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Notebook[{

Cell[CellGroupData[{
Cell["\<\
Finite Difference and Finite Element solution to flow in a \
Rectangular Duct\
\>", "Title",
  Evaluatable->False,
  TextAlignment->Left,
  AspectRatioFixed->True],

Cell[TextData[{
  "This notebook has been written in ",
  StyleBox["Mathematica ",
    FontSlant->"Italic"],
  "by \n\nMark J. McCready\nProfessor and Chair of Chemical Engineering\n\
University of Notre Dame\nNotre Dame IN 46556\nUSA\n\nmjm@nd.edu\n\n",
  ButtonBox["http://www.nd.edu/~mjm/",
    ButtonData:>{
      URL[ "http://www.nd.edu/~mjm/"], None},
    ButtonStyle->"Hyperlink"],
  "\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \
the World Wide Web and the Internet.\n\nI would hope that as a professional \
courtesy that this notice remain visible to other users. \nThere is no charge \
for copying and dissemination   \n\nVersion:  11/1/00\n"
}], "Text"],

Cell[TextData[{
  "The numerical code for this problem is adapted from a Fortran program and \
discussion given by\nC. A. J. Fletcher (1991) ",
  StyleBox["Computational Techniques for Fluid Dynamics",
    FontSlant->"Italic"],
  ", Springer vol.1 p 112.  "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell["Homework problem ", "Subtitle",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
1.  Plot the solution for different values of the the aspect ratio, \
ba.  Explain what is happening.
2.  What do you have to do to get accuate answers as ba is changed ??
3.  If you use the finite element method with nx=6 and ba>=6, there will be \
two cells of flow.  \t\t            
How do you know that this is incorrect ?  One of the important issues that \
you may face when using numerical codes is that they may not be correct !!
4.  It is rather significant that the solution depends only on the geometric \
parameter, ba.  There is no qualitative change in the flow field with flow \
rate.  The group , (-eta/b^2/DPDZ) scales the velocity to make this possible, \
discuss this physical meaning of this group. \
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
This program solves steady laminar flow of a Newtonian fluid in a \
rectangular duct\
\>", "Subtitle",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
Most flows of real interest are in geometries that do not give \
simple ode's that have analytical solutions.  This program addresses the flow \
in a rectanular duct that is 2a wide (x direction) by 2b high (y direction).  \
The flow is in the z direction.  As with the infinitely wide channel, the \
pressure gradient is constant as are the viscous stress terms.  \
\>", "Text",\

  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell["Equation set up", "SectionFirst"],

Cell[CellGroupData[{

Cell["here is the differential equation", "Subsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{
        RowBox[{"\[Eta]", " ", 
          RowBox[{"(", 
            RowBox[{
              
              FractionBox[\(\[PartialD]\^2\( VZ(X, 
                    Y)\)\), \(\[PartialD]X\^2\),
                MultilineFunction->None], "+", 
              
              FractionBox[\(\[PartialD]\^2\( VZ(X, 
                    Y)\)\), \(\[PartialD]Y\^2\),
                MultilineFunction->None]}], ")"}]}], "-", "DPDZ"}], 
      TraditionalForm]], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    FormBox[
      RowBox[{
        RowBox[{"\[Eta]", " ", 
          RowBox[{"(", 
            RowBox[{
              RowBox[{
                SuperscriptBox["VZ", 
                  TagBox[\((0, 2)\),
                    Derivative],
                  MultilineFunction->None], "(", \(X, Y\), ")"}], "+", 
              RowBox[{
                SuperscriptBox["VZ", 
                  TagBox[\((2, 0)\),
                    Derivative],
                  MultilineFunction->None], "(", \(X, Y\), ")"}]}], ")"}]}], 
        "-", "DPDZ"}], TraditionalForm]], "Output"]
}, Open  ]]
}, Open  ]],

Cell[TextData[
"the boundary conditions are that the velocity is 0 on the wall.  This gives\n\
VZ[-a,Y] = 0, VZ[a,Y] = 0, VZ[X,-b] = 0, VZ[X,b] = 0\n\nA problem with \
numerical solutions is that everytime a variable is changed, the solution \
must be recalculated.  Thus it is convenient to rescale the variables.  This \
will reduce the number of things that can vary.  This idea is similar to the \
dimensional analysis that was employed in the beginning of the course.\nx = \
X/a, \ny = Y/b\nvz[x,y] = VZ[X,Y]*(-\[Eta]/b^2/DPDZ)\n\nIf the equation is \
divided by -DPDZ, the X,and Y derivatives are replaced by x and y derivatives \
(creating a 1/a^2 by the x term and 1/b^2 by the dy^2 term because \
\[PartialD]^2/\[PartialD]y^2 = b^2 \[PartialD]^2/\[PartialD]Y^2), then with \
the substitution of the new velocity we get with b/a = ba,"], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{
        RowBox[{"ba", " ", 
          FractionBox[\(\[PartialD]\^2\( vz(x, y)\)\), \(\[PartialD]x\^2\),
            MultilineFunction->None]}], "+", 
        FractionBox[\(\[PartialD]\^2\( vz(x, y)\)\), \(\[PartialD]y\^2\),
          MultilineFunction->None], "+", "1"}], TraditionalForm]], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    FormBox[
      RowBox[{
        RowBox[{
          SuperscriptBox["vz", 
            TagBox[\((0, 2)\),
              Derivative],
            MultilineFunction->None], "(", \(x, y\), ")"}], "+", 
        RowBox[{"ba", " ", 
          RowBox[{
            SuperscriptBox["vz", 
              TagBox[\((2, 0)\),
                Derivative],
              MultilineFunction->None], "(", \(x, y\), ")"}]}], "+", "1"}], 
      TraditionalForm]], "Output"]
}, Open  ]],

Cell[TextData[
"the boundary conditions become\n\nvz[-1,y] = 0, vz[1,y] = 0, vz[x,-1] = 0, \
vz[x,1] = 0"], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"The equation separates and can be solved analytically.  You can learn to do \
it at some later time.  Below I give a program that solves the problem \
numerically.  "], "Text",
  Evaluatable->False,
  AspectRatioFixed->True]
}, Closed]],

Cell[CellGroupData[{

Cell["Here is the numerical solution.", "SectionFirst",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(nx = 20;\)\), "\n", 
    \(\(me = 2;\)\), "\n", 
    \(\(nem = 12;\)\), "\n", 
    \(\(bar = 8;\)\), "\n", 
    \(\(eps =  .00001;\)\), "\n", 
    \(\(imax = 30;\)\), "\n", 
    \(\(om = 1.5;\)\), "\n", 
    \(\(rms = 1;\)\)}], "Input",
  AspectRatioFixed->True],

Cell[BoxData[{
    \(\(nh = nx\/2 + 1;\)\), "\n", 
    \(\(ny = nx;\)\), "\n", 
    \(\(nxp = nx - 1;\)\), "\n", 
    \(\(nyp = nxp;\)\), "\n", 
    \(\(nypp = nyp - 1;\)\), "\n", 
    \(\(anx = nxp;\)\), "\n", 
    \(\(dx = 2. \/anx;\)\), "\n", 
    \(\(dxs = dx\ dx;\)\), "\n", 
    \(\(dy = dx;\)\), "\n", 
    \(\(dys = dxs;\)\), "\n", 
    \(\(bas = \(bar\ bar\)\/dxs;\)\), "\n", 
    \(\(par1 = bas + 1. \/dys;\)\), "\n", 
    \(\(par2 = 1\/3\ \((2. \/dys - bas)\);\)\), "\n", 
    \(\(par3 = 1\/3\ \((2. \ bas - 1. \/dys)\);\)\), "\n", 
    \(\(n = nem;\)\)}], "Input",
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[TextData["This part generates the analytical solution"], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(\(Do[x = \(-1. \) + \((j - 1)\)\ dx; 
        Do[y = \(-1. \) + \((k - 1)\)\ dy; xan = 0.5\ N[\[Pi]]\ x; 
          yan = 0.5\ N[\[Pi]]\ y; w = 0.0; ql = 0.0; 
          Do[al = l; cx = N[Cos[al\ xan]]; 
            Do[am = m; cy = N[Cos[am\ yan]]; lm = \(m + l\)\/2 - 1. ; 
              If[lm == 0, sig = 1]; If[lm > 0, sig = \((\(-1. \))\)\^lm]; 
              den = al\ am\ \((\((al\ bar)\)\^2 + am\ am)\); 
              dum = N[\(sig\ cx\ cy\)\/den]; w = N[w + dum]; 
              ql = ql + 1. \/\(den\ al\ am\), {m, 1, n, 2}], {l, 1, n, 2}]; 
          w = N[\((8. \/\[Pi]\^2)\)\^2]\ w; 
          ql = \(16. \ \((8. \/\(\[Pi]\ \[Pi]\))\)\^2\ ql\)\/\(\[Pi]\ \
\[Pi]\); wex[j, k] = N[w]; wa[j, k] = N[w], {k, 1, ny}], {j, 1, 
          nx}];\)\)], "Input",
  AspectRatioFixed->True]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[
"This section displays the exact solution.  Note that if too few terms are \
used, it is not very accurate !!"], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(N[Table[wex[i, j], {i, 1, nx}, {j, 1, ny}]];\)\)], "Input",
  AspectRatioFixed->True],

Cell[TextData[
"The contour plot makes the fastest velocity white and slower velocities \
darker.  "], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(ListContourPlot[Table[wex[i, j], {i, 1, nx}, {j, 1, ny}], 
        AspectRatio \[Rule] 1\/bar];\)\)], "Input",
  AspectRatioFixed->True],

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.77834 .00464 L
.82895 .00474 L
.87955 .00464 L
.93016 .00535 L
.96169 .00873 L
.97119 .01506 L
.97396 .02138 L
.9753 .02771 L
.97605 .03403 L
.97649 .04036 L
.97678 .04669 L
.97695 .05301 L
.97702 .05934 L
.97702 .06566 L
.97695 .07199 L
.97678 .07831 L
.97649 .08464 L
.97605 .09097 L
.9753 .09729 L
.97396 .10362 L
.97119 .10994 L
.96169 .11627 L
.93016 .11965 L
.87955 .12036 L
.82895 .12026 L
.77834 .12036 L
.72773 .12043 L
.67713 .12034 L
.62652 .12036 L
.57591 .12042 L
.5253 .12035 L
.4747 .12035 L
.42409 .12042 L
.37348 .12036 L
.32287 .12034 L
.27227 .12043 L
.22166 .12036 L
.17105 .12026 L
.12045 .12036 L
.06984 .11965 L
.03831 .11627 L
.02881 .10994 L
.02604 .10362 L
.0247 .09729 L
.02395 .09097 L
.02351 .08464 L
.02322 .07831 L
.02305 .07199 L
.02298 .06566 L
.02298 .05934 L
.02305 .05301 L
.02322 .04669 L
.02351 .04036 L
.02395 .03403 L
.0247 .02771 L
.02604 .02138 L
.02881 .01506 L
.03831 .00873 L
F
0 g
.06984 .00535 m
.12045 .00464 L
.17105 .00474 L
.22166 .00464 L
.27227 .00457 L
.32287 .00466 L
.37348 .00464 L
.42409 .00458 L
.4747 .00465 L
.5253 .00465 L
.57591 .00458 L
.62652 .00464 L
.67713 .00466 L
.72773 .00457 L
.77834 .00464 L
.82895 .00474 L
.87955 .00464 L
.93016 .00535 L
.96169 .00873 L
.97119 .01506 L
.97396 .02138 L
.9753 .02771 L
.97605 .03403 L
.97649 .04036 L
.97678 .04669 L
.97695 .05301 L
.97702 .05934 L
.97702 .06566 L
.97695 .07199 L
.97678 .07831 L
.97649 .08464 L
.97605 .09097 L
.9753 .09729 L
.97396 .10362 L
.97119 .10994 L
.96169 .11627 L
.93016 .11965 L
.87955 .12036 L
.82895 .12026 L
.77834 .12036 L
.72773 .12043 L
.67713 .12034 L
.62652 .12036 L
.57591 .12042 L
.5253 .12035 L
.4747 .12035 L
.42409 .12042 L
.37348 .12036 L
.32287 .12034 L
.27227 .12043 L
Mistroke
.22166 .12036 L
.17105 .12026 L
.12045 .12036 L
.06984 .11965 L
.03831 .11627 L
.02881 .10994 L
.02604 .10362 L
.0247 .09729 L
.02395 .09097 L
.02351 .08464 L
.02322 .07831 L
.02305 .07199 L
.02298 .06566 L
.02298 .05934 L
.02305 .05301 L
.02322 .04669 L
.02351 .04036 L
.02395 .03403 L
.0247 .02771 L
.02604 .02138 L
.02881 .01506 L
.03831 .00873 L
.06984 .00535 L
Mfstroke
.2 g
.12045 .00778 m
.17105 .008 L
.22166 .00776 L
.27227 .00759 L
.32287 .00782 L
.37348 .00776 L
.42409 .00761 L
.4747 .00778 L
.5253 .00778 L
.57591 .00761 L
.62652 .00776 L
.67713 .00782 L
.72773 .00759 L
.77834 .00776 L
.82895 .008 L
.87955 .00778 L
.91813 .00873 L
.93016 .00958 L
.95558 .01506 L
.96356 .02138 L
.96716 .02771 L
.96913 .03403 L
.97029 .04036 L
.97101 .04669 L
.97145 .05301 L
.97163 .05934 L
.97163 .06566 L
.97145 .07199 L
.97101 .07831 L
.97029 .08464 L
.96913 .09097 L
.96716 .09729 L
.96356 .10362 L
.95558 .10994 L
.93016 .11542 L
.91813 .11627 L
.87955 .11722 L
.82895 .117 L
.77834 .11724 L
.72773 .11741 L
.67713 .11718 L
.62652 .11724 L
.57591 .11739 L
.5253 .11722 L
.4747 .11722 L
.42409 .11739 L
.37348 .11724 L
.32287 .11718 L
.27227 .11741 L
.22166 .11724 L
.17105 .117 L
.12045 .11722 L
.08187 .11627 L
.06984 .11542 L
.04442 .10994 L
.03644 .10362 L
.03284 .09729 L
.03087 .09097 L
.02971 .08464 L
.02899 .07831 L
.02855 .07199 L
.02837 .06566 L
.02837 .05934 L
.02855 .05301 L
.02899 .04669 L
.02971 .04036 L
.03087 .03403 L
.03284 .02771 L
.03644 .02138 L
.04442 .01506 L
.06984 .00958 L
.08187 .00873 L
F
0 g
.12045 .00778 m
.17105 .008 L
.22166 .00776 L
.27227 .00759 L
.32287 .00782 L
.37348 .00776 L
.42409 .00761 L
.4747 .00778 L
.5253 .00778 L
.57591 .00761 L
.62652 .00776 L
.67713 .00782 L
.72773 .00759 L
.77834 .00776 L
.82895 .008 L
.87955 .00778 L
.91813 .00873 L
.93016 .00958 L
.95558 .01506 L
.96356 .02138 L
.96716 .02771 L
.96913 .03403 L
.97029 .04036 L
.97101 .04669 L
.97145 .05301 L
.97163 .05934 L
.97163 .06566 L
.97145 .07199 L
.97101 .07831 L
.97029 .08464 L
.96913 .09097 L
.96716 .09729 L
.96356 .10362 L
.95558 .10994 L
.93016 .11542 L
.91813 .11627 L
.87955 .11722 L
.82895 .117 L
.77834 .11724 L
.72773 .11741 L
.67713 .11718 L
.62652 .11724 L
.57591 .11739 L
.5253 .11722 L
.4747 .11722 L
.42409 .11739 L
.37348 .11724 L
.32287 .11718 L
.27227 .11741 L
.22166 .11724 L
Mistroke
.17105 .117 L
.12045 .11722 L
.08187 .11627 L
.06984 .11542 L
.04442 .10994 L
.03644 .10362 L
.03284 .09729 L
.03087 .09097 L
.02971 .08464 L
.02899 .07831 L
.02855 .07199 L
.02837 .06566 L
.02837 .05934 L
.02855 .05301 L
.02899 .04669 L
.02971 .04036 L
.03087 .03403 L
.03284 .02771 L
.03644 .02138 L
.04442 .01506 L
.06984 .00958 L
.08187 .00873 L
.12045 .00778 L
Mfstroke
.3 g
.06984 .01428 m
.12045 .0111 L
.17105 .01145 L
.22166 .01105 L
.27227 .01076 L
.32287 .01114 L
.37348 .01104 L
.42409 .0108 L
.4747 .01108 L
.5253 .01108 L
.57591 .0108 L
.62652 .01104 L
.67713 .01114 L
.72773 .01076 L
.77834 .01105 L
.82895 .01145 L
.87955 .0111 L
.93016 .01428 L
.93435 .01506 L
.9512 .02138 L
.95793 .02771 L
.96146 .03403 L
.9635 .04036 L
.96476 .04669 L
.9655 .05301 L
.96583 .05934 L
.96583 .06566 L
.9655 .07199 L
.96476 .07831 L
.9635 .08464 L
.96146 .09097 L
.95793 .09729 L
.9512 .10362 L
.93435 .10994 L
.93016 .11072 L
.87955 .1139 L
.82895 .11355 L
.77834 .11395 L
.72773 .11424 L
.67713 .11386 L
.62652 .11396 L
.57591 .1142 L
.5253 .11392 L
.4747 .11392 L
.42409 .1142 L
.37348 .11396 L
.32287 .11386 L
.27227 .11424 L
.22166 .11395 L
.17105 .11355 L
.12045 .1139 L
.06984 .11072 L
.06565 .10994 L
.0488 .10362 L
.04207 .09729 L
.03854 .09097 L
.0365 .08464 L
.03524 .07831 L
.0345 .07199 L
.03417 .06566 L
.03417 .05934 L
.0345 .05301 L
.03524 .04669 L
.0365 .04036 L
.03854 .03403 L
.04207 .02771 L
.0488 .02138 L
.06565 .01506 L
F
0 g
.06984 .01428 m
.12045 .0111 L
.17105 .01145 L
.22166 .01105 L
.27227 .01076 L
.32287 .01114 L
.37348 .01104 L
.42409 .0108 L
.4747 .01108 L
.5253 .01108 L
.57591 .0108 L
.62652 .01104 L
.67713 .01114 L
.72773 .01076 L
.77834 .01105 L
.82895 .01145 L
.87955 .0111 L
.93016 .01428 L
.93435 .01506 L
.9512 .02138 L
.95793 .02771 L
.96146 .03403 L
.9635 .04036 L
.96476 .04669 L
.9655 .05301 L
.96583 .05934 L
.96583 .06566 L
.9655 .07199 L
.96476 .07831 L
.9635 .08464 L
.96146 .09097 L
.95793 .09729 L
.9512 .10362 L
.93435 .10994 L
.93016 .11072 L
.87955 .1139 L
.82895 .11355 L
.77834 .11395 L
.72773 .11424 L
.67713 .11386 L
.62652 .11396 L
.57591 .1142 L
.5253 .11392 L
.4747 .11392 L
.42409 .1142 L
.37348 .11396 L
.32287 .11386 L
.27227 .11424 L
.22166 .11395 L
.17105 .11355 L
Mistroke
.12045 .1139 L
.06984 .11072 L
.06565 .10994 L
.0488 .10362 L
.04207 .09729 L
.03854 .09097 L
.0365 .08464 L
.03524 .07831 L
.0345 .07199 L
.03417 .06566 L
.03417 .05934 L
.0345 .05301 L
.03524 .04669 L
.0365 .04036 L
.03854 .03403 L
.04207 .02771 L
.0488 .02138 L
.06565 .01506 L
.06984 .01428 L
Mfstroke
.4 g
.12045 .0147 m
.16511 .01506 L
.17105 .01516 L
.18429 .01506 L
.22166 .01459 L
.27227 .01417 L
.32287 .0147 L
.37348 .01457 L
.42409 .01421 L
.4747 .01462 L
.5253 .01462 L
.57591 .01421 L
.62652 .01457 L
.67713 .0147 L
.72773 .01417 L
.77834 .01459 L
.81571 .01506 L
.82895 .01516 L
.83489 .01506 L
.87955 .0147 L
.89161 .01506 L
.93016 .01971 L
.93552 .02138 L
.94714 .02771 L
.9528 .03403 L
.95596 .04036 L
.95789 .04669 L
.95901 .05301 L
.9595 .05934 L
.9595 .06566 L
.95901 .07199 L
.95789 .07831 L
.95596 .08464 L
.9528 .09097 L
.94714 .09729 L
.93552 .10362 L
.93016 .10529 L
.89161 .10994 L
.87955 .1103 L
.83489 .10994 L
.82895 .10984 L
.81571 .10994 L
.77834 .11041 L
.72773 .11083 L
.67713 .1103 L
.62652 .11043 L
.57591 .11079 L
.5253 .11038 L
.4747 .11038 L
.42409 .11079 L
.37348 .11043 L
.32287 .1103 L
.27227 .11083 L
.22166 .11041 L
.18429 .10994 L
.17105 .10984 L
.16511 .10994 L
.12045 .1103 L
.10839 .10994 L
.06984 .10529 L
.06448 .10362 L
.05286 .09729 L
.0472 .09097 L
.04404 .08464 L
.04211 .07831 L
.04099 .07199 L
.0405 .06566 L
.0405 .05934 L
.04099 .05301 L
.04211 .04669 L
.04404 .04036 L
.0472 .03403 L
.05286 .02771 L
.06448 .02138 L
.06984 .01971 L
.10839 .01506 L
F
0 g
.12045 .0147 m
.16511 .01506 L
.17105 .01516 L
.18429 .01506 L
.22166 .01459 L
.27227 .01417 L
.32287 .0147 L
.37348 .01457 L
.42409 .01421 L
.4747 .01462 L
.5253 .01462 L
.57591 .01421 L
.62652 .01457 L
.67713 .0147 L
.72773 .01417 L
.77834 .01459 L
.81571 .01506 L
.82895 .01516 L
.83489 .01506 L
.87955 .0147 L
.89161 .01506 L
.93016 .01971 L
.93552 .02138 L
.94714 .02771 L
.9528 .03403 L
.95596 .04036 L
.95789 .04669 L
.95901 .05301 L
.9595 .05934 L
.9595 .06566 L
.95901 .07199 L
.95789 .07831 L
.95596 .08464 L
.9528 .09097 L
.94714 .09729 L
.93552 .10362 L
.93016 .10529 L
.89161 .10994 L
.87955 .1103 L
.83489 .10994 L
.82895 .10984 L
.81571 .10994 L
.77834 .11041 L
.72773 .11083 L
.67713 .1103 L
.62652 .11043 L
.57591 .11079 L
.5253 .11038 L
.4747 .11038 L
.42409 .11079 L
Mistroke
.37348 .11043 L
.32287 .1103 L
.27227 .11083 L
.22166 .11041 L
.18429 .10994 L
.17105 .10984 L
.16511 .10994 L
.12045 .1103 L
.10839 .10994 L
.06984 .10529 L
.06448 .10362 L
.05286 .09729 L
.0472 .09097 L
.04404 .08464 L
.04211 .07831 L
.04099 .07199 L
.0405 .06566 L
.0405 .05934 L
.04099 .05301 L
.04211 .04669 L
.04404 .04036 L
.0472 .03403 L
.05286 .02771 L
.06448 .02138 L
.06984 .01971 L
.10839 .01506 L
.12045 .0147 L
Mfstroke
.5 g
.12045 .01866 m
.17105 .01921 L
.22166 .01845 L
.27227 .01788 L
.32287 .01859 L
.37348 .01841 L
.42409 .01794 L
.4747 .01848 L
.5253 .01848 L
.57591 .01794 L
.62652 .01841 L
.67713 .01859 L
.72773 .01788 L
.77834 .01845 L
.82895 .01921 L
.87955 .01866 L
.91227 .02138 L
.93016 .02603 L
.93384 .02771 L
.94272 .03403 L
.94743 .04036 L
.95022 .04669 L
.95182 .05301 L
.95252 .05934 L
.95252 .06566 L
.95182 .07199 L
.95022 .07831 L
.94743 .08464 L
.94272 .09097 L
.93384 .09729 L
.93016 .09897 L
.91227 .10362 L
.87955 .10634 L
.82895 .10579 L
.77834 .10655 L
.72773 .10712 L
.67713 .10641 L
.62652 .10659 L
.57591 .10706 L
.5253 .10652 L
.4747 .10652 L
.42409 .10706 L
.37348 .10659 L
.32287 .10641 L
.27227 .10712 L
.22166 .10655 L
.17105 .10579 L
.12045 .10634 L
.08773 .10362 L
.06984 .09897 L
.06616 .09729 L
.05728 .09097 L
.05257 .08464 L
.04978 .07831 L
.04818 .07199 L
.04748 .06566 L
.04748 .05934 L
.04818 .05301 L
.04978 .04669 L
.05257 .04036 L
.05728 .03403 L
.06616 .02771 L
.06984 .02603 L
.08773 .02138 L
F
0 g
.12045 .01866 m
.17105 .01921 L
.22166 .01845 L
.27227 .01788 L
.32287 .01859 L
.37348 .01841 L
.42409 .01794 L
.4747 .01848 L
.5253 .01848 L
.57591 .01794 L
.62652 .01841 L
.67713 .01859 L
.72773 .01788 L
.77834 .01845 L
.82895 .01921 L
.87955 .01866 L
.91227 .02138 L
.93016 .02603 L
.93384 .02771 L
.94272 .03403 L
.94743 .04036 L
.95022 .04669 L
.95182 .05301 L
.95252 .05934 L
.95252 .06566 L
.95182 .07199 L
.95022 .07831 L
.94743 .08464 L
.94272 .09097 L
.93384 .09729 L
.93016 .09897 L
.91227 .10362 L
.87955 .10634 L
.82895 .10579 L
.77834 .10655 L
.72773 .10712 L
.67713 .10641 L
.62652 .10659 L
.57591 .10706 L
.5253 .10652 L
.4747 .10652 L
.42409 .10706 L
.37348 .10659 L
.32287 .10641 L
.27227 .10712 L
.22166 .10655 L
.17105 .10579 L
.12045 .10634 L
.08773 .10362 L
.06984 .09897 L
Mistroke
.06616 .09729 L
.05728 .09097 L
.05257 .08464 L
.04978 .07831 L
.04818 .07199 L
.04748 .06566 L
.04748 .05934 L
.04818 .05301 L
.04978 .04669 L
.05257 .04036 L
.05728 .03403 L
.06616 .02771 L
.06984 .02603 L
.08773 .02138 L
.12045 .01866 L
Mfstroke
.6 g
.12045 .02304 m
.17105 .02366 L
.22166 .02267 L
.27227 .02195 L
.32287 .02285 L
.37348 .02262 L
.42409 .02202 L
.4747 .0227 L
.5253 .0227 L
.57591 .02202 L
.62652 .02262 L
.67713 .02285 L
.72773 .02195 L
.77834 .02267 L
.82895 .02366 L
.87955 .02304 L
.91559 .02771 L
.93016 .03387 L
.93041 .03403 L
.93746 .04036 L
.94146 .04669 L
.94371 .05301 L
.94468 .05934 L
.94468 .06566 L
.94371 .07199 L
.94146 .07831 L
.93746 .08464 L
.93041 .09097 L
.93016 .09113 L
.91559 .09729 L
.87955 .10196 L
.82895 .10134 L
.77834 .10233 L
.72773 .10305 L
.67713 .10215 L
.62652 .10238 L
.57591 .10298 L
.5253 .1023 L
.4747 .1023 L
.42409 .10298 L
.37348 .10238 L
.32287 .10215 L
.27227 .10305 L
.22166 .10233 L
.17105 .10134 L
.12045 .10196 L
.08441 .09729 L
.06984 .09113 L
.06959 .09097 L
.06254 .08464 L
.05854 .07831 L
.05629 .07199 L
.05532 .06566 L
.05532 .05934 L
.05629 .05301 L
.05854 .04669 L
.06254 .04036 L
.06959 .03403 L
.06984 .03387 L
.08441 .02771 L
F
0 g
.12045 .02304 m
.17105 .02366 L
.22166 .02267 L
.27227 .02195 L
.32287 .02285 L
.37348 .02262 L
.42409 .02202 L
.4747 .0227 L
.5253 .0227 L
.57591 .02202 L
.62652 .02262 L
.67713 .02285 L
.72773 .02195 L
.77834 .02267 L
.82895 .02366 L
.87955 .02304 L
.91559 .02771 L
.93016 .03387 L
.93041 .03403 L
.93746 .04036 L
.94146 .04669 L
.94371 .05301 L
.94468 .05934 L
.94468 .06566 L
.94371 .07199 L
.94146 .07831 L
.93746 .08464 L
.93041 .09097 L
.93016 .09113 L
.91559 .09729 L
.87955 .10196 L
.82895 .10134 L
.77834 .10233 L
.72773 .10305 L
.67713 .10215 L
.62652 .10238 L
.57591 .10298 L
.5253 .1023 L
.4747 .1023 L
.42409 .10298 L
.37348 .10238 L
.32287 .10215 L
.27227 .10305 L
.22166 .10233 L
.17105 .10134 L
.12045 .10196 L
.08441 .09729 L
.06984 .09113 L
.06959 .09097 L
.06254 .08464 L
Mistroke
.05854 .07831 L
.05629 .07199 L
.05532 .06566 L
.05532 .05934 L
.05629 .05301 L
.05854 .04669 L
.06254 .04036 L
.06959 .03403 L
.06984 .03387 L
.08441 .02771 L
.12045 .02304 L
Mfstroke
.7 g
.22166 .02738 m
.27227 .02642 L
.32287 .02759 L
.37348 .02729 L
.42409 .02651 L
.4747 .0274 L
.5253 .0274 L
.57591 .02651 L
.62652 .02729 L
.67713 .02759 L
.72773 .02642 L
.77834 .02738 L
.78954 .02771 L
.82895 .02866 L
.87955 .02798 L
.91386 .03403 L
.92521 .04036 L
.93016 .04534 L
.93109 .04669 L
.93428 .05301 L
.93564 .05934 L
.93564 .06566 L
.93428 .07199 L
.93109 .07831 L
.93016 .07966 L
.92521 .08464 L
.91386 .09097 L
.87955 .09702 L
.82895 .09634 L
.78954 .09729 L
.77834 .09762 L
.72773 .09858 L
.67713 .09741 L
.62652 .09771 L
.57591 .09849 L
.5253 .0976 L
.4747 .0976 L
.42409 .09849 L
.37348 .09771 L
.32287 .09741 L
.27227 .09858 L
.22166 .09762 L
.21046 .09729 L
.17105 .09634 L
.12045 .09702 L
.08614 .09097 L
.07479 .08464 L
.06984 .07966 L
.06891 .07831 L
.06572 .07199 L
.06436 .06566 L
.06436 .05934 L
.06572 .05301 L
.06891 .04669 L
.06984 .04534 L
.07479 .04036 L
.08614 .03403 L
.12045 .02798 L
.17105 .02866 L
.21046 .02771 L
F
0 g
.22166 .02738 m
.27227 .02642 L
.32287 .02759 L
.37348 .02729 L
.42409 .02651 L
.4747 .0274 L
.5253 .0274 L
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Cell[TextData[
"Here is the first numerical calculation which is by finite difference \
because me =2"], "Text",
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Cell[BoxData[
    \(\(rms = 1;\)\)], "Input",
  AspectRatioFixed->True],

Cell[TextData["Set the velocity to 0 on the boundary"], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(\(Table[wa[j, k] = 0, {j, 1, nx}, {k, 1, nx}];\)\)], "Input",
  AspectRatioFixed->True],

Cell[TextData["set the velocity elsewhere to a constant"], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(\(Table[wa[j, k] =  .2, {j, 2, nx - 1}, {k, 2, nx - 1}];\)\)], "Input",
  AspectRatioFixed->True]
}, Open  ]],

Cell[BoxData[
    \(While[rms > eps, sum = 0; 
      Do[jm = j - 1; jp = j + 1; 
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          If[me == 1, 
            dum = 1.  + \(1\/6. \) \((par1\ \((wa[jm, kp] + wa[jp, kp] + 
                          wa[jm, km] + wa[jp, km])\))\); 
            dum = dum + par2\ \((wa[j, kp] + wa[j, km])\) + 
                par3\ \((wa[jm, k] + wa[jp, k])\); wdd = \(0.75\ dum\)\/par1, 
            dum = 1.  + 
                bas\ \((wa[jm, k] + 
                      wa[jp, k])\) + \(wa[j, km] + wa[j, kp]\)\/dys; 
            wdd = \(0.5\ dum\)\/par1]; dif = wdd - wa[j, k]; 
          sum = sum + dif\ dif; 
          wa[j, k] = wa[j, k] + om\ dif, {k, 2, nyp}], {j, 2, nxp}]; 
      rms = \@\(sum\/\(\((anx - 1. )\)\ \((anx - 1. )\)\)\)]\)], "Input",
  AspectRatioFixed->True],

Cell[TextData["Display the solution"], "Text",
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  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(ListContourPlot[Table[wa[i, j], {i, 1, nx}, {j, 1, nx}], 
        AspectRatio \[Rule] 1\/bar];\)\)], "Input",
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F
.5 Mabswid
.1 g
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F
0 g
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Mistroke
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Mfstroke
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F
0 g
.12045 .00808 m
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.22166 .00763 L
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.37348 .00758 L
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Mistroke
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Mfstroke
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F
0 g
.06984 .01437 m
.12045 .01164 L
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.27227 .01081 L
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Mistroke
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Mfstroke
.4 g
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F
0 g
.17105 .01461 m
.22166 .01434 L
.27227 .01425 L
.32287 .01422 L
.37348 .0142 L
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Mistroke
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.17105 .01461 L
Mfstroke
.5 g
.12045 .01971 m
.17105 .01845 L
.22166 .01808 L
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.37348 .01788 L
.42409 .01788 L
.4747 .01787 L
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.0516 .04036 L
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F
0 g
.12045 .01971 m
.17105 .01845 L
.22166 .01808 L
.27227 .01795 L
.32287 .0179 L
.37348 .01788 L
.42409 .01788 L
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.93016 .02623 L
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.94331 .03403 L
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.9511 .04669 L
.95269 .05301 L
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.95264 .07199 L
.95102 .07831 L
.94814 .08464 L
.94311 .09097 L
.93337 .09729 L
.93016 .0986 L
.90485 .10362 L
.87955 .10504 L
.82895 .10628 L
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.72773 .10675 L
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.5253 .10683 L
.4747 .10683 L
.42409 .10682 L
.37348 .10682 L
.32287 .1068 L
.27227 .10676 L
.22166 .10665 L
.17105 .10631 L
.12045 .10508 L
.0947 .10362 L
.06984 .09866 L
Mistroke
.06647 .09729 L
.05677 .09097 L
.05176 .08464 L
.04889 .07831 L
.04727 .07199 L
.04652 .06566 L
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.04879 .04669 L
.0516 .04036 L
.05652 .03403 L
.066 .02771 L
.06984 .02613 L
.09285 .02138 L
.12045 .01971 L
Mfstroke
.6 g
.12045 .02441 m
.17105 .02267 L
.22166 .02217 L
.27227 .022 L
.32287 .02193 L
.37348 .02191 L
.42409 .0219 L
.4747 .0219 L
.5253 .0219 L
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.62652 .02191 L
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.82895 .02274 L
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.9114 .02771 L
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.93016 .03422 L
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.92945 .09097 L
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.87955 .10027 L
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.5253 .10274 L
.4747 .10274 L
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.37348 .10273 L
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.08929 .09729 L
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.06984 .09068 L
.0623 .08464 L
.05798 .07831 L
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.05453 .06566 L
.0545 .05934 L
.05552 .05301 L
.05782 .04669 L
.06204 .04036 L
.06984 .03406 L
.06989 .03403 L
.08804 .02771 L
F
0 g
.12045 .02441 m
.17105 .02267 L
.22166 .02217 L
.27227 .022 L
.32287 .02193 L
.37348 .02191 L
.42409 .0219 L
.4747 .0219 L
.5253 .0219 L
.57591 .0219 L
.62652 .02191 L
.67713 .02194 L
.72773 .02202 L
.77834 .02221 L
.82895 .02274 L
.87955 .0245 L
.9114 .02771 L
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.93016 .03422 L
.93776 .04036 L
.94201 .04669 L
.94433 .05301 L
.94536 .05934 L
.94534 .06566 L
.94426 .07199 L
.94189 .07831 L
.93755 .08464 L
.93016 .09057 L
.92945 .09097 L
.91033 .09729 L
.87955 .10027 L
.82895 .10199 L
.77834 .10247 L
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.67713 .1027 L
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.5253 .10274 L
.4747 .10274 L
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.37348 .10273 L
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.22166 .1025 L
.17105 .10203 L
.12045 .10033 L
.08929 .09729 L
.07035 .09097 L
.06984 .09068 L
.0623 .08464 L
Mistroke
.05798 .07831 L
.05561 .07199 L
.05453 .06566 L
.0545 .05934 L
.05552 .05301 L
.05782 .04669 L
.06204 .04036 L
.06984 .03406 L
.06989 .03403 L
.08804 .02771 L
.12045 .02441 L
Mfstroke
.7 g
.17105 .02742 m
.22166 .02673 L
.27227 .02649 L
.32287 .02641 L
.37348 .02637 L
.42409 .02636 L
.4747 .02636 L
.5253 .02636 L
.57591 .02636 L
.62652 .02638 L
.67713 .02642 L
.72773 .02653 L
.77834 .02678 L
.82895 .0275 L
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.87955 .02993 L
.90905 .03403 L
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.93016 .04599 L
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.93016 .07873 L
.92349 .08464 L
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.87955 .09479 L
.82895 .09716 L
.82329 .09729 L
.77834 .09783 L
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.4747 .0982 L
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.17408 .09729 L
.17105 .09722 L
.12045 .09487 L
.09148 .09097 L
.07625 .08464 L
.06984 .079 L
.06931 .07831 L
.06573 .07199 L
.06414 .06566 L
.0641 .05934 L
.06559 .05301 L
.06905 .04669 L
.06984 .04565 L
.07578 .04036 L
.09035 .03403 L
.12045 .02981 L
.15774 .02771 L
F
0 g
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Cell[TextData["Set the velocity to 0 on the boundary"], "Text",
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Cell[TextData["set the velocity elsewhere to a constant"], "Text",
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