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Notebook[{
Cell["\<\
ChEg 355
Homework #7\
\>", "Title"],

Cell["Due November 17, 2000", "SectionFirst"],

Cell[CellGroupData[{

Cell["1.  Making a differential equation dimensionless.", "Subtitle"],

Cell[CellGroupData[{

Cell["Background", "Subsubsection"],

Cell[TextData[{
  "We saw in the rotating cylinder problem that there were non-zero terms in \
the ",
  StyleBox["r",
    FontSlant->"Italic"],
  "-direction equation, even though we claimed that the flow was only in the \
\[Theta] direction.  The conclusion was that for sufficiently large rotation \
rates, an outward flow in the ",
  StyleBox["r",
    FontSlant->"Italic"],
  " direction would occur.  Once this happens, we might expect that a ",
  StyleBox["z",
    FontSlant->"Italic"],
  " direction flow will also be occurring (see the continuity equation) and \
then many more terms in all three equations will be nonzero.\n\nSo the \
question becomes, is there a general criterion that tells us that if we can \
assume the simplest possible flow?\n\nThe answer is, of course ",
  StyleBox["yes",
    FontWeight->"Bold"],
  ".  ",
  StyleBox["If the Reynolds number is small compared to 1",
    FontVariations->{"Underline"->True}],
  ", then the fluid does not have any significant inertia.  This means we can \
ignore all of the left hand side terms, even if they are non-zero, because we \
can make an argument that they are small.\n\nHow can we possibly do this? \
(you ask!!)\n\nThere are two crucial steps, (please think about these and \
commit them to memory.)\n\n1.  We need to make sure that none of the terms in \
the Navier-Stokes equations are any larger than unity.  \n\n2.  We then can \
neglect all terms that are multiplied by a small number.\n\n\nSo now the \
question is, ",
  StyleBox["how",
    FontWeight->"Bold",
    FontVariations->{"Underline"->True}],
  " do we do this?\n\n",
  StyleBox["Glad you asked!!!",
    FontWeight->"Bold"]
}], "Text"]
}, Open  ]],

Cell["\<\
1.  Choose a length and a velocity (for more complex problems also \
a time) that are characteristic of the problem.   \
\>", "Subsubsection"],

Cell["\<\
For flow inside something, you would probably choose the diameter \
or radius.  For flow around something, you would choose a length scale that \
best describes the extent to which the object interferes with the flow (e.g., \
radius/diameter for a sphere). 

For the velocity, you would like a typical value.  This would be the average \
velocity of the flow inside something or the undisturbed flow far away from \
an object.  An interesting point is that for problems where the Reynolds \
number is small compared to one, there will be no place where the velocity \
has accelerated above this value to any significant extent, because the fluid \
has no inertia.   \
\>", "Text"],

Cell["\<\
2.  Make all of the terms of the differential equation \
dimensionless.\
\>", "Subsubsection"],

Cell[TextData[{
  "Here is where the work starts for you.  You now need to make all of the \
terms in the differential equation dimensionless.  The ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " notebook, ",
  ButtonBox["http://www.nd.edu/~mjm/dimensionless.nb,",
    ButtonData:>{
      URL[ "http://www.nd.edu/~mjm/dimensionless.nb"], None},
    ButtonStyle->"Hyperlink"],
  "will show you how.  You can use it for other terms as well."
}], "Text"],

Cell[TextData[{
  StyleBox["a.  For the ",
    FontWeight->"Bold"],
  StyleBox["x",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["-direction component of the Cartesian N-S equation, assume that \
the characteristic length is ",
    FontWeight->"Bold"],
  StyleBox["R",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox[" and the characteristic velocity is ",
    FontWeight->"Bold"],
  StyleBox["U",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox[".  Show how each term is made dimensionless.  (For each different \
type of term, write down by hand how you did this.)    \n\nb.  What \
dimensionless groups can be formed by division or multiplication to get the \
extra variables together?  \n\nc.  What did you use to nondimensionalize \
Pressure?  For the limit we are doing, switch to \n",
    FontWeight->"Bold"],
  Cell[BoxData[
      \(TraditionalForm\`\(\[Mu]\ U\)\/R\)],
    FontWeight->"Bold"],
  StyleBox[" to make the pressure dimensionless.\n\nd.  Suppose that the \
Reynolds number is small compared to one.  What significant terms remain?  \
Give the names of the types of terms and write these down.  Recall that after \
nondimensionalization, none of the individual terms is larger than order 1.   \
",
    FontWeight->"Bold"],
  "\n\nThe limit where the Reynolds number is much less than unity, is called \
\"creeping flow\" by some.  In the chemical engineering world, a large group \
of people will call these \"viscous\" flows.  The equations are often \
referred to as just the \"Stokes\" equations.  ",
  StyleBox["The big point to remember is if the Reynolds number is small \
compared to 1, you can neglect all of the left hand side terms. ",
    FontWeight->"Bold"],
  " "
}], "Text"],

Cell[CellGroupData[{

Cell["Verify that nondimensional terms are indeed of order unity.", \
"Subsubsection"],

Cell[TextData[{
  "The ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " notebook that you have been using also shows how big a terms in a \
dimensional equation can be. \n\n",
  StyleBox["e.  Try some other numbers in the example to show that the \
conclusion given is generally true.",
    FontWeight->"Bold"],
  "\n\nNote that if you have chosen an inappropriate characteristic velocity \
or length, you will find that the nondimensional terms are not of order 1 (or \
smaller) and you could completely mess up the problem.   \n\nf.  ",
  StyleBox["Just think this one through",
    FontWeight->"Bold"],
  ", what if instead of ",
  StyleBox["r",
    FontSlant->"Italic"],
  " you chose ",
  StyleBox["r",
    FontSlant->"Italic"],
  "/100.   What would have messed up?   "
}], "Text"]
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Cell[CellGroupData[{

Cell["2.  Solving a PDE, as in the rotating sphere problem.", "Subtitle"],

Cell[TextData[{
  "You have become familiar with the equations that govern the slowly \
rotating sphere problem.\n\nAs of now, we can make an argument that if the \
Reynolds number is small, we are able to neglect the inertia terms.\n\n\
However, wouldn't you like to solve for the flow field for this one?\n\n",
  StyleBox["Here is your chance!!  ",
    FontWeight->"Bold"],
  "\n\nI have done the problem in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " ",
  ButtonBox["http://www.nd.edu/~mjm/rotatingsphere.nb",
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  " and all you need to do is work through it and understand it.\n",
  StyleBox["\na.  What is the general strategy for getting an analytic \
solution to a PDE?\nb.  What two methods are used to effect this strategy?\n\
c.  Why is the method that we use, the only possible choice for this problem.\
\nd.  Show by substitution that the Euler differential equation can be solved \
by a function of the form ",
    FontWeight->"Bold"],
  Cell[BoxData[
      \(TraditionalForm\`r\^\[Alpha]\)],
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  StyleBox[".  Find the values of \[Alpha] that solve the equation.  Note you \
have just used the common general strategy for solving an ODE, \"turn it into \
an algebraic equation\"!  This is used for both analytic (if you don't know \
the functional form of the solution) and (all) numerical solutions to ODEs.  \
",
    FontWeight->"Bold"],
  "  "
}], "Text"]
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Cell[CellGroupData[{

Cell["\<\
3.  Understanding the solution to creeping flow past a stationary \
sphere.\
\>", "Subtitle"],

Cell[TextData[{
  "This problem, which is solved in the book on pages 166-172 and in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " at ",
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        ButtonBox[
          
          ButtonBox[\(\(\(http\)\(:\)\) // \(www . nd . 
                    edu/\(\(~\)\(mjm\)\)\)/creeping_sphere . nb\),
            ButtonData:>{
              URL[ "http://www.nd.edu/~creeping_sphere.nb"], None},
            ButtonStyle->"Hyperlink"],
          ButtonData:>{
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          ButtonStyle->"Hyperlink"], TraditionalForm]]],
  ", is ",
  StyleBox["the",
    FontWeight->"Bold"],
  " classic problem of low Reynolds number flows.  It has the practical \
implication of telling the drag force on a settling solid sphere in a liquid, \
(will the pigment in the paint settle before the paint dries?), it is \
simplest problem related to properties of suspensions of solid particles, and \
it has a particularly interesting \"perturbation\" solution.  Further if the \
particle is not a perfect sphere, its drag is unlikely to differ very much \
from the value for a sphere.\n\nYou will want to understand the following \
issues."
}], "Text"],

Cell[TextData[{
  StyleBox["1.  For the situation where the Reynolds number is not much \
smaller than one, find the governing equations if the fluid flows smoothly \
past the sphere.  Explain why the 0 terms are 0.\n\n2.  Show with a sketch \
why the boundary conditions on the velocity are given by equation 4.4.113 in \
the book.\n\n3.  Explain why the solution is expected to have the form of \
equation 4.4.114 in the book.\n\n4.  Write down the Euler differential \
equation that arises in this problem. \n\n5.  Explain how you can calculate \
the pressure field from the solution for the velocities.\n\n6.  Show with \
some calculation that the velocity never increases as the fluid goes past the \
sphere.  Note that for high Reynolds number flows you expect the fluid to \
accelerate if the area available for flow is decreased.    \n\n7.  Show to \
calculate the skin drag and form drag and get these results.\n\n8.  Explain \
which term(s) in the velocity field dominate as",
    FontWeight->"Bold"],
  StyleBox[" r",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox["--> \[Infinity].\n\n9.  Use the solution for ",
    FontWeight->"Bold"],
  Cell[BoxData[
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  StyleBox[" and ",
    FontWeight->"Bold"],
  Cell[BoxData[
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    FontWeight->"Bold"],
  StyleBox[" to get the stream function \[Psi] for this problem.  That is use \
the relations in equation 8.2.1 (yup, chapter 8) with the velocity relations \
to get eq. 8.2.2.  Look at the plots of the stream function, or make some of \
your own. ",
    FontWeight->"Bold"]
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["4.  What is the behavior for very high Reynolds number?", "Subtitle"],

Cell[TextData[{
  "You must be curious about the limit when the Reynolds number is very much \
greater than 1.  This problem will help satisfy this curiosity.  Some help is \
given in ",
  ButtonBox["http://www.nd.edu/~mjm/InviscidSphere.nb",
    ButtonData:>{
      URL[ "http://www.nd.edu/~mjm/InviscidSphere.nb"], None},
    ButtonStyle->"Hyperlink"],
  "."
}], "Text"],

Cell[CellGroupData[{

Cell["Exposition", "Subsubsection"],

Cell[TextData[{
  "If you return to problem 1 above, and use the nondimensionalization for \
pressure that you might have naturally used, p ~ \[Rho] ",
  Cell[BoxData[
      \(TraditionalForm\`U\^2\)]],
  ", which is an inertial formulation, you can take the limit of Re--> \
\[Infinity] and see the terms that you have remaining."
}], "Text"]
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Cell[CellGroupData[{

Cell["First question", "Subsubsection"],

Cell[TextData[{
  StyleBox["a.  Do this, and write down, for say the ",
    FontWeight->"Bold"],
  StyleBox["x",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox[" direction Cartesian equation, the terms that are important at \
very high Reynolds number.  Neglect gravity for simplicity.  ",
    FontWeight->"Bold"]
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Back to exposition", "Subsubsection"],

Cell[TextData[{
  "You will note that you have only first order derivatives, but there are \
many nonlinear terms.  It is not obvious that you could obtain solutions for \
these equations, which are called the Euler equations (not to be confused \
with Euler's differential equations which we also talked about in this \
homework set).  Oh well, I guess we are stuck.\n\n\nBut wait, what are the \
physical consequences of there being no viscous stresses?  (Which is the \
limit that your equations tell that you are examining.)  There is no \
\"friction\".  Tangential impacts do not result in any transmission of \
momentum.  This is the \"ideal\" case from physics and what G. I. Taylor \
called an \"ideal\" flow in the movie.    \n\nWe could think more, but this \
means that there it is not possible to cause a fluid packet to rotate.  (Can \
you make a hockey puck spin without friction?)\n\nWe could think still more, \
but if fluid packets are not rotating, the flow can be called irrotational \
and it will not have any ",
  StyleBox["vorticity",
    FontSlant->"Italic"],
  ".  Recall that we can figure out what part of the fluid deformation rate \
is the rotational, ",
  Cell[BoxData[
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  "= (",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]u\_i\/\[PartialD]x\_j\)]],
  "- ",
  Cell[BoxData[
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  ").  For the fluid rotation to be 0, every term of ",
  Cell[BoxData[
      \(TraditionalForm\`\[Omega]\_ij\)]],
  "must be  0. \n\n\nWhen you studied elementary physics, the absence of \
friction made many problems a lot simpler.  The same is true here.  The \
question is just, how do we use this useful physical information that we \
have? \n\nStart with a slightly different representation of fluid rotation.  \
The rotation of the fluid at any point can be represented by a vector, ",
  StyleBox["\[Omega]",
    FontWeight->"Bold"],
  " = ",
  StyleBox["\[Del]",
    FontWeight->"Bold"],
  "\[Times]",
  StyleBox["u",
    FontWeight->"Bold"],
  ".  \n\nfor each component to be 0,\n\n ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]u\_3\/\[PartialD]x\_2\)]],
  "- ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]u\_2\/\[PartialD]x\_3\)]],
  "  = 0 for rotation about the 1 axis,\n \n ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]u\_1\/\[PartialD]x\_3\)]],
  "- ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]u\_3\/\[PartialD]x\_1\)]],
  " = 0  for rotation about the 2 axis.\n \n ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]u\_2\/\[PartialD]x\_1\)]],
  "- ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]u\_1\/\[PartialD]x\_2\)]],
  " = 0  for rotation about the 3 axis.  \n \n Now suppose that we define, ",
  StyleBox["u ",
    FontWeight->"Bold"],
  "= \[Del]\[CapitalPhi], where \[CapitalPhi] is a scalar function of \
(x,y,z,t), will the equations for fluid rotation be satisfied?  Note that \n \
\n ",
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      \(TraditionalForm\`u\_1\)]],
  StyleBox[" = ",
    FontWeight->"Bold"],
  Cell[BoxData[
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  ",  ",
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      \(TraditionalForm\`u\_2\)]],
  StyleBox[" = ",
    FontWeight->"Bold"],
  Cell[BoxData[
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  ",  ",
  Cell[BoxData[
      \(TraditionalForm\`u\_3\)]],
  StyleBox[" = ",
    FontWeight->"Bold"],
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]\[CapitalPhi]\/\[PartialD]x\_3\)]],
  "\n \n We can substitute into\n \n  ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]\(\[PartialD]\[CapitalPhi]\/\[PartialD]x\_\
3\)\/\[PartialD]x\_2\)]],
  "- ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]\(\[PartialD]\[CapitalPhi]\/\[PartialD]x\_\
2\)\/\[PartialD]x\_3\)]],
  " = ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]\^2 \[CapitalPhi]\/\(\[PartialD]x\_2 \
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  "- ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]\^2 \[CapitalPhi]\/\(\[PartialD]x\_3 \
\[PartialD]x\_2\)\)]],
  " which equals 0.  (You can check the other two terms.)\n  \n  Thus, \
representing the velocity field as a gradient of a scalar is consistent with \
an irrotational flow.  \n \n \n Now if we use the definition of the velocity \
in the continuity equation for an incompressible fluid,  ",
  StyleBox["\[Del]",
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  "\[CenterDot]",
  StyleBox["u = 0",
    FontWeight->"Bold"],
  ", we get\n \n  ",
  Cell[BoxData[
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1\)\/\[PartialD]x\_1\)]],
  "+ ",
  Cell[BoxData[
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  " +  ",
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  Cell[BoxData[
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  "+ ",
  Cell[BoxData[
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  "+ ",
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  " = 0  or\n\n",
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  "\[CapitalPhi]=0.  \n\n",
  StyleBox["This last equation is LaPlace's equation and will give the \
velocity field for every ideal fluid flow",
    FontWeight->"Bold"],
  ".  \n\nYou cannot get the pressure field from this equation.  You will \
need to go back to the complete equations for the momentum of an inviscid \
fluid to get it or use the Bernoulli equation.  "
}], "Text"]
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Cell[TextData[StyleBox["b.  Name another physical situation that is governed \
by the gradient of a scalar function. \n\nc.  For the inviscid flow past a \
solid sphere, what are the boundary conditions?\n\nd.  What is the boundary \
condition that you have commonly been using that is \"missing\"? \n\ne.  Does \
the absence of this boundary condition make sense?\n\n(note that this problem \
is also solved with separation of variables and an Euler differential \
equation is produced.)\n\nf.  What equation is the stream function based on?\n\
\ng.  What equations do you need to use to calculate the pressure field for \
this problem?\n\nNote that lines of constant potential function are \
perpendicular to the streamlines.  Doesn't the plot of this look nice?\n\nh.  \
Explain why there is no drag on the sphere in an inviscid flow.  ",
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