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Notebook[{
Cell["Linear Stability of Pressure Driven Channel Flow", "Title"],

Cell[TextData[{
  "This notebook has been written in ",
  StyleBox["Mathematica ",
    FontSlant->"Italic"],
  "by \n\n",
  StyleBox["Mark J. McCready\nProfessor and Chair of Chemical Engineering\n\
University of Notre Dame\nNotre Dame IN 46556\nUSA",
    FontSize->14],
  "\n\nMark.J.McCready.1@nd.edu\nhttp://www.nd.edu/~mjm/\n\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 \
if you use it, that this notice remain visible to other users. \nThere is no \
charge for copying and dissemination   \n\nVersion:  6/20/99\nMore recent \
versions of this notebook should be available at the web site:\n\
http://www.nd.edu/~mjm/channelflow.nb"
}], "Text"],

Cell[CellGroupData[{

Cell["Summary", "Subtitle"],

Cell[TextData[{
  "This notebook gives the Orszag-tau spectral linear stability calculation \
for channel flow.\n\n",
  StyleBox["Reference:  S. A. Orszag (1971) ", "SmallText"],
  StyleBox["\"Accurate solution of the Orr-Sommerfeld stability equation\", \
Journal of Fluid Mechanics", "SmallText",
    FontWeight->"Plain"],
  StyleBox[", ", "SmallText"],
  StyleBox["50", "SmallText",
    FontWeight->"Bold"],
  StyleBox[" ", "SmallText"],
  StyleBox["pp 689-703", "SmallText",
    FontWeight->"Plain"],
  StyleBox[".", "SmallText"],
  "\n\nThe coefficients of the algebraic equations are computed directly from \
the orthogonality property of the Chebyshev polynomials using integration. \n\
\n",
  StyleBox["Reference:  R. Miesen and B. J. Boersma (1995) \"Hydrodynamic \
stability of a sheared liquid film\",", "SmallText"],
  " ",
  StyleBox["Journal of Fluid Mechanics", "SmallText",
    FontWeight->"Plain"],
  StyleBox[", 301 ", "SmallText"],
  StyleBox["pp 175-202", "SmallText",
    FontWeight->"Plain"],
  StyleBox[".\n\n", "SmallText"],
  "In the tau method, the solution is approximated with Orthogonal \
polynomials that are valid over the entire domain, as in the Galerkin \
methods.  However, in many problems because of the complexity of the boundary \
conditions, it is not possible to make a convenient set of functions exactly \
fit the boundary conditions.  To solve this problem, the tau method enforces \
the boundary conditions as additional algebraic equations along with those \
that originate with the differential equation.  To obtain the extra degrees \
of freedom that this requires, a requisite number of the highest \
\"frequency\"  (i.e. highest n) modes generated from the differential \
equations are not enforced.  As long as the number of modes can be made large \
(>10-20) this method will probably work fine.  For the long wave region, \
numerical resolution limits the number of modes that can be used in this \
formulation and so there could be some problem getting the tau method to give \
reliable results.  "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

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Cell[TextData[{
  StyleBox["Mathematica",
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Cell["\<\
I have used a mix of input notation to show how it can be done. \
\
\>", "Text"],

Cell["\<\
I would enter the dynamic boundary condition from a key pad \
as\
\>", "Text"],

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Cell["\<\
Note that you can convert any \"cryptic\" input expression by \
selecting the cell, going up to the cell menu and selecting \"convert to \
traditional form\".  This is \"shift curly t\" on the keypad.  \
\>", "Text"]
}, Closed]]
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Cell["\<\
This section expands the terms of the Orr-Sommerfeld equation in \
terms of the Chebyshev polynomials and gets the coefficients using the \
orthogonality.\
\>", "Subtitle",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
We need use only even modes because the boundary conditions are \
symmetric.  There are then two boundary conditions, no slip and no flow \
through the wall, thus we will drop off the two highest equations produced \
from substitution of the Chebyshev's in the ODE.\
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
To do some of the integrals, we need to extend the limit of \
internal recursion.  I don't know why!!\
\>", "Text"],

Cell[BoxData[
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  CellLabel->"In[212]:="],

Cell["\<\
Here is the start of the calculations.  
In this example we use nn/2 +1 modes.    \
\>", "Text"],

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  CellLabel->"In[213]:=",
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Cell["\<\
Here is the second derivative term of the Orr-Sommerfeld equation \
\
\>", "Subsubsection",
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  AspectRatioFixed->True],

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Here is the fourth derivative term of the Orr-Sommerfeld equation \
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Here is the zero derivative term of the Orr-Sommerfeld equation \
\
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Cell[BoxData[
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Cell["Here are the boundary conditions.", "Subsubsection",
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Cell[BoxData[
    \(bc1 = vexpand /. {y \[Rule] \(-1\)}\)], "Input",
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Cell[BoxData[
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Cell["Here is where we use orthogonality", "Subsubsection",
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Cell["First get all of the pieces that we need to integrate", "Text"],

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Cell["\<\
Now form the products that would need to be integrated.  First in \
terms of c[i]\
\>", "Text"],

Cell[BoxData[
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Cell["Now make all of the possible combinations.", "Text"],

Cell[BoxData[
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Cell["Let's get rid of the c[i]'s", "Text"],

Cell[BoxData[
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Cell[TextData[{
  "At this point, we could just integrate \"temp7\" from -1 to 1 and we would \
have the desired algebraic equations.  However the integration is very slow \
for a lot of terms and we need to speed it up.  To do this, we factor out the \
common terms.  These are all of the form ",
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Cell[TextData[{
  "Here is where we factor them out.  The Transpose is needed to put the \
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  "'s.   "
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orthog6=Transpose[Table[Coefficient[orthog5,y,i],{i,0,2 \
nn,2}]];\
\>", "Input",
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Cell["Here is what we integrate:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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      y\^6\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^8\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^10\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^12\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^14\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^16\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^18\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^20\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^22\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^24\/\(\[Pi]\ \@\(1 - y\^2\)\), 
      y\^26\/\(\[Pi]\ \@\(1 - y\^2\)\), y\^28\/\(\[Pi]\ \@\(1 - y\^2\)\), 
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Cell["Here is the integration showing the time it takes.  ", "Text"],

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Cell[BoxData[
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        46189\/262144, 88179\/524288, 676039\/4194304, 1300075\/8388608, 
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Cell["\<\
Now multiply the coefficients times the integrated polynomial \
pieces.   \
\>", "Text"],

Cell["orthog9= Simplify[Expand[orthog6.orthog8]]; ", "Input",
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Cell["Tau solution method", "Subsubsection"],

Cell["\<\
We solve for the two highest coefficients because the two highest \
modes are dropped from the matrix of the differential equation mode \
coefficients (the Tau method).  The boundary conditions replace these modes \
assure that the BC's are satisfied.  However, the BC equations do not contain \
the eigen value and thus must be eliminated algebraically before solving the \
eigenvalue problem.\
\>", "Text",
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Cell[TextData[
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"Here is where we use the boundary conditions to replace the two highest \
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Cell[TextData["Here we pick out the number of valid equations"], "Text",
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Here we construct the matrix that comprises the generalized \
eigenvalue problem 
AB = (AA  -   B  \[Lambda])\
\>", "Text",
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Cell["Now get numbers, so we finish it numerically.", "Subsubsection"],

Cell["\<\
At this point we start doing things numerically.
First choose \[Alpha] and Reynolds number\
\>", "Text",
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Cell["Here is the calculation of Binverse to solve the GEP.", "Text",
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Cell["Binv.AA = AZ", "Text",
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Cell[BoxData[
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Cell["Get the eigenvalues.", "Text",
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Cell["\<\
Here is the same calculation but made to run faster numerically by \
using recursion formulas for the coefficients.\
\>", "Subtitle",
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Cell["\<\
Here is the second derivative term of the Orr-Sommerfeld equation \
\
\>", "Text",
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Cell[TextData[{
  "The recursion formulas to construct the Matrix eigenvalue problem are \
available in Orszag's paper. These are several more that are useful when \
solving the two-layer interfacial eigenvalue problem are given in:\n\n",
  StyleBox["Primary and Secondary Interfacial Disturbances in Horizontal \
Cocurrent Flows",
    FontSlant->"Italic"],
  "\nPh. D. Thesis, \nWilliam C. Kuru\nDepartment of Chemical Engineering\n\
University of Notre Dame\n1995.\n\nNote how these match the ones obtained by \
integration."
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Cell["\<\
Here is the fourth derivative term of the Orr-Sommerfeld equation \
\
\>", "Text",
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Cell["\<\
Here is the zero derivative term of the Orr-Sommerfeld equation.  I \
now have all of the zero derivative terms here.  \
\>", "Text",
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Cell[TextData["Here are the boundary conditions"], "Text",
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\\!\\(TraditionalForm\\`\\((\[NoBreak] \\(\[LeftSkeleton] 1 \
\[RightSkeleton]\\) \[NoBreak])\\)\\) may contain significant numerical \
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Cell["\<\
We get a warning that the matrix is badly conditioned, but the \
answers are not bad.  You can look at the numbers to see how bad.  The \
contour plot is not too scary since the change is spread over the entire \
range. \
\>", "Text"],

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Here is the growth curve in Im[\[Lambda]] vs. \[Alpha] space\
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Initialize functions to use in the calculations.  If this is done, \
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Cell["\<\
Here is the stability boundary in \[Alpha]-Re space\
\>", "Subtitle",
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  "This module employs the secant method (adapted from Press, Teukolsky, \
Vetterling and Flannery  ",
  StyleBox["Numerical Recipes in Fortran, ",
    FontSlant->"Italic"],
  "2nd Ed. Cambridge Press, 1992, p 351.)  (Since there is no ",
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  " ",
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  ". "
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          If[Abs[fl] < Abs[f], {rtsec = x1; x1 = x2; swap = fl; fl = f; 
              f = swap}, {xl = x1; rtsec = x2}]; 
          While[Abs[dx] > xacc, {dx = \(\((xl - rtsec)\)\ f\)\/\(f - fl\); 
              xl = rtsec; fl = f; rtsec = rtsec + dx; rrx = rtsec; 
              f = Im[lambda]; Print["\<rrx= \>", rrx, "\< f= \>", f]}]; 
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}, Closed]]
},
FrontEndVersion->"4.0 for Macintosh",
ScreenRectangle->{{0, 1152}, {0, 850}},
ScreenStyleEnvironment->"Working",
WindowToolbars->{},
CellGrouping->Manual,
WindowSize->{500, 710},
WindowMargins->{{10, Automatic}, {Automatic, 16}},
PrintingCopies->1,
PrintingPageRange->{1, Automatic},
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ShowCellLabel->True,
ShowCellTags->False,
RenderingOptions->{"ObjectDithering"->True,
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CharacterEncoding->"MacintoshAutomaticEncoding",
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StyleDefinitions -> Notebook[{
  
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Modify the definitions below to change the default appearance of \
all cells in a given style.
Make modifications to any definition using commands in the Format menu.\
\>", 
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    FontSlant->"Italic",
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    CounterIncrements->"Subsection",
    CounterAssignments->{{"Subsubsection", 0}},
    FontSize->14,
    FontWeight->"Bold",
    Background->RGBColor[0.970001, 0.890013, 0.850004]],
  
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    FontWeight->"Bold",
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  Cell[StyleData["Subsubsection", "Printout"],
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  Cell[CellGroupData[{
  
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    LineSpacing->{1, 2},
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    LineSpacing->{1, 3},
    FontSize->10]
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    FontSize->10],
  
  Cell[StyleData["Caption", "Printout"],
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  Cell[CellGroupData[{
  
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*)




(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)