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Notebook[{
Cell["Boundary-layer flow over a flat plate and wedge", "Title",
  Evaluatable->False,
  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\n\
Mark.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 this notice \
remain visible to other users. \nThere is no charge for copying and \
dissemination   \n\nVersion:  11/26/00"
}], "Text"],

Cell[CellGroupData[{

Cell["Summary", "Subtitle"],

Cell["\<\
This notebook examines the boundary-layer flow over a flat plate \
and a wedge.  For the flat plate, a similarity variable is used to reduce the \
continuity and momentum equations to a single nonlinear ODE. (A slightly more \
complex transformation and corresponding ODE exist for the flow over a \
wedge.)  For both cases, the solution is given using a numerical scheme and \
several important physical aspects of boundary layers are elucidated.\
\>", \
"Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["References", "Subtitle"],

Cell[TextData[{
  "The references for this NoteBook are:\nM. M. Denn\n",
  StyleBox["Process Fluid Mechanics",
    FontSlant->"Italic"],
  "\nPrentice Hall\n1978\n\nR. L. Panton\nIncompressible Flow\nWiley\n1986"
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Homework problem for this note book", "Subtitle"],

Cell[TextData[StyleBox["1.  How does the boundary layer thickness change with \
Reynolds number ??\n2.  How does the wall shear stress change with Reynolds \
number ??\n3.  What are two defining physical characteristics of boundary \
layers ??\n4.  Run the code to find is the approximate location of the outer \
edge of the boundary layer.\n5.  Run the code to find the value of the stress \
at the wall for a flat plate.  This value is related to any heat or mass \
transfer coefficient for flow past a solid surface. \n6.  Run the code to \
show how the boundary-layer thickness and shear stress change when the \
imposed pressure gradient changes (i.e. the wedge problem) \n7.  For the case \
where the angle changes, explain how the pressure gradient acts to change the \
momentum within the boundary layer.  ",
  FontSize->12,
  FontWeight->"Plain"]], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Flow past a flat plate", "Subtitle"],

Cell[CellGroupData[{

Cell["Derivation of the ODE's from the original PDE's.", "SectionFirst",
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  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell["Do background work to set up the problem", "Subsection"],

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Cell["Existence of a similarity variable.  ", "Subsubsection"],

Cell[TextData[{
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  ",  Butterworth.)  \n\nIf you encounter a fluid flow problem in a \
rectangular channel, you expect the characteristic length to be the channel \
height.  For a circular pipe, it will be the diameter or radius.  For flow \
past a sphere, the radius is a reasonable length.  Thus we always expect to \
find a characteristic length, that is contained in the physics of the flow.  \
However, for a boundary-layer flow, where the boundary-layer is much smaller \
than the scale of the container, or a boundary-layer on the outside of an \
object where again the layer is much (say ~1/100) smaller than the device, \
what now is the relevant length scale?  Would nature leave us with no \
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  " length scale.  It is just not a ",
  StyleBox["geometric",
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  " length scale.  In the absence of a reasonable geometric length scale we \
look to define a \"similarity\" variable.  The implication is that the \
velocity or temperature profiles are \"self-similar\", and with this proper \
scaling can be collapsed onto the same curve.  The length scale for the \
normal direction within a boundary - layer is constructed from the physical \
properties of the flow and any other length or time variables that come to \
play in the problem.\n\nFor example, for the \"suddenly started\" plate \
problem, where a long and wide flat plate is started in motion, the \
boundary-layer thickness scales as ",
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  " is time and \[Nu] is the kinematic viscosity. (The boundary layer gets \
thicker in time as ",
  Cell[BoxData[
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  ".) \n\nTo wrap this up, you always want to investigate the scaling of your \
problems.  If there is no geometric length scale, (i.e., problem has a \
infinite spatial domain), you expect the existence of a similarity variable \
which is a combination of the other variables and physical properties.  \n\n\
Use of a similarity variable can lead to transformations that turn the PDE's \
into ODE's.  This can work for both linear and nonlinear equations.      "
}], "Text"]
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Cell["\<\
The  mass and momentum  equations for a forced boundary - layer \
flow past a flat plate\
\>", "Subsubsection"],

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this\ region . \ \ The\ scaling\ arguments\ on\ pp\ 395\), "-", 
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Cell["The continuity equation is just what it always is.", "Text"],

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Cell["Stream function formulation to satisfy continuity", "Subsubsection"],

Cell["\<\
We want to turn these two coupled PDE's into a single ODE.  It will \
still be nonlinear, but it is much more easy to solve that the original 2 \
equations.

We will define a stream function from the continuity equation, define a \
similarity variable involving x and y.  Then transform the equations.  \
\>", \
"Text"],

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Cell["\<\
Transformation of the momentum equation in terms of the stream \
function\
\>", "Subsubsection"],

Cell["\<\
As mentioned before, the stream function \[Psi], can represent both \
the x and y directions velocities, this gives for the momentum equations, \
\
\>", "Text"],

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Cell["\<\
We won't use this equation because the substitutions are \
harder.\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Similarity variable formulation", "Subsubsection"],

Cell["\<\
Because the only length scale is the one measured along the plate \
and because we expect that the velocity profiles will be self similar (the \
same shape scaled by just the local boundary-layer thickness), we look for \
similarity variables.  \
\>", "Text"],

Cell[TextData[{
  "Start by defining a similarity variable that will be dimensionless.  We \
will just choose the correct one (see Panton or Denn), there are formal \
procedures for finding these.  (See Leal's book for how to do this.)  We will \
also be changing the dimensional equation into a dimensionless ODE.  When we \
do this, there will not be a parameter in this resulting equation because the \
length scale for y is the boundary-layer thickness (which scales as ",
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Reynolds number removes the Reynolds number in the equation. "
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Cell["Likewise for the stream function,", "Text"],

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Cell["\<\
We use the Greek, in the equations, but for substitution purposes, \
it is convenient to have a different name, eta and psi.\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Define the velocities in terms of the stream function and \
similarity variable\
\>", "Subsubsection"],

Cell[TextData[{
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  "We want the velocify in terms of our definition of the stream function.  \
",
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Note again that U shows up because the f(\[Eta]) is dimensionless \
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\>", "Text"],

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Now get the y derivative of \[Eta] and substitute. \
\>", "Text"],

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For further substitutions below we it is easiest to use vee2, \
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\>", "Text"],

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Cell["Now derive the nonlinear ODE that describes the flow field.", \
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Cell[TextData[ButtonBox["Return to conclusions",
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  ButtonStyle->"Hyperlink"]], "Text"],

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This step replaces derivatives of \[Eta].\
\>", "Text"],

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Now that we are done taking derivatives we can make the \[Eta](x,y) \
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\>", "Text"],

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Cell["Boundary conditions", "Subsection"],

Cell["\<\
The boundary conditions are that there is no slip on the surface, \
no flow through the surface and that the tangential velocity matches the free \
stream far away. 

The are in terms of the newest notation:

u(x,y=0)=0, \[Eta] = 0, f'(\[Eta]= 0)=0,
v(x,y=0)=0, \[Eta] = 0, f(\[Eta]= 0=0,
y->\[Infinity], u(x,y->\[Infinity])=U, f'(\[Eta]=\[Infinity]) = 1,  
x=0, u(x,y->\[Infinity])=U, f'(\[Eta]=\[Infinity]) = 1,  \
\>", "Text"]
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Cell["\<\
The primes (') denote derivatives with respect to the similarity \
variable.  This nonlinear ode does not have a known analytical solution.  \
However, it can be easily solved numerically.  
\t\t
Numerical solutions for ode's are often done by creating a system of first \
order odes.  This is done by defining
\t\t
\t\ty1 = f,
\t\ty2 = f'
\t\ty3 = f'',
\t\t
These have to be related. 
\t\t
\t\tThus we have
\t\ty1' = y2 (by definition)
\t\ty2' = y3 (by definition)
\t\ty3' = -1/2*y1*y3   (which is from the original ode.)
\t\t
In the first part I solve the flat plate problem separately using a crude \
numerical scheme that usually converges, although not real fast.  It is a \
shooting method.  I have picked y =15 as the end of the integrate.  The \
NDSolve routine uses a Runge-Kutta method with built in step size adjustment. \
 \
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  "From Denn the flow past a wedge (figure 15-1) is given by\nU(x) = A x",
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  StyleBox["Thus, the flow past a wedge is a model problem for telling how \
the boundary layer will change as the pressure gradient changes",
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  ". *****\n\nDenn tells us that the equation can be reduced to a nonlinear \
ode again taking advantage of similarity of solution profiles.  This gives\n\n\
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  ")  = 0.\n\t\t\n\t\tIf m=0, the equation is the same as for the flat plate. \
 \n\n\nWe again need to make a system of first order ODE's .  These are\n\n\
y1' = y2\ny2' = y3\ny3' = - (m+1)/2 y1 y3 + m*(1-y2",
  StyleBox["2",
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  ")   \n\nThis time I use a real shooting method, with a pseudo \
Newton-Raphson iteration.  It works for most cases.  If the angle is too \
negative, the stress will go through 0, this means that there is no layer and \
thus it is not surprising that the answer blows up.  Of course if we look at \
the physical situation, we might be surprised that we can get any solution \
for negative \[Beta].  The trend is correct and it is interesting to examine \
the case of a diverging flow. \n\n"
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Cell[TextData[
"Note that you will want to run this several times changing the value of \
\[Beta] over the greatest possible range.  If \[Beta]=0, the result is the \
same as a flat plate.    "], "Text"],

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Cell["Plots of the results at different angles", "Subsection"],

Cell["\<\
Here I save some old results, which I must do after a run. (You \
need to do these yourself, they will not just run !!).  Just make up a name \
for the result that you wish to save and then equate this to \"zz\" as I show \
below.  Your newly chosen name will store all of the useful results of the \
calculation.\
\>", "Text",
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  AspectRatioFixed->True],

Cell["\<\
For example, for -\[Pi]i/5.05 I did\
\>", "Text"],

Cell[BoxData[
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  AspectRatioFixed->True],

Cell["For -\[Pi]i/8 I did ", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(\(zzmpis8 = zz;\)\)], "Input",
  AspectRatioFixed->True],

Cell["For 0", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(\(zz0 = zz;\)\)], "Input",
  AspectRatioFixed->True],

Cell[TextData["\[Pi]//2"], "Text"],

Cell[BoxData[
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  AspectRatioFixed->True],

Cell[TextData["\[Pi]/1.3"], "Text"],

Cell[BoxData[
    \(\(zzpis1p3 = zz;\)\)], "Input",
  AspectRatioFixed->True],

Cell["\<\
Here I plot some of them together.  The middle one is the flow \
plate.  The two to the left of it are for a converging flow (more stress, \
thinner layer), the two to the right are for diverging flow (thicker layer, \
less stress).\
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
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            zzmpis5p05\[LeftDoubleBracket]1\[RightDoubleBracket], 
          y2[x] /. 
            zzmpis8\[LeftDoubleBracket]1\[RightDoubleBracket], \n\t\ty2[x] /. 
            zz0\[LeftDoubleBracket]1\[RightDoubleBracket]}, {x, 0, youter}, 
        PlotRange \[Rule] All, AxesLabel \[Rule] {"\<zeta\>", "\<f'\>"}, 
        PlotStyle \[Rule] {Dashing[{ .02,  .02}], 
            Dashing[{ .05,  .03}], \[IndentingNewLine]Dashing[{1, 0}], 
            Dashing[{ .03,  .01}], \[IndentingNewLine]Dashing[{ .02,  .04,  \
.06,  .04}]}];\)\)], "Input",
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The important result is that in a converging region, m>0, the \
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