Lecture P2: Integral Momentum Equation


General comments

In this lecture I developed the integral momentum equation for a control volume of fixed mass. It is an expression of Newton's Second Law relating changes in momentum to forces. The integral momentum equation is the same as that presented in Unified Fluids, however I specialized it to allow application in a non-inertial reference frame (an accelerating reference frame). When the equations are solved in an accelerating reference frame the momentum of the control volumer mass changes in response to the acceleration. This is represented in the equation with an extra term (control volume mass times the acceleration of the reference frame) which is often called a psuedo-force or an inertial force. There is an introductory physics text that you can review to brush-up on these concepts (Hudson and Nelson, University Physics, ©1982 by Harcourt Brace and Jovanovich, Inc.). After discussing the various terms in the equation (it is important that you understand what each of the terms represents), I gave a conceptual problem (Q1). This was intended to reinforce that thrust is primarily a balance between momentum flux into an engine and momentum flux out of an engine. After discussing this conceptually, we used the integral control volume equation to arrive at the same answer. The key points to remember: 1) you must evaluate the dot product in the flux term (not just set it to u_sub_x), and 2) draw the outward unit normal verctor on each surface of the control volume -- this will help alleviate sign errors. We will spend more time on using the integral momentum equation in the next lecture and on the homeworks.

Responses to 'Muddiest Part of the Lecture Cards'

(15 respondents, 57 attending class)

1) How do cars fit into the two options to propell a vehicle, or do they? (1 student) They don't really. Cars grab onto a fixed object (the road) and push back on it to propel themselves forward.

2) The unsteady term in the integral momentum equation is important in rocket propulsion because of the change in mass of the rocket right? This is true for the inertial force also right? How do we account for the loss of mass in the rocket? The equations that we derived in class are for a control volume of FIXED mass. In the next lecture I will show you how the two terms you refer to apply for certain problems. In a few lectures we will talk about the rocket problem specifically. If you like you can read ahead in the notes. Since we are taking 16.05 this semester, will 16.50 be available this coming fall? (1 student). 16.50 will still be offered in the spring (to limit conflicts with 16.06 and 16.07 and 16.100 and 16.20).

3) What does inertial reference frame mean? (1 student). A frame that is not accelerating.

4) What is the purpose of using the substantial derivative instead of du/dt -- kind of shaky from fluids? (1 student). The change in velocity at a fixed point in an unsteady flow has two components, the change in velocity at that fixed point with respect to time only, and the convective component -- that due to the fact that there are spatial gradients in velocity. To help understand the two terms it may be helpful to think of the two cases: 1) the velocity field is uniform everywhere (no spatial gradients) but varies with time, and 2) the velocity field is constant in time, but varies from point to point. If one wants to evaluate the change in velocity of a fluid particle it is necessary to account for both components.

5) I actually didn't get the rock throwing problem right away. (1 student). Do you understand it now? If not, please see me and we can discuss it further.

6) What are body forces? (1 student). Body forces are forces that act on each element of mass in the c.v. (like gravity) versus acting at the boundary (like pressure forces and shear forces).

7) If the faster you go, the smaller the output force is, doesn't this imply that to fly faster you need to reduce your applied force? How can you reduce your applied force to speed up?(1 student). IF mass flow is constant (it isn't) and the exit velocity is constant (it isn't), then the faster you fly, the smaller the force. But the part of the equation you are missing in your reasoning is the drag, which goes up with the square of the velocity. So you need a larger force to fly faster. You do this by putting more fuel into the engine, changing the mass flow throught the engine and changing the exhaust velocity.

8) No mud (8 students). Good.