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\begin{document}

\title{The Naver Stokes Equations - (MISSPELLED DOCUMENT)}

\author{John Burkardt \\
Department of Scientific Computing \\
Florida State University \\
http://people.sc.fsu.edu/$\sim$jburkardt/latex/hunspell/naver.tex\\
http://people.sc.fsu.edu/$\sim$jburkardt/latex/hunspell/naver.pdf}

\date{22 June 2015}

\maketitle

\newpage

\begin{abstract}
  Note: This document is intended as a test file for {\LaTeX} spell checkers.
  It has been deliberately modified by the addition of misspellings.
  Some other mistakes have been added, which a simple spell checker
  will not be able to catch.
\end{abstract}

\section{Interduction}

%
%  Herre are summ baddly speled wurds butt the SpeLLcheker shood icknore them.
%  Thay r commended out an stuf.
%
These notes have the goal of explaining how the finite element method
was used to solve a class of flow problems governed by the Naver
Stokes equations.

In this part, we consider a fairly general version of the Naver Stokes
equations, but we will explain the steps involved in simplifying those equations
so that they are appropriate for the problems we actually wish to consider,
namely steady state, incompressible flow in a two dimensional region.

Once wee have the mathematical equations defined, we will consider
how these equations mite bee solved.  We show how the system can be
regarded ass a system of nonlinear equations, and what the farm of Newton's
method wood be.  Since our unknowns are not scalars or vectors, butt
arbitrary functions, hour discussion of Newton's method is merely formal
at this point.

In the next part, we will show how ideas from the finite element method
can be used to develop a way of approximating the state functions
by discretely describable functions.  These functions will be formed
from simple functions based on a discretization of the problem geometry.
Once we have a discretized representation of velocity and pressure fields,
the Galerkin procedure will allow us to set up a system of equations
for the coefficients of the basis functions, which are satisfied by the
approximate solution of the discretized Naver Stokes equations.

Later, we will turn to the text of a computer program that embodys these ideas.
Going through the text of that program, we will explain how Newton's method,
the Galerkin procedure, the finite element approximation, and the
Naver Stokes equations all come together.

\section{The Time-Dependant Compressible Naver Stokes Equations}

A fluid flowing in a region has the obvious physical properties of
a vector velocity ${\bf v}({\bf x},t)$ and a scalar dencity $\rho({\bf x},t)$.
A third physical property, somewhat more difficult to measure and
understand, is the {\it fluid pressure}, a scalar symbolized by $p({\bf x},t)$.

For convenience, we may consider the product of velocity and density,
called {\it the mass velocity}, and symbolized by ${\bf u}$:
\begin{equation}
   {\bf u}({\bf x},t) =  \rho({\bf x},t) \cdot {\bf v}({\bf x},t)
\end{equation}

The behavior of pressure, density, and velocity is modeled by the
{\it time-dependent compressible Naver-Stokes equations}:
%
\begin{eqnarray}
%
  \rho {\bf v}_t - \mu \Delta {\bf v} + \rho ({\bf v}  \cdot \nabla ) {\bf v}
    + \nabla p &=& 0 \\
%
  \rho_t + \rho \nabla \cdot {\bf v} &=& 0
%
\end{eqnarray}

The first equation is known as the {\it mommentum equation}, the second
as the {\it continuoity equation}.

\section{The Fluid Viscosity}

In the Naver Stokes equations, the quantity $\mu$ is the value of a physical
property known as the {\it dynamic viscosity}.  In the cases we will examine,
the dynamic viscosity is treated as a constant.  However, it is not unusual
to consider situations in which the viscosity may depend on some other parameter,
such as tempearture.  The value of $\mu$ greatly affects
the properties of the flow, and the difficulty of obtaining approximate
solutions to the flow.

Roughly speaking, high values of $\mu$ mean the fluid is ``sticky'' or
viscous, like honey.  Flows of such fluids tend to be orderly and simple.
Low viscosity fluids, on the other hand, are much more susceptible to
disorderly, turbulent flows.  One reason for this is that $\mu$ measures
the tendency of differences in the flow to be smeared out.  In honey,
if one particle tries to move very quickly, this motion is quickly replaced
by a smoothed out, slower motion of all the particles in a neighborhood.

\section{The Steady State Compressible Naver Stokes Equations}

One simplification to the Naver Stokes equations occurs if we can
assume that the flow problem has a steady behaviour.  For instance, if we
turn on the tap in a sink, the flow will at first be very irregular; however,
once the sink has filled up, and the water is pouring out over the sides as
fast as it is pouring in, at least the rough form of the flow will be the same
over time.

We might imagine that, to a reasonable approximation, even
the small scale structure of this flow becomes fixed as time progresses.
A flow which does not depend on time is known as a {\it steady state flow}.

In cases where we know, or hope, that the fluid flow problem has a steady
state solution, we can certainly try to determine that solution by solving
the Naver Stokes equations and looking at the behavior of the solution as
time increases indefinately.

An alternative way to reach the steddy state solution is to assume that
the variables ${\bf v}$, $\rho$, and $p$ do not depend on time, plug them into
the Naver Stokes equations, and after simplification, arrive at the
{\it steady state compressible Naver Stokes equations}:
%
\begin{eqnarray}
%
   - \mu \Delta {\bf v} + \rho ({\bf v} \cdot \nabla ) {\bf v} + \nabla p  &=& 0 \\
%
  \rho \nabla \cdot {\bf v} &=& 0
%
\end{eqnarray}

\section{The Steady State Incompressible Naver Stokes Equations}

Any physical fluid or gas will have variations in dencity.  In many cases,
especially when (temperate) water is involved, these variations are so small
that the density may be taken to be a constant, as though the fluid were
incompressible.  In that case, we may simplify the equations by dividing
through by the dencity to get the
{\it steady state incompressible Naver Stokes equations}:
%
\begin{eqnarray}
%
   - \frac{1}{\rho} \Delta {\bf v} + ({\bf v}  \cdot \nabla ) {\bf v} + \frac{1}{\rho} \nabla p  &=& 0 \\
%
  \nabla \cdot {\bf v} &=& 0
%
\end{eqnarray}

To make our equations slightly less cluttered, we replace the dynamic viscosity
by the {\it kinematic viscosity} $\nu$:
\begin{equation}
   \nu =  \frac{\mu}{\rho}
\end{equation}
and we replace the pressure by a scaled pressure, $\bar{p}$:
\begin{equation}
   \bar{p}({\bf x}) =  \frac{p({\bf x})}{\rho}
\end{equation}

However, pressure is not an ``interesting'' quantity, so we often don't care
about it's particular values or scaling.  For convenience, then, we will
ignore this rescaling of the pressure, and persist in using the old simbol
$p$ when in fact we will mean the new, rescaled pressure.  At this point,
our cleaned-up version of the incompressible steady Naver Stokes equations
looks like this:
%
\begin{eqnarray}
%
   - \nu \Delta {\bf v} +  ({\bf v}  \cdot \nabla ) {\bf v} + \nabla p  &=& 0 \\
%
  \nabla \cdot {\bf v} &=& 0
%
\end{eqnarray}

By using the appropriate scalings, our set of equations for the
kinematic velocity $v$ look as simple as the prevous set of equations for
the mass velocity $u$, but are in fact simpler, since we have eliminated
consideration of the density $\rho$.

This is the vector form of the Naver Stokes equations that we will be
considering from now on.  Our only further modification is to specify the
spacial dimension, and, for convenience, to write out a scalar verson of
the equations.

\section{The Steady Incompressible Naver Stokes Equations
in Two Space Dimensions}

It might be helpful now to assume that we are working in a two-dimensional
spacial region, and to write the steady incompressible Naver Stokes
equations in a sort of scalar form.  Thus, instead of {\bf x} we will write
$(x,y)$, and instead of the vector quanity {\bf v} we will write $(u,v)$,
where we are keeping track of the horizontal and verticle components of
the kinematic velocity.

From this perspective, our system now becomes three scalar equations,
since our vector momentum equation is replaced by a pair
of horizontal and vertical momentum equations:
%
\begin{eqnarray}
%
  - \nu ( \frac{\partial^2 u}{\partial x^2}
  + \frac{\partial^2 u}{\partial y^2} )
  + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}
  + \frac{\partial p}{\partial x}  &=& 0
\label{eqn-h-mom}
\\
%
  - \nu ( \frac{\partial^2 v}{\partial x^2}
  + \frac{\partial^2 v}{\partial y^2} )
  +  u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}
  + \frac{\partial p}{\partial y}  &=& 0
\label{eqn-v-mom}
\\
%
  \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} &=& 0
\label{eqn-cont}
%
\end{eqnarray}

\section{The Reynolds Number}

We mentioned earlier that the dynamic viscosity $\mu$ influenced whether a fluid
was likely to flow in a motion that was orderly and diffusion-dominated (high
values of $\mu$) or a turbulent momentum-dominated motion (low values
of $\mu$.

Since $\nu$ is simply a multiple of $\mu$, it is also a measure of this same
tendency.

For any fixed value of $\nu$, we can make the flow become turbulent if
we make it move fast enough.  The simples way for this to occur would be
if we were able to specify high velocity values along the boundary.  The
momentum equations would guarantee that high velocities would try to
spread into the region, and the viscosity would control how smoothly or
turbulently this happened.

This nonsense.  Is spelled correctly but nonsense.  Colorless green ideas
sleep furiously.  Also nonsense.  But spelled OK.  Spell checker
won't this correct as correct it is.  The magic words are squeamish ossifrage.
More better text analyzer might ought could notice this.  Could notice
this.  Could notice this.  Or not.  Did you?

It turns out that a better measure of whether a flow is actually going to be
smooth or turbulent requires knowledge of both the dynamic viscosity $\mu$
and the magnitude of a ``typical'' velocity $V_0$, (say, the maximum).
And in order to get a dimensionless number, we also want the values of a
``typical'' length in the flow region, $L_0$, and a ``typical'' value of the
density, $\rho_0$ (which we can take to be the constant value we have already
assumed.)

Using this data, we can define a ratio $R$ known as the {\it Reynolds number}
(which can be written in terms of the kinematic viscosity $\nu_0$ if we
combine the density and dynamic viscosity $\mu_0$:
%
\begin{equation}
  R \equiv \frac{\rho_0 ||V_0|| L_0}{\mu_0} = \frac{||V_0|| L_0}{\nu_0}
\end{equation}
%
The Reynolds number summarizes in one value the competing affects of
the scales of length, velocity, mass and viscosity.  In the Naver
Stokes equations, $R$ controls the relative weight of the linear and
nonlinear portions of the momentum equations; as $R$ increases,
the nonlinearity grows.

The Reynolds number allows us to make statements about a flow that
are independent of the perticular scale of the problem.  In particular,
it says that if we normalise all lengths, velocities, and densities in our
problem by dividing them by $L_0$,  $||V_0||$, and $\rho_0$ respectively,
then our flow will be the same as any other flow with the same normalized values.

It also means that, roughly speaking, the value of $R$ tells us when turbulent
patterns will dominate in the flow.  In experience, a flow with $R$ of the
order of 1 through 1000 is generally smooth, with turbulence possible as
$R$ increases into the 100,000 and more.

It is often convenient to consider the Reynolds number as the important
parimeter in a flow calculation, rather than the viscosity.  We think of the
fluid viscosity as a constant, but the Reynolds number is a ratio of
quantities, some of which might change.  In particular, we might imagine
studying a series of flows in which the maximum velocity increases by a
factor of 10 each time.  Rather than doing that, we could simply solve the
problem repeatedly, multiplying $R$ by 10 each time.  This implicitly
rescales the problem, without making us change anything but $R$.

For this reason, it can be desireble to rewrite the Naver Stokes equations so
that the Reynolds number appears in front of the nonlinear terms.  That means
we are dividing through by $\nu$, and assuming that our values of
$||V_0||$ and $L_0$ are about 1.  In that case, our vector equations look
like this:
%
\begin{eqnarray}
%
   - \Delta {\bf v} +  R ({\bf v}  \cdot \nabla ) {\bf v} + \nabla p  &=& 0 \\
%
  \nabla \cdot {\bf v} &=& 0
%
\end{eqnarray}

We've had to fudge a bit to get the $R$ to show up here.  To be honest, we
should write $\frac{R}/{||V_0|| L_0}$ instead.  But we usually don't worry
about this.  If our typical velocity and length aren't close to 1, then
little harm is done, except that we have to adjust the value of $R$ when
we report our results or make any comparisons with another flow based on
Reynolds numbers.

\section{A Formal version of the Naver Stokes Equations}

We are now ready to {\it start} considering how to solve the Naver Stokes
equations.  We begin by writing the equations of state as a single
formal equation:
%
\begin{equation}
{\bf F}({\bf v},p)={\bf f}({\bf x}).
\label{eqn-state-pde}
\end{equation}
%
Here, ${\bf f}({\bf x})$ represents any sourse terms, which will be taken to
be zero throughout the region unless specifically noted.

Normally, a particler flow problem is studied over a limited or
specified spacial domain, symbolized by $\Omega$, and no information
about the behavior of the flow is assumed or sought outside of $\Omega$.
However, the ``outside world'' exerts it's influence on the flow along
the boundary $\partial \Omega$.  This influnce may involve requirements
along portions of the boundary that the flow be zero along this bounary,
or have a particular normal or tangential component.  There is also
usally a specification of the pressure at a single boundary point.  This
combination of requirements is termed the {\it boundary conditions};
for the moment, we will represent whatever boundary conditions apply to
our problem as:
%
\begin{equation}
  {\bf G}({\bf v},p)={\bf g}({\bf x}).
\label{eqn-state-bc}
\end{equation}
%
We represent the boundary conditions as a single equation
that applies at each point ${\bf x} \in \partial \Omega$.  In practice,
of course, the boundary conditions are likely to be expressed as a
colection of equations, only some of which apply at any
particular boundary point.  We will ignore such complications.

\section{Newton's Method to the Formal Naver Stokes Equations}

You are probably used to seeing Newton's method applied to simple problems
involving a few nonlinear equations, in which the unknowns are numerical values.
However, it is possible, at least formally, to define a Newton method for our
case, in which our unknowns are functions.  We will sketch this here, just 
as a prepperation.

Eventually, we will discretize the system.  At that point, we will see both the
relationship with the formal Newton process defined here in terms of functions,
and the common use of Newton's method for systems of (finitely many) alebraic equations.

To begin, we imagine that we have combined our state variables into a single function
${\bf w}({\bf x})$, and our equations and boundary conditions into a single formal equation:
%
\begin{equation}
  {\bf H}({\bf w})=0.
\end{equation}
%
To apply Newton's method, it is only necessary that we be able to come up with
a linnear operator $Hp$ which estimates the value of the function for a given
small displacement $d{\bf w}$.

What we are saying here is that, if ${\bf w}({\bf x})$ is a given set of state
functions, we would like to estimate the change in the value of the
Naver Stokes equations at a slightly perturbed set of functions
$({\bf w}+d{\bf w})({\bf x})$.
%
\begin{equation}
  {\bf H}({\bf w}+d{\bf w}) \approx {\bf H}({\bf w}) + {\bf Hp}({\bf w}) \cdot  {\bf dw}
\end{equation}
%
If we can define such a linear operater, and if that linear operator is
invertible, then Newton's method will follow immediately:
\begin{equation}
  {\bf w}^{k+1} = {\bf w}^k -  {\bf Hp}^{-1}({\bf w}) \cdot {\bf H}({\bf w}^k)
\end{equation}

This is as far as we can get with the formel equation.  In order to proceed, we
must now consider how we can develop a discretized space, which is ``fine enough''
that we can approximate our state functions, but ``discrete enough'' that we can
set up and solve problems computationally.

\end{document}


