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Notebook[{
Cell["Introduction to hydrodynamic stability", "Title"],

Cell[TextData[{
  "This notebook has been written in ",
  StyleBox["Mathematica ",
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  "by \n\n",
  StyleBox[
  "Mark J. McCready\nProfessor and Chair of Chemical Engineering\nUniversity \
of Notre Dame\nNotre Dame IN 46556\nUSA",
    FontSize->14],
  "\n\nMark.J.McCready.1@nd.edu\n",
  ButtonBox["http://www.nd.edu/~mjm/",
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  "\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:  5/14/99\nA more recent version \
of this notebook may be available at\n",
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Cell["Summary", "Subtitle"],

Cell[TextData[{
  "This notebook is intended to give a first introduction to hydrodynamic \
instability.  This includes both the physical concepts and several useful \
mathematical manipulations.\n\nThere are three parts.  The first discusses \
the concept of linear instability theory and uses a simple wave equation to \
demonstrate the linearization and calculation of temporal and spatial growth. \
 \n\nThe second part derives the stability relation for a two-layer inviscid \
flow, the ",
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    FontSlant->"Italic"],
  " instability.\n\nThe third part shows how to derive the basic equation of \
hydrodynamic stability for Newtonian fluids, the ",
  StyleBox["Orr-Sommerfeld",
    FontSlant->"Italic"],
  " equation.  "
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  "In ",
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  ", it is convenient to give all expressions a \"name\".  I try to pick ones \
that are consistent with what is being done (but sometimes \"temp\" is used). \
 This assignment is done with an \"=\" sign.  To make an equation, a \"==\" \
is used.  This distinction is very useful in computer algebra and is employed \
in all of the packages with which I am familiar ."
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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["How to use this notebook", "Subtitle"],

Cell[TextData[{
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  " and work through the examples and make changes in parameters or \
procedural steps to ",
  StyleBox["explore",
    FontSlant->"Italic"],
  " these problems.  On line help is available in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " 3 so that a definition and in most cases examples for any unknown command \
can be obtained.  If you do not have a license for ",
  StyleBox["Mathematica",
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  ", you can download ",
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  " (http://www.wolfram.com/mathreader/) free of charge.  It does not let you \
change anything or run the calculations, but it does allow full access to the \
notebook.  \n\nOther notebooks that cover a range of fluid dynamics problems \
are available at:\n",
  ButtonBox["http://www.nd.edu/~mjm/",
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      URL[ "http://www.nd.edu/~mjm/"], None},
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about using ",
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  ".  \n      "
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Cell[CellGroupData[{

Cell["Motivation: Importance of waves", "Subtitle",
  Background->RGBColor[0.8, 0.919997, 0.919997]],

Cell[TextData[{
  "Waves form as the result of wind on natural bodies of water and contribute \
to increased gas transfer and droplet production.  Despite the high Reynolds \
numbers involved, these originate as a hydrodynamic instability that requires \
a fully-viscous solution of the linearized Navier-Stokes equations to \
characterize.  You may also observe waves on the windshield of   your car \
during a rain storm or in the car wash or on the wing of an airplane -- \
perhaps as the deicing fluid is sheared of.  These examples involve much \
lower liquid Reynolds numbers and  (with some additional simplifications) can \
be analyzed with wave equations similar to the example used in the first part \
below.  \n\nA big area of interest in interfacial waves are those that occur \
in gas-liquid (i.e., two-phase flow) pipelines and process equipment.  In \
these situations they greatly increase the pressure drop and interface \
transport rates, and can lead to flow regime transitions.  Some examples of \
these kinds of waves are available as still pictures and movies at ",
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Cell["The general idea of flow instability", "Subtitle",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[{
  "It has been observed in nature, that the steady state solutions for \
different systems can become unstable to infinitesimal disturbances which \
should be expected to always be present, (the ground is always vibrating, \
buildings breath and bend, etc. ...) and possibly because of molecular \
motions.  A common example is the formation of waves on bodies of water owing \
to the action of wind.  The \"Taylor- Couette Flow\" instability is a popular \
laboratory instabilility that arises due to centrifugal force, and \
Rayleigh-Benard convection, which arises because of density differences is \
important both in nature and in laboratories.\n\nEach of these instabilities \
has a precise, although not necessarily well understood, physical mechanism.  \
The common feature of an instability is that ",
  StyleBox["infinitesimal velocity or density perturbances are amplified ",
    FontVariations->{"Underline"->True}],
  "(by the base flow or global forces) and thus grow to finite size.  Growth \
of distrubances could be algebraic or exponential.  Typical analysis (such as \
those shown below) assume an exponential growth because it is expected that \
this would overwhelm any algebraic growth.  However, algebraic analyses have \
been used in some situations where exponential models did not match data.  It \
is not clear that these have matched any better, but this discussion is \
beyond the point of this introductory module.\n\nInfinitesimal perturbations \
are expected to be in the form of noise.  The question is, how to represent \
this when we want to model instability.  Fortunately, the noise is \
infinitesimal, which means its amplitude is small compared to any length \
scale such as its wavelength.  This allows the nonlinear governing equations \
to be well-approximated by linearized versions.  The linear equations are \
amenable to a Fourier mode  analysis that can be used to represent any noice \
signal as a linear combination of independent modes.   \n \nIf we assume \
exponential growth (because is the strongest possible and what is observed in \
nature), and if the growth is in time (which could be how it occurs)  an \
equation for the amplitude, ",
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show some evidence of growing disturbances, unless the residence time is \
short.  "
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Note that if the amplitude is growing exponentially, at some point \
nonlinear processes will become important.  Nonlinear analysis is beyond the \
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\>", "Text"],

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Cell["Analysis of instability", "Section"],

Cell[TextData[{
  "To do mathematical analysis of an instability, we need to chose a \
\"basestate\" that is the base flow in the absence of an instability.  This \
could be 0 velocity or it could be a falling film with no waves or a \
stratified flow with no waves, etc..  \n\nSince we expect that linear \
equations should govern the initial growth of instability, we will linearize \
the complete governing equations around the base flow.  This is done by \
taking the baseflow, say ",
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disturbance and will contain ",
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linear (we just linearized them with this procedure).  Any higher powers of \
\[CurlyEpsilon] will be ignored and saved for when we want to do nonlinear \
analysis.\n\nTo determine the response of the equation to an arbitrary noise \
signal, we choose a mode that represents the kind of disturbance that we \
expect to see.  If the domain is fixed and there is no flow through, (e.g., a \
solid beam), we might expect to use fixed spatially periodic modes that grow \
in time.  For waves on water, we would use (traveling) spatially and \
temporally periodic disturbances that could grow in space and/or time.     \n\
\nSince the system is linear, we can examine the response of any separate \
mode without worrying about the effect of other modes. This linearity allows \
us to decompose an arbitrary disturbance into a sum of Fourier modes; each \
mode will ultimately satisfy the equations and boundary conditions.  By \
scanning the entire frequency or wavenumber range, we can be sure that we \
understand the effect of any initial (infintesimal) disturbance."
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Cell["Example: Waves on a falling liquid film", "Section"],

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[",
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form of the Kuramoto-Sivshinsky equation which is derived for a falling film \
of water by (e.g., H. -C. Chang, ",
  StyleBox["Chemical Eng. Sci",
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value, t is time and x is the flow direction.\nThis is a \"wave\" equation in \
which all of the complexities of the flow have been reduced to a  single \
equation for one dependent variable, the liquid height, h.\n\nIn this \
equation, all of the variables are real.\n\nClearly a constant value of h is \
a solution to this equation.  We will choose\nh=h0 and begin by linearizing \
the equation around h =h0."
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Cell["The linearized equation", "Subsubsection",
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Cell[TextData["The powers of \[CurlyEpsilon] give:"], "Text",
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Which is the base state solution; any constant value of h0 is a \
solution.  \
\>", "Text",
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  ".  "
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  ButtonData:>"conclusions",
  ButtonStyle->"Hyperlink"]], "Text"],

Cell["This is the nonlinear term that is neglected.", "Text"],

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Cell["Traveling wave mode expansion", "Subsubsection"],

Cell["\<\
Now we substitute a traveling wave disturbance into the linearized \
equation.  Note that the substitution is done in a way that works for any \
arbitrary derivative.\
\>", "Text",
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    \(TraditionalForm
    \`a\ \[ExponentialE]\^\(\[ImaginaryI]\ \((k\ x - t\ \[Omega])\)\)\ 
        \[Sigma]\ k\^4 - 
      \[ImaginaryI]\ a\ 
        \[ExponentialE]\^\(\[ImaginaryI]\ \((k\ x - t\ \[Omega])\)\)\ 
        \[Gamma]\ k\^3 - 
      a\ \[ExponentialE]\^\(\[ImaginaryI]\ \((k\ x - t\ \[Omega])\)\)\ 
        \[Beta]\ k\^2 + 
      \[ImaginaryI]\ a\ c0\ 
        \[ExponentialE]\^\(\[ImaginaryI]\ \((k\ x - t\ \[Omega])\)\)\ k + 
      \[ImaginaryI]\ a\ 
        \[ExponentialE]\^\(\[ImaginaryI]\ \((k\ x - t\ \[Omega])\)\)\ h0\ 
        \[Alpha]\ k - 
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We now divide out the original disturbance, which is possible since \
we used an exponential form.\
\>", "Text",
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      \[ImaginaryI]\ \[Omega]\)], "Output"],

Cell[TextData[ButtonBox["Back to conclusions",
  ButtonData:>"conclusions",
  ButtonStyle->"Hyperlink"]], "Text"],

Cell["\<\
The result is the complete dispersion relation for waves in this \
system.  The relation between speed and frequency and wavenumber is given in \
this relation as is the linear growth.\
\>", "Text"],

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    \`k\ \((\[Sigma]\ k\^3 - \[ImaginaryI]\ \[Gamma]\ k\^2 - \[Beta]\ k + 
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Cell["Temporal growth", "Subsubsection"],

Cell[TextData[{
  "Consider first, spatially periodic disturbances that grow in time. This is \
the easy case (only 1 time derivative and it is a first derivative) and the \
one that most people naturally do.  Of course waves on a falling film seem to \
grow with distance (!) \n For this case, ",
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  " is real and \[Omega] is complex.  "
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\[Omega].  (It takes this format to do it)"], "Text"],

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Cell[TextData[
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of \[Omega] and others only to the real part.  Let's look at the real part \
which is the frequency and relates to the wave speed.  "], "Text"],

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We might as well use the canned program for Real and Imaginary \
variables.\
\>", "Text"],

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of speed change due to the film thickness, h0 \[Alpha].  The wave speed is \
dependent upon the wavenumber and this is given by the term, ",
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surface tension parameter), we have the first term trying to cause the \
instability, (fluid inertia) and the second one countering it.\n\nIt is \
instructive to plot the growth rate.  We see that at low wavenumber (long \
waves) the growth rate is increasing.  However at higher wavenumber the \
stabilization becomes stronger and eventually the waves become stable .  "], 
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Cell[TextData[{
  "So that there is a region for ",
  StyleBox["k",
    FontSlant->"Italic"],
  " < ~ .85 where the waves are unstable.  The fastest growing wave is at ",
  StyleBox["k",
    FontSlant->"Italic"],
  " =~ 0.6.  \n\nNow for a linear system, we can never produce a mode that is \
not initially excited.  Thus the only waves that grow are those that were \
initially present.  If the initial disturbance is noise, then all modes are \
present and we would expect to see the fastest growing mode dominate.  \
However, this will not usually be a pure mode but a band of waves in the \
vicinity of the peak. \n\nIf the initial disturbance does not include modes \
near the peak, then others may dominate.  "
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Cell["Spatial growth", "Subsubsection"],

Cell[TextData[{
  "We could have also considered spatially growing waves, which are most \
likely to be the physically relevant case for a falling film, by defining \
\[Omega] as real and then allowing for complex ",
  StyleBox["k",
    FontSlant->"Italic"],
  ".  Even for this simple equation, the analysis is more complicated because \
the spatial derivatives are of higher order than the time derivatives.     "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp5 = Expand[temp4]\)], "Input",
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Cell[BoxData[
    \(TraditionalForm
    \`\[Sigma]\ k\^4 - \[ImaginaryI]\ \[Gamma]\ k\^3 - \[Beta]\ k\^2 + 
      \[ImaginaryI]\ c0\ k + \[ImaginaryI]\ h0\ \[Alpha]\ k - 
      \[ImaginaryI]\ \[Omega]\)], "Output"]
}, Open  ]],

Cell[BoxData[
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Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " knows how to solve quartic equations analytically, so it can get the 4 \
roots.   "
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  AspectRatioFixed->True],

Cell["Here is the first one:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Table[
      N[kays\[LeftDoubleBracket]1\[RightDoubleBracket] /. {h0 \[Rule] 1.5, 
            \[Alpha] \[Rule] 1, c0 \[Rule] 0, \[Beta] \[Rule] 3, 
            \[Gamma] \[Rule] \(-3\), \[Sigma] \[Rule] 4}], {\[Omega], 0, 
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Cell[BoxData[
    \(TraditionalForm
    \`{{k \[Rule] 
          \(-0.905684798885634201`\) - 
            0.543123952696074674`\ \[ImaginaryI]}, {
        k \[Rule] 
          \(-0.950731698876204056`\) - 
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Cell["\<\
This root, [[1]], corresponds to a wave that is going upstream.  \
This would not happen.

Here is the second one:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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      N[kays\[LeftDoubleBracket]2\[RightDoubleBracket] /. {h0 \[Rule] 1.5, 
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            \[Gamma] \[Rule] \(-3\), \[Sigma] \[Rule] 4}], {\[Omega], 0, 
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Cell["\<\
This roo,t [[2]], also corresponds to a wave that is going \
upstream.  This also would not happen.

Here is the third one:\
\>", "Text"],

Cell[CellGroupData[{

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Cell["\<\
This root, [[3]], seems to be more and more unstable as the \
frequency increases, this is not physical for this system.  However, it \
travels down stream and so we cannot ignore it.  When an \"unphysical mode\" \
appears, this is cause for concern that the derivation has neglected \
something important or that our solution procedure has flaws in it.  In this \
case, I think that the equation, which has been \"generalized\" to make it \
more interesting to study, is being used with parameters that are outside \
where it behaves well.

A more interesting consequence of this analysis is that you might find it \
very difficult to integrate this equation numerically!  

To proceed, consider both, [[3]], and [[4]].   \
\>", "Text"],

Cell["\<\
Let's plot the two parts to see what happens.
Here we extract the 3rd root.\
\>", "Text"],

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Cell[TextData[{
  "We can plot the real wave speed as  \[Omega]/",
  StyleBox["k.",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell["\<\
The lower mode, which starts out dashing and then becomes solid, is \
behaving more like a real wave than the top one.  We do not see how a wave \
can have unbounded growth for very short waves when surface tension is \
present.  The speed is better also as it more closely matches the qualitative \
behavior of the temporal analysis.  

We will leave the issue of the \"unphysical\" mode because we can do nothing \
else with it right now (Although you could vary some of the parameters to see \
how it behaves.) but in a real problem, you would have to find out if arises \
because of the limitations of the derviation or the solution -- or if the \
instability is really there!\
\>", "Text"]
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Cell["Relation between spatial and temporal growth", "Subsubsection"],

Cell[TextData[{
  "The spatial and temporal growth rates can be related by a Gaster \
Transformation (M. Gaster, ",
  StyleBox["J. Fluid Mech",
    FontSlant->"Italic"],
  ", ",
  StyleBox["14",
    FontWeight->"Bold"],
  ", p222, 1962).  This is done by using the group velocity, \n\[PartialD]\
\[Omega]/\[PartialD]",
  StyleBox["k",
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  ", to convert temporal to spatial growth, \n\n",
  StyleBox["spatial growth",
    FontSlant->"Italic"],
  "  =  ",
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    FontSlant->"Italic"],
  "/ (\[PartialD]\[Omega]/\[PartialD]",
  StyleBox["k)",
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Cell["The group velocity is defined to be real.  ", "Text"],

Cell[CellGroupData[{

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Cell[BoxData[
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Cell["\<\
Here is a plot that shows the spatial growth obtained from the \
temporal growth.\
\>", "Text"],

Cell[CellGroupData[{

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              c0 \[Rule] 0, \[Beta] \[Rule] 2, \[Gamma] \[Rule] \(-3\), 
              \[Sigma] \[Rule] 8}}, \n\t{k, 0, 1}, 
        AxesLabel -> {"\<frequency\>", "\<converted spatial growth\>"}]; 
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Now we show the real spatial and the converted spatial growths. \
They agree well for positive growth but not well as the amplitude of the \
negative growth gets larger.

Generally, it is possible to make this conversion for regions close to \
neutral stability.   \
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Cell["\<\
Why can we use a complex representation for a real disturbance \
?\
\>", "Subsection"],

Cell[TextData[ButtonBox["Back to conclusions",
  ButtonData:>"conclusions",
  ButtonStyle->"Hyperlink"]], "Text"],

Cell[TextData[{
  "An important question could be that the disturbance that we used was not a \
real number.   To make the disturbance a real number, we can use a0 Exp[I(",
  StyleBox["k ",
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  "x - \[Omega] t)] + c.c. or\na0 Exp[I(",
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  "x - \[Omega]t)] + a0cc Exp[-I(",
  StyleBox["k",
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  "x - \[Omega] t)].  If this is substituted into the equation we will get \
twice as many terms.  One set will have Exp[I(",
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  "x - \[Omega]t)] as a factor and the other set will have Exp[-I(",
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  "x - \[Omega]t)] as a factor.  For this expression to always be true (i.e. \
for any x and t ), these sets must each be equal to 0 separately.  The \
analysis would then be identical to the one done above.  Note that the \
appearance of an \[ImaginaryI] in an equation denotes a \[Pi]/2 phase shift \
(i.e., Sine versus Cosine).  \n\nNote that if we are doing a nonlinear \
problem, we would have to keep track of the complex conjugate part as it may \
contain additional information.  "
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Cell["Kelvin-Helmholtz Instability for finite depth", "Subtitle",
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Cell["Problem set up", "Subsubsection"],

Cell["\<\
A good reference for this section is R. L. Panton, Incompressible \
flow, Wiley, 1984\
\>", "Text",
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Cell[TextData[{
  "The generation of water waves by wind has puzzled and fascinated \
scientists for centuries.  It is a classic stability problem that, as it \
turns out, was not really completely understood until about the 1970's.  \n\n\
This problem was attacked very early in the study of hydrodyanmic stability \
for the idealized case of inviscid flows.  This solutions is known as the ",
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  " instability and it ",
  StyleBox["does not",
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  " predict the observed behavior of water wave generation!\nIt does not work \
for any two layer system where the viscosities of the two fluids are very \
different, no matter how high the Reynolds numbers are.  Still it is a useful \
instability for study as it provides a nice physical mechanism to an easily \
understood situation.\n"
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  "Consider a stratified horizontal flow of two inviscid fluids.  The top \
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  StyleBox["2",
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  "."
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Cell[CellGroupData[{

Cell["Laplace equation solution", "Subsubsection"],

Cell[TextData[{
  "First we should find solutions for ",
  StyleBox["\[CapitalPhi]",
    FontFamily->"Symbol"],
  " in each phase that fits the necessary boundary conditions.  This would be \
that there is no flow through the top and bottom walls.  \nThe other boundary \
conditions will be introduced along the way.\n \nThe waves will be traveling \
in the x direction, grow and oscillate in time and must decay away from the \
interface in y.    \n\nFor the lower phase, why don't we guess a form as\n\n",
  
  StyleBox["\[CapitalPhi]",
    FontFamily->"Symbol"],
  StyleBox["1",
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  "[x,y,t] = U1 x + ",
  StyleBox["\[Phi]",
    FontFamily->"Symbol"],
  StyleBox["1",
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    FontVariations->{"CompatibilityType"->"Subscript"}],
  "[y] Exp[I ",
  StyleBox["k",
    FontSlant->"Italic"],
  "(x - ",
  StyleBox["c",
    FontSlant->"Italic"],
  " t)]\n\n",
  StyleBox["\[Phi]",
    FontFamily->"Symbol"],
  StyleBox["1",
    FontFamily->"Symbol",
    FontVariations->{"CompatibilityType"->"Subscript"}],
  "[y] = B1  Cosh[k (y+h1)]\n\nthis gives the base flow plus the disturbance. \
  \n\nFor the upper phase, how about\n\n",
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    FontFamily->"Symbol"],
  StyleBox["2",
    FontVariations->{"CompatibilityType"->"Subscript"}],
  "[x,y,t] = U2 x + ",
  StyleBox["\[Phi]",
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  StyleBox["2",
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    FontVariations->{"CompatibilityType"->"Subscript"}],
  "[y] Exp[I k(x - c t)]\n\n",
  StyleBox["\[Phi]",
    FontFamily->"Symbol"],
  StyleBox["2",
    FontFamily->"Symbol",
    FontVariations->{"CompatibilityType"->"Subscript"}],
  "[y] = A2  Cosh[k (y-h2)]\n\nIn these expressions, ",
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    FontSlant->"Italic"],
  " is wavenumber, ",
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  " is wave speed, x is the direction of travel and t is time.  \n\nLet's see \
how well we have guessed!\n "
}], "Text"],

Cell[" For the upper phase", "Text",
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Cell["Check in Laplace's equation and it works", "Text"],

Cell[CellGroupData[{

Cell["(first see it in traditional form)", "Text"],

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Cell[TextData[ButtonBox["Back to conclusions",
  ButtonData:>"conclusions",
  ButtonStyle->"Hyperlink"]], "Text"],

Cell["Now check the lower boundary condition, it works also", "Text"],

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Cell["now try the upper phase.", "Text"],

Cell[CellGroupData[{

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Cell["test in LaPlaces' equation,", "Text"],

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Cell["Test the boundary condition,", "Text"],

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Cell["OK so far, so good.  ", "Text"]
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Cell[CellGroupData[{

Cell["Kinematic boundary condition", "Subsubsection"],

Cell[TextData[{
  "Whenever there is an interface between fluids that has waves on it, we \
need a relation between the velocity field and the motion of the interface.  \
This can formulated by using the substantial derivative of the surface \
position function, ",
  StyleBox["\[Eta]",
    FontFamily->"Symbol"],
  "(x,t) and equating this to the normal velocity of the fluid at the \
interface.\n\n",
  Cell[BoxData[
      \(\(D\[Eta] \((x, t)\)\)\/Dt\)]],
  "  = normal velocity at the interface = ",
  Cell[BoxData[
      \(\(\[PartialD]\ \[Phi] \((x, y, t)\)\)\/\[PartialD]y\)]],
  "  (note that this potential is just for the perturbation velocity because \
there is no average normal velocity)\n\nthis becomes\n\n\[PartialD]",
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  "/\[PartialD]t + U \[PartialD]",
  StyleBox["\[Eta]",
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  "/\[PartialD]x + W \[PartialD]",
  StyleBox["\[Eta]",
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  "/\[PartialD]z = \[PartialD] ",
  StyleBox["\[Phi]",
    FontFamily->"Symbol"],
  "/\[PartialD]y  @ y= ",
  StyleBox["\[Eta]",
    FontFamily->"Symbol"],
  ". \n\nIn this example we will not consider the transverse direction,  so W \
= 0 and \[PartialD] ()/\[PartialD]z = 0.  In its present form, the second \
term on the left is nonlinear.  We just need a linearized version for this \
linear stability problem.  This is\n\n \[PartialD]",
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  "/\[PartialD]t + U1 \[PartialD]",
  StyleBox["\[Eta]",
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  "/\[PartialD]x  = \[PartialD] ",
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  "/\[PartialD]x \[PartialD]",
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  "/\[PartialD]x is order ",
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  ", which can be ignored.\n\nThere is still one problem remaining.  It would \
be exceedingly inconvenient to evaluate this condition at y = ",
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    FontFamily->"Symbol"],
  ", since this changes with x and t.  It turns out that this problem can be \
solved by domain perturbation.  We would like to evaluate these equations at \
y = 0 and we need the real values of the variables at this point.  We can get \
them by expanding the velocities in a Taylor series.  The surface variable is \
not a function of y and thus does not require any expanding. \n\nU(y=h) = \
U(y=0) + ",
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  " \[PartialD]U/\[PartialD]y(y=0) + H.O.T.\n\nv(y=h)  =  v(y=0) + ",
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  " \[PartialD]v/\[PartialD]y (y=0) + H.O.T \n\nThe correction terms are of \
order ",
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    FontSlant->"Italic"],
  " and would not enter in this linear problem.  Further, for this inviscid \
flow, there is no velocity gradient in the y direction so there is no \
correction for the flow direction velocity.  \n\nAs a result of this \
discussion, we have this \"kinematic\" condition for both phases related to \
the interfacial shape.     \n\n \[PartialD]",
  StyleBox["\[Eta]",
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  "/\[PartialD]t + U1 \[PartialD]",
  StyleBox["\[Eta]",
    FontFamily->"Symbol"],
  "/\[PartialD]x  = \[PartialD] \[Phi]",
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  StyleBox["0",
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  ".   \n \n  \[PartialD]",
  StyleBox["\[Eta]",
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  "/\[PartialD]t + U2 \[PartialD]",
  StyleBox["\[Eta]",
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  "/\[PartialD]x  = \[PartialD] \[Phi]",
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  StyleBox["0",
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  ".   \n"
}], "Text",
  CellTags->"boundarycondition"],

Cell[TextData[ButtonBox["Back to conclusions",
  ButtonData:>"conclusions",
  ButtonStyle->"Hyperlink"]], "Text"],

Cell["Here is the kinematic condition, first in traditional form,", "Text"],

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  "Here we impose the kinematic condition for phase 1 by  substituting for \
\[Phi]1 and \[Eta].  Note that ",
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        \ U1\ \(\[Eta]\&^\) + 
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  "So now we know B1 in terms of ",
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  ". \nNow do the second kinematic condition"
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Cell[CellGroupData[{

Cell["Here it is is in traditional form", "Text"],

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Cell["Now we can get A2", "Text"],

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Cell["Dynamic Boundary condition", "Subsubsection"],

Cell[TextData[{
  "The pressure will not be constant along the interface, if waves are \
present.  We can find out the variation by applying the Bernoulli equation \
for each phase at the interface.  The pressure must be the same on each side \
of the interface, except for the effect of durface tension.  This is a \
\"dynamic condition\" (pressure changes with velocity) and is referred to in \
this way.  \n\nWe can write the Bernoulli equation for each phase as:\n\n\
\[PartialD]",
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  StyleBox["1",
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  ")",
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  "/",
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  StyleBox["1",
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  "(x,t)  = 0  \n\n\[PartialD]",
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  StyleBox["2",
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  ")",
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  "/",
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  StyleBox["2",
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  " + g ",
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  "(x,t)  = 0      \n\nWe will linearize the (",
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  ")",
  StyleBox["2 ",
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  "term.  The difference in the pressure can be written as\n(P",
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  "-P",
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  ") = - ",
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  " ",
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              StyleBox["2",
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  "\n\nwhere \[Gamma] is the surface tension coefficient.  "
}], "Text"],

Cell["Here is the dynamic boundary condition.", "Text"],

Cell[CellGroupData[{

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                RowBox[{"-", 
                  
                  FractionBox[\(\[PartialD]\[Phi]\_2[x, y, t]\), 
                    \(\[PartialD]t\),
                    MultilineFunction->None]}], "-", 
                RowBox[{"U2", " ", 
                  
                  FractionBox[\(\[PartialD]\[Phi]\_2[x, y, t]\), 
                    \(\[PartialD]x\),
                    MultilineFunction->None]}], "-", 
                \(g\ \(\[Eta](x, t)\)\)}], ")"}]}]}]}], TraditionalForm]], 
  "Input",
  Evaluatable->False,
  CellLabelAutoDelete->True]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(dc = 
      \(-\((\[Gamma]\ \[PartialD]\_{x, 2}\[Eta][x, t])\)\) - 
        \[Rho]\_1\ 
          \((\(-\[PartialD]\_t \[Phi]\_1[x, y, t]\) - 
              U1\ \[PartialD]\_x \[Phi]\_1[x, y, t] - g\ \[Eta][x, t])\) + 
        \[Rho]\_2\ 
          \((\(-\[PartialD]\_t \[Phi]\_2[x, y, t]\) - 
              U2\ \[PartialD]\_x \[Phi]\_2[x, y, t] - g\ \[Eta][x, t])\)\)], 
  "Input"],

Cell[BoxData[
    FormBox[
      RowBox[{
        RowBox[{\(-\[Gamma]\), " ", 
          RowBox[{
            SuperscriptBox["\[Eta]", 
              TagBox[\((2, 0)\),
                Derivative],
              MultilineFunction->None], "(", \(x, t\), ")"}]}], "-", 
        RowBox[{\(\[Rho]\_1\), " ", 
          RowBox[{"(", 
            RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", 
              RowBox[{
                SubsuperscriptBox["\[Phi]", "1", 
                  TagBox[\((0, 0, 1)\),
                    Derivative],
                  MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", 
              RowBox[{"U1", " ", 
                RowBox[{
                  SubsuperscriptBox["\[Phi]", "1", 
                    TagBox[\((1, 0, 0)\),
                      Derivative],
                    MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], 
            ")"}]}], "+", 
        RowBox[{\(\[Rho]\_2\), " ", 
          RowBox[{"(", 
            RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", 
              RowBox[{
                SubsuperscriptBox["\[Phi]", "2", 
                  TagBox[\((0, 0, 1)\),
                    Derivative],
                  MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", 
              RowBox[{"U2", " ", 
                RowBox[{
                  SubsuperscriptBox["\[Phi]", "2", 
                    TagBox[\((1, 0, 0)\),
                      Derivative],
                    MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], 
            ")"}]}]}], TraditionalForm]], "Output"]
}, Open  ]],

Cell["Substitute all that we know", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{"temp6", "=", 
      RowBox[{"dc", "/.", 
        RowBox[{"{", 
          RowBox[{
          \(\[Eta][x, t] \[Rule] \[Eta]\&^\ Exp[I\ k\ \((x - c\ t)\)]\), ",", 
            \(\[Phi]\_1[x, y, t] \[Rule] 
              B1\ Cosh[k\ \((h1 + y)\)]\ Exp[I\ k\ \((x - c\ t)\)]\), ",", 
            \(\[Phi]\_2[x, y, t] \[Rule] 
              A2\ Cosh[k\ \((y - h2)\)]\ Exp[I\ k\ \((x - c\ t)\)]\), ",", 
            RowBox[{
              RowBox[{
                SuperscriptBox["\[Eta]", 
                  TagBox[\((a1_, a2_)\),
                    Derivative],
                  MultilineFunction->None], "[", \(x, t\), "]"}], 
              "\[RuleDelayed]", 
              \(\[PartialD]\_\({x, a1}, {t, a2}\)\((
                  \[Eta]\&^\ Exp[I\ k\ \((x - c\ t)\)])\)\)}], ",", 
            RowBox[{
              RowBox[{
                SubsuperscriptBox["\[Phi]", "1", 
                  TagBox[\((a1_, a2_, a3_)\),
                    Derivative],
                  MultilineFunction->None], "[", \(x, y, t\), "]"}], 
              "\[RuleDelayed]", 
              \(\[PartialD]\_\({x, a1}, {y, a2}, {t, a3}\)\((
                  B1\ Cosh[k\ \((h1 + y)\)]\ Exp[I\ k\ \((x - c\ t)\)])
                  \)\)}], ",", 
            RowBox[{
              RowBox[{
                SubsuperscriptBox["\[Phi]", "2", 
                  TagBox[\((a1_, a2_, a3_)\),
                    Derivative],
                  MultilineFunction->None], "[", \(x, y, t\), "]"}], 
              "\[RuleDelayed]", 
              \(\[PartialD]\_\({x, a1}, {y, a2}, {t, a3}\)\((
                  A2\ Cosh[k\ \((y - h2)\)]\ Exp[I\ k\ \((x - c\ t)\)])
                  \)\)}]}], "}"}]}]}]], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\[ExponentialE]\^\(\[ImaginaryI]\ k\ \((x - c\ t)\)\)\ \[Gamma]\ 
        \(\[Eta]\&^\)\ k\^2 - 
      \((\[ImaginaryI]\ B1\ c\ 
              \[ExponentialE]\^\(\[ImaginaryI]\ k\ \((x - c\ t)\)\)\ k\ 
              \(cosh(k\ \((h1 + y)\))\) - 
            \[ImaginaryI]\ B1\ 
              \[ExponentialE]\^\(\[ImaginaryI]\ k\ \((x - c\ t)\)\)\ k\ U1\ 
              \(cosh(k\ \((h1 + y)\))\) - 
            \[ExponentialE]\^\(\[ImaginaryI]\ k\ \((x - c\ t)\)\)\ g\ 
              \(\[Eta]\&^\))\)\ \[Rho]\_1 + 
      \((\[ImaginaryI]\ A2\ c\ 
              \[ExponentialE]\^\(\[ImaginaryI]\ k\ \((x - c\ t)\)\)\ k\ 
              \(cosh(k\ \((y - h2)\))\) - 
            \[ImaginaryI]\ A2\ 
              \[ExponentialE]\^\(\[ImaginaryI]\ k\ \((x - c\ t)\)\)\ k\ U2\ 
              \(cosh(k\ \((y - h2)\))\) - 
            \[ExponentialE]\^\(\[ImaginaryI]\ k\ \((x - c\ t)\)\)\ g\ 
              \(\[Eta]\&^\))\)\ \[Rho]\_2\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp7 = 
      Cancel[Expand[temp6\/Exp[I\ k\ \((x - c\ t)\)]]] /. y \[Rule] 0\)], 
  "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\[Gamma]\ \(\[Eta]\&^\)\ k\^2 - 
      \[ImaginaryI]\ B1\ c\ \(cosh(h1\ k)\)\ \[Rho]\_1\ k + 
      \[ImaginaryI]\ B1\ U1\ \(cosh(h1\ k)\)\ \[Rho]\_1\ k + 
      \[ImaginaryI]\ A2\ c\ \(cosh(h2\ k)\)\ \[Rho]\_2\ k - 
      \[ImaginaryI]\ A2\ U2\ \(cosh(h2\ k)\)\ \[Rho]\_2\ k + 
      g\ \(\[Eta]\&^\)\ \[Rho]\_1 - g\ \(\[Eta]\&^\)\ \[Rho]\_2\)], "Output"]
}, Open  ]],

Cell["Substitute for A2 and B1", "Text"],

Cell["This command substitutes for A2 from above", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp8 = temp7 /. A2 \[Rule] AY2\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\[Gamma]\ \(\[Eta]\&^\)\ k\^2 - 
      \[ImaginaryI]\ B1\ c\ \(cosh(h1\ k)\)\ \[Rho]\_1\ k + 
      \[ImaginaryI]\ B1\ U1\ \(cosh(h1\ k)\)\ \[Rho]\_1\ k + 
      g\ \(\[Eta]\&^\)\ \[Rho]\_1 - g\ \(\[Eta]\&^\)\ \[Rho]\_2 + 
      \[ImaginaryI]\ c\ \(coth(h2\ k)\)\ 
        \((\[ImaginaryI]\ c\ k\ \(\[Eta]\&^\) - 
            \[ImaginaryI]\ k\ U2\ \(\[Eta]\&^\))\)\ \[Rho]\_2 - 
      \[ImaginaryI]\ U2\ \(coth(h2\ k)\)\ 
        \((\[ImaginaryI]\ c\ k\ \(\[Eta]\&^\) - 
            \[ImaginaryI]\ k\ U2\ \(\[Eta]\&^\))\)\ \[Rho]\_2\)], "Output"]
}, Open  ]],

Cell["This command substitutes for B1 from above", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp9 = temp8 /. B1 \[Rule] BEE1\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\[Gamma]\ \(\[Eta]\&^\)\ k\^2 + g\ \(\[Eta]\&^\)\ \[Rho]\_1 + 
      \[ImaginaryI]\ c\ \(coth(h1\ k)\)\ 
        \((\[ImaginaryI]\ c\ k\ \(\[Eta]\&^\) - 
            \[ImaginaryI]\ k\ U1\ \(\[Eta]\&^\))\)\ \[Rho]\_1 - 
      \[ImaginaryI]\ U1\ \(coth(h1\ k)\)\ 
        \((\[ImaginaryI]\ c\ k\ \(\[Eta]\&^\) - 
            \[ImaginaryI]\ k\ U1\ \(\[Eta]\&^\))\)\ \[Rho]\_1 - 
      g\ \(\[Eta]\&^\)\ \[Rho]\_2 + 
      \[ImaginaryI]\ c\ \(coth(h2\ k)\)\ 
        \((\[ImaginaryI]\ c\ k\ \(\[Eta]\&^\) - 
            \[ImaginaryI]\ k\ U2\ \(\[Eta]\&^\))\)\ \[Rho]\_2 - 
      \[ImaginaryI]\ U2\ \(coth(h2\ k)\)\ 
        \((\[ImaginaryI]\ c\ k\ \(\[Eta]\&^\) - 
            \[ImaginaryI]\ k\ U2\ \(\[Eta]\&^\))\)\ \[Rho]\_2\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp10 = Expand[temp9]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(-k\)\ \(coth(h1\ k)\)\ \(\[Eta]\&^\)\ \[Rho]\_1\ c\^2 - 
      k\ \(coth(h2\ k)\)\ \(\[Eta]\&^\)\ \[Rho]\_2\ c\^2 + 
      2\ k\ U1\ \(coth(h1\ k)\)\ \(\[Eta]\&^\)\ \[Rho]\_1\ c + 
      2\ k\ U2\ \(coth(h2\ k)\)\ \(\[Eta]\&^\)\ \[Rho]\_2\ c + 
      k\^2\ \[Gamma]\ \(\[Eta]\&^\) + g\ \(\[Eta]\&^\)\ \[Rho]\_1 - 
      k\ U1\^2\ \(coth(h1\ k)\)\ \(\[Eta]\&^\)\ \[Rho]\_1 - 
      g\ \(\[Eta]\&^\)\ \[Rho]\_2 - 
      k\ U2\^2\ \(coth(h2\ k)\)\ \(\[Eta]\&^\)\ \[Rho]\_2\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Dispersion relation and growth expression", "Subsubsection"],

Cell[TextData[{
  "By substituting, we have obtained a solution for ",
  StyleBox["c",
    FontSlant->"Italic"],
  " in terms of known variables.  Note that ",
  Cell[BoxData[
      \(\(\[Eta]\&^\)\)]],
  ", which is arbitrary, is multiplying every term linearly and will cancel."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp11 = Collect[temp10, c]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\((\(-k\)\ \(coth(h1\ k)\)\ \(\[Eta]\&^\)\ \[Rho]\_1 - 
            k\ \(coth(h2\ k)\)\ \(\[Eta]\&^\)\ \[Rho]\_2)\)\ c\^2 + 
      \((2\ k\ U1\ \(coth(h1\ k)\)\ \(\[Eta]\&^\)\ \[Rho]\_1 + 
            2\ k\ U2\ \(coth(h2\ k)\)\ \(\[Eta]\&^\)\ \[Rho]\_2)\)\ c + 
      k\^2\ \[Gamma]\ \(\[Eta]\&^\) + g\ \(\[Eta]\&^\)\ \[Rho]\_1 - 
      k\ U1\^2\ \(coth(h1\ k)\)\ \(\[Eta]\&^\)\ \[Rho]\_1 - 
      g\ \(\[Eta]\&^\)\ \[Rho]\_2 - 
      k\ U2\^2\ \(coth(h2\ k)\)\ \(\[Eta]\&^\)\ \[Rho]\_2\)], "Output"]
}, Open  ]],

Cell["This can be written more concisely as", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(temp12 = \ FullSimplify[%]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(\[Eta]\&^\)\ 
      \((\[Gamma]\ k\^2 + 
          \((g - k\ \((c - U1)\)\^2\ \(coth(h1\ k)\))\)\ \[Rho]\_1 - 
          \((k\ \(coth(h2\ k)\)\ \((c - U2)\)\^2 + g)\)\ \[Rho]\_2)\)\)], 
  "Output"]
}, Open  ]],

Cell["\<\
This is the dispersion relation for the wave speed.  
The solutions are:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(cees = Solve[temp12 == 0, c]; \)\)], "Input"],

Cell[BoxData[
    \(Short[cees]\)], "Input"],

Cell[BoxData[
    FormBox[
      TagBox[
        \({{c \[Rule] 
              \(\(-k\)\ \((\[LeftSkeleton]1\[RightSkeleton])\) - 
                  \[LeftSkeleton]1\[RightSkeleton]\)\/\(2\ k\ 
                  \((\(coth(h1\ k)\)\ \[Rho]\_1 + 
                      \[LeftSkeleton]1\[RightSkeleton])\)\)}, {
            c \[Rule] 
              \[LeftSkeleton]1\[RightSkeleton]\/\[LeftSkeleton]1
                  \[RightSkeleton]}}\),
        Short], TraditionalForm]], "Output"]
}, Open  ]],

Cell["\<\
Which should be a set of upstream and downstream traveling waves.  
c1 will be the upstream wave and will not appear in a flow system
c2 will be the downstream wave that we would like to study the behavior \
of\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(c1 = c /. cees\[LeftDoubleBracket]1\[RightDoubleBracket]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\((\(-k\)\ 
            \((\(-2\)\ U1\ \(coth(h1\ k)\)\ \[Rho]\_1 - 
                2\ U2\ \(coth(h2\ k)\)\ \[Rho]\_2)\) - 
          \[Sqrt]\((
              k\^2\ \((
                      \(-2\)\ U1\ \(coth(h1\ k)\)\ \[Rho]\_1 - 
                        2\ U2\ \(coth(h2\ k)\)\ \[Rho]\_2)\)\^2 - 
                4\ k\ \((
                    \(coth(h1\ k)\)\ \[Rho]\_1 + \(coth(h2\ k)\)\ \[Rho]\_2)
                    \)\ \((\(-\[Gamma]\)\ k\^2 + 
                      U1\^2\ \(coth(h1\ k)\)\ \[Rho]\_1\ k + 
                      U2\^2\ \(coth(h2\ k)\)\ \[Rho]\_2\ k - g\ \[Rho]\_1 + 
                      g\ \[Rho]\_2)\))\))\)/
      \((2\ k\ \((\(coth(h1\ k)\)\ \[Rho]\_1 + \(coth(h2\ k)\)\ \[Rho]\_2)\))
        \)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(c2 = c /. cees\[LeftDoubleBracket]2\[RightDoubleBracket]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\((\[Sqrt]\((
              k\^2\ \((
                      \(-2\)\ U1\ \(coth(h1\ k)\)\ \[Rho]\_1 - 
                        2\ U2\ \(coth(h2\ k)\)\ \[Rho]\_2)\)\^2 - 
                4\ k\ \((
                    \(coth(h1\ k)\)\ \[Rho]\_1 + \(coth(h2\ k)\)\ \[Rho]\_2)
                    \)\ \((\(-\[Gamma]\)\ k\^2 + 
                      U1\^2\ \(coth(h1\ k)\)\ \[Rho]\_1\ k + 
                      U2\^2\ \(coth(h2\ k)\)\ \[Rho]\_2\ k - g\ \[Rho]\_1 + 
                      g\ \[Rho]\_2)\))\) - 
          k\ \((\(-2\)\ U1\ \(coth(h1\ k)\)\ \[Rho]\_1 - 
                2\ U2\ \(coth(h2\ k)\)\ \[Rho]\_2)\))\)/
      \((2\ k\ \((\(coth(h1\ k)\)\ \[Rho]\_1 + \(coth(h2\ k)\)\ \[Rho]\_2)\))
        \)\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Example calculations", "Subsubsection"],

Cell[CellGroupData[{

Cell["\<\
We are using cgs units for the air-water system.  The wind speed is \
300 cm/s and the layers are deep.   The system is stable. \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(c2numb = 
      c2 /. {\[Rho]\_2 \[Rule]  .00125, \[Rho]\_1 \[Rule] 1.020, 
          g \[Rule] 980, \[Gamma] \[Rule] 74, U1 \[Rule] 1, U2 \[Rule] 300, 
          h1 \[Rule] 10, h2 \[Rule] 10}\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(1\/k\((
      0.489596083231334144`\ 
        \((2.79000000000000003`\ k\ \(coth(10\ k)\) + 
            \[Sqrt]\((
                7.78410000000000046`\ k\^2\ \(\(coth\^2\)(10\ k)\) - 
                  4.08499999999999996`\ k\ \(coth(10\ k)\)\ 
                    \((\(-74\)\ k\^2 + 
                        113.519999999999998`\ \(coth(10\ k)\)\ k - 
                        998.375000000000056`)\))\))\)\ \(tanh(10\ k)\))
      \)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(kh1 = 
      Plot[Re[N[c2numb]], {k,  .01, 30}, 
        AxesLabel \[Rule] {"\<wavenumber\>", "\<wavespeed\>"}]; \)\)], "Input"],

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Gravity waves (long waves) get faster as they get longer.  In contrast, \
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\>", "Text"],

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Here is the growth curve which is always 0.  This means that the \
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\>", "Text"],

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Cell["\<\
We see that the behavior of the speed in the low wavenumber region \
is different.  This is what the model predicts when the waves become \
unstable.  This does not seem correct and is not predicted by more accurate \
models.\
\>", "Text"],

Cell["\<\
Now the growth curve is not always 0.  Wavenumbers less than about \
5/cm are predicted to be unstable.\
\>", "Text"],

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You would find that the velocity difference will need to be about \
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Now make both layers deep.  We need to make the velocity \
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Cell[TextData[{
  "Which shows the temporal grow rate for an interfacial disturbance.  \nThis \
calculation is easy and neat, however there is one big problem, it does not \
match the linear behavior for the air water system.  The predicted point of \
instability is too high; the real value would be more like 3 m/s.  \
Furthermore, note the speed of unstable waves.  They are predicted to travel \
at the speed (",
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  "), which is not correct.   "
}], "Text"],

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The upstream wave does have a negative growth region that \
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It is worthwhile to get some additional physical feel for this \
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what if  there is no surface tension. \
\>", "Text"],

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The wave speed looks peculiar.  Without surface tension to \
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Cell["\<\
Without surface tension, all waves shorter than some cutoff are \
unstable.\
\>", "Text"],

Cell["\<\
So surface tension stabilizes the short waves, which is reasonable. \
 \
\>", "Text"]
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Cell["\<\
What is the point of neutral stability and what is the mechanism? \
\
\>", "Subsubsection"],

Cell[TextData[{
  "Let's consider the simplest case, both layers deep, air-water ( ",
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      \(\[Rho]\_2\)]],
  " <<< ",
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      \(\[Rho]\_1\)]],
  ") and that the velocity, U1 is much less than U2. "
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          \[Sqrt]\((
              4\ k\^2\ U2\^2\ \(\(coth\^2\)(h1\ k)\)\ \[Rho]\_2\%2 - 
                4\ k\ \((
                    \(coth(h1\ k)\)\ \[Rho]\_1 + \(coth(h1\ k)\)\ \[Rho]\_2)
                    \)\ \((\(-\[Gamma]\)\ k\^2 + 
                      U2\^2\ \(coth(h1\ k)\)\ \[Rho]\_2\ k - g\ \[Rho]\_1 + 
                      g\ \[Rho]\_2)\))\))\)/
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        \)\)], "Output"],

Cell["\<\
We know (that is I know from looking at the expression that this \
can be done), by definition, for h1->Infinity ...\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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        \@\(4\ k\^2\ U2\^2\ \[Rho]\_2\%2 - 
            4\ k\ \((\[Rho]\_1 + \[Rho]\_2)\)\ 
              \((\(-\[Gamma]\)\ k\^2 + U2\^2\ \[Rho]\_2\ k - g\ \[Rho]\_1 + 
                  g\ \[Rho]\_2)\)\)\)\/\(2\ k\ 
        \((\[Rho]\_1 + \[Rho]\_2)\)\)\)], "Output"]
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Cell["\<\
For instability, we need the expression inside the root to be \
negative.  Let's look at it.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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            4\ k\ \((\[Rho]\_1 + \[Rho]\_2)\)\ 
              \((\(-\[Gamma]\)\ k\^2 + U2\^2\ \[Rho]\_2\ k - g\ \[Rho]\_1 + 
                  g\ \[Rho]\_2)\)\)\/\(2\ k\ 
          \((\[Rho]\_1 + \[Rho]\_2)\)\)\)], "Output"]
}, Open  ]],

Cell["Grab the part inside the radical,", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(nuet4 = \ \(Numerator[nuet3[\([2]\)]]\)[\([1]\)]\)], "Input"],

Cell[BoxData[
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      4\ k\ \((\[Rho]\_1 + \[Rho]\_2)\)\ 
        \((\(-\[Gamma]\)\ k\^2 + U2\^2\ \[Rho]\_2\ k - g\ \[Rho]\_1 + 
            g\ \[Rho]\_2)\)\)], "Output"]
}, Open  ]],

Cell["Do a little more manipulation on it,", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(nuet5\  = 
      Collect[Collect[ExpandAll[\(nuet4/\[Rho]\_1\)/4], \[Rho]\_2], k]\)], 
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Cell[BoxData[
    \(TraditionalForm
    \`\((\(\[Rho]\_2\ \[Gamma]\)\/\[Rho]\_1 + \[Gamma])\)\ k\^3 - 
      U2\^2\ \[Rho]\_2\ k\^2 + 
      \((g\ \[Rho]\_1 - \(g\ \[Rho]\_2\%2\)\/\[Rho]\_1)\)\ k\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Now we see that for each power of k, how important the",
  Cell[BoxData[
      \(\(\ \[Rho]\_2\)\)]],
  "terms are.  We choose the important ones by making the others 0.  This is \
inelegant, but the easiest way to do it.  "
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(neutral1\  = 
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Cell[BoxData[
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}, Open  ]],

Cell[TextData[{
  "If this quantity is greater than 0, the flow is unstable.  The first term \
is the restoring force of surface tension, the third term is the restoring \
force of gravity.  The second term is the action of the gas flow which is \
causing deformation because of the low pressure region at the wave crest and \
high pressure regions in the wave trough.\n\nThus the mechanism for \
instability is the pressure caused by a Bernoulli effect becoming stronger \
than the restoring forces of gravity and / or surface tension. \n\nFor flows \
involving real fluids this effect is definitely present.  However, this \
mechanism is not as efficient as another mechanism (which involves a shift in \
the pressure minimum, see Jurman, Deutsch and McCready, JFM, ",
  StyleBox["238",
    FontWeight->"Bold"],
  ", 178-219, 1992) and hence is not responsible for the initial formation of \
waves in air-water flows and other sytems where there is a significant \
viscosity difference between the phases.  However this general mechanism may \
be important for growth of ",
  StyleBox["large",
    FontSlant->"Italic"],
  " amplitude waves.    "
}], "Text"]
}, Open  ]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Orr-Sommerfeld Equation", "Subtitle",
  Evaluatable->False,
  PageBreakAbove->True,
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Cell["\<\
A good reference for this section is R. L. Panton, Incompressible \
flow, Wiley, 1984\
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
Here we derive the Orr-Sommerfeld equation which is a 4th order ODE \
that describes the growth on infinitesimal periodic distrubances that are \
governed by the Navier-Stokes equations.\
\>", "Text"],

Cell["\<\
The linearized x-momentum equations for a nearly-parallel flow are
x direction (first in traditional form)\
\>", "Text",
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Cell[TextData[ButtonBox["Back to conclusions",
  ButtonData:>"conclusions",
  ButtonStyle->"Hyperlink"]], "Text"],

Cell["y direction (first in traditional form)", "Text",
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Cell[TextData["The continuity equation is"], "Text",
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Cell[TextData[{
  "We need to expand these in terms of normal modes where the variable will \
be assumed to have the form \[Xi][x,y,t]=",
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  "[y] Exp[I(\[Alpha] x -\[Alpha] ",
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  " t)] based on the idea that we need functions that can describe spatially \
and temporally varying disturbances"
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