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Cell["Flat earth investigation into the concentric cylinder flow ", "Title"],

Cell["\<\
This little exercise is how this issue can be thought of.

Main points:

1.  We should investigate our solutions to see what they mean.
2.  Looking at limits can help with this understanding and can help us make \
connections with other things that we understand.
3.  A series expansion is necessary (Series expansions are th greatest \
thing!!!)
4.  Let the computer do the series expansion!!
5.  Sometimes it is hard to interpret!!
6.  In this case the expansion in the r coordinate, tells us the most.  \
(which is not what we usually see.)\
\>", "Text"],

Cell["\<\
If we nondimensionalize the solution, r->r R, u-> u \[CapitalOmega] \
R, we see that the velocity profile is\
\>", "Text"],

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Cell["\<\
With the flow region between \[Kappa] and 1.  \
\>", "Text"],

Cell["Let's look at some profiles...", "Text"],

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Cell["\<\
We should try a series expansion in \[Kappa].  \[Kappa] is close to \
1 and I keep up to second order terms.\
\>", "Text"],

Cell[CellGroupData[{

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  CellLabel->"In[71]:="],

Cell[BoxData[
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          "+", \(7\/8\ \((r - 1\/r)\)\ \((\[Kappa] - 1)\)\), 
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            Plus[ 
              Times[ -1, 
                Power[ r, -1]], r]], 
          Times[ 
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            Plus[ 
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          Times[ 
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            Plus[ 
              Times[ -1, 
                Power[ r, -1]], r]], 
          Times[ 
            Rational[ 1, 16], 
            Plus[ 
              Times[ -1, 
                Power[ r, -1]], r]]}, -1, 3, 1]], TraditionalForm]], "Output",\

  CellLabel->"Out[71]="]
}, Open  ]],

Cell["We can make this a little less ugly", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[72]:="],

Cell[BoxData[
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      7\/8\ \((r - 1\/r)\)\ \((\[Kappa] - 1)\) + 
      5\/4\ \((r - 
            1\/r)\) + \(r - 1\/r\)\/\(2\ \((\[Kappa] - 1)\)\)\)], "Output",
  CellLabel->"Out[72]="]
}, Open  ]],

Cell["This is hard to see, try something else", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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      vprofile /. \[Kappa] \[Rule] 1 - \[Delta], {\[Delta], 0, 2}]\)], "Input",\

  CellLabel->"In[73]:="],

Cell[BoxData[
    FormBox[
      InterpretationBox[
        RowBox[{\(\(1\/r - r\)\/\(2\ \[Delta]\)\), 
          "+", \(5\/4\ \((r - 1\/r)\)\), 
          "-", \(7\/8\ \((r - 1\/r)\)\ \[Delta]\), 
          "+", \(1\/16\ \((r - 1\/r)\)\ \[Delta]\^2\), "+", 
          InterpretationBox[\(O(\[Delta]\^3)\),
            SeriesData[ \[Delta], 0, {}, -1, 3, 1]]}],
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          Times[ 
            Rational[ 1, 2], 
            Plus[ 
              Power[ r, -1], 
              Times[ -1, r]]], 
          Times[ 
            Rational[ 5, 4], 
            Plus[ 
              Times[ -1, 
                Power[ r, -1]], r]], 
          Times[ 
            Rational[ -7, 8], 
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              Times[ -1, 
                Power[ r, -1]], r]], 
          Times[ 
            Rational[ 1, 16], 
            Plus[ 
              Times[ -1, 
                Power[ r, -1]], r]]}, -1, 3, 1]], TraditionalForm]], "Output",\

  CellLabel->"Out[73]="]
}, Open  ]],

Cell["\<\
No good luck so far.

Here is the first term, we\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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        Normal[%], \[Delta], \(-1\)]\)\(\ \)\)\)], "Input",
  CellLabel->"In[74]:="],

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    \(TraditionalForm\`1\/2\ \((1\/r - r)\)\)], "Output",
  CellLabel->"Out[74]="]
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Cell["\<\
From this we cannot tell the shape of the velocity profile in any \
limit.  

Thus we should try to find the limit for r close to 1.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[75]:="],

Cell[BoxData[
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            SeriesData[ r, 1, {}, 1, 3, 1]]}],
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          Rational[ 1, 2]}, 1, 3, 1]], TraditionalForm]], "Output",
  CellLabel->"Out[75]="]
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Cell["\<\
We see that if r is close to 1, then the profile is linear!  

Maybe we should just to an expansion of the original solution in r, \
\>", \
"Text"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[86]:="],

Cell[BoxData[
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        RowBox[{\(\(2\ \[Kappa]\^2\ \((r - 1)\)\)\/\(\[Kappa] - 
                1\/\[Kappa]\)\), 
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                1\/\[Kappa]\)\), "+", 
          InterpretationBox[\(O(\((r - 1)\)\^3)\),
            SeriesData[ r, 1, {}, 1, 3, 1]]}],
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          Times[ 2, 
            Power[ \[Kappa], 2], 
            Power[ 
              Plus[ 
                Times[ -1, 
                  Power[ \[Kappa], -1]], \[Kappa]], -1]], 
          Times[ -1, 
            Power[ \[Kappa], 2], 
            Power[ 
              Plus[ 
                Times[ -1, 
                  Power[ \[Kappa], -1]], \[Kappa]], -1]]}, 1, 3, 1]], 
      TraditionalForm]], "Output",
  CellLabel->"Out[86]="]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[88]:="],

Cell[BoxData[
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\[Kappa]\) - \(\((r - 1)\)\^2\ \[Kappa]\^2\)\/\(\[Kappa] - 1\/\[Kappa]\)\)], \
"Output",
  CellLabel->"Out[88]="]
}, Open  ]],

Cell["\<\
So this seems justified.  Note that the coefficients are about the \
same magnitude with the lowest order being a little bigger.\
\>", "Text"],

Cell[CellGroupData[{

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