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Cell["Creeping flow past a stationary sphere", "Title",
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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 if you use \
it, that this notice remain visible to other users. \nThere is no charge for \
copying and dissemination   \n\nVersion:  3/19/99\nMore recent versions of \
this notebook should be available at the web site:\n\
http://www.nd.edu/~mjm/sphereleal.nb"
}], "Text"],

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Cell[TextData[{
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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\
\>",
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Note that you can convert any \"cryptic\" input expression by \
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Cell[TextData[{
  "This notebook shows how to solve creeping flow past a stationary sphere \
(Stokes's Problem)\n\n",
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  "Consider the flow past a stationary sphere where the Reynolds number is \
significantly less than unity.  Inertia can be neglected.  The far away \
velocity field is a constant straight flow.  To make the problem as simple as \
possible, we will use spherical coordinates with the \[Phi] axis oriented \
parallel to the flow.  Thus we have \[Phi] symmetry and no ",
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form of the solution of problems in the creeping flow regime.  From the \
boundary conditions at the far field, the following forms are suggested for \
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Substitute this form of the pressure into r equation to check (but \
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Which is the r dependence of the theta equation.  Now look at \
continuity\
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The easiest way to solve these is to eliminate variables to leave \
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p[r], then use the vth equation, (after taking one more r derivative) to \
eliminate pp[r] from the vr equation.  This will be a simple ode.  (Leal has \
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Now if we multiply by r, then take an r derivative we can add these \
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  "We get a single ode for f[r] which is the same as Leal's eq 4-55.  \
Amazingly, ",
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Now we look at r=1.  We must have f[1]=0 so that vr is 0 and by \
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is not the case.  To check this we write  \n"], "Text",
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The result is shown here with the sphere blacked out (and \
unfortunately some frame).  Note that the contours smoothly change as we \
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\>",
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%!
%%Creator: Mathematica
%%AspectRatio: 1 
MathPictureStart
/Mabs {
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1 g
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0 g
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0 g
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0 g
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s
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F
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The code is as the shading gets darker, the velocity is slower.  We \
see that coming in the y=0 axis from x=-Infinity, the velocity is decreasing \
to agree with the: no flow through the surface boundary condition.  Looking \
at the x=0 axis, coming in from y=Infinity, the velocity is increasing \
reflecting the deflection of fluid around the sphere.  This results in a \
Bernoulli effect with local fluid acceleration.  Below we check some numbers \
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The same question can be asked about the pressure field.  We will \
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Substitute for the pressure field equation, \
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Mfstroke
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F
0 g
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s
.5 g
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F
0 g
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.5 .43197 L
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s
.4 g
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s
.3 g
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F
0 g
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.23325 .5 L
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.36395 .33997 L
s
.2 g
.36395 .35199 m
.43197 .35531 L
.44292 .36395 L
.48225 .43197 L
.5 .43197 L
.51775 .43197 L
.55708 .36395 L
.56803 .35531 L
.63605 .35199 L
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.5 .56803 L
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F
0 g
.36395 .35199 m
.43197 .35531 L
.44292 .36395 L
.48225 .43197 L
.5 .43197 L
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s
.1 g
.36395 .36068 m
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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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s
1 g
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F
0 g
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s
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F
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s
1 g
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F
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s
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Cell["\<\
In contrast, if a high velocity fluid is impinging on a surface, \
there will be a pressure increase.  For an inviscid flow, from Bernoulli's \
equation, we find that dp ~ (U0^2-u^2)/2, where we have already plotted u^2 \
\
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s
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s
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F
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So that the pressure is highest at y=0, x=1,-1 (i.e. the stagnation \
point) and decreases out to infinity.  The pressure is lowest, consistent \
with the Bernoulli effect, at y=1,-1,x=0, and increases out to y=infinity.    \
\
\>", "Text",
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stream function, the Re = 0 solution is -1/2 f[r] r^2 sin[\[Theta]]^2 "], 
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We can make a plot of this if we transform from polar to Cartesian \
coordinates\
\>", "Text",
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*)




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