same direction 00:05:17.340 --> 00:05:24.440 align:middle line:90% as the barycentric velocity. 00:05:24.440 --> 00:05:27.950 align:middle line:84% And so you can take it also as the barycentric velocity 00:05:27.950 --> 00:05:32.005 align:middle line:84% times the density of property A. So it's this vector here. 00:05:32.005 --> 00:05:46.275 align:middle line:90% 00:05:46.275 --> 00:05:51.170 align:middle line:84% Then we have defined this extension 00:05:51.170 --> 00:05:56.510 align:middle line:84% or generalization of the idea of a heat interaction to when-- 00:05:56.510 --> 00:06:00.470 align:middle line:84% in the same way as we did heat, we 00:06:00.470 --> 00:06:07.940 align:middle line:84% approached the interactions between systems that are almost 00:06:07.940 --> 00:06:12.710 align:middle line:84% in mutual equilibrium, so exactly or almost exactly 00:06:12.710 --> 00:06:16.220 align:middle line:84% the same temperature and the same chemical potentials. 00:06:16.220 --> 00:06:20.270 align:middle line:84% And they exchange not only energy but also 00:06:20.270 --> 00:06:23.730 align:middle line:90% amounts of constituents. 00:06:23.730 --> 00:06:27.270 align:middle line:84% And the relation Q over T for heat generalizes 00:06:27.270 --> 00:06:32.710 align:middle line:84% to this relation between the flux of entropy, 00:06:32.710 --> 00:06:39.130 align:middle line:84% the flux of energy, the flux of constituents. 00:06:39.130 --> 00:06:46.330 align:middle line:84% Here we are introducing also these variables 00:06:46.330 --> 00:06:49.180 align:middle line:90% that are shorthand. 00:06:49.180 --> 00:06:52.970 align:middle line:84% And instead of carrying all the times one over the temperature, 00:06:52.970 --> 00:06:55.620 align:middle line:84% we just call it tau, which I know 00:06:55.620 --> 00:06:58.500 align:middle line:84% it's a bit unfortunate nomenclature 00:06:58.500 --> 00:07:01.680 align:middle line:84% because then we will use tau double underline also 00:07:01.680 --> 00:07:02.890 align:middle line:90% for the stress tensor. 00:07:02.890 --> 00:07:07.960 align:middle line:84% But that double underline will keep the difference. 00:07:07.960 --> 00:07:13.260 align:middle line:84% And also lambda-- lambda j denotes 00:07:13.260 --> 00:07:18.420 align:middle line:84% minus the potential over T So we use lambda or lambda tot 00:07:18.420 --> 00:07:23.610 align:middle line:84% for the chemical potential or the total potential. 00:07:23.610 --> 00:07:34.880 align:middle line:84% And so the relation between energy, entropy, and n 00:07:34.880 --> 00:07:38.590 align:middle line:90% can be written this way. 00:07:38.590 --> 00:07:41.000 align:middle line:84% So once we have this relation between vectors, 00:07:41.000 --> 00:07:43.450 align:middle line:84% we are going to use it, for example, 00:07:43.450 --> 00:07:44.950 align:middle line:84% in the balance equations where there 00:07:44.950 --> 00:07:50.770 align:middle line:90% is a divergence of the fluxes. 00:07:50.770 --> 00:07:54.650 align:middle line:84% Well, we can take the divergence of these, 00:07:54.650 --> 00:07:57.570 align:middle line:90% and that's how it goes. 00:07:57.570 --> 00:07:59.410 align:middle line:90% It's typically a product. 00:07:59.410 --> 00:08:03.960 align:middle line:84% So the divergence of a product of a scalar times a vector 00:08:03.960 --> 00:08:06.390 align:middle line:90% is given by these two terms. 00:08:06.390 --> 00:08:08.220 align:middle line:84% The gradient of the scalar dotted 00:08:08.220 --> 00:08:11.820 align:middle line:84% into the vector times the scalar times the divergence 00:08:11.820 --> 00:08:12.425 align:middle line:90% of the vector. 00:08:12.425 --> 00:08:17.390 align:middle line:90% 00:08:17.390 --> 00:08:25.690 align:middle line:84% Now, remember that energy in our model has several components. 00:08:25.690 --> 00:08:32.350 align:middle line:84% One is the internal energy, proper of the stable equilibrium 00:08:32.350 --> 00:08:35.450 align:middle line:84% state that is underlying, that is there. 00:08:35.450 --> 00:08:41.650 align:middle line:84% That would be the only thing if we view our fluid element 00:08:41.650 --> 00:08:44.230 align:middle line:84% from a reference frame that moves 00:08:44.230 --> 00:08:47.320 align:middle line:84% with the barycentric velocity, and therefore, 00:08:47.320 --> 00:08:49.160 align:middle line:84% with respect to that reference frame, 00:08:49.160 --> 00:08:51.340 align:middle line:90% there is no kinetic energy. 00:08:51.340 --> 00:08:56.080 align:middle line:84% And also an accelerated frame where you don't also 00:08:56.080 --> 00:08:59.240 align:middle line:84% see the potential gravitational energy. 00:08:59.240 --> 00:09:01.390 align:middle line:84% And then here you have the additional contribution 00:09:01.390 --> 00:09:05.340 align:middle line:90% for the electrostatic energy. 00:09:05.340 --> 00:09:08.700 align:middle line:90% 00:09:08.700 --> 00:09:11.520 align:middle line:84% And if you multiply this by rho-- 00:09:11.520 --> 00:09:16.500 align:middle line:84% and I put all the stars just to be pedantic, but at least 00:09:16.500 --> 00:09:21.300 align:middle line:84% we know what we're talking about. 00:09:21.300 --> 00:09:23.900 align:middle line:90% We get this expression here. 00:09:23.900 --> 00:09:26.410 align:middle line:90% 00:09:26.410 --> 00:09:30.520 align:middle line:84% The total potential, therefore, is defined this way. 00:09:30.520 --> 00:09:33.730 align:middle line:84% It's the derivative of E with respect to n, 00:09:33.730 --> 00:09:36.010 align:middle line:84% is given by these contributions to the chemical 00:09:36.010 --> 00:09:44.020 align:middle line:84% potential plus this potential for kinetic, gravitational, 00:09:44.020 --> 00:09:48.210 align:middle line:90% and electrostatic potentials. 00:09:48.210 --> 00:09:50.810 align:middle line:90% 00:09:50.810 --> 00:09:53.430 align:middle line:84% So this is where we were, I guess. 00:09:53.430 --> 00:10:02.480 align:middle line:84% And, of course, the next step is to substitute these assumptions 00:10:02.480 --> 00:10:05.660 align:middle line:84% about the interactions between fluid elements 00:10:05.660 --> 00:10:11.890 align:middle line:84% into the energy, entropy, and constituent balance. 00:10:11.890 --> 00:10:14.280 align:middle line:84% And here, I've also done, although I'm not 00:10:14.280 --> 00:10:18.030 align:middle line:84% going to talk too much about it, but I've also 00:10:18.030 --> 00:10:23.100 align:middle line:84% written the momentum balance, that bases the fluid mechanics 00:10:23.100 --> 00:10:26.180 align:middle line:90% part, for reference. 00:10:26.180 --> 00:10:29.960 align:middle line:84% And as you remember, I have written the balance equation 00:10:29.960 --> 00:10:34.730 align:middle line:84% in the two equivalent forms that we have derived. 00:10:34.730 --> 00:10:38.540 align:middle line:84% One in terms of the Lagrangian fluxes 00:10:38.540 --> 00:10:41.650 align:middle line:84% with the partial derivative of rho e with respect 00:10:41.650 --> 00:10:46.030 align:middle line:84% to time, the other with respect to the diffusive fluxes 00:10:46.030 --> 00:10:54.580 align:middle line:84% with the material derivative of the mass specific properties. 00:10:54.580 --> 00:10:59.000 align:middle line:84% We proved the last time that these are equivalent. 00:10:59.000 --> 00:11:03.620 align:middle line:90% 00:11:03.620 --> 00:11:10.610 align:middle line:84% From these equations here and possibly-- 00:11:10.610 --> 00:11:11.660 align:middle line:90% well, no. 00:11:11.660 --> 00:11:16.520 align:middle line:84% From these equations here, you can obtain also 00:11:16.520 --> 00:11:21.190 align:middle line:84% these other equations below that, I am skipping the details. 00:11:21.190 --> 00:11:26.220 align:middle line:84% But it's simply, for example, the equation 00:11:26.220 --> 00:11:29.830 align:middle line:84% for the conservation of mass which 00:11:29.830 --> 00:11:33.340 align:middle line:84% derives from this one of the amounts multiplied 00:11:33.340 --> 00:11:35.690 align:middle line:90% by the molecular masses. 00:11:35.690 --> 00:11:40.890 align:middle line:84% This is the so-called continuity equation of fluid mechanics 00:11:40.890 --> 00:11:42.650 align:middle line:84% which you can also write this way. 00:11:42.650 --> 00:11:47.460 align:middle line:90% 00:11:47.460 --> 00:11:49.830 align:middle line:84% And so this is the divergence of v 00:11:49.830 --> 00:11:57.480 align:middle line:84% represents the rate of change of the volume of your fluid element 00:11:57.480 --> 00:11:59.360 align:middle line:84% as you follow it during the motion, 00:11:59.360 --> 00:12:03.310 align:middle line:90% during the flow along its path. 00:12:03.310 --> 00:12:11.050 align:middle line:84% And typically, unless the flow is very speedy 00:12:11.050 --> 00:12:17.350 align:middle line:84% and unless the Mach number is high, 00:12:17.350 --> 00:12:19.610 align:middle line:84% you can assume incompressible flow. 00:12:19.610 --> 00:12:24.290 align:middle line:84% And so this divergence is equal to zero. 00:12:24.290 --> 00:12:28.200 align:middle line:84% But, for example, from the momentum balance equation, 00:12:28.200 --> 00:12:33.090 align:middle line:84% if you dot it into vm, the velocity, 00:12:33.090 --> 00:12:38.280 align:middle line:84% you can obtain a balance for the kinetic energy. 00:12:38.280 --> 00:12:39.960 align:middle line:84% And then you can also get a balance 00:12:39.960 --> 00:12:43.560 align:middle line:84% for the gravitational potential energy and a balance 00:12:43.560 --> 00:12:46.600 align:middle line:84% for the electrostatic potential energy, 00:12:46.600 --> 00:13:02.140 align:middle line:84% so that you can subtract from the overall energy of your fluid 00:13:02.140 --> 00:13:05.080 align:middle line:90% the internal energy. 00:13:05.080 --> 00:13:08.020 align:middle line:84% And so that difference is made up 00:13:08.020 --> 00:13:11.110 align:middle line:84% of just kinetic, potential, gravitational, 00:13:11.110 --> 00:13:12.050 align:middle line:90% and electrostatic. 00:13:12.050 --> 00:13:17.055 align:middle line:84% So the sum of these three gives you this. 00:13:17.055 --> 00:13:20.430 align:middle line:90% 00:13:20.430 --> 00:13:26.310 align:middle line:84% And it's the rate of change of the mechanical energy 00:13:26.310 --> 00:13:30.110 align:middle line:90% content of your fluid. 00:13:30.110 --> 00:13:40.670 align:middle line:90% 00:13:40.670 --> 00:13:42.675 align:middle line:84% And now we can put them all together. 00:13:42.675 --> 00:13:47.030 align:middle line:90% 00:13:47.030 --> 00:13:53.590 align:middle line:84% So we have assumed the local simple system equilibrium 00:13:53.590 --> 00:14:02.370 align:middle line:84% so that we can write the energy this way that we just described. 00:14:02.370 --> 00:14:06.150 align:middle line:84% The simple system model includes the Euler relations 00:14:06.150 --> 00:14:09.300 align:middle line:84% so that together, it allows you to write the Gibbs 00:14:09.300 --> 00:14:11.910 align:middle line:90% relation in this way-- 00:14:11.910 --> 00:14:18.490 align:middle line:84% remember, tau is 1 over T, and lambda is minus mu over T-- 00:14:18.490 --> 00:14:26.820 align:middle line:84% which, as you recall, gets rid of the pressure. 00:14:26.820 --> 00:14:29.080 align:middle line:84% So that if you need the pressure, 00:14:29.080 --> 00:14:31.500 align:middle line:84% you need to go to the Gibbs-Duhem relation 00:14:31.500 --> 00:14:37.920 align:middle line:84% to get you the pressure from the concentrations and temperature 00:14:37.920 --> 00:14:38.680 align:middle line:90% gradients. 00:14:38.680 --> 00:14:42.410 align:middle line:90% 00:14:42.410 --> 00:14:46.160 align:middle line:84% So we are going to assume this relation between the properties 00:14:46.160 --> 00:14:47.210 align:middle line:90% of the particles. 00:14:47.210 --> 00:14:50.150 align:middle line:84% And notice that this is the internal energy, just 00:14:50.150 --> 00:14:52.780 align:middle line:90% this part of the energy. 00:14:52.780 --> 00:15:00.370 align:middle line:84% So we assume this relation holds as you follow the fluid particle 00:15:00.370 --> 00:15:02.950 align:middle line:90% along its path. 00:15:02.950 --> 00:15:10.040 align:middle line:84% So we assume that it is valid for the material derivatives. 00:15:10.040 --> 00:15:14.240 align:middle line:90% 00:15:14.240 --> 00:15:19.600 align:middle line:84% That means that the left hand side of the balance 00:15:19.600 --> 00:15:24.160 align:middle line:84% equations that we have in the previous slide where 00:15:24.160 --> 00:15:29.050 align:middle line:84% we have these material derivatives, they are related. 00:15:29.050 --> 00:15:32.380 align:middle line:84% We can essentially substitute the balance equations 00:15:32.380 --> 00:15:38.480 align:middle line:84% into this one, and we get an extra relation. 00:15:38.480 --> 00:15:45.200 align:middle line:84% Plus, we are going to assume the mode of interaction 00:15:45.200 --> 00:15:55.740 align:middle line:84% that we call heat and diffusion to be valid 00:15:55.740 --> 00:16:05.390 align:middle line:84% and holding between fluid elements as they move along. 00:16:05.390 --> 00:16:10.970 align:middle line:84% So it's the diffusive fluxes that are related in that way 00:16:10.970 --> 00:16:17.720 align:middle line:84% that we have discussed with the exclusion, though, of the flux 00:16:17.720 --> 00:16:22.020 align:middle line:90% of energy, which is related-- 00:16:22.020 --> 00:16:25.010 align:middle line:90% which is work. 00:16:25.010 --> 00:16:31.260 align:middle line:84% Because when you have the fluid element that deforms, especially 00:16:31.260 --> 00:16:35.930 align:middle line:84% it changes in volume, that change 00:16:35.930 --> 00:16:42.680 align:middle line:84% in volume against the normal components of the stress 00:16:42.680 --> 00:16:43.890 align:middle line:90% imply work. 00:16:43.890 --> 00:16:45.380 align:middle line:84% And that is a kind of interaction 00:16:45.380 --> 00:16:48.890 align:middle line:84% between adjacent fluid elements that is not 00:16:48.890 --> 00:16:51.150 align:middle line:90% of the heat and diffusion kind. 00:16:51.150 --> 00:16:53.190 align:middle line:90% So we exclude it. 00:16:53.190 --> 00:16:58.920 align:middle line:84% That's why I call these nw non-work. 00:16:58.920 --> 00:17:01.730 align:middle line:84% So the non-work component of the heat transfer 00:17:01.730 --> 00:17:09.980 align:middle line:84% between the fluid elements as they move along the path line 00:17:09.980 --> 00:17:15.349 align:middle line:84% is related to entropy flux, and constituent flux, 00:17:15.349 --> 00:17:20.790 align:middle line:84% as in the heat and diffusion mode of interaction. 00:17:20.790 --> 00:17:23.380 align:middle line:84% But energy has an additional contribution, 00:17:23.380 --> 00:17:28.720 align:middle line:84% which is the work component, given by the stress tensor 00:17:28.720 --> 00:17:31.190 align:middle line:90% dotted into the velocity. 00:17:31.190 --> 00:17:37.760 align:middle line:90% 00:17:37.760 --> 00:17:41.210 align:middle line:90% This one, as we said, we take-- 00:17:41.210 --> 00:17:43.340 align:middle line:84% since in the balance equations there 00:17:43.340 --> 00:17:48.950 align:middle line:84% appear divergences, so all this stuff, all these fluxes, 00:17:48.950 --> 00:17:55.380 align:middle line:84% we can combine those with the divergence of this relation, 00:17:55.380 --> 00:17:57.640 align:middle line:90% which is here. 00:17:57.640 --> 00:18:01.000 align:middle line:84% And a few things will cancel out, 00:18:01.000 --> 00:18:06.270 align:middle line:84% and here are the terms that survive. 00:18:06.270 --> 00:18:10.870 align:middle line:84% And we write what survives as an expression-- 00:18:10.870 --> 00:18:12.340 align:middle line:84% We took on the left hand side only 00:18:12.340 --> 00:18:15.430 align:middle line:84% the rate of entropy production, the density 00:18:15.430 --> 00:18:17.170 align:middle line:84% of entropy production, so the rate 00:18:17.170 --> 00:18:21.070 align:middle line:84% of entropy production per unit volume, sigma. 00:18:21.070 --> 00:18:23.860 align:middle line:84% And we have on the right hand side 00:18:23.860 --> 00:18:31.400 align:middle line:84% all the terms that contribute to that entropy production. 00:18:31.400 --> 00:18:35.490 align:middle line:84% And this is the most general form. 00:18:35.490 --> 00:18:38.720 align:middle line:90% It includes all sorts of terms. 00:18:38.720 --> 00:18:42.633 align:middle line:84% One is an old friend from chemical kinetics. 00:18:42.633 --> 00:18:46.490 align:middle line:90% 00:18:46.490 --> 00:18:50.160 align:middle line:84% This term here will give rise, as you will see, 00:18:50.160 --> 00:18:53.900 align:middle line:84% to the dissipation function of fluid mechanics, which 00:18:53.900 --> 00:18:56.760 align:middle line:90% you will see in a moment. 00:18:56.760 --> 00:19:08.790 align:middle line:84% The first term is the one that gives the Fourier expression 00:19:08.790 --> 00:19:13.800 align:middle line:84% when you have only heat and not also diffusion. 00:19:13.800 --> 00:19:16.590 align:middle line:84% And these are terms that we have to see 00:19:16.590 --> 00:19:20.460 align:middle line:84% what they do because they do important effects 00:19:20.460 --> 00:19:22.346 align:middle line:90% for many applications. 00:19:22.346 --> 00:19:45.180 align:middle line:90% 00:19:45.180 --> 00:19:51.920 align:middle line:84% Now, the other time we also have seen how to 00:19:51.920 --> 00:19:59.260 align:middle line:90% recognize in the heat-- 00:19:59.260 --> 00:20:03.670 align:middle line:84% sorry, energy, in the energy flux, a portion of the energy 00:20:03.670 --> 00:20:08.080 align:middle line:84% flux that we can still recognize and call heat, 00:20:08.080 --> 00:20:12.400 align:middle line:84% or we call it the measurable heat, more properly. 00:20:12.400 --> 00:20:14.680 align:middle line:84% But essentially, it's the part that we 00:20:14.680 --> 00:20:19.210 align:middle line:84% can identify with the heat of heat transfer so, 00:20:19.210 --> 00:20:21.465 align:middle line:84% again, with this q double prime notation. 00:20:21.465 --> 00:20:24.660 align:middle line:90% 00:20:24.660 --> 00:20:28.720 align:middle line:84% So if we take the non-work part of the energy flux, 00:20:28.720 --> 00:20:34.200 align:middle line:84% it's given by the heat plus a portion, which 00:20:34.200 --> 00:20:37.530 align:middle line:90% is related to the flux. 00:20:37.530 --> 00:20:48.530 align:middle line:84% So this is the energy which is carried by the particles 00:20:48.530 --> 00:20:52.220 align:middle line:90% just because they are diffusing. 00:20:52.220 --> 00:20:57.060 align:middle line:84% This is not convective because this is a diffusive part. 00:20:57.060 --> 00:21:02.480 align:middle line:84% So this is particles that are moving at a different velocity 00:21:02.480 --> 00:21:04.680 align:middle line:90% than the barycentric velocity. 00:21:04.680 --> 00:21:08.460 align:middle line:84% So they diffuse away or into our fluid element. 00:21:08.460 --> 00:21:11.090 align:middle line:84% And as they do so, they carry with them 00:21:11.090 --> 00:21:14.250 align:middle line:84% their backpack of internal energy. 00:21:14.250 --> 00:21:20.240 align:middle line:84% And plus, they also have to advance and face 00:21:20.240 --> 00:21:22.440 align:middle line:90% the pressure that is present. 00:21:22.440 --> 00:21:24.320 align:middle line:84% And therefore, there is also this-- 00:21:24.320 --> 00:21:27.380 align:middle line:90% 00:21:27.380 --> 00:21:29.180 align:middle line:84% there is this pulsion work, which 00:21:29.180 --> 00:21:34.160 align:middle line:84% is p times the partial volume, which is hidden here 00:21:34.160 --> 00:21:37.460 align:middle line:84% because remember, the enthalpy is u plus pv 00:21:37.460 --> 00:21:40.110 align:middle line:90% So this is partial enthalpy. 00:21:40.110 --> 00:21:45.230 align:middle line:84% So it's partial energy plus p times partial volume. 00:21:45.230 --> 00:21:51.720 align:middle line:84% And that p times partial volume is the pulsion work 00:21:51.720 --> 00:21:55.630 align:middle line:84% equivalent to what you would have in a bulk flow situation. 00:21:55.630 --> 00:21:57.380 align:middle line:90% But here we are in diffusion. 00:21:57.380 --> 00:22:03.050 align:middle line:90% 00:22:03.050 --> 00:22:11.010 align:middle line:84% So we can substitute these terms here into our expression. 00:22:11.010 --> 00:22:13.740 align:middle line:84% And this is a very typical procedure 00:22:13.740 --> 00:22:23.600 align:middle line:84% that we like somehow to express, to rewrite. 00:22:23.600 --> 00:22:26.930 align:middle line:84% Every time you formulate a problem in non-equilibrium 00:22:26.930 --> 00:22:29.420 align:middle line:84% thermodynamics, you typically start-- 00:22:29.420 --> 00:22:32.180 align:middle line:84% and it's the safest way to do it-- 00:22:32.180 --> 00:22:34.620 align:middle line:84% start from the entropic point of view. 00:22:34.620 --> 00:22:37.250 align:middle line:90% So you write it in this way. 00:22:37.250 --> 00:22:44.330 align:middle line:84% You identify these gradients as the degrees of non-equilibrium 00:22:44.330 --> 00:22:47.670 align:middle line:84% related to the various transport of the various properties. 00:22:47.670 --> 00:22:51.180 align:middle line:90% 00:22:51.180 --> 00:22:55.030 align:middle line:84% But then you like to translate this 00:22:55.030 --> 00:22:58.360 align:middle line:84% into a more common and familiar language. 00:22:58.360 --> 00:23:02.720 align:middle line:84% And since we consider more familiar, 00:23:02.720 --> 00:23:07.090 align:middle line:84% talking about heat because it's related to Q and Q over T, 00:23:07.090 --> 00:23:11.620 align:middle line:84% we often go back to these and then also 00:23:11.620 --> 00:23:19.840 align:middle line:84% the fluxes of particles and the chemical potentials. 00:23:19.840 --> 00:23:24.960 align:middle line:84% So we substitute this into there. 00:23:24.960 --> 00:23:30.510 align:middle line:84% We also use a relation that I think we've 00:23:30.510 --> 00:23:35.010 align:middle line:90% discussed that you can write. 00:23:35.010 --> 00:23:37.600 align:middle line:84% When you write a differential of the chemical potential, 00:23:37.600 --> 00:23:39.990 align:middle line:84% you can separate the part that depends 00:23:39.990 --> 00:23:46.990 align:middle line:84% on temperature and the part that changes at constant temperature. 00:23:46.990 --> 00:23:49.270 align:middle line:84% So it depends on pressure and concentrations. 00:23:49.270 --> 00:23:55.950 align:middle line:84% And so we keep that part written this way. 00:23:55.950 --> 00:23:57.930 align:middle line:84% And so the bottom line is that we 00:23:57.930 --> 00:24:05.370 align:middle line:84% can write the entropy production density sigma in this way. 00:24:05.370 --> 00:24:09.630 align:middle line:84% So here, we have more explicitly, the Fourier kind 00:24:09.630 --> 00:24:14.520 align:middle line:90% of entropy production. 00:24:14.520 --> 00:24:18.870 align:middle line:84% Here, you have a similar term related to diffusion. 00:24:18.870 --> 00:24:22.110 align:middle line:84% Here, we still have the usual term 00:24:22.110 --> 00:24:25.260 align:middle line:90% related to chemical kinetics. 00:24:25.260 --> 00:24:30.180 align:middle line:84% And what is left here is the dissipation function-- 00:24:30.180 --> 00:24:33.060 align:middle line:84% 1 over the temperature times the dissipation function. 00:24:33.060 --> 00:24:36.550 align:middle line:84% And here are the details of the dissipation functions, 00:24:36.550 --> 00:24:40.230 align:middle line:84% if you like, if you are a fluid mechanicist. 00:24:40.230 --> 00:24:41.920 align:middle line:90% It has two contributions. 00:24:41.920 --> 00:24:47.430 align:middle line:84% It's not just the deviatoric part of the stress tensor 00:24:47.430 --> 00:24:55.530 align:middle line:84% and the deviatoric component of the strain tensor. 00:24:55.530 --> 00:24:58.830 align:middle line:84% But it has also this term, which is related 00:24:58.830 --> 00:25:02.850 align:middle line:84% to the divergence of v. And it's true 00:25:02.850 --> 00:25:08.230 align:middle line:84% that for most flows like low Mach number flows, this is zero. 00:25:08.230 --> 00:25:10.350 align:middle line:90% So this is really not important. 00:25:10.350 --> 00:25:17.770 align:middle line:84% But in principle, this is one of the dissipative terms. 00:25:17.770 --> 00:25:23.550 align:middle line:84% And in fact, notice also that in our doing the procedure here, 00:25:23.550 --> 00:25:32.430 align:middle line:84% there was a pressure which came in here. 00:25:32.430 --> 00:25:35.620 align:middle line:84% And there is a pm, which is the mechanical pressure. 00:25:35.620 --> 00:25:40.335 align:middle line:84% That is the trace over one-third of the trace of the stress 00:25:40.335 --> 00:25:40.835 align:middle line:90% tensor. 00:25:40.835 --> 00:25:45.890 align:middle line:90% 00:25:45.890 --> 00:25:48.865 align:middle line:84% They are not necessarily the same. 00:25:48.865 --> 00:25:51.710 align:middle line:90% 00:25:51.710 --> 00:25:59.405 align:middle line:84% And so their difference gives rise to this entropy production 00:25:59.405 --> 00:25:59.905 align:middle line:90% term. 00:25:59.905 --> 00:26:04.690 align:middle line:90% 00:26:04.690 --> 00:26:09.250 align:middle line:84% You can still view each of these terms 00:26:09.250 --> 00:26:13.800 align:middle line:84% as a product of a rate and an affinity 00:26:13.800 --> 00:26:15.430 align:middle line:90% or a degree of disequilibrium. 00:26:15.430 --> 00:26:20.860 align:middle line:84% So the gradient in temperature, more properly minus 1 00:26:20.860 --> 00:26:23.200 align:middle line:84% over T squared times the gradient in temperature, 00:26:23.200 --> 00:26:25.960 align:middle line:84% can be interpreted as the degree of disequilibrium 00:26:25.960 --> 00:26:32.530 align:middle line:90% associated with heat transport. 00:26:32.530 --> 00:26:35.590 align:middle line:84% Similarly, minus 1 over 3 times the gradient 00:26:35.590 --> 00:26:40.180 align:middle line:84% of the chemical potential done at constant T 00:26:40.180 --> 00:26:44.530 align:middle line:84% is the degree of disequilibrium with respect 00:26:44.530 --> 00:26:46.836 align:middle line:90% to diffusion of particles. 00:26:46.836 --> 00:26:50.030 align:middle line:90% 00:26:50.030 --> 00:26:52.520 align:middle line:90% The affinities in chemistry-- 00:26:52.520 --> 00:26:55.460 align:middle line:84% 1 over r (I meant, Y sub k) have already 00:26:55.460 --> 00:26:59.030 align:middle line:84% been recognized as the degrees of disequilibrium. 00:26:59.030 --> 00:27:02.810 align:middle line:84% And here for the fluid part, we have a degree 00:27:02.810 --> 00:27:07.100 align:middle line:84% of disequilibrium which is related to the rate of strain, 00:27:07.100 --> 00:27:10.280 align:middle line:84% just the symmetric part of the rate of strain 00:27:10.280 --> 00:27:13.970 align:middle line:90% of the local stress tensor-- 00:27:13.970 --> 00:27:17.150 align:middle line:90% I'm sorry, of the local 00:27:17.150 --> 00:27:19.730 align:middle line:90% gradient of the velocity field. 00:27:19.730 --> 00:27:24.980 align:middle line:84% And whereas, the conjugated flux is 00:27:24.980 --> 00:27:30.080 align:middle line:84% the flux of momentum, which is related to the stress tensor. 00:27:30.080 --> 00:27:32.360 align:middle line:90% But then there is also this one. 00:27:32.360 --> 00:27:38.250 align:middle line:84% So the divergence of velocity, which, as we said, 00:27:38.250 --> 00:27:44.090 align:middle line:84% is the rate of change of volume as you follow the fluid element. 00:27:44.090 --> 00:27:54.800 align:middle line:84% That is also an affinity for a dissipation for which the flux 00:27:54.800 --> 00:27:58.550 align:middle line:84% is given by this difference between 00:27:58.550 --> 00:28:02.425 align:middle line:84% the thermodynamic pressure and the mechanical pressure. 00:28:02.425 --> 00:28:10.360 align:middle line:90% 00:28:10.360 --> 00:28:17.790 align:middle line:84% If we also have charged particles-- 00:28:17.790 --> 00:28:24.700 align:middle line:84% so if some of the constituents of our fluid are charged 00:28:24.700 --> 00:28:30.050 align:middle line:84% particles, and we're talking electrostatics, not-- 00:28:30.050 --> 00:28:33.190 align:middle line:84% electrodynamics is even more complicated 00:28:33.190 --> 00:28:36.160 align:middle line:84% because, of course, then you get the Maxwell 00:28:36.160 --> 00:28:40.990 align:middle line:84% equations, the magnetic field, the vector potential, and so on. 00:28:40.990 --> 00:28:43.870 align:middle line:90% Too complicated to cover. 00:28:43.870 --> 00:28:48.820 align:middle line:84% But in any case, so here we have just electrostatics. 00:28:48.820 --> 00:28:53.460 align:middle line:84% And so phi is the electrostatic field. 00:28:53.460 --> 00:29:05.625 align:middle line:84% Jq is the diffusive component of the electric current density. 00:29:05.625 --> 00:29:09.140 align:middle line:90% 00:29:09.140 --> 00:29:14.150 align:middle line:84% Because if this fluid element contains charges, 00:29:14.150 --> 00:29:19.820 align:middle line:84% and these fluid elements moves with the barycentric velocity, 00:29:19.820 --> 00:29:23.240 align:middle line:84% there is a convective part of the charge which 00:29:23.240 --> 00:29:25.850 align:middle line:90% produces an electric current. 00:29:25.850 --> 00:29:27.960 align:middle line:84% We're not talking about this part. 00:29:27.960 --> 00:29:33.110 align:middle line:84% We are talking about the part of charge exchange 00:29:33.110 --> 00:29:38.840 align:middle line:84% between adjacent fluid elements due to the fact that they have-- 00:29:38.840 --> 00:29:41.760 align:middle line:90% 00:29:41.760 --> 00:29:46.500 align:middle line:84% the diffusion with respect to the barycentric velocity. 00:29:46.500 --> 00:29:50.710 align:middle line:84% So this is a diffusion flux of charge. 00:29:50.710 --> 00:29:53.290 align:middle line:84% And in fact, so the total current density 00:29:53.290 --> 00:29:57.510 align:middle line:84% may be denoted with I double prime. 00:29:57.510 --> 00:30:01.700 align:middle line:84% And it is the flux of charge with respect to the laboratory 00:30:01.700 --> 00:30:02.200 align:middle line:90% frame. 00:30:02.200 --> 00:30:06.290 align:middle line:84% So it's the flux with a zero on top. 00:30:06.290 --> 00:30:08.520 align:middle line:84% And it's made up of the diffusive 00:30:08.520 --> 00:30:10.390 align:middle line:90% plus the convective term. 00:30:10.390 --> 00:30:13.430 align:middle line:90% 00:30:13.430 --> 00:30:18.080 align:middle line:84% And rho q is the density of charges, 00:30:18.080 --> 00:30:19.570 align:middle line:90% the local density of charges. 00:30:19.570 --> 00:30:23.500 align:middle line:90% 00:30:23.500 --> 00:30:31.420 align:middle line:84% So the non-work part of the diffusive energy flux 00:30:31.420 --> 00:30:36.320 align:middle line:84% is related to heat, to the diffusion of particles, 00:30:36.320 --> 00:30:41.140 align:middle line:90% and to the diffusion of charge. 00:30:41.140 --> 00:30:43.740 align:middle line:84% So we have this extra term, which 00:30:43.740 --> 00:30:50.940 align:middle line:84% we can single out in here when you do the same procedure 00:30:50.940 --> 00:30:53.610 align:middle line:84% as before to write this expression in terms 00:30:53.610 --> 00:30:57.260 align:middle line:90% of the familiar variables. 00:30:57.260 --> 00:31:08.080 align:middle line:84% So here, the entropy production is related also to minus 1 00:31:08.080 --> 00:31:10.900 align:middle line:84% over T times the gradient in the electrostatic field 00:31:10.900 --> 00:31:17.260 align:middle line:84% is the degree of disequilibrium with respect to moving charges. 00:31:17.260 --> 00:31:21.780 align:middle line:84% And the associated or the conjugated flux 00:31:21.780 --> 00:31:24.030 align:middle line:84% is this difference here, which is just 00:31:24.030 --> 00:31:26.165 align:middle line:90% the diffusive flux of charge. 00:31:26.165 --> 00:31:33.710 align:middle line:90% 00:31:33.710 --> 00:31:37.600 align:middle line:84% That's, again, the same equation. 00:31:37.600 --> 00:31:41.680 align:middle line:84% Now, the next step to introduce the next notion 00:31:41.680 --> 00:31:49.780 align:middle line:84% is to group these terms in order of 00:31:49.780 --> 00:31:52.280 align:middle line:90% increasing tensorial character. 00:31:52.280 --> 00:31:58.670 align:middle line:84% So some of these are scalars like chemistry. 00:31:58.670 --> 00:32:03.020 align:middle line:84% And also inside the dissipation function, 00:32:03.020 --> 00:32:05.890 align:middle line:84% there is a scalar term, which is the one that we described 00:32:05.890 --> 00:32:08.720 align:middle line:84% before, and then there is a tensorial term, 00:32:08.720 --> 00:32:13.630 align:middle line:84% which is the one related to the contraction 00:32:13.630 --> 00:32:17.575 align:middle line:84% between the deviatoric parts of stress and (rate of) 00:32:17.575 --> 00:32:20.460 align:middle line:90% strain tensors. 00:32:20.460 --> 00:32:24.970 align:middle line:84% And the rest are vectorial terms-- 00:32:24.970 --> 00:32:32.580 align:middle line:84% heat, flux, electric current, flux of particles. 00:32:32.580 --> 00:32:33.460 align:middle line:90% These are vectors. 00:32:33.460 --> 00:32:36.340 align:middle line:90% 00:32:36.340 --> 00:32:37.850 align:middle line:90% Why is that important? 00:32:37.850 --> 00:32:44.620 align:middle line:90% 00:32:44.620 --> 00:32:49.060 align:middle line:84% All right, now, so I started with some coloring 00:32:49.060 --> 00:32:51.793 align:middle line:90% here just to single out the-- 00:32:51.793 --> 00:32:54.511 align:middle line:90% 00:32:54.511 --> 00:32:57.950 align:middle line:84% the degrees of disequilibrium are the red stuff, 00:32:57.950 --> 00:33:01.900 align:middle line:90% and the fluxes are the blue. 00:33:01.900 --> 00:33:04.100 align:middle line:84% And so you see each one is written 00:33:04.100 --> 00:33:07.845 align:middle line:84% as a product of a degree of disequilibrium times a flux. 00:33:07.845 --> 00:33:10.400 align:middle line:90% 00:33:10.400 --> 00:33:12.960 align:middle line:84% And if we recall what we said about exergy, 00:33:12.960 --> 00:33:16.130 align:middle line:84% you can also read this expression 00:33:16.130 --> 00:33:24.560 align:middle line:84% as a representation of how locally your fluid or continuum 00:33:24.560 --> 00:33:27.005 align:middle line:90% is dissipating exergy. 00:33:27.005 --> 00:33:29.580 align:middle line:90% 00:33:29.580 --> 00:33:33.360 align:middle line:84% Multiply that by the temperature of the environment with respect 00:33:33.360 --> 00:33:36.570 align:middle line:84% to which you want to-- or the reservoir with respect to which 00:33:36.570 --> 00:33:40.410 align:middle line:90% you want to measure exergy. 00:33:40.410 --> 00:33:46.530 align:middle line:84% And so these terms tell you that you have dissipation whenever 00:33:46.530 --> 00:33:53.520 align:middle line:84% you drive, like I read here, you drive a chemical reaction 00:33:53.520 --> 00:33:56.730 align:middle line:84% down the direction of a decreasing affinity. 00:33:56.730 --> 00:34:00.630 align:middle line:84% And that gives you entropy production. 00:34:00.630 --> 00:34:04.710 align:middle line:84% You drive a heat flux down a temperature gradient. 00:34:04.710 --> 00:34:08.070 align:middle line:84% Also you drive a diffusion flux down 00:34:08.070 --> 00:34:09.900 align:middle line:90% a chemical potential gradient. 00:34:09.900 --> 00:34:13.790 align:middle line:84% Or an electric current down a voltage drop. 00:34:13.790 --> 00:34:18.590 align:middle line:84% Or a momentum flux down a velocity gradient, 00:34:18.590 --> 00:34:19.889 align:middle line:90% which is the rate of strain. 00:34:19.889 --> 00:34:25.920 align:middle line:90% 00:34:25.920 --> 00:34:30.370 align:middle line:84% You may find, if you look into papers that deal with 00:34:30.370 --> 00:34:38.820 align:middle line:84% non-equilibrium thermodynamics, that expression up there-- 00:34:38.820 --> 00:34:41.340 align:middle line:90% this one with all these terms-- 00:34:41.340 --> 00:34:44.429 align:middle line:84% written in a compact form in this way 00:34:44.429 --> 00:34:51.000 align:middle line:84% as a sum of products of fluxes and-- 00:34:51.000 --> 00:34:53.320 align:middle line:84% instead of degrees of disequilibrium, 00:34:53.320 --> 00:34:55.949 align:middle line:90% they are called forces-- 00:34:55.949 --> 00:34:57.370 align:middle line:90% fluxes and forces. 00:34:57.370 --> 00:35:00.150 align:middle line:84% So this is a force flux representation 00:35:00.150 --> 00:35:03.490 align:middle line:84% of the same or shorthand notation. 00:35:03.490 --> 00:35:08.490 align:middle line:84% And this symbol here is whatever is 00:35:08.490 --> 00:35:11.880 align:middle line:84% appropriate to use when you have to multiply 00:35:11.880 --> 00:35:17.820 align:middle line:84% a scalar flux times a scalar degree of disequilibrium. 00:35:17.820 --> 00:35:23.630 align:middle line:84% It's just a simple multiplication like here. 00:35:23.630 --> 00:35:28.800 align:middle line:84% When you have to multiply a vector times another vector, 00:35:28.800 --> 00:35:30.200 align:middle line:90% it is the dot-- 00:35:30.200 --> 00:35:31.680 align:middle line:90% it's the vector product. 00:35:31.680 --> 00:35:35.000 align:middle line:84% I'm sorry, the dot product, because the vector product 00:35:35.000 --> 00:35:36.020 align:middle line:90% is something else. 00:35:36.020 --> 00:35:42.410 align:middle line:84% And if you have to multiply the tensor, two tensors, 00:35:42.410 --> 00:35:45.540 align:middle line:84% and get a scalar, it's the contraction. 00:35:45.540 --> 00:35:48.690 align:middle line:84% So if you take the two matrices of the two tensors, 00:35:48.690 --> 00:35:50.760 align:middle line:84% you multiply them, and then you take the trace. 00:35:50.760 --> 00:35:56.430 align:middle line:84% And that's the equivalent of this double dot operation. 00:35:56.430 --> 00:35:59.360 align:middle line:84% So we will proceed now for a while-- 00:35:59.360 --> 00:36:01.930 align:middle line:90% 00:36:01.930 --> 00:36:04.340 align:middle line:84% I mean, for today, for the rest of today, 00:36:04.340 --> 00:36:11.790 align:middle line:84% I guess, with these more abstract and shorthand notation. 00:36:11.790 --> 00:36:20.130 align:middle line:84% Because we want to discuss some geometry of the relations 00:36:20.130 --> 00:36:27.000 align:middle line:84% that we can write in order to model how a flux is related 00:36:27.000 --> 00:36:29.480 align:middle line:90% to a force or vice versa. 00:36:29.480 --> 00:36:34.910 align:middle line:90% 00:36:34.910 --> 00:36:39.775 align:middle line:84% Yeah, actually, maybe you should go back for a moment. 00:36:39.775 --> 00:36:43.880 align:middle line:90% 00:36:43.880 --> 00:36:47.270 align:middle line:84% I mean, there was one slide that I 00:36:47.270 --> 00:36:48.720 align:middle line:90% don't know where where it went. 00:36:48.720 --> 00:36:53.520 align:middle line:84% But in any case, maybe it will come up later. 00:36:53.520 --> 00:36:56.390 align:middle line:84% But what I need to say at this moment 00:36:56.390 --> 00:37:04.140 align:middle line:84% is that this relation is what I call an extrinsic relation. 00:37:04.140 --> 00:37:14.460 align:middle line:84% Extrinsic means that it has been obtained using only assumptions 00:37:14.460 --> 00:37:17.580 align:middle line:84% about the continuum, the balance equations, things 00:37:17.580 --> 00:37:19.140 align:middle line:90% that hold in general. 00:37:19.140 --> 00:37:21.630 align:middle line:84% It doesn't contain anything that has to do 00:37:21.630 --> 00:37:24.420 align:middle line:90% with how the material behaves. 00:37:24.420 --> 00:37:27.780 align:middle line:84% It doesn't yet contain any property of the material. 00:37:27.780 --> 00:37:31.210 align:middle line:84% For example, here, the property of the material, 00:37:31.210 --> 00:37:33.090 align:middle line:84% when you do heat transfer, comes in when 00:37:33.090 --> 00:37:34.660 align:middle line:90% you write the Fourier law. 00:37:34.660 --> 00:37:40.390 align:middle line:84% And therefore, you insert the thermal conductivity tensor, 00:37:40.390 --> 00:37:44.160 align:middle line:90% which is the description of-- 00:37:44.160 --> 00:37:47.620 align:middle line:84% which contains how your system behaves. 00:37:47.620 --> 00:37:49.355 align:middle line:90% And so that takes me here. 00:37:49.355 --> 00:37:52.290 align:middle line:90% 00:37:52.290 --> 00:37:56.950 align:middle line:84% In other words, once we have written 00:37:56.950 --> 00:38:00.560 align:middle line:84% the extrinsic general relation for sigma, 00:38:00.560 --> 00:38:03.610 align:middle line:84% then we have to go into the detail of the particular problem 00:38:03.610 --> 00:38:06.280 align:middle line:84% and the particular material or materials 00:38:06.280 --> 00:38:08.260 align:middle line:90% that we are dealing with. 00:38:08.260 --> 00:38:11.800 align:middle line:84% And each material has its own way 00:38:11.800 --> 00:38:20.350 align:middle line:84% of reacting to fluxes and forces. 00:38:20.350 --> 00:38:22.660 align:middle line:84% The principle in general is that the material 00:38:22.660 --> 00:38:32.440 align:middle line:84% tends to spontaneously go, if it can, towards stable equilibrium. 00:38:32.440 --> 00:38:39.230 align:middle line:84% But sometimes, if it is subjected to permanent forces, 00:38:39.230 --> 00:38:42.620 align:middle line:90% it will reach a steady state. 00:38:42.620 --> 00:38:51.530 align:middle line:84% Because those fluxes and forces that are imposed by the exterior 00:38:51.530 --> 00:38:54.900 align:middle line:84% or by the adjacent fluid elements-- 00:38:54.900 --> 00:38:58.520 align:middle line:84% by the flow situation, by the boundary conditions-- 00:38:58.520 --> 00:39:03.480 align:middle line:84% those prevent the system from relaxing spontaneously 00:39:03.480 --> 00:39:04.150 align:middle line:90% to equilibrium. 00:39:04.150 --> 00:39:09.375 align:middle line:84% So it will keep the state off equilibrium. 00:39:09.375 --> 00:39:12.770 align:middle line:90% 00:39:12.770 --> 00:39:19.870 align:middle line:84% And typically, boundary conditions, most of the times, 00:39:19.870 --> 00:39:23.295 align:middle line:84% impose either the forces or the fluxes. 00:39:23.295 --> 00:39:25.950 align:middle line:90% 00:39:25.950 --> 00:39:32.820 align:middle line:84% But sometimes you also impose a relation between force and flux. 00:39:32.820 --> 00:39:35.430 align:middle line:84% Like when you do convective heat transfer, 00:39:35.430 --> 00:39:40.985 align:middle line:84% that is a boundary condition of Robin type. 00:39:40.985 --> 00:39:47.280 align:middle line:90% 00:39:47.280 --> 00:39:51.150 align:middle line:90% In any case, you can view-- 00:39:51.150 --> 00:39:54.720 align:middle line:90% there is a dual point of view. 00:39:54.720 --> 00:40:02.190 align:middle line:84% You can describe your material resistance to flux 00:40:02.190 --> 00:40:14.020 align:middle line:84% by saying how it reacts when you impose some fluxes in it. 00:40:14.020 --> 00:40:18.970 align:middle line:84% And the material typically reacts to a flux 00:40:18.970 --> 00:40:23.820 align:middle line:84% by building up a degree of disequilibrium. 00:40:23.820 --> 00:40:26.400 align:middle line:84% Remember, Le Chatelier-Braun theorem-- 00:40:26.400 --> 00:40:31.320 align:middle line:84% degree of disequilibrium that will enhance its tendency 00:40:31.320 --> 00:40:36.820 align:middle line:84% to contrast that flux and return to equilibrium. 00:40:36.820 --> 00:40:40.970 align:middle line:90% 00:40:40.970 --> 00:40:43.640 align:middle line:84% So your model for the material will 00:40:43.640 --> 00:40:48.620 align:middle line:84% be a relation that gives you the force as a function of the flux. 00:40:48.620 --> 00:40:51.490 align:middle line:84% And the gammas are the other properties, 00:40:51.490 --> 00:40:54.370 align:middle line:84% like equilibrium properties, if you are near equilibrium, 00:40:54.370 --> 00:41:02.490 align:middle line:84% like temperature, chemical potentials, electrostatic field. 00:41:02.490 --> 00:41:05.900 align:middle line:84% But of course, you can also view it in the opposite way. 00:41:05.900 --> 00:41:10.200 align:middle line:84% It's just the inverse function or the dual point of view. 00:41:10.200 --> 00:41:13.210 align:middle line:90% You impose a force, for example. 00:41:13.210 --> 00:41:16.170 align:middle line:84% You impose a gradient in temperature on a specimen 00:41:16.170 --> 00:41:19.740 align:middle line:90% of a piece of metal. 00:41:19.740 --> 00:41:30.120 align:middle line:84% And the material reacts by building up a flux which will 00:41:30.120 --> 00:41:35.850 align:middle line:84% try to go in the direction of re-establishing a flat-- 00:41:35.850 --> 00:41:40.740 align:middle line:84% of killing that temperature gradient that you are imposing. 00:41:40.740 --> 00:41:46.470 align:middle line:84% Again, in this case, you write the relation flux 00:41:46.470 --> 00:41:48.050 align:middle line:90% as a function of force. 00:41:48.050 --> 00:41:51.090 align:middle line:90% 00:41:51.090 --> 00:41:54.600 align:middle line:84% And clearly, then, you have-- you put it 00:41:54.600 --> 00:42:00.420 align:middle line:84% into the extrinsic formula for the entropy production 00:42:00.420 --> 00:42:08.280 align:middle line:84% that we just obtained, and here comes 00:42:08.280 --> 00:42:12.150 align:middle line:90% a relation in this picture. 00:42:12.150 --> 00:42:14.760 align:middle line:84% The entropy production is related to the fluxes 00:42:14.760 --> 00:42:19.560 align:middle line:84% that you have imposed, or this one is related to the forces 00:42:19.560 --> 00:42:20.480 align:middle line:90% that you have imposed. 00:42:20.480 --> 00:42:27.400 align:middle line:90% 00:42:27.400 --> 00:42:34.590 align:middle line:84% And then you start (with) particular models-- 00:42:34.590 --> 00:42:36.095 align:middle line:90% particular, but pretty general. 00:42:36.095 --> 00:42:40.080 align:middle line:90% 00:42:40.080 --> 00:42:44.010 align:middle line:84% For example, here, if you want to model 00:42:44.010 --> 00:42:48.640 align:middle line:84% the relation between force and flux in heat transfer, 00:42:48.640 --> 00:42:55.310 align:middle line:84% you adopt the Fourier law, which for an isotropic material, 00:42:55.310 --> 00:42:57.210 align:middle line:90% doesn't have a tensor here. 00:42:57.210 --> 00:43:00.550 align:middle line:90% This is just a scalar. 00:43:00.550 --> 00:43:02.590 align:middle line:84% But if it is anisotropic, we have 00:43:02.590 --> 00:43:06.300 align:middle line:84% seen that this should become a tensor-- 00:43:06.300 --> 00:43:11.540 align:middle line:84% a symmetric tensor-- due to the Onsager reciprocity. 00:43:11.540 --> 00:43:16.340 align:middle line:90% 00:43:16.340 --> 00:43:20.570 align:middle line:84% The relation between the difference 00:43:20.570 --> 00:43:23.300 align:middle line:84% between mechanical and thermodynamic pressure 00:43:23.300 --> 00:43:27.650 align:middle line:84% and the divergence of the velocity 00:43:27.650 --> 00:43:33.590 align:middle line:84% is called the Lamb law for a Newtonian fluid. 00:43:33.590 --> 00:43:37.350 align:middle line:84% And the constant between the two-- 00:43:37.350 --> 00:43:41.870 align:middle line:84% I mean, the-- the constant of proportionality between the two 00:43:41.870 --> 00:43:43.636 align:middle line:90% is called the bulk viscosity. 00:43:43.636 --> 00:43:50.370 align:middle line:90% 00:43:50.370 --> 00:43:59.210 align:middle line:84% In fluid mechanics, the relation between the deviatoric part 00:43:59.210 --> 00:44:02.180 align:middle line:84% of the stress tensor and the deviatoric part 00:44:02.180 --> 00:44:07.390 align:middle line:84% of the rate of strain tensor is what we call-- 00:44:07.390 --> 00:44:09.020 align:middle line:90% is this linear relation. 00:44:09.020 --> 00:44:14.350 align:middle line:84% And the proportionality relation is called the shear viscosity. 00:44:14.350 --> 00:44:16.970 align:middle line:84% We typically just call it viscosity. 00:44:16.970 --> 00:44:19.940 align:middle line:90% 00:44:19.940 --> 00:44:27.190 align:middle line:84% And with this plus Lamb, you may rewrite the stress tensor 00:44:27.190 --> 00:44:36.210 align:middle line:84% in this way where, notice here that I'm 00:44:36.210 --> 00:44:38.340 align:middle line:84% using the thermodynamic pressure, not 00:44:38.340 --> 00:44:42.900 align:middle line:84% the mechanical pressure, because that difference between the two 00:44:42.900 --> 00:44:52.670 align:middle line:84% has been shifted on this side to use the Lamb law. 00:44:52.670 --> 00:44:54.770 align:middle line:84% And so here you have the standard stuff, 00:44:54.770 --> 00:44:57.230 align:middle line:84% and here you have the stuff that you usually neglect 00:44:57.230 --> 00:45:03.600 align:middle line:90% because this is typically zero. 00:45:03.600 --> 00:45:06.540 align:middle line:84% But this is the second coefficient of viscosity. 00:45:06.540 --> 00:45:09.730 align:middle line:90% 00:45:09.730 --> 00:45:15.590 align:middle line:84% And then, if you have electricity conduction, 00:45:15.590 --> 00:45:21.250 align:middle line:90% diffusion, you have Ohm's law. 00:45:21.250 --> 00:45:23.770 align:middle line:84% The coefficient of proportionality 00:45:23.770 --> 00:45:29.095 align:middle line:84% between the charge diffusive flux and the-- 00:45:29.095 --> 00:45:33.220 align:middle line:90% 00:45:33.220 --> 00:45:35.470 align:middle line:84% minus the gradient in the electrostatic field 00:45:35.470 --> 00:45:38.680 align:middle line:90% is the electric field. 00:45:38.680 --> 00:45:43.800 align:middle line:84% So this is-- you could also write it as E. And in any case, 00:45:43.800 --> 00:45:47.720 align:middle line:84% this is the inverse of the electrical resistivity 00:45:47.720 --> 00:45:48.475 align:middle line:90% of your material. 00:45:48.475 --> 00:45:52.650 align:middle line:90% 00:45:52.650 --> 00:45:55.030 align:middle line:84% And this is one way of proceeding, and it has-- 00:45:55.030 --> 00:45:58.380 align:middle line:90% 00:45:58.380 --> 00:46:06.350 align:middle line:84% it has been the way for most of the applications. 00:46:06.350 --> 00:46:12.500 align:middle line:84% However, there is some arbitrariness 00:46:12.500 --> 00:46:20.630 align:middle line:84% in saying, as we've seen here, that, for example, the heat 00:46:20.630 --> 00:46:29.090 align:middle line:84% transfer is not related to some other degrees of disequilibrium. 00:46:29.090 --> 00:46:32.450 align:middle line:84% And in fact, there is this possibility, 00:46:32.450 --> 00:46:38.390 align:middle line:84% and the matrix of possibilities is this one. 00:46:38.390 --> 00:46:42.500 align:middle line:84% Here is all your degrees of disequilibrium, possible ones, 00:46:42.500 --> 00:46:45.400 align:middle line:90% and here is all the fluxes. 00:46:45.400 --> 00:46:48.730 align:middle line:84% And those laws that we have seen in the previous slide 00:46:48.730 --> 00:46:56.730 align:middle line:84% are just the diagonal terms in this matrix of coupling. 00:46:56.730 --> 00:47:01.480 align:middle line:84% They presume a direct coupling between-- 00:47:01.480 --> 00:47:05.950 align:middle line:84% I mean, a chemical reaction rate and the affinity 00:47:05.950 --> 00:47:13.090 align:middle line:84% of the reaction, a coupling between these mechanical rate 00:47:13.090 --> 00:47:14.890 align:middle line:90% of change of velocity-- 00:47:14.890 --> 00:47:17.050 align:middle line:84% I'm sorry, of volume of the fluid elements 00:47:17.050 --> 00:47:24.720 align:middle line:84% and this difference in pressures or these and the heat transfer 00:47:24.720 --> 00:47:25.290 align:middle line:90% and so on. 00:47:25.290 --> 00:47:30.580 align:middle line:84% But as you see, at least part of this matrix 00:47:30.580 --> 00:47:35.600 align:middle line:84% is filled with other possibilities that have names. 00:47:35.600 --> 00:47:41.770 align:middle line:84% So they correspond to effects that have been observed. 00:47:41.770 --> 00:47:43.190 align:middle line:90% And long ago, as you see. 00:47:43.190 --> 00:47:45.615 align:middle line:84% We're talking about the 19th century still. 00:47:45.615 --> 00:47:52.000 align:middle line:90% 00:47:52.000 --> 00:47:56.470 align:middle line:84% These are typically called cross effects 00:47:56.470 --> 00:48:02.860 align:middle line:84% because they happen in the off diagonal terms of this coupling 00:48:02.860 --> 00:48:04.240 align:middle line:90% matrix. 00:48:04.240 --> 00:48:06.280 align:middle line:84% And also you may have noticed that I haven't 00:48:06.280 --> 00:48:09.640 align:middle line:90% put anything in these parts. 00:48:09.640 --> 00:48:14.230 align:middle line:84% So why don't I fill completely this matrix? 00:48:14.230 --> 00:48:20.210 align:middle line:84% Well, in principle, you should, except that there 00:48:20.210 --> 00:48:28.260 align:middle line:84% are many applications in which your fluid is isotropic. 00:48:28.260 --> 00:48:33.630 align:middle line:84% If the fluid is isotropic, if the symmetry of your problem, 00:48:33.630 --> 00:48:41.270 align:middle line:84% the material, as well as the boundary conditions 00:48:41.270 --> 00:48:48.160 align:middle line:84% are isotropic, then there is this principle. 00:48:48.160 --> 00:49:01.500 align:middle line:84% Symmetry takes-- it's saying and it 00:49:01.500 --> 00:49:06.780 align:middle line:84% implements what Pierre Curie described with this words, 00:49:06.780 --> 00:49:08.910 align:middle line:84% that the symmetry of the cause must 00:49:08.910 --> 00:49:12.840 align:middle line:90% be preserved into its effects. 00:49:12.840 --> 00:49:18.730 align:middle line:84% So what this means is that if you are in isotropic conditions, 00:49:18.730 --> 00:49:23.280 align:middle line:84% there is no way you can couple a phenomenon which is 00:49:23.280 --> 00:49:27.390 align:middle line:84% of, say, a vectorial character with one which 00:49:27.390 --> 00:49:31.720 align:middle line:84% is of a different character, like a scalar or a tensor. 00:49:31.720 --> 00:49:38.350 align:middle line:84% So tensors can couple only with tensors and scalars 00:49:38.350 --> 00:49:41.340 align:middle line:90% only with scalars-- 00:49:41.340 --> 00:49:43.590 align:middle line:90% and sorry here are the scalars-- 00:49:43.590 --> 00:49:45.245 align:middle line:90% and vectors only with vectors. 00:49:45.245 --> 00:49:49.820 align:middle line:90% 00:49:49.820 --> 00:49:51.980 align:middle line:84% But remember also that this is true only 00:49:51.980 --> 00:49:53.700 align:middle line:90% for isotropic conditions. 00:49:53.700 --> 00:49:57.910 align:middle line:90% 00:49:57.910 --> 00:50:04.870 align:middle line:84% Because, for example-- there is an important aspect 00:50:04.870 --> 00:50:14.150 align:middle line:84% of when you have, for example, an interface. 00:50:14.150 --> 00:50:19.130 align:middle line:84% So this is supposed to represent an interface between two phases, 00:50:19.130 --> 00:50:27.700 align:middle line:84% for example, let's say f on one side and g on the other side-- 00:50:27.700 --> 00:50:33.151 align:middle line:84% two different phases separated by an interface. 00:50:33.151 --> 00:50:37.360 align:middle line:84% In an interface,-- ok, let's view it also 00:50:37.360 --> 00:50:38.710 align:middle line:90% in this other point of view-- 00:50:38.710 --> 00:50:41.270 align:middle line:90% 00:50:41.270 --> 00:50:45.960 align:middle line:90% f-- and let's plot the density. 00:50:45.960 --> 00:50:58.350 align:middle line:84% Now, as we know, if we plot the density of a liquid 00:50:58.350 --> 00:51:03.320 align:middle line:84% and the density of a vapor, there is a big difference. 00:51:03.320 --> 00:51:05.130 align:middle line:90% The vapor is about-- 00:51:05.130 --> 00:51:11.130 align:middle line:84% for water, is about 1,000 times less dense than the liquid. 00:51:11.130 --> 00:51:15.370 align:middle line:84% So in the interface, things are not sharp. 00:51:15.370 --> 00:51:19.822 align:middle line:90% Things are smooth. 00:51:19.822 --> 00:51:21.970 align:middle line:90% There is this smooth interface. 00:51:21.970 --> 00:51:28.010 align:middle line:84% There is a layer-- there is a small layer in which there is 00:51:28.010 --> 00:51:33.750 align:middle line:90% this huge gradient in density. 00:51:33.750 --> 00:51:37.800 align:middle line:84% And this is at equilibrium, so this is not the kind of gradient 00:51:37.800 --> 00:51:42.000 align:middle line:90% that would cause fluxes. 00:51:42.000 --> 00:51:44.170 align:middle line:84% It's not a degree of disequilibrium. 00:51:44.170 --> 00:51:46.980 align:middle line:84% This happens at equilibrium, But still, you 00:51:46.980 --> 00:51:49.470 align:middle line:90% have a gradient there. 00:51:49.470 --> 00:51:57.840 align:middle line:84% And the description of this situation 00:51:57.840 --> 00:52:00.240 align:middle line:90% can be done in several ways. 00:52:00.240 --> 00:52:06.270 align:middle line:84% Gibbs did it by assuming a sharp equivalent jump from this value 00:52:06.270 --> 00:52:10.390 align:middle line:84% to that value, which is a good approach. 00:52:10.390 --> 00:52:15.960 align:middle line:84% And I will direct you to a very nice (1976) paper 00:52:15.960 --> 00:52:21.220 align:middle line:84% by Bedeaux, Mazur and (Albano) another author-- 00:52:21.220 --> 00:52:23.580 align:middle line:90% I forget the name now-- 00:52:23.580 --> 00:52:29.830 align:middle line:84% that writes exactly the equations 00:52:29.830 --> 00:52:36.190 align:middle line:84% that we have written here, these extrinsic relation 00:52:36.190 --> 00:52:46.975 align:middle line:84% for sigma in the case of a jump description between two phases. 00:52:46.975 --> 00:52:51.517 align:middle line:90% 00:52:51.517 --> 00:52:58.440 align:middle line:84% In this situation here, definitely there is no isotropy 00:52:58.440 --> 00:53:07.260 align:middle line:84% because one thing is to move across this interface, 00:53:07.260 --> 00:53:10.960 align:middle line:84% and one thing is to move along the interface. 00:53:10.960 --> 00:53:15.990 align:middle line:84% So if you sit on the interface and move 00:53:15.990 --> 00:53:23.250 align:middle line:84% on the interfacial region, maybe that 00:53:23.250 --> 00:53:30.030 align:middle line:84% is isotropic in the tangent direction. 00:53:30.030 --> 00:53:34.800 align:middle line:84% But it's not in the orthogonal direction. 00:53:34.800 --> 00:53:44.700 align:middle line:84% So for example, in this case, the stress tensor which is-- 00:53:44.700 --> 00:53:49.620 align:middle line:84% you can represent it as a 3 by 3 matrix-- 00:53:49.620 --> 00:53:50.120 align:middle line:90% splits. 00:53:50.120 --> 00:53:53.230 align:middle line:90% 00:53:53.230 --> 00:54:00.130 align:middle line:84% And so you have a pressure, an orthogonal pressure 00:54:00.130 --> 00:54:05.690 align:middle line:84% and a tangent pressure-- and a tangent tensor field. 00:54:05.690 --> 00:54:09.010 align:middle line:84% So you have a two dimensional tensor 00:54:09.010 --> 00:54:17.800 align:middle line:84% field, which is isotropic, and it represents the stress 00:54:17.800 --> 00:54:25.090 align:middle line:84% component in the tangential part plus a scalar-- 00:54:25.090 --> 00:54:31.870 align:middle line:84% a scalar for-- the orthogonal pressure becomes a scalar. 00:54:31.870 --> 00:54:35.470 align:middle line:84% And so in this situation, this is 00:54:35.470 --> 00:54:37.330 align:middle line:84% very interesting because that scalar 00:54:37.330 --> 00:54:42.130 align:middle line:84% now, since it is a scalar, it can couple 00:54:42.130 --> 00:54:45.910 align:middle line:90% with all these scalars here. 00:54:45.910 --> 00:54:52.330 align:middle line:84% So you may have the possibility at interfaces 00:54:52.330 --> 00:54:59.530 align:middle line:84% to couple chemistry and the flow field, 00:54:59.530 --> 00:55:03.015 align:middle line:84% and the pressure field, the orthogonal pressure. 00:55:03.015 --> 00:55:09.080 align:middle line:90% 00:55:09.080 --> 00:55:18.060 align:middle line:84% And that is I think used in models also 00:55:18.060 --> 00:55:29.110 align:middle line:84% for biological systems because that coupling is important. 00:55:29.110 --> 00:55:35.930 align:middle line:84% So it's mechanical and chemical coupling. 00:55:35.930 --> 00:55:40.340 align:middle line:84% But you need to break the isotropy in order 00:55:40.340 --> 00:55:47.256 align:middle line:84% to have this kind of, let's say, exotic couplings. 00:55:47.256 --> 00:55:51.550 align:middle line:90% 00:55:51.550 --> 00:55:57.870 align:middle line:84% Now, we have already talked about Onsager relations. 00:55:57.870 --> 00:56:02.010 align:middle line:84% And you remember that we did it in the context of chemical 00:56:02.010 --> 00:56:04.500 align:middle line:90% reactions, and we did it-- 00:56:04.500 --> 00:56:08.730 align:middle line:90% we proved the reciprocity-- 00:56:08.730 --> 00:56:10.860 align:middle line:90% Onsager reciprocity theorem. 00:56:10.860 --> 00:56:12.690 align:middle line:84% We have shown that in the linear regime, 00:56:12.690 --> 00:56:15.030 align:middle line:84% we could prove it as a consequence 00:56:15.030 --> 00:56:19.835 align:middle line:84% of the principle of maximum entropy production by Ziegler. 00:56:19.835 --> 00:56:22.830 align:middle line:90% 00:56:22.830 --> 00:56:29.240 align:middle line:84% Here, I'm not discussing this slide a lot 00:56:29.240 --> 00:56:32.360 align:middle line:84% because, essentially, it is exactly the same slide 00:56:32.360 --> 00:56:37.820 align:middle line:84% that we had then where I changed some of the symbols. 00:56:37.820 --> 00:56:42.130 align:middle line:84% Here, instead of sigma, we had entropy production-- 00:56:42.130 --> 00:56:46.520 align:middle line:90% S dot irreversible. 00:56:46.520 --> 00:56:51.990 align:middle line:84% Here, we had the reaction rate, so the epsilon dots. 00:56:51.990 --> 00:56:55.110 align:middle line:84% And here, we had the affinities of the reactions. 00:56:55.110 --> 00:57:04.020 align:middle line:84% And instead of these super dot symbol, 00:57:04.020 --> 00:57:06.205 align:middle line:84% we just had nothing because they were scalars. 00:57:06.205 --> 00:57:08.980 align:middle line:90% 00:57:08.980 --> 00:57:11.390 align:middle line:84% And the procedure was just the same. 00:57:11.390 --> 00:57:20.780 align:middle line:84% We proved-- we assumed that the material in the linear regime-- 00:57:20.780 --> 00:57:24.460 align:middle line:90% so we assume a linear relation-- 00:57:24.460 --> 00:57:27.610 align:middle line:84% and the linear relation is, therefore, 00:57:27.610 --> 00:57:36.610 align:middle line:84% in terms of this tensor here, L. It's actually a super tensor 00:57:36.610 --> 00:57:40.990 align:middle line:84% because it has as many entries as that matrix 00:57:40.990 --> 00:57:47.990 align:middle line:84% that we just had in the previous slide with some zeros around. 00:57:47.990 --> 00:57:52.000 align:middle line:84% But it's not a full because of the Curie principal, 00:57:52.000 --> 00:57:55.095 align:middle line:90% but it's still-- 00:57:55.095 --> 00:58:00.710 align:middle line:84% OK, so this is what we assume, a linear relation. 00:58:00.710 --> 00:58:03.920 align:middle line:84% The bottom line will be that these super tensor 00:58:03.920 --> 00:58:05.340 align:middle line:90% has to be symmetric. 00:58:05.340 --> 00:58:08.280 align:middle line:90% 00:58:08.280 --> 00:58:13.900 align:middle line:84% And the assumption, in order to obtain this result, 00:58:13.900 --> 00:58:17.170 align:middle line:84% is that the entropy production should 00:58:17.170 --> 00:58:20.980 align:middle line:84% be maximal subject to the constraint given 00:58:20.980 --> 00:58:25.590 align:middle line:90% by the extrinsic relation. 00:58:25.590 --> 00:58:30.720 align:middle line:84% And in passing, we obtain this orthogonality-- 00:58:30.720 --> 00:58:35.250 align:middle line:84% Ziegler orthogonality relation that we also 00:58:35.250 --> 00:58:39.560 align:middle line:84% described graphically in one of the previous lectures. 00:58:39.560 --> 00:58:43.890 align:middle line:90% 00:58:43.890 --> 00:58:46.730 align:middle line:84% So fine, you can proceed this way. 00:58:46.730 --> 00:58:49.940 align:middle line:90% 00:58:49.940 --> 00:58:53.690 align:middle line:84% Zeigler actually introduced this idea 00:58:53.690 --> 00:59:00.170 align:middle line:84% of maximum entropy production because his objective was to-- 00:59:00.170 --> 00:59:03.720 align:middle line:84% he was a material science, I guess, person. 00:59:03.720 --> 00:59:09.836 align:middle line:84% And he wanted to be able to describe things like plasticity. 00:59:09.836 --> 00:59:12.840 align:middle line:90% 00:59:12.840 --> 00:59:17.160 align:middle line:84% And plasticity doesn't have a linear regime. 00:59:17.160 --> 00:59:22.650 align:middle line:84% So we wanted to extend Onsager reciprocity 00:59:22.650 --> 00:59:25.800 align:middle line:90% to nonlinear regimes. 00:59:25.800 --> 00:59:29.790 align:middle line:84% And in fact, here is how you do it, again, 00:59:29.790 --> 00:59:35.330 align:middle line:84% just to skip the details, but just to give you a flavor. 00:59:35.330 --> 00:59:37.660 align:middle line:84% All it takes is to-- instead of assuming, 00:59:37.660 --> 00:59:40.790 align:middle line:84% like we did in the previous slide, a linear relation 00:59:40.790 --> 00:59:50.160 align:middle line:84% between forces and fluxes, we assume a generic relation 00:59:50.160 --> 00:59:53.500 align:middle line:84% and still apply the principle of maximum entropy production. 00:59:53.500 --> 00:59:59.130 align:middle line:84% So maximize sigma subject to the extrinsic relation 00:59:59.130 --> 01:00:04.010 align:middle line:90% between the forces and fluxes. 01:00:04.010 --> 01:00:08.310 align:middle line:84% And you come up with an orthogonality condition 01:00:08.310 --> 01:00:10.600 align:middle line:84% in terms of the Lagrange multiplier, 01:00:10.600 --> 01:00:16.120 align:middle line:84% which is still only 1 because it's only one constraint. 01:00:16.120 --> 01:00:18.820 align:middle line:90% The only problem is that now-- 01:00:18.820 --> 01:00:20.500 align:middle line:90% remember, when you have-- 01:00:20.500 --> 01:00:23.230 align:middle line:84% in the method of Lagrange multipliers, 01:00:23.230 --> 01:00:26.090 align:middle line:84% in order to find the value of the Lagrange multiplier, 01:00:26.090 --> 01:00:29.210 align:middle line:84% you have to substitute back into the constraint. 01:00:29.210 --> 01:00:32.050 align:middle line:84% So you take this orthogonality relation this way, 01:00:32.050 --> 01:00:36.380 align:middle line:84% put it back in the constraint-- and this is done here-- 01:00:36.380 --> 01:00:41.360 align:middle line:90% and then solve for lambda. 01:00:41.360 --> 01:00:45.030 align:middle line:84% In the case of linear case, we found that lambda is equal to 2. 01:00:45.030 --> 01:00:47.780 align:middle line:84% Here, it's more complicated because lambda 01:00:47.780 --> 01:00:53.330 align:middle line:84% is this function of whatever functional 01:00:53.330 --> 01:00:58.610 align:middle line:84% form you have chosen for your relations between forces 01:00:58.610 --> 01:01:02.840 align:middle line:84% and fluxes, which is non linear, in general. 01:01:02.840 --> 01:01:05.030 align:middle line:84% So you can substitute these into here, 01:01:05.030 --> 01:01:08.930 align:middle line:84% and you get this complicated stuff, which nevertheless, 01:01:08.930 --> 01:01:11.120 align:middle line:90% is an orthogonality. 01:01:11.120 --> 01:01:15.530 align:middle line:84% It's a condition that must be satisfied by the functional 01:01:15.530 --> 01:01:19.520 align:middle line:90% relation of your-- 01:01:19.520 --> 01:01:21.150 align:middle line:90% that you have chosen here. 01:01:21.150 --> 01:01:24.320 align:middle line:84% So it tells you that you don't have complete freedom 01:01:24.320 --> 01:01:27.320 align:middle line:90% in choosing these functions. 01:01:27.320 --> 01:01:30.010 align:middle line:90% They have to satisfy this. 01:01:30.010 --> 01:01:34.420 align:middle line:84% For example, if you choose the linear relation, which 01:01:34.420 --> 01:01:40.760 align:middle line:84% is this one, this expression reduces 01:01:40.760 --> 01:01:47.170 align:middle line:84% to the one that required the symmetry of the Onsager 01:01:47.170 --> 01:01:51.500 align:middle line:90% conductivity tensor. 01:01:51.500 --> 01:01:55.910 align:middle line:84% In a general case, you get something else, 01:01:55.910 --> 01:01:59.240 align:middle line:84% which is a generalization to the nonlinear 01:01:59.240 --> 01:02:05.960 align:middle line:84% case of Onsager relations, à la Zeigler, so to speak. 01:02:05.960 --> 01:02:10.990 align:middle line:90% 01:02:10.990 --> 01:02:15.910 align:middle line:84% There is another way of viewing things. 01:02:15.910 --> 01:02:17.836 align:middle line:90% They are all sort of equivalent. 01:02:17.836 --> 01:02:24.220 align:middle line:90% 01:02:24.220 --> 01:02:35.050 align:middle line:84% And let me try to explain it in this heat transfer example. 01:02:35.050 --> 01:02:37.730 align:middle line:84% They are all equivalent, and there's so much. 01:02:37.730 --> 01:02:39.110 align:middle line:90% And as you see-- 01:02:39.110 --> 01:02:44.200 align:middle line:84% I'll say a little bit later about this-- 01:02:44.200 --> 01:02:49.750 align:middle line:84% I collected this point of view in a paper a couple of years 01:02:49.750 --> 01:02:50.250 align:middle line:90% ago. 01:02:50.250 --> 01:02:53.490 align:middle line:90% 01:02:53.490 --> 01:03:03.620 align:middle line:84% And let's do it for the heat transfer case. 01:03:03.620 --> 01:03:10.210 align:middle line:84% So consider this an anisotropic material made of-- 01:03:10.210 --> 01:03:12.160 align:middle line:84% so that the microstructure is made 01:03:12.160 --> 01:03:19.180 align:middle line:84% of layers sandwiched of high conductivity 01:03:19.180 --> 01:03:21.110 align:middle line:90% and low conductivity materials. 01:03:21.110 --> 01:03:24.320 align:middle line:90% 01:03:24.320 --> 01:03:33.040 align:middle line:84% And so suppose you subject these material to a-- 01:03:33.040 --> 01:03:36.210 align:middle line:84% you impose a gradient in temperature. 01:03:36.210 --> 01:03:42.050 align:middle line:90% The material, then, is under a-- 01:03:42.050 --> 01:03:45.720 align:middle line:84% I shouldn't say stress-- but in any case, it has to decide. 01:03:45.720 --> 01:03:51.470 align:middle line:84% All right, how do I react to that gradient in temperature 01:03:51.470 --> 01:03:52.210 align:middle line:90% that you impose? 01:03:52.210 --> 01:03:55.640 align:middle line:90% 01:03:55.640 --> 01:04:03.694 align:middle line:84% Definitely I want to react in a way that will allow me to, 01:04:03.694 --> 01:04:09.020 align:middle line:84% like Le Chatelier-Braun says, to reduce 01:04:09.020 --> 01:04:15.220 align:middle line:84% that gradient in temperature and try to reestablish equilibrium. 01:04:15.220 --> 01:04:24.030 align:middle line:84% But you see here, what I am showing 01:04:24.030 --> 01:04:26.590 align:middle line:84% is that there are many possibilities. 01:04:26.590 --> 01:04:31.890 align:middle line:84% It could decide to move heat in one direction or in another. 01:04:31.890 --> 01:04:36.510 align:middle line:84% And how come it chooses this particular direction? 01:04:36.510 --> 01:04:49.320 align:middle line:84% In some sense, it's like, suppose you are in a crowd. 01:04:49.320 --> 01:04:53.450 align:middle line:84% You're in the middle of a lot of people, maybe at a concert 01:04:53.450 --> 01:04:58.650 align:middle line:84% or going into or out of a soccer stadium. 01:04:58.650 --> 01:05:00.530 align:middle line:84% Lots of people, and everybody wants 01:05:00.530 --> 01:05:05.540 align:middle line:84% to reach or get next to the podium or to the entrance 01:05:05.540 --> 01:05:07.970 align:middle line:90% or go to your car. 01:05:07.970 --> 01:05:10.740 align:middle line:84% But you have a crowd in front of you. 01:05:10.740 --> 01:05:16.520 align:middle line:84% And so even though the direct path 01:05:16.520 --> 01:05:22.200 align:middle line:84% could be apparently shorter, but it would require less effort 01:05:22.200 --> 01:05:25.440 align:middle line:84% for you to move in another direction 01:05:25.440 --> 01:05:30.100 align:middle line:84% where you see less resistance to your motion. 01:05:30.100 --> 01:05:35.410 align:middle line:90% So this is what happens here. 01:05:35.410 --> 01:05:42.750 align:middle line:84% So there is-- you evaluate, and the material does a local 01:05:42.750 --> 01:05:47.230 align:middle line:90% evaluation sensing-- 01:05:47.230 --> 01:05:52.240 align:middle line:84% OK, how much would it cost me to move in terms of effort 01:05:52.240 --> 01:05:54.100 align:middle line:90% or in terms of-- 01:05:54.100 --> 01:05:57.640 align:middle line:84% yes, effort in order to move in the crowd? 01:05:57.640 --> 01:06:00.760 align:middle line:84% How much would it cost me to go in this direction 01:06:00.760 --> 01:06:02.130 align:middle line:90% or in that direction? 01:06:02.130 --> 01:06:08.850 align:middle line:84% And so there is this tensor G, which I would call a metric. 01:06:08.850 --> 01:06:12.720 align:middle line:90% It's the driver of that logic. 01:06:12.720 --> 01:06:19.710 align:middle line:84% It measures how dissipative is moving 01:06:19.710 --> 01:06:21.840 align:middle line:90% in the various directions. 01:06:21.840 --> 01:06:31.290 align:middle line:84% And if you write an equation like this, 01:06:31.290 --> 01:06:38.460 align:middle line:84% the quadratic form with that G for given heat 01:06:38.460 --> 01:06:43.870 align:middle line:84% fluxes and set it equal to some constant. 01:06:43.870 --> 01:06:48.870 align:middle line:84% That gives you an equation of an ellipsoid. 01:06:48.870 --> 01:06:52.050 align:middle line:84% And that ellipsoid represents the locus 01:06:52.050 --> 01:06:58.740 align:middle line:84% of all the tips of the vectors that 01:06:58.740 --> 01:07:06.796 align:middle line:84% would cost the material the same effort, so the same dissipation. 01:07:06.796 --> 01:07:11.340 align:middle line:90% 01:07:11.340 --> 01:07:19.850 align:middle line:84% And so it turns out that the entropy production 01:07:19.850 --> 01:07:22.580 align:middle line:84% is the scalar product between the blue vector 01:07:22.580 --> 01:07:25.990 align:middle line:90% and the red vector. 01:07:25.990 --> 01:07:30.160 align:middle line:84% And the scalar product here, between two vectors, 01:07:30.160 --> 01:07:34.790 align:middle line:84% is given by the projection of one vector onto the other one. 01:07:34.790 --> 01:07:36.940 align:middle line:84% So for example, this blue one, you 01:07:36.940 --> 01:07:41.980 align:middle line:84% see that this blue one, the actual full, not 01:07:41.980 --> 01:07:45.910 align:middle line:84% the dotted ones, is the one that has the maximum projection 01:07:45.910 --> 01:07:46.970 align:middle line:90% onto the red vector. 01:07:46.970 --> 01:07:49.570 align:middle line:90% So it maximizes-- 01:07:49.570 --> 01:07:55.330 align:middle line:84% Among all these possibilities that have equal effort, 01:07:55.330 --> 01:08:00.735 align:middle line:84% the one that maximizes sigma is this blue one, 01:08:00.735 --> 01:08:02.610 align:middle line:84% and that's the one that the material chooses. 01:08:02.610 --> 01:08:09.690 align:middle line:90% 01:08:09.690 --> 01:08:12.360 align:middle line:84% It turns out that G is the inverse 01:08:12.360 --> 01:08:16.460 align:middle line:90% of the conductivity tensor. 01:08:16.460 --> 01:08:23.620 align:middle line:84% So you can read Fourier law for an anisotropic material 01:08:23.620 --> 01:08:28.779 align:middle line:84% in this way, that the material is choosing 01:08:28.779 --> 01:08:37.004 align:middle line:84% the direction of maximal entropy production for a given effort. 01:08:37.004 --> 01:08:40.260 align:middle line:90% 01:08:40.260 --> 01:08:43.080 align:middle line:84% Like in your crowd, you choose the direction 01:08:43.080 --> 01:08:50.520 align:middle line:84% with which you feel that you are moving farther away with a given 01:08:50.520 --> 01:08:52.800 align:middle line:84% effort, for a given amount of effort 01:08:52.800 --> 01:08:54.394 align:middle line:90% that you put into that movement. 01:08:54.394 --> 01:08:58.310 align:middle line:90% 01:08:58.310 --> 01:09:05.380 align:middle line:84% So these are words just to explain what I just said. 01:09:05.380 --> 01:09:08.050 align:middle line:90% 01:09:08.050 --> 01:09:15.075 align:middle line:90% And so if you-- 01:09:15.075 --> 01:09:20.520 align:middle line:90% 01:09:20.520 --> 01:09:24.149 align:middle line:84% so you could have an isotropic material. 01:09:24.149 --> 01:09:26.700 align:middle line:84% For an isotropic material, that ellipsoid 01:09:26.700 --> 01:09:30.810 align:middle line:90% becomes a sphere, a hypersphere. 01:09:30.810 --> 01:09:33.819 align:middle line:90% All directions are equivalent. 01:09:33.819 --> 01:09:38.240 align:middle line:84% And if all directions are equivalent, 01:09:38.240 --> 01:09:42.950 align:middle line:84% which one is the direction of steepest entropy ascent? 01:09:42.950 --> 01:09:45.950 align:middle line:84% It's the same as the gradient of T 01:09:45.950 --> 01:09:55.250 align:middle line:84% because the vector q, has the maximal projection 01:09:55.250 --> 01:10:02.880 align:middle line:84% on the red vector when they are co-linear. 01:10:02.880 --> 01:10:09.830 align:middle line:84% And so here is the construction in that same diagram 01:10:09.830 --> 01:10:17.983 align:middle line:84% for which we have seen the orthogonality Ziegler relation. 01:10:17.983 --> 01:10:21.380 align:middle line:90% 01:10:21.380 --> 01:10:30.920 align:middle line:84% So we have an x and y or 1 and 2 versors. 01:10:30.920 --> 01:10:34.220 align:middle line:90% So think of it as x and y. 01:10:34.220 --> 01:10:38.100 align:middle line:84% I impose a gradient in temperature, 01:10:38.100 --> 01:10:39.786 align:middle line:90% which has two components. 01:10:39.786 --> 01:10:44.580 align:middle line:90% 01:10:44.580 --> 01:10:50.150 align:middle line:90% And that places me-- 01:10:50.150 --> 01:10:56.100 align:middle line:84% the two components places me somewhere, not at equilibrium. 01:10:56.100 --> 01:10:58.080 align:middle line:84% If they are zero, I'm at the center. 01:10:58.080 --> 01:11:00.840 align:middle line:84% So I'm already in an equilibrium state, and nothing happens. 01:11:00.840 --> 01:11:03.800 align:middle line:84% But if I'm off equilibrium, these 01:11:03.800 --> 01:11:07.640 align:middle line:84% are the lines of contours of constant entropy. 01:11:07.640 --> 01:11:10.830 align:middle line:84% So the higher the gradient, the further I am, 01:11:10.830 --> 01:11:13.443 align:middle line:90% and the highest is the entropy. 01:11:13.443 --> 01:11:18.890 align:middle line:90% 01:11:18.890 --> 01:11:25.570 align:middle line:84% I'm sorry-- the highest is the degree of disequilibrium. 01:11:25.570 --> 01:11:27.130 align:middle line:90% This is the maximum entropy. 01:11:27.130 --> 01:11:31.260 align:middle line:84% So the entropy goes decreasing as I go farther. 01:11:31.260 --> 01:11:34.120 align:middle line:90% 01:11:34.120 --> 01:11:44.610 align:middle line:84% So if I am here and that degree of disequilibrium has been 01:11:44.610 --> 01:11:46.990 align:middle line:84% generated at one instant of time, 01:11:46.990 --> 01:11:50.340 align:middle line:84% and then the system is allowed to evolve by itself, 01:11:50.340 --> 01:11:54.690 align:middle line:84% as if it were isolated-- nothing else, no constraints, 01:11:54.690 --> 01:11:58.630 align:middle line:90% just you can go to equilibrium-- 01:11:58.630 --> 01:12:01.730 align:middle line:90% 01:12:01.730 --> 01:12:07.330 align:middle line:84% it doesn't go to equilibrium in the straight path. 01:12:07.330 --> 01:12:13.760 align:middle line:84% It follows the path that is dictated by its local metric. 01:12:13.760 --> 01:12:17.965 align:middle line:84% Now, true, the isotropic material is-- 01:12:17.965 --> 01:12:20.500 align:middle line:90% 01:12:20.500 --> 01:12:22.220 align:middle line:90% doesn't favor any direction. 01:12:22.220 --> 01:12:26.600 align:middle line:84% So these, although it doesn't show exactly as a sphere, 01:12:26.600 --> 01:12:28.180 align:middle line:90% it should be a circle. 01:12:28.180 --> 01:12:34.370 align:middle line:84% In any case, it has to evolve, though, 01:12:34.370 --> 01:12:40.000 align:middle line:84% in a scenario in which the entropy contour lines may not 01:12:40.000 --> 01:12:41.590 align:middle line:90% be also circles. 01:12:41.590 --> 01:12:45.010 align:middle line:84% If they were also circles, you would go down straight. 01:12:45.010 --> 01:12:50.830 align:middle line:84% But if they are not-- like here, they are ellipses-- then 01:12:50.830 --> 01:12:58.810 align:middle line:84% it would choose, among the ones that are all equivalent effort, 01:12:58.810 --> 01:13:02.530 align:middle line:84% it chooses the one that takes you to the highest entropy 01:13:02.530 --> 01:13:03.320 align:middle line:90% possible. 01:13:03.320 --> 01:13:06.270 align:middle line:84% So it's the one that is tangent here, 01:13:06.270 --> 01:13:10.490 align:middle line:84% and that is also for this other metric. 01:13:10.490 --> 01:13:15.680 align:middle line:84% And then, of course, you can proceed and go on as you-- 01:13:15.680 --> 01:13:20.560 align:middle line:84% at each point, you insert, stop, and pause. 01:13:20.560 --> 01:13:23.000 align:middle line:84% It doesn't happen, of course, but you 01:13:23.000 --> 01:13:25.610 align:middle line:84% stop and consider which one is the direction in which it 01:13:25.610 --> 01:13:27.600 align:middle line:90% should go the next step. 01:13:27.600 --> 01:13:31.690 align:middle line:84% And that's how it gets to equilibrium. 01:13:31.690 --> 01:13:33.930 align:middle line:84% They all get to equilibrium, of course, 01:13:33.930 --> 01:13:35.900 align:middle line:90% but along different paths. 01:13:35.900 --> 01:13:39.630 align:middle line:84% And if I start even from this exactly same state, 01:13:39.630 --> 01:13:48.460 align:middle line:84% but with a material that is oriented differently, 01:13:48.460 --> 01:13:59.140 align:middle line:84% and therefore, for it the metric and these ellipsoids 01:13:59.140 --> 01:14:03.310 align:middle line:84% of equal dissipation, of equal effort 01:14:03.310 --> 01:14:05.200 align:middle line:90% is oriented in a different way. 01:14:05.200 --> 01:14:07.540 align:middle line:84% And therefore, it would move there, 01:14:07.540 --> 01:14:10.960 align:middle line:90% and then it will search again. 01:14:10.960 --> 01:14:15.380 align:middle line:84% And it will move, maybe even overshoots a little bit, 01:14:15.380 --> 01:14:18.010 align:middle line:84% and then goes back to equilibrium. 01:14:18.010 --> 01:14:31.270 align:middle line:84% So this is the role of the metric that describes-- 01:14:31.270 --> 01:14:33.020 align:middle line:84% This is for a heat transfer problem, 01:14:33.020 --> 01:14:37.690 align:middle line:84% but you can also view it in a more general context. 01:14:37.690 --> 01:14:39.740 align:middle line:90% And that is what-- 01:14:39.740 --> 01:14:45.910 align:middle line:84% I like to call it the Fourth Law, 01:14:45.910 --> 01:14:52.710 align:middle line:84% simply because it is something that has emerged. 01:14:52.710 --> 01:14:53.980 align:middle line:90% Let me see, do I have-- 01:14:53.980 --> 01:14:56.660 align:middle line:90% I think I had-- 01:14:56.660 --> 01:14:59.540 align:middle line:90% keep missing slides today. 01:14:59.540 --> 01:15:04.070 align:middle line:84% There will be a slide sometime with a list 01:15:04.070 --> 01:15:11.030 align:middle line:84% of many different theories of non equilibrium over the last 40 01:15:11.030 --> 01:15:16.410 align:middle line:84% years in which people in different languages, 01:15:16.410 --> 01:15:20.300 align:middle line:84% different mathematics, different frameworks. 01:15:20.300 --> 01:15:22.320 align:middle line:84% Often, they don't talk to each other. 01:15:22.320 --> 01:15:24.470 align:middle line:84% They completely ignore each other. 01:15:24.470 --> 01:15:26.850 align:middle line:84% They wouldn't admit that they're doing the same thing, 01:15:26.850 --> 01:15:28.910 align:middle line:84% but they are doing the same thing. 01:15:28.910 --> 01:15:35.900 align:middle line:84% And that is what I was trying to point out to the community 01:15:35.900 --> 01:15:38.100 align:middle line:84% here by calling it the Fourth Law. 01:15:38.100 --> 01:15:41.240 align:middle line:90% And why do I-- 01:15:41.240 --> 01:15:46.370 align:middle line:84% why do I dare elevating it to such a high level 01:15:46.370 --> 01:15:50.840 align:middle line:84% in the hierarchy of physical laws? 01:15:50.840 --> 01:15:54.410 align:middle line:84% That's because of the generality. 01:15:54.410 --> 01:16:00.960 align:middle line:84% Remember, the great conservation principles are great-- 01:16:00.960 --> 01:16:03.940 align:middle line:84% the greatness comes from the generality. 01:16:03.940 --> 01:16:08.760 align:middle line:84% It should hold for every model of physical reality. 01:16:08.760 --> 01:16:11.040 align:middle line:84% Well, it turns out that in every model 01:16:11.040 --> 01:16:15.150 align:middle line:84% of non-equilibrium dynamics, there 01:16:15.150 --> 01:16:19.170 align:middle line:84% is something that, maybe you don't call it a metric, 01:16:19.170 --> 01:16:20.350 align:middle line:90% but it is. 01:16:20.350 --> 01:16:22.720 align:middle line:84% It has the properties of a metric. 01:16:22.720 --> 01:16:29.860 align:middle line:84% And therefore, it appears to be an essential feature 01:16:29.860 --> 01:16:32.420 align:middle line:90% of any model. 01:16:32.420 --> 01:16:37.730 align:middle line:84% And also, I like also the fact that it 01:16:37.730 --> 01:16:41.930 align:middle line:90% is a statement of existence. 01:16:41.930 --> 01:16:44.200 align:middle line:84% Like, the first law is a statement 01:16:44.200 --> 01:16:46.570 align:middle line:90% of existence of energy. 01:16:46.570 --> 01:16:50.080 align:middle line:84% The second law is a statement of existence of entropy. 01:16:50.080 --> 01:16:52.870 align:middle line:84% This is a statement of existence of a metric 01:16:52.870 --> 01:16:54.820 align:middle line:90% in non-equilibrium states. 01:16:54.820 --> 01:17:02.360 align:middle line:84% And it's a property, like this thermal conductivity, that 01:17:02.360 --> 01:17:06.650 align:middle line:84% has to be defined at every state, 01:17:06.650 --> 01:17:09.500 align:middle line:84% every non-equilibrium state of your system for it 01:17:09.500 --> 01:17:13.400 align:middle line:84% to decide in which direction to move when you let it alone 01:17:13.400 --> 01:17:17.118 align:middle line:84% and want to let it go towards equilibrium. 01:17:17.118 --> 01:17:23.600 align:middle line:90% 01:17:23.600 --> 01:17:30.920 align:middle line:84% Yeah, here, this is to show that the same construction can 01:17:30.920 --> 01:17:38.240 align:middle line:84% be also done more formally and mathematically, again, 01:17:38.240 --> 01:17:44.220 align:middle line:84% with the method of Lagrange multipliers. 01:17:44.220 --> 01:17:50.250 align:middle line:84% You see-- you maximize the scalar product between the X 01:17:50.250 --> 01:17:55.210 align:middle line:90% and the J, subject-- 01:17:55.210 --> 01:17:59.050 align:middle line:84% so that's the Lagrange multiplier-- 01:17:59.050 --> 01:18:04.490 align:middle line:84% to the constraint given by an equal effort ellipsoid. 01:18:04.490 --> 01:18:09.430 align:middle line:84% So if-- so it expresses exactly mathematically 01:18:09.430 --> 01:18:13.690 align:middle line:84% the word description that I gave. 01:18:13.690 --> 01:18:16.810 align:middle line:84% There is a metric that describes this ellipsoid. 01:18:16.810 --> 01:18:20.300 align:middle line:84% It tells you, for a given distance 01:18:20.300 --> 01:18:23.410 align:middle line:84% I want to move, which one is the best direction. 01:18:23.410 --> 01:18:26.620 align:middle line:84% And best is in terms of maximizing 01:18:26.620 --> 01:18:30.730 align:middle line:90% the rate of entropy production. 01:18:30.730 --> 01:18:35.840 align:middle line:84% From this one, using the method of Lagrange multiplier, 01:18:35.840 --> 01:18:39.370 align:middle line:84% setting the derivative equal to zero, 01:18:39.370 --> 01:18:41.270 align:middle line:84% substituting back into constraint, 01:18:41.270 --> 01:18:49.060 align:middle line:84% and so on, you get a relation between forces and fluxes. 01:18:49.060 --> 01:18:52.120 align:middle line:84% You can always write it in this way, which 01:18:52.120 --> 01:18:54.820 align:middle line:84% looks like a linear relation, but it's not 01:18:54.820 --> 01:19:04.150 align:middle line:84% because these R, which is a resistance, is not a constant, 01:19:04.150 --> 01:19:07.000 align:middle line:84% in general, but it's just a function 01:19:07.000 --> 01:19:10.060 align:middle line:84% of the fluxes which is given by the solution of this problem 01:19:10.060 --> 01:19:10.560 align:middle line:90% here. 01:19:10.560 --> 01:19:13.160 align:middle line:90% 01:19:13.160 --> 01:19:14.815 align:middle line:84% So this is a non linear relation. 01:19:14.815 --> 01:19:18.570 align:middle line:90% 01:19:18.570 --> 01:19:25.990 align:middle line:84% And of course, you can do the exact same in the force picture, 01:19:25.990 --> 01:19:31.720 align:middle line:84% and impose the X. And there is a metric 01:19:31.720 --> 01:19:35.080 align:middle line:84% also from that point of view, and the two 01:19:35.080 --> 01:19:36.570 align:middle line:90% are related to one another. 01:19:36.570 --> 01:19:39.880 align:middle line:90% They are the inverse somehow. 01:19:39.880 --> 01:19:43.690 align:middle line:84% So you also maximize these projections 01:19:43.690 --> 01:19:47.050 align:middle line:84% subject to these other blue ellipsoid, which 01:19:47.050 --> 01:19:50.820 align:middle line:90% also represents the effort. 01:19:50.820 --> 01:19:53.710 align:middle line:84% At this time, it's the effort for the system 01:19:53.710 --> 01:20:00.080 align:middle line:84% to set up a gradient in temperature 01:20:00.080 --> 01:20:05.980 align:middle line:84% so as to contrast the flux of heat 01:20:05.980 --> 01:20:09.100 align:middle line:90% that you have imposed in it. 01:20:09.100 --> 01:20:16.930 align:middle line:84% And of course, when you get close to equilibrium, 01:20:16.930 --> 01:20:20.070 align:middle line:84% then you go to the linear regime, 01:20:20.070 --> 01:20:21.820 align:middle line:90% you get Onsager relations. 01:20:21.820 --> 01:20:25.770 align:middle line:90% And of course, these two are-- 01:20:25.770 --> 01:20:35.280 align:middle line:84% the metric becomes the Onsager generalized conductivity 01:20:35.280 --> 01:20:42.037 align:middle line:84% for the Ls or resistivity for the Rs. 01:20:42.037 --> 01:20:43.370 align:middle line:90% One is the inverse of the other. 01:20:43.370 --> 01:20:46.110 align:middle line:90% 01:20:46.110 --> 01:20:58.510 align:middle line:84% OK, here is the list of theories that converge to the Fourth Law. 01:20:58.510 --> 01:21:04.300 align:middle line:90% 01:21:04.300 --> 01:21:07.390 align:middle line:84% Of course, I've-- we contributed to it. 01:21:07.390 --> 01:21:12.670 align:middle line:84% But in the same period, already you 01:21:12.670 --> 01:21:20.110 align:middle line:84% have Grmela, Morrison, Kaufman, who did the-- 01:21:20.110 --> 01:21:23.270 align:middle line:84% completely-- from completely different point of view. 01:21:23.270 --> 01:21:30.220 align:middle line:84% So, in fact, I met Miroslav Grmela 01:21:30.220 --> 01:21:34.015 align:middle line:90% only maybe 30 years later. 01:21:34.015 --> 01:21:36.560 align:middle line:90% 01:21:36.560 --> 01:21:47.290 align:middle line:84% And so these all are independent contributions. 01:21:47.290 --> 01:21:52.680 align:middle line:84% GENERIC has become very popular because some communities 01:21:52.680 --> 01:21:58.300 align:middle line:84% are more effective than others into propagating and publishing. 01:21:58.300 --> 01:22:00.900 align:middle line:90% 01:22:00.900 --> 01:22:03.160 align:middle line:84% I will say a few words also about the constructal law 01:22:03.160 --> 01:22:03.750 align:middle line:90% of Bejan. 01:22:03.750 --> 01:22:12.270 align:middle line:84% Although, he would be unhappy if I say that he belongs here 01:22:12.270 --> 01:22:16.470 align:middle line:84% because I think he believes that his point of view 01:22:16.470 --> 01:22:20.310 align:middle line:84% is more general, and I think I agree with him. 01:22:20.310 --> 01:22:25.560 align:middle line:84% I'll explain in a moment, I mean, later, why. 01:22:25.560 --> 01:22:29.480 align:middle line:84% This other also-- some mathematicians, 01:22:29.480 --> 01:22:32.870 align:middle line:84% from the point of view of certain classes of partial 01:22:32.870 --> 01:22:36.140 align:middle line:84% differential equations, have developed-- 01:22:36.140 --> 01:22:40.340 align:middle line:90% now it's already 25 years ago-- 01:22:40.340 --> 01:22:42.265 align:middle line:90% this theory of gradient flows. 01:22:42.265 --> 01:22:46.570 align:middle line:90% 01:22:46.570 --> 01:22:54.880 align:middle line:84% They-- in order to do that, they have introduced a particular 01:22:54.880 --> 01:22:55.930 align:middle line:90% metric-- 01:22:55.930 --> 01:22:58.450 align:middle line:84% pretty complicated mathematical-- but it 01:22:58.450 --> 01:23:03.490 align:middle line:84% is a metric that regularizes those differential equations 01:23:03.490 --> 01:23:08.020 align:middle line:84% and makes them fit into this same scheme. 01:23:08.020 --> 01:23:10.370 align:middle line:90% So that's very interesting. 01:23:10.370 --> 01:23:13.990 align:middle line:90% 01:23:13.990 --> 01:23:16.570 align:middle line:84% Then there is also more modern stuff 01:23:16.570 --> 01:23:20.670 align:middle line:90% like large deviation theory. 01:23:20.670 --> 01:23:23.290 align:middle line:84% And not only that, but also there 01:23:23.290 --> 01:23:33.900 align:middle line:84% is some people that are applying this notion 01:23:33.900 --> 01:23:41.710 align:middle line:84% to solve in original way complicated problems 01:23:41.710 --> 01:23:50.090 align:middle line:84% in a much simpler way than the traditional descriptions. 01:23:50.090 --> 01:23:56.630 align:middle line:84% Because this view, geometrical view, you know, 01:23:56.630 --> 01:23:58.970 align:middle line:84% the power of mathematics and of geometry, 01:23:58.970 --> 01:24:03.410 align:middle line:84% is if you can view something from a geometrical point 01:24:03.410 --> 01:24:06.380 align:middle line:84% of view, you have captured the essence. 01:24:06.380 --> 01:24:09.860 align:middle line:84% And you have also often simplified 01:24:09.860 --> 01:24:17.560 align:middle line:84% even the numerical solutions of your problem. 01:24:17.560 --> 01:24:21.610 align:middle line:84% I'll give you some references here just for fun, 01:24:21.610 --> 01:24:23.860 align:middle line:90% if you're interested in this. 01:24:23.860 --> 01:24:34.280 align:middle line:84% Now regarding Bejan constructal law, if you haven't come across, 01:24:34.280 --> 01:24:39.560 align:middle line:84% Adrian Bejan was a graduate of this department two 01:24:39.560 --> 01:24:42.140 align:middle line:84% or three years before me, I guess. 01:24:42.140 --> 01:24:47.090 align:middle line:84% He did a PhD in mechanical engineering here. 01:24:47.090 --> 01:25:02.020 align:middle line:84% And his-- you see, his focus is on design and evolution 01:25:02.020 --> 01:25:05.500 align:middle line:90% in nature, the role of design. 01:25:05.500 --> 01:25:12.820 align:middle line:84% Design, for example, the structure of-- 01:25:12.820 --> 01:25:18.560 align:middle line:84% the tree structure of lungs or even the structure 01:25:18.560 --> 01:25:25.580 align:middle line:84% of trees, which are sorts of the result of optimization 01:25:25.580 --> 01:25:32.420 align:middle line:90% of two contrasting tendencies. 01:25:32.420 --> 01:25:36.190 align:middle line:84% And the system is trying to find the best direction 01:25:36.190 --> 01:25:40.660 align:middle line:90% in which to build and to-- 01:25:40.660 --> 01:25:46.810 align:middle line:84% but in which to make things flow. 01:25:46.810 --> 01:25:52.000 align:middle line:84% And things flow, to me, is also translated 01:25:52.000 --> 01:26:00.760 align:middle line:84% into choose the way in which you can produce as much as possible 01:26:00.760 --> 01:26:06.040 align:middle line:84% your entropy at the fastest rate compatible with the constraints. 01:26:06.040 --> 01:26:08.440 align:middle line:84% And so for this part, I think that we're 01:26:08.440 --> 01:26:09.650 align:middle line:90% saying the same thing. 01:26:09.650 --> 01:26:12.800 align:middle line:84% So it's also fitting into the Fourth Law. 01:26:12.800 --> 01:26:19.300 align:middle line:84% But what's beyond that, and I think he's right, 01:26:19.300 --> 01:26:27.850 align:middle line:84% is that he's saying that in nature, things evolve. 01:26:27.850 --> 01:26:35.210 align:middle line:84% And also the design of natural things, 01:26:35.210 --> 01:26:39.820 align:middle line:84% including the design, for example, of the wings of animals 01:26:39.820 --> 01:26:56.080 align:middle line:84% and how they are related to evolution, so how from the-- 01:26:56.080 --> 01:27:03.940 align:middle line:84% I forget the names of the ancient birds 01:27:03.940 --> 01:27:10.690 align:middle line:84% when you had the T-Rex and how that evolved. 01:27:10.690 --> 01:27:13.270 align:middle line:84% And that evolution is an evolution 01:27:13.270 --> 01:27:19.490 align:middle line:84% in the design of how structures are built in nature. 01:27:19.490 --> 01:27:24.230 align:middle line:84% They always have the purpose of favoring flows. 01:27:24.230 --> 01:27:28.150 align:middle line:84% That means favoring entropy generation 01:27:28.150 --> 01:27:32.270 align:middle line:84% and making it as effective as possible. 01:27:32.270 --> 01:27:37.060 align:middle line:84% Because as you know, life is impossible 01:27:37.060 --> 01:27:40.060 align:middle line:84% without entropy generation, and we need that. 01:27:40.060 --> 01:27:42.170 align:middle line:90% That is the driving force. 01:27:42.170 --> 01:27:46.540 align:middle line:84% While you produce entropy, you also produce useful things. 01:27:46.540 --> 01:27:51.400 align:middle line:84% So entropy production is not completely negative. 01:27:51.400 --> 01:27:55.520 align:middle line:90% It's not just a waste of exergy. 01:27:55.520 --> 01:27:58.270 align:middle line:84% You need it, because if you didn't have it, 01:27:58.270 --> 01:28:01.610 align:middle line:90% nothing useful would happen. 01:28:01.610 --> 01:28:08.930 align:middle line:84% But what Bejan says is that also structures evolve, 01:28:08.930 --> 01:28:14.190 align:middle line:90% and the choice-- 01:28:14.190 --> 01:28:21.950 align:middle line:84% there is some randomness in how a particular structure is 01:28:21.950 --> 01:28:24.230 align:middle line:90% given birth. 01:28:24.230 --> 01:28:27.690 align:middle line:84% It was nonexistent, and then all of a sudden, 01:28:27.690 --> 01:28:34.400 align:middle line:84% perhaps because of a fluctuation or of perturbation, 01:28:34.400 --> 01:28:37.850 align:middle line:90% it becomes possible. 01:28:37.850 --> 01:28:44.940 align:middle line:84% Now, regarding structure, though, for example, 01:28:44.940 --> 01:28:47.910 align:middle line:90% there is this problem of the-- 01:28:47.910 --> 01:28:54.150 align:middle line:90% it's called Benard conduction. 01:28:54.150 --> 01:29:01.250 align:middle line:84% It's a problem in which you heat-- 01:29:01.250 --> 01:29:03.770 align:middle line:90% you have a fluid-- 01:29:03.770 --> 01:29:09.200 align:middle line:84% like-- you have a pan with some oil, and it's heated from below 01:29:09.200 --> 01:29:10.900 align:middle line:90% and cooled from above. 01:29:10.900 --> 01:29:17.360 align:middle line:90% 01:29:17.360 --> 01:29:26.100 align:middle line:84% Now, typically, if your heating is not strong enough, 01:29:26.100 --> 01:29:29.400 align:middle line:90% you would have just conduction. 01:29:29.400 --> 01:29:33.210 align:middle line:84% But then, there is also the possibility, 01:29:33.210 --> 01:29:36.750 align:middle line:84% because of the instability, due to the fact 01:29:36.750 --> 01:29:39.310 align:middle line:84% that the stratification-- since you 01:29:39.310 --> 01:29:45.030 align:middle line:84% get hot stuff below and cold above, and they like to switch. 01:29:45.030 --> 01:29:52.990 align:middle line:84% So this movement, which requires moving a flow, 01:29:52.990 --> 01:29:55.120 align:middle line:84% therefore, it involves a friction. 01:29:55.120 --> 01:30:05.130 align:middle line:84% So if you have enough motivation for doing it, 01:30:05.130 --> 01:30:07.170 align:middle line:84% eventually, that flow will start. 01:30:07.170 --> 01:30:10.810 align:middle line:84% And therefore, you'll get convective-- 01:30:10.810 --> 01:30:13.110 align:middle line:84% it will switch from conductive heat transfer 01:30:13.110 --> 01:30:18.780 align:middle line:84% to convective rolls of heat conduction, which will 01:30:18.780 --> 01:30:21.450 align:middle line:90% enhance the heat conduction. 01:30:21.450 --> 01:30:22.850 align:middle line:90% It's more effective. 01:30:22.850 --> 01:30:29.330 align:middle line:84% So you see, here, it is a situation in which structure 01:30:29.330 --> 01:30:35.390 align:middle line:84% gets built in order to make the entropy production more 01:30:35.390 --> 01:30:38.440 align:middle line:90% effective. 01:30:38.440 --> 01:30:45.100 align:middle line:84% And in this paper, you view it from the point of view 01:30:45.100 --> 01:30:50.050 align:middle line:90% of maximum entropy production. 01:30:50.050 --> 01:30:54.690 align:middle line:84% So you see, for example, here you 01:30:54.690 --> 01:30:59.255 align:middle line:84% have a situation in which you are-- 01:30:59.255 --> 01:31:04.300 align:middle line:90% 01:31:04.300 --> 01:31:06.680 align:middle line:90% for example, you have the rolls. 01:31:06.680 --> 01:31:10.160 align:middle line:84% So you have the high non-equilibrium situation, 01:31:10.160 --> 01:31:14.230 align:middle line:84% high flux, in which you have the Rayleigh Benard rolls. 01:31:14.230 --> 01:31:17.890 align:middle line:84% And that structure is the one that maximizes 01:31:17.890 --> 01:31:19.220 align:middle line:90% the entropy (production). 01:31:19.220 --> 01:31:21.670 align:middle line:84% So among all the possibilities, the system 01:31:21.670 --> 01:31:26.180 align:middle line:84% chooses that one, maximizing the entropy production. 01:31:26.180 --> 01:31:35.710 align:middle line:84% But as you turn the heat off, and here, as time goes on, 01:31:35.710 --> 01:31:39.540 align:middle line:90% you reduce the heat from below. 01:31:39.540 --> 01:31:42.900 align:middle line:90% 01:31:42.900 --> 01:31:54.340 align:middle line:84% That tendency goes down until it becomes comparable or even less 01:31:54.340 --> 01:31:58.570 align:middle line:84% efficient than the direct simple conduction. 01:31:58.570 --> 01:32:00.880 align:middle line:84% And then what happens, the system switches. 01:32:00.880 --> 01:32:06.760 align:middle line:84% And the convection gets stopped, and the system 01:32:06.760 --> 01:32:11.530 align:middle line:90% switches to another maximum. 01:32:11.530 --> 01:32:15.640 align:middle line:84% So this is also a jump in configuration. 01:32:15.640 --> 01:32:18.660 align:middle line:90% So it's a construction of-- 01:32:18.660 --> 01:32:20.620 align:middle line:90% it's a design, so to speak. 01:32:20.620 --> 01:32:23.130 align:middle line:84% It's a design choice that your system 01:32:23.130 --> 01:32:27.330 align:middle line:84% has made in order to move from one maximum to another maximum. 01:32:27.330 --> 01:32:30.930 align:middle line:90% 01:32:30.930 --> 01:32:35.965 align:middle line:84% And this, of course, corresponds to critical 01:32:35.965 --> 01:32:39.130 align:middle line:84% a value of the Rayleigh number that 01:32:39.130 --> 01:32:49.770 align:middle line:84% is the onset number for this Rayleigh-Benard instability. 01:32:49.770 --> 01:32:53.325 align:middle line:84% It is a very famous problem in fluid mechanics and heat 01:32:53.325 --> 01:32:53.825 align:middle line:90% transfer. 01:32:53.825 --> 01:33:00.120 align:middle line:90% 01:33:00.120 --> 01:33:06.610 align:middle line:84% And by the way, Adrian Bejan has written hundreds of papers, 01:33:06.610 --> 01:33:10.870 align:middle line:84% but also many books, including one at the beginning, 01:33:10.870 --> 01:33:16.680 align:middle line:84% which is entropy production in heat transfer 01:33:16.680 --> 01:33:19.850 align:middle line:90% which is very good. 01:33:19.850 --> 01:33:23.500 align:middle line:90% 01:33:23.500 --> 01:33:31.870 align:middle line:84% So let me conclude today with this last idea 01:33:31.870 --> 01:33:34.965 align:middle line:84% just to add to the confusion, if you want. 01:33:34.965 --> 01:33:41.770 align:middle line:90% 01:33:41.770 --> 01:33:45.340 align:middle line:84% The idea that-- there is a theorem, famous theorem, 01:33:45.340 --> 01:33:50.620 align:middle line:84% proved by Glansdorff and Prigogine 01:33:50.620 --> 01:33:54.320 align:middle line:84% called the minimum entropy production principle. 01:33:54.320 --> 01:33:56.880 align:middle line:90% 01:33:56.880 --> 01:34:03.645 align:middle line:84% So many people facing simply the words in a superficial way, 01:34:03.645 --> 01:34:05.645 align:middle line:84% they say, well, is it minimum, or is it maximum? 01:34:05.645 --> 01:34:09.960 align:middle line:90% 01:34:09.960 --> 01:34:10.460 align:middle line:90% All right. 01:34:10.460 --> 01:34:14.430 align:middle line:84% So the difference and the key is local and global. 01:34:14.430 --> 01:34:17.440 align:middle line:90% 01:34:17.440 --> 01:34:23.460 align:middle line:90% So the system locally chooses-- 01:34:23.460 --> 01:34:26.190 align:middle line:90% 01:34:26.190 --> 01:34:28.650 align:middle line:84% the material locally, chooses to react 01:34:28.650 --> 01:34:34.230 align:middle line:84% in a direction that goes towards the steepest entropy ascent. 01:34:34.230 --> 01:34:40.600 align:middle line:84% So it's the local maximum entropy production. 01:34:40.600 --> 01:34:45.190 align:middle line:84% This theorem instead looks at a global situation 01:34:45.190 --> 01:34:50.500 align:middle line:84% like an integral over a volume of many pieces of material. 01:34:50.500 --> 01:34:53.993 align:middle line:84% And it does the integral not of sigma, 01:34:53.993 --> 01:34:55.660 align:middle line:84% which is the rate of entropy production, 01:34:55.660 --> 01:34:57.510 align:middle line:84% but of the rate of change of sigma. 01:34:57.510 --> 01:35:00.280 align:middle line:90% 01:35:00.280 --> 01:35:10.070 align:middle line:84% And it proves that the rate of change of sigma 01:35:10.070 --> 01:35:13.910 align:middle line:84% under a broad set of boundary conditions and assumptions 01:35:13.910 --> 01:35:16.510 align:middle line:90% that are listed there-- 01:35:16.510 --> 01:35:19.660 align:middle line:90% very large set of conditions-- 01:35:19.660 --> 01:35:24.210 align:middle line:90% is negative. 01:35:24.210 --> 01:35:28.185 align:middle line:84% So that means that sigma, if you start-- 01:35:28.185 --> 01:35:29.910 align:middle line:84% like in the previous slide, if you 01:35:29.910 --> 01:35:38.890 align:middle line:84% start from a situation in which in your control volume, 01:35:38.890 --> 01:35:47.220 align:middle line:84% there is a lot of entropy production, and you let it go. 01:35:47.220 --> 01:35:50.880 align:middle line:84% If it is not already in a steady state, 01:35:50.880 --> 01:35:58.230 align:middle line:84% it will change, evolve, towards a lesser and lesser 01:35:58.230 --> 01:36:04.470 align:middle line:84% entropy production situation until it gets to zero. 01:36:04.470 --> 01:36:07.950 align:middle line:84% And the zero will be either a steady state, 01:36:07.950 --> 01:36:10.890 align:middle line:84% if it is sustained by the boundary conditions, 01:36:10.890 --> 01:36:12.460 align:middle line:90% or the equilibrium state. 01:36:12.460 --> 01:36:16.340 align:middle line:90% 01:36:16.340 --> 01:36:18.200 align:middle line:84% Instead of doing the details here 01:36:18.200 --> 01:36:20.430 align:middle line:84% that are not really essential for us, 01:36:20.430 --> 01:36:22.820 align:middle line:90% let me give you this example. 01:36:22.820 --> 01:36:27.270 align:middle line:84% Consider a one dimensional heat transfer problem. 01:36:27.270 --> 01:36:31.740 align:middle line:84% And suppose that at time equals zero, 01:36:31.740 --> 01:36:35.335 align:middle line:84% you have imposed this temperature profile. 01:36:35.335 --> 01:36:39.540 align:middle line:90% 01:36:39.540 --> 01:36:42.300 align:middle line:84% And your boundary conditions are that the temperature here 01:36:42.300 --> 01:36:44.280 align:middle line:90% is given and fixed. 01:36:44.280 --> 01:36:49.570 align:middle line:84% So you fix the temperatures left and right. 01:36:49.570 --> 01:36:51.680 align:middle line:84% Of course, that is not a steady state. 01:36:51.680 --> 01:36:53.450 align:middle line:90% The system will evolve. 01:36:53.450 --> 01:36:58.330 align:middle line:84% You can solve this with the Fourier equation 01:36:58.330 --> 01:37:01.630 align:middle line:84% or any software that does this solution. 01:37:01.630 --> 01:37:05.620 align:middle line:84% And you see it will evolve and at the end, 01:37:05.620 --> 01:37:12.250 align:middle line:84% it will reach the steady linear solution of the typical heat 01:37:12.250 --> 01:37:14.190 align:middle line:90% transfer problem. 01:37:14.190 --> 01:37:19.890 align:middle line:84% Now, if you compute the entropy production rate here, 01:37:19.890 --> 01:37:21.020 align:middle line:90% it's this much. 01:37:21.020 --> 01:37:24.900 align:middle line:90% 01:37:24.900 --> 01:37:27.850 align:middle line:84% If you compute it here, it's this much. 01:37:27.850 --> 01:37:28.415 align:middle line:90% It's a half. 01:37:28.415 --> 01:37:31.310 align:middle line:90% 01:37:31.310 --> 01:37:37.210 align:middle line:84% So the system, as it moves from here to there, 01:37:37.210 --> 01:37:40.720 align:middle line:84% has an overall global entropy production 01:37:40.720 --> 01:37:43.600 align:middle line:84% that goes down in time until it reaches 01:37:43.600 --> 01:37:47.180 align:middle line:90% a minimum at steady state. 01:37:47.180 --> 01:37:48.963 align:middle line:84% If you start-- this is another example 01:37:48.963 --> 01:37:51.380 align:middle line:84% in which you have different boundary conditions that would 01:37:51.380 --> 01:37:53.820 align:middle line:90% allow an equilibrium state. 01:37:53.820 --> 01:37:57.330 align:middle line:84% So you start from this one, and you have an entropy production 01:37:57.330 --> 01:37:59.050 align:middle line:90% rate, which is given by this. 01:37:59.050 --> 01:38:03.960 align:middle line:84% You let it go, it will reach a flat temperature profile, which 01:38:03.960 --> 01:38:06.735 align:middle line:84% is equilibrium, zero entropy production, again, a minimum. 01:38:06.735 --> 01:38:09.730 align:middle line:90% 01:38:09.730 --> 01:38:15.440 align:middle line:90% So globally, it has gone down. 01:38:15.440 --> 01:38:21.080 align:middle line:84% But locally, at every instant, at every local, 01:38:21.080 --> 01:38:27.230 align:middle line:84% each material element here has followed Fourier's law. 01:38:27.230 --> 01:38:34.400 align:middle line:84% And therefore, has at any time chosen the direction of flow 01:38:34.400 --> 01:38:40.120 align:middle line:84% is compatible with the gradient imposed. 01:38:40.120 --> 01:38:47.020 align:middle line:84% All right, so we're just on time and just finished. 01:38:47.020 --> 01:38:58.000 align:middle line:90%