\documentstyle[10pt]{report}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%TCIDATA{Created=Mon Jul 28 13:12:06 1997}
%TCIDATA{LastRevised=Wed Nov 12 09:18:04 1997}
%TCIDATA{Language=American English}

\topmargin=-0.5in
\parskip=0
\baselineskip
\parindent=.5in
\oddsidemargin  0truein
\evensidemargin 0truein
\topmargin      -.5truein
\textheight     8.5in
\textwidth      6.5in
\setcounter{page}{31}
\setcounter{chapter}{4}
\makeindex
\input{tcilatex}
\begin{document}


\chapter{Compressible fluids}

All real fluids are compressible, and almost all fluids expand when heated.
Compression waves can propagate in most fluids: these are the familiar sound
waves in the audible frequency range, and ultrasound at higher frequencies.
Thermal expansion gives rise to heat convection, especially in the presence
of a gravitational field: hot air rises and cold air sinks.

In general, heat transfers and fluid motions are coupled and should be
treated together by using the equations of fluid dynamics along with those
of thermodynamics and heat diffusion. However, the coupled equations are
complicated, and we will start with the simplified assumption that fluid
motions occur either {\it isothermally }(at constant temperature) or {\it %
adiabatically }(with negligible heat transfer), as a first approximation.

In order to use thermodynamics, it must be possible to define a temperature $%
T({\bf r},t)$ that varies with position ${\bf r}$ and time $t$, in the same
way as one defines other hydrodynamic variables such as the mass density $%
\rho ({\bf r},t)$, the pressure $p({\bf r},t),$ and the fluid velocity ${\bf %
v}({\bf r},t)$. One must be able to consider a volume $V$ that is large
enough to be macroscopic (it contains many particles) and small enough to be
infinitesimal with respect to variations of $T$; in addition, the particle
velocities within $V$ must be given by the thermal equilibrium distribution,
when vieved in a frame moving along with the fluid with the local velocity $%
{\bf v}({\bf r,}t{\bf )}$.

We will not treat variations in the chemical composition of the fluid, so
that in effect we can suppose that the volume $V$ contains $N$ particles of
average mass $m$, and we also assume that the fluid is uncharged and
non-magnetic. Then the first law of thermodynamics (just energy balance) can
be stated as follows: the heat transfer $dQ$ to the fluid element containing 
$N$ particles causes a change of the internal energy $E$ and of the volume $%
V $ according to 
\begin{equation}
dE=dQ-pdV  \label{dE}
\end{equation}

We recall that an infinitesimal heat transfer $dQ$ corresponds to a change
of the entropy $S$, according to $dQ=TdS$: thus an adiabatic volume change ($%
dQ=0,$ no heat transfer) is also isentropic ($dS=0).$ In thermodynamics, the
advantage of introducing $S.$is that a system contains a well defined amount
of entropy, but not a definite amount of heat (since work can be turned into
heat and, to some extent, heat can be used to do work). The entropy $S.$also
has the important property that it never decreases for a closed system
(second law of thermodynamics). For simplicity, we do not use the second law
in this chapter, although we will often refer to the entropy content of the
fluid. When we need to carry out a derivation involving $S$, we will assume
that the fluid is an ideal gas, work up to the final formula, and then
simply quote its general form for any fluid.

\section{Compressibility}

When the density $\rho $ changes, both the pressure $p$ and the temperature $%
T$ will change, in general. The usual way to describe these changes in
thermodynamics is to change the volume $V$ occupied by a fixed number $N$ of
particles, so that 
\begin{equation}
dV=\left( \dfrac{\partial V}{\partial p}\right) _{T}dp+\left( \dfrac{%
\partial V}{\partial T}\right) _{p}dT  \label{dV}
\end{equation}
Since $dV$ is proportional to $V$, it is convenient to consider the
fractional volume change $dV/V$ and to define the {\it isothermal
compressibility} 
\begin{equation}
\kappa _{T}=-\frac{1}{V}\left( \dfrac{\partial V}{\partial p}\right) _{T}
\label{KT}
\end{equation}
and the {\it thermal expansion coefficient} 
\[
\beta =\frac{1}{V}\left( \dfrac{\partial V}{\partial T}\right) _{p} 
\]
which are positive quantities characteristic of the material (fluid or
solid). To convert from $dV/V$ to a change of density, we note that a fixed $%
N$ implies that $\rho V$ is a constant (it is the mass contained in $V)$.
Then $\rho dV+Vd\rho =0$ and we get 
\begin{equation}
\frac{d\rho }{\rho }=-\frac{dV}{V}=\kappa _{T}dp-\beta dT  \label{drho}
\end{equation}
As an example, consider an ideal gas. From $pV=Nk_{B}T,$ get $%
pdV+Vdp=Nk_{B}dT$, hence 
\begin{equation}
\kappa _{T}=1/p\quad \quad \quad \quad {\rm (ideal\;gas)}  \label{KTi}
\end{equation}
\begin{equation}
\beta =1/T\quad \quad \quad \quad {\rm (ideal\;gas)}
\end{equation}

As we said already, there are two limiting cases where heat transfer and
fluid motions are effectively decoupled, and then the dynamics of the fluid
is fully determined by the appropriate {\it compressibility coefficient:}

\begin{itemize}
\item  If the density changes slowly, so that heat conduction keeps the
temperature $T$ constant, eq. (\ref{drho}) reduces to $d\rho /\rho =\kappa
_{T}dp$, where $\kappa _{T}$ is the {\it isothermal compressibility } (\ref
{KT}). As we have already noted, $\kappa _{T}$ is positive (because a
compression corresponds to a positive $dp$ and a negative $dV$), and does
not depend on the size of $V$ (because for a larger $V$ we get a
correspondingly larger $dV$ for the same $dp$). However, $\kappa _{T}$ can
depend on $p$ and on the temperature $T$. For example, in an ideal gas, $%
\kappa _{T}=1/p$.

\item  If the density changes rapidly, so that no heat can flow, we have
instead $d\rho /\rho =\kappa _{S}dp$, where 
\begin{equation}
\kappa _{S}=-\dfrac{1}{V}\left( \dfrac{\partial V}{\partial p}\right) _{S}
\label{KS}
\end{equation}
is the {\it adiabatic compressibility, }which we now discuss. The subscript $%
S$ denotes a process at constant entropy, in accordance with the fact that
an adiabatic volume change ($dQ=0,$ no heat transfer) is also isentropic ($%
dS=0)$. In such a process all the work done in changing the volume goes into
the internal energy $E$ of the fluid and eq. (\ref{dE}) reduces to 
\[
dE=-pdV\quad \quad \quad {\rm (adiabatic\;process)}
\]
This condition allows us to compute $\kappa _{S}$. For example, in an ideal
gas $dE=C_{V}dT$ and for an adiabatic volume change $-pdV=$ $C_{V}dT$. 
\footnote{%
In general we would proceed by using eq. (), but for an ideal gas it is more
direct to use the gas law; the two methods are equivalent, since $\beta
=1/T. $} On the other hand, from $pV=Nk_{B}T$ we get $Vdp+pdV=Nk_{B}dT,$ and
eliminating $dT$ we find 
\[
Vdp=-p(1+Nk_{B}/C_{V})dV
\]
\[
\kappa _{S}=\dfrac{1}{p}\,\dfrac{C_{V}}{C_{V}+Nk_{B}}\quad \quad \quad \quad 
{\rm (ideal\;gas),}
\]
Recalling that $C_{p}=C_{V}+Nk_{B}$ is the specific heat at constant pressure%
\footnote{%
In general, the specific heat is $dQ/dT$, or $TdS/dT$. This can be rewritten
using the basic equation of thermodynamics (just energy balance), $%
dE=TdS-pdV.$ At constant volume, $dE=TdS$ and 
\[
C_{V}=T\left( \dfrac{\partial S}{\partial T}\right) _{V}=\left( \dfrac{%
\partial E}{\partial T}\right) _{V} 
\]
At constant pressure, it is convenient to work with the quantity $H=E+pV$,
called the {\it enthalpy}. We find $dH=dE+pdV+Vdp=TdS+Vdp,$ and so 
\[
C_{p}=T\left( \dfrac{\partial S}{\partial T}\right) _{p}=\left( \dfrac{%
\partial H}{\partial T}\right) _{p} 
\]
In an ideal gas, $dE=C_{V}dT$ and $pV=Nk_{B}T$, so that $%
dH=dE+d(pV)=(C_{V}+Nk_{B})dT$ and $C_{p}=C_{V}+Nk_{B}.$} and comparing with
equation (\ref{KTi}), we see that 
\begin{equation}
\dfrac{\kappa _{S}}{\kappa _{T}}=\dfrac{C_{V}}{C_{p}}  \label{Kratio}
\end{equation}
This relation is true in general (as can be proved by thermodynamics). It is
almost always true that $C_{p}>C_{V}$, so that $\kappa _{S}<\kappa _{T}$: it
is harder to compress a fluid if heat does not flow out of it.
\end{itemize}

In a typical liquid, there is little difference between $\kappa _{S}$ and $%
\kappa _{T}$, but in a monatomic ideal gas the difference is large: $%
C_{V}=(3/2)Nk_{B}$ and $C_{p}=(5/2)Nk_{B}$. The ratio $C_{p}/C_{V}$ appears
often in the physics of gases and is called $\gamma $. For a monatomic ideal
gas, $\gamma =5/3=1.667$; for air near room temperature $\gamma $ it is very
close to 1.4, which is the value for a diatomic ideal gas when the molecules
can be regarded as rigid (at higher temperatures, molecular vibrations
increase $C_{V}$, so that $\gamma $ decreases).

The inverse of the compressibility coefficient is called the {\it bulk
modulus} $B.$ We can define $B_{T}=1/\kappa _{T}$ and $B_{S}=1/\kappa _{S}$,
but we can use simply $B$ and let the context distinguish what we mean
(keeping in mind that for liquids, and solids, the distinction is
unimportant). We also want to relate pressure changes to density changes,
rather than volume changes, as in eq. (\ref{drho}). We note that the
definitions (\ref{KT}) and (\ref{KS}) contain the fractional volume change, $%
dV/V,$ rather than just $V,$ so that the compressibilities and $B$ depend
only on the particle density $N/V,$ and not on $V$ and $N$ separately. In
fluid dynamics, we use the mass density $\rho $ as a variable and use the
formula 
\begin{equation}
dp=\dfrac{B}{\rho }\,d\rho  \label{B}
\end{equation}
to eliminate the pressure (or the density) from the Navier-Stokes equation.

Values of $1/B$ are given in Table 7.16 of {\it PQRG}. We see that for
common organic liquids $B$ is about 1 GigaPascal (10$^{9}\,$N/m$^{2}$),
increasing slowly with pressure. Water is a little less compressible: $%
B=2.17\,\,$GPa at $T=22{{}^\circ}$C and $p=0.1\,$MPa. Mercury is a lot less
compressible: with $B=25\,\,$GPa it is getting close to the range of solid
metals.

\section{Sound waves}

Small-amplitude sound is a linear disturbance of the medium, which means
that we can write, for instance, $\rho =\rho _{0}+\delta \rho ({\bf r},t)$
and keep only terms linear in $\delta {\bf v},$ $\delta \rho ,$ and $\delta p
$. We will also assume that ${\bf v}$ itself is small, which still allows us
to treat sound in drifting fluids by going to a frame moving with the fluid
(this works as long as the sound's wavelength is smaller than the size of
the drift). The effect of gravity is included to leading order by letting
the undisturbed density $\rho _{0}$ vary with altitude. Neglecting
viscosity, the Navier-Stokes equation reduces to the linearized Euler
equation 
\begin{equation}
\rho _{0}\,\dfrac{\partial {\bf v}}{\partial t}={\bf -\nabla }p  \label{NS1}
\end{equation}
and the linearized continuity equation is simply 
\begin{equation}
\dfrac{\partial \rho }{\partial t}+\rho _{0}{\bf \nabla }\cdot {\bf v}=0
\label{C1}
\end{equation}
(For brevity, we write for instance $\partial \rho /\partial t$ instead $%
\partial (\delta \rho )/\partial t$, since the two are equal). To determine
everything, we need an equation relating pressure changes to density
changes. Assuming that these changes occur either adiabatically or
isothermally, we can write, from eq.(\ref{B}), 
\[
\dfrac{\partial p}{\partial x}=\dfrac{B}{\rho }\dfrac{\partial \rho }{%
\partial x}
\]
where $x$ is any coordinate. In vector notation 
\begin{equation}
{\bf \nabla }p=\dfrac{B}{\rho }{\bf \nabla }\rho   \label{gradp}
\end{equation}
The ratio $B/\rho $ can be taken as constant because we have assumed small
density changes; for a liquid, $B$ and $\rho $ vary very little anyhow and
for an ideal gas $B_{T}/\rho $ depends only on $T,$ and so is constant under
isothermic conditions.

From equations (\ref{NS1}), (\ref{C1}), and (\ref{gradp}), it is follows
that $\rho $, $p$, and ${\bf v}$ obey the wave equation. Let us show this
for $\rho $, for instance. From (\ref{NS1}) and (\ref{gradp}) we get 
\[
\rho _{0}\,\dfrac{\partial {\bf v}}{\partial t}=-\dfrac{B}{\rho _{0}}{\bf %
\nabla }\rho 
\]
and taking the divergence of both sides: 
\[
\rho _{0}\,\dfrac{\partial }{\partial t}\,{\bf \nabla }\cdot {\bf v}=-\dfrac{%
B}{\rho _{0}}\nabla ^{2}\rho 
\]
On the other hand, the time derivative of (\ref{C1}) gives 
\begin{equation}
\dfrac{\partial ^{2}\rho }{\partial t^{2}}+\rho _{0}\dfrac{\partial }{%
\partial t}{\bf \nabla }\cdot {\bf v}=0  \label{dC1}
\end{equation}
and from the last two equations it follows that 
\begin{equation}
\dfrac{\partial ^{2}\rho }{\partial t^{2}}=\dfrac{B}{\rho _{0}}\nabla
^{2}\rho 
\end{equation}
As is well known, this equation has plane wave solutions of the form $\rho
=\rho _{0}+\delta \rho ,$ with 
\[
\delta \rho =|A|\,\cos (kx-\omega t+\alpha )
\]
where

\begin{itemize}
\item  $|A|$ is the amplitude, i.e., the maximum value of $\delta \rho $

\item  $k$ is the wave number, positive for a wave propagating in the
positive $x$ direction

\item  $\omega $ is the angular frequency, always taken to be positive. It
is related to $k$ by $\omega ^{2}=c^{2}k^{2},$ where (dropping the subscript
on $\rho _{0}$) 
\[
c=\sqrt{\dfrac{B}{\rho }}\quad \quad \quad {\rm is\;the\;speed\;of\;sound}
\]

\item  $\alpha $ is the phase of the wave at $x=0,$ $t=0$.
\end{itemize}

Intuition suggests that one should use $B_{S}$ for low frequencies and $B_{T}
$ for high frequency, with the crossover at $\omega =Dk^{2},$ or $\omega
=c^{2}/D,$ where $D$ is the thermal diffusivity (more about this later). For
audible sound in air, $\omega /2\pi $ is between 40\thinspace Hz and
20\thinspace kHz (decreasing with age), while $c^{2}/D=5\times 10^{9}$Hz.
Thus in practice one should use $B_{S}$ even for ultrasound, and one finds 
\[
c_{{\rm air}}=\sqrt{\dfrac{\gamma p}{\rho }}=\sqrt{\dfrac{\gamma k_{B}T}{m}}%
=340\sqrt{\dfrac{T}{293\,{\rm K}}}\,\dfrac{{\rm m}}{{\rm \sec }}
\]

\end{document}