(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 7046, 220]*) (*NotebookOutlinePosition[ 7832, 245]*) (* CellTagsIndexPosition[ 7788, 241]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Time-Dependent Solution to the Diffusion Equation in the Plane with \ a Point Source at the Origin.\ \>", "Subtitle"], Cell[BoxData[ StyleBox[\(<< Graphics`\), FontWeight->"Bold"]], "Input"], Cell["\<\ These notes are based on notes written by Prof. W.C. Carter for MIT \ subject 3.016. These are excerpts with some additional comments and \ extensions for MIT subject 3.21. \ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell["\<\ Note: This notebook will produce some neat animations but they each \ take some time to compute. And, if you save the notebook with the graphics, \ it will require 30MB or so of hard disk space.\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell["\<\ You should definitely take the time to learn how to make your own \ animations because the results can be really instructive. You can even save \ animations as Quicktime files and import and play them back in other programs \ such as PowerPoint.\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell["Scalar field: concentration", "Subsubtitle"], Cell["\<\ As an example we will look at the concentration field arising from \ a line source in an isotropic medium (e.g. think of a Cu wire embedded in a \ block of Ni; allow interdiffusion to occur for time t > 0).\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell["\<\ Define the concentration c(x,y,t) by the following (we will soon \ learn where this equation comes from):\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[BoxData[ \(concentration\ = Exp[\(-\((x^2\ + \ y^2)\)\)/t]/\((\(\@\[Pi]\) t)\)\)], "Input"], Cell["\<\ Illustrate the time dependence of this scalar field with a seqence \ of plots...\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[BoxData[ StyleBox[\(<< Graphics`Animation`\), FontWeight->"Bold"]], "Input"], Cell["\<\ To see animations,use the menu command to group all the graphics \ cells together, and then animate\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[BoxData[ \(MoviePlot3D[ concentration, {x, \(-4\), 4}, {y, \(-4\), 4}, {t, 0.01, 2.51, .05}, PlotPoints \[Rule] 40, PlotRange \[Rule] {0, 2}, DisplayFunction \[Rule] Identity]\)], "Input"], Cell["\<\ Gradient of a scalar field and relation to flux in a concentration \ field\ \>", "Subsubtitle"], Cell[TextData[{ "Now look at the ", StyleBox["gradient", FontSlant->"Italic"], " of this scalar field. The gradient at a point is a vector directed \ toward the fastest rate of change (\"steepest ascent\") of the scalar field." }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell["\<\ Flux is a vector that points in the direction of the flow and is a \ measure of how much is flowing per unit time. This illustration is a simple \ but common situation in which the flux is antiparallel to the concentration \ gradient, so the resulting vector field also illustrates the negative of the \ gradient of the concentration field.\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[BoxData[ \(<< Graphics`PlotField`\)], "Input"], Cell[BoxData[ \(flux\ = \ {\(-D[concentration, x]\), \(-D[concentration, y]\)}\)], "Input"], Cell[BoxData[ \(This\ is\ an\ example\ of\ a\ time - dependent\ vector\ field\ \(j\&\[LongRightArrow]\) \((x, y, t)\)\)], "Text"], Cell[BoxData[ \({\(2\ \[ExponentialE]\^\(\(\(-x\^2\) - y\^2\)\/t\)\ x\)\/\(\@\[Pi]\ \ t\^2\), \(2\ \[ExponentialE]\^\(\(\(-x\^2\) - y\^2\)\/t\)\ y\)\/\(\@\[Pi]\ \ t\^2\)}\)], "Output"], Cell[BoxData[ \(Animate[ PlotVectorField[flux, {x, \(-2\), 2}, {y, \(-2\), 2}, PlotPoints \[Rule] 20, ScaleFunction \[Rule] \((If[# < .05, 0, \((100.0 #)\)] &)\), ColorFunction \[Rule] \((Hue[1 - 0.75 #] &)\)], {t, 0.01, 4.01, .05}]\)], "Input"], Cell["\<\ Divergence and relation to accumulation in a concentration \ field\ \>", "Subsubtitle"], Cell["\<\ Now look at the accumulation which is the negative of the \ divergence of the flux.\ \>", "Text"], Cell["\<\ Define a function that takes a two-dimensional vector function of \ (x,y) as an argument and returns its divergence\ \>", "Text"], Cell[BoxData[ \(divergence[{xcomp_\ , \ ycomp_}]\ := \ Simplify[D[xcomp, x]\ + \ D[ycomp, y]]\)], "Input"], Cell[BoxData[ \(divgradptsource\ = \ divergence[\(-flux\)]\)], "Input"], Cell[BoxData[ \(Plotting\ the\ divergence\ of\ the\ gradient\ \((\[Del]\(\(\[CenterDot]\ \)\((\[Del]\ f)\)\)\ is\ the\ ``Laplacian''\ \[Del]\^2\ f, \ sometimes\ indicated\ with\ symbol\ \[CapitalDelta]f)\)\)], "Text", FontFamily->"Helvetica"], Cell[BoxData[ \(MoviePlot3D[ divgradptsource, {x, \(-4\), 4}, {y, \(-4\), 4}, {t, 0.01, 2.51, .05}, PlotPoints \[Rule] 40, PlotRange \[Rule] {\(-2\), 2}, DisplayFunction \[Rule] Identity]\)], "Input"], Cell["\<\ You should review these animations and ensure that you understand \ their behavior, particularly the time and position dependence of the flux \ vector, and the time and position dependence of the accumulation.\ \>", "Text"] }, FrontEndVersion->"5.2 for Macintosh", ScreenRectangle->{{0, 1148}, {0, 746}}, ScreenStyleEnvironment->"Presentation", WindowSize->{890, 631}, WindowMargins->{{4, Automatic}, {Automatic, 1}}, WindowTitle->"Lecture 11 MIT 3.016 \[Copyright] W. Craig Carter 2003", StyleDefinitions -> "3016_Carter.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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