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Notebook[{
Cell["Inviscid flow past a stationary sphere", "Title"],

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",
  ButtonBox["Mark.J.McCready.1@nd.edu",
    ButtonData:>{
      URL[ "mailto:mjm@nd.edu"], None},
    ButtonStyle->"Hyperlink"],
  "\n",
  ButtonBox["http://www.nd.edu/~mjm/",
    ButtonData:>{
      URL[ "http://www.nd.edu/~mjm/"], None},
    ButtonStyle->"Hyperlink"],
  "\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:  11/06/00\n",
  StyleBox["More recent versions of this notebook should be available at the \
web site:\n",
    FontSize->12],
  ButtonBox["http://www.nd.edu/~mjm/InviscidSphere.nb",
    ButtonData:>{
      URL[ "http://www.nd.edu/~mjm/InviscidSphere.nb"], None},
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Cell["\<\
Reference:  M. M. Denn, Process Fluid Dynamics, Prentice Hall, 1978 \
\
\>", "Subsubsection"],

Cell[CellGroupData[{

Cell["Summary", "Subtitle",
  CellLabelAutoDelete->True],

Cell["\<\
This notebook considers the problem of an inviscid flow past a \
stationary sphere.  Solutions are obtained using both a potential function \
and a stream function.  The pressure field around the sphere and the drag, \
which is 0 (d'Alembert's paradox) are found.

This solution is a nice example of how for an inviscid flow (the idealized \
limit of a very high Reynolds number flow), the flow field can be computed \
even though the governing Euler equations are nonlinear.  This is because the \
flow can be considered irrotational (vorticity=0) so that the flow field can \
be defined as the gradient of a scalar potential function.  

Such a formulation should give useful results for high Reynolds number flows \
away from boundaries where vorticity is produced.  Perhaps the best example \
of the utility of inviscid flow theory is for flow around airfoils to give \
the pressure field and thus the lift.  The drag is obtained from \
boundary-layer theory -- where viscous effects are important.       \
\>", \
"Text",
  Evaluatable->False,
  CellLabelAutoDelete->True,
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Cell[TextData[{
  StyleBox["Mathematica",
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  " aside"
}], "Subsection",
  CellLabelAutoDelete->True],

Cell[TextData[{
  "I have used a mix of input notation to show how it can be done.   Also, \
some of the major equations are shown in input form and also in typeset form \
for clarity.  The typeset input equations are usually marked as inactive and \
so are not read in by ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " .. but they could be.  "
}], "Text",
  CellLabelAutoDelete->True],

Cell[TextData[{
  "I would enter the dynamic boundary condition from a key pad in ",
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  " form as"
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Cell[CellGroupData[{

Cell[BoxData[
    \(dc = \(-\ \[Gamma]\)\ D[\[Eta][x, t], {x, 
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              U1\ D[\[Phi]\_1[x, y, t], x] - 
              g\ \[Eta][x, 
                  t])\) + \n\t\t\t\(\[Rho]\_2\) \((\(-D[\[Phi]\_2[x, y, t], 
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Cell[TextData[{
  "In ",
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  ", which is a little shorter, you would need the type set window to make \
this practical.  "
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Cell[TextData[{
  "Finally it may be more easily ",
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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",
  CellLabelAutoDelete->True]
}, Closed]]
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Cell[CellGroupData[{

Cell["Keywords", "Subtitle",
  CellLabelAutoDelete->True],

Cell["\<\
Inviscid flow, Ideal flow theory, Irrotational flow, Transport \
phenomena,  Fluid Dynamics\
\>", "Text",
  CellLabelAutoDelete->True]
}, Open  ]],

Cell[CellGroupData[{

Cell["Homework questions for this notebook:", "Subtitle",
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  CellLabelAutoDelete->True,
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Cell["\<\
Run the file, read the exposition and then answer these questions.
 
1.\t  For an inviscid flow, what physical observation is made that leads to \
the governing equation for the flow field in terms of a scalar function?

2.\tWhy is there a substantial simplification of the original problem in \
terms of the N-S equation with \[Eta] =0., Re--> \[Infinity]?

3.\tFlow separation occurs readily in the flow past a sphere problem if Re is \
sufficiently high.  Explain why it does not occur for an inviscid flow.  \
\
\>", "Text",
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  CellLabelAutoDelete->True,
  AspectRatioFixed->True]
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Cell[CellGroupData[{

Cell["Problem statement", "Subtitle",
  CellLabelAutoDelete->True],

Cell[TextData[{
  "The problem of interest is a uniform, inviscid flow, with velocity U that \
has a sphere of radius R placed in the flow.  There is a obvious axis of \
symmetry, the \[Phi] direction, (not to be confused with \[Phi][",
  StyleBox["r",
    FontSlant->"Italic"],
  ",\[Theta]], the velocity potential) which is oriented parallel to the \
flow.  The problem will then be solved in the ",
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    FontSlant->"Italic"],
  " and \[Theta] coordinate directions.  The angle \[Theta] = 0, is at the \
three o'clock position on the sphere and \[Theta]=\[Pi] at nine o'clock \
(which is the entire range of \[Theta] since the angle \[Phi] would go from 0 \
to 2 \[Pi] if necessary -- but not in this problem).        "
}], "Text",
  CellLabelAutoDelete->True],

Cell[CellGroupData[{

Cell["Boundary conditions", "Subsection",
  CellLabelAutoDelete->True],

Cell[TextData[{
  "Since this is an inviscid flow, the boundary conditions for this flow are \
quite simple.  As r-> \[Infinity], the velocity must match the free stream \
flow.  The other boundary condition is that no flow can go through the \
surface of the sphere.  Our flow is left to right and the angle \[Theta] is \
measured starting at 3 o'clock and increases counter clockwise on the circle. \
 We can figure out the correct sign for the far away flow by considering that \
at 9 O'Clock, the flow is purely radial and is coming inward which is \
negative.  At 3 o'clock the flow is going straight out, which is positive.  \
Similarly, at 12 o'clock, the flow is purely tangential, in the direction of \
decreasing \[Theta] value.  Thus we use a ",
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  " in front of the radial velocity and a ",
  StyleBox["-Sin",
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  " in front of the tangential velocity.  (I could have drawn a picture, but \
this is a useful visualization exercise.)    \n\nThus\n\nFor r->\[Infinity],\n\
",
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  " = U Cos(\[Theta])\n",
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Cell[CellGroupData[{

Cell["Solution using the velocity potential", "Subtitle",
  CellLabelAutoDelete->True],

Cell[CellGroupData[{

Cell["\<\
Governing equation for the flow field in the form of a velocity \
potential\
\>", "Subsection",
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  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell["\<\
The governing equation for inviscid flows in three dimensions is \
Laplace's equation for the potential function, \[Phi][r,\[Theta]]  
\
\>", "Text",
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  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell["\<\
Note that this first cell is inactive, but shows the equation, the \
next one really works\
\>", "Text",
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Cell[CellGroupData[{

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Cell["\<\
laplaceeq = Simplify[1/(r^2*Sin[\[Theta]])*
   (D[r^2*Sin[\[Theta]]*D[\[Phi][r, \[Theta]], r], r] + 
     D[Sin[\[Theta]]*D[\[Phi][r, \[Theta]], \[Theta]], \[Theta]])]\
\>", \
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Cell[TextData[{
  "Linear pde's can often be solved using a \"separation of variables\" \
technique. This is done by assuming \[Phi][r,\[Theta]]  = A[r] G[\[Theta]].  \
As for Re=0 flows, we can often find the form of the angle function from the \
boundary conditions, in this case, the free stream flow.   \n\nThe boundary \
conditions for the sphere at infinity are ",
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  " ",
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\[Theta]]\)\)\/\[PartialD]\[Theta]\)]],
  " this suggests that the boundary conditions on  \[Phi][r,\[Theta]] are  \n\
\[Phi][r->\[Infinity],\[Theta]] = -U r Cos[\[Theta]],  \nat r=R, we have ",
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Cell[BoxData[
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Cell["\<\
We need to start the solution by substituting the trial solution \
into
the laplace equation\
\>", "Text",
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  CellLabelAutoDelete->True,
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Cell[CellGroupData[{

Cell["\<\
laplacetemp = Simplify[laplaceeq /. 
   {Derivative[a1_, a2_][\[Phi]][r, \[Theta]] :> 
     D[phitrial, {r, a1}, {\[Theta], a2}], \[Phi][r, \[Theta]] -> \
phitrial}]\
\>", "Input",
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Cell["\<\
Things are going well, the \[Theta] dependence divides out!  Let's \
get rid of it and rearrange the equation.\
\>", "Text",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[TextData[{
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Now solve for the constants 
\
\>", "Text",
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Cell[CellGroupData[{

Cell["Plot of velocity potential for flow past a sphere", "Subsection",
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Cell["Now we can plot the answer in a contour plot.  ", "Text",
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Cell[BoxData[
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We see that potential lines get closer together as the velocity \
increases.  Further, the potential lines are curved when the velocity \
gradient is non zero.  \
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  "Now we would like to get the identical answer by using the stream \
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which = 0 if there is no vorticity.  Thus with the restriction of \
no vorticity, which should not be produced by an inviscid flow, then a simple \
PDE can be obtained for the stream function.

Note that this is NOT the Laplace equation (It would be for a planar flow!!), \
even though it looks similar.  Compare to the Laplace equation from above.  \
\
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Here is the far away boundary condition, we match the r velocity to \
the free stream , @ r-->\[Infinity]\
\>", "Text",
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Now solve for the constants 
\
\>", "Text",
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Now replace the constants in the answer to get the expression for \
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Now check the answer from the first method compared to this one.  \
Here is the radial velocity.\
\>", "Text",
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Cell["Calculation of the pressure field", "Subsection",
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Now we can calculate the pressure field.  This could be from the \
Bernoulli equation or from the r component of the N-S equation.  Lets use the \
latter.  \
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dpdr = -(vrpsi*D[vrpsi, r] + (vthpsi*D[vrpsi, \[Theta]])/r - 
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dpdrtemp = Simplify[dpdr /. 
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  "If we form the definite integral for pressure as a function of r, we would \
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It is clear that these die off in distance.  Let's make the far \
away pressure = 0, i.e., rwayway ==> really far away.  If this is the case, \
then the second and 4th terms are 0.   The pressure as a function of r and \
theta is:\
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It is interesting to see what the pressure on the surface of the \
sphere is.\
\>", "Text",
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s
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s
.9 g
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F
0 g
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s
1 g
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0 g
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s
1 g
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F
0 g
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s
1 g
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F
0 g
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Cell[CellGroupData[{

Cell["Other issues", "Subtitle",
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Cell["Flow separation", "Subsection",
  Evaluatable->False,
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Cell["\<\
The pressure field is symmetric.  The highest pressure is in front \
of and behind the flow in the stagnation point region.  The lowest pressure \
is at the top where the velocity is fastest.  From the pressure field, we can \
see that separation is a likely possibility for a real fluid.  Consider a \
left to right flow.  For x<0, the value of the pressure is always decreasing, \
this is a \"favorable\" pressure gradient.  If you were a fluid particle, you \
would feel more push behind than in front.  For x>0, the pressure is \
increasing.  The fluid encounters an \"adverse\" pressure gradient and the \
fluid is feeling more push in front than behind and this slowing down.  If we \
remember that there is really drag caused by the solid surface, a fluid \
particle will not have enough momentum to get from the \[Pi]/2 point back \
down to \[Theta]=0 (as it is expected to from the no drag case).  Thus the \
fluid has no choice but to go off at some angle up to the right.      \
\>", \
"Text",
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  CellLabelAutoDelete->True,
  AspectRatioFixed->True]
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Cell[CellGroupData[{

Cell["Drag on the sphere", "Subsection",
  Evaluatable->False,
  CellLabelAutoDelete->True,
  AspectRatioFixed->True],

Cell["\<\
There are two kinds of drag that are defined.  One is \"skin drag\" \
which is the integral of the tangential force on the surface of the sphere. \
This is defined to be 0 for inviscid fluids.  The \"form drag\" is the \
integral of the total normal stress over the entire sphere.  Because the \
pressure is the only normal stress for an inviscid fluid, we can guess that \
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(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)