
\begin{spacing}{0}
\chapter{Conducting Flow in a Magnetic Field}
\label{chap:mhd_math}
\end{spacing}
\
\par

In this chapter, the governing equations for magnetohydrodynamics (MHD), based on electrodynamic relations of Maxwell's equation and hydrodynamic Navier-Stokes equation, are given and the effects of Lorenz force induced by magnetic field are discussed. Considering the energy equation for MHD, the roles of energy deposition induced
by proton beam interaction and energy of state equation are discussed.
Hartmann considered the flow between two parallel, infinite, non-conducting walls,
with magnetic field applied normal to the walls \cite{Hartmann}.
An exact solution was obtained for this case. Shercliff solved the more general problem of three dimensional flow
in a rectangular duct \cite{Shercliff1953}. Exact solutions demonstrated the fact that
for large Hartmann number, the velocity distribution consists of a
uniform core with a boundary layer near the walls. This result enabled the solution of the corresponding problem
for a circular pipe in an approximate manner for large Hartmann numbers,
assuming walls of zero conductivities \cite{Shercliff1956}. Chang \cite{Chang} considered
the effects of wall conductivity for the same problem.
Gold \cite{Gold} considered a steady one-dimensional flow of an incompressible, viscous, electrically conducting
fluid through a circular pipe in the presence of a uniform transverse field.
A no-slip condition is assumed at the electrically non-conducting
wall because if the walls are conducting, there is a electromagnetic force on the wall
and a corresponding force on the fluid. The solution is exact and valid for all values of the Hartmann number.
Conducting liquid jet under the influence of a strong magnetic field raises several MHD issues,
such as the possible deformation of the jet's shape and trajectory, as well as the effect
of the magnetic field on the beam-induced dispersal of the jet. The electrically conducting flow moving in a magnetic field experiences induced currents \cite{Gallardo2002}. These induced currents cause the jet to experience
anisotropic pressure distribution with respect to the major and minor axis of jet
cross section normal to the jet axis while the jet penetrates the nonuniform magnetic
field \cite{Gallardo2002}. In addition, axial currents are induced if the jet axis does not coincide
with the magnetic field axis. These currents in turn produce transverse elliptical distortions of
the Hg jet.
Finally, the liquid jet can develop surface instabilities and jet breakup during liquid motion
in a inhomogeneous magnetic field and after interaction with an intense proton beam.
These phenomena can change the shape of the jet to a form that is less efficient target for pion production.

\section{Governing Equations for MHD Flow}

\subsection{Electromagnetic equations}

In this section, we describe the electromagnetic relations that have been used in the derivation
of the MHD governing equations. The following parameters are defined as follows:
\begin{itemize}
  \item polarization density \textbf{P}: the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. It is defined as the dipole moment per unit volume.
  \item magnetization density \textbf{M}: the magnetic dipole moment per unit volume.
  \item electrical susceptibility $\chi_{e}$: a measure of how easily a dielectric material polarizes in response to an electric field. This determines the electric permittivity of the material. It is defined as the constant of proportionality when relating an electric field \textbf{E} to the induced dielectric polarization density \textbf{P}.
  \item magnetic susceptibility $\chi_{m}$: the degree of magnetization of a material in response to an applied magnetic field.
  \item electric displacement field \textbf{D}: It accounts for the effects of bound charges
  within materials. It is the macroscopic field average of electric fields from charged particles
  that make up otherwise electrically neutral material. It can be considered the field after
  taking into account the response of a medium to an external field such as reorientation of
  electric dipoles.
  \item magnetic field strength \textbf{H}: A vector field that permeates space which can
  exert a magnetic force on a moving electric charge and on magnetic dipoles such as permanent
  magnets.
  \item electric field \textbf{E}: the electric force per unit charge. The direction of the field is taken to be the direction of the force it would exert on a positive test charge.
\end{itemize}

\subsubsection{\textit{electromagnetic relations in a linear material}}

In a linear material, the polarization density  \textbf{P}, magnetization density  \textbf{M}, and
the magnetic field strength are related by

\begin{equation}
 \label{e:eq21}
\textbf{P} = \chi_{e} \varepsilon_{o} \textbf{E} \
\end{equation}

and

\begin{equation}
 \label{e:eq22}
\textbf{M} = \chi_{m} \textbf{H} \ ,
\end{equation}

where $\chi_{e} $ is the electrical susceptibility and $ \chi_{m} $ is the magnetic susceptibility of the material.
Electric displacement field \textbf{D} and magnetic induction field \textbf{B} are related to the electric field
\textbf{E} and magnetic field \textbf{H} by

\begin{equation}
 \label{e:eq23}
\textbf{D}= \varepsilon_{o} \textbf{E} + \textbf{P} = \varepsilon \textbf{E} \
\end{equation}

and

\begin{equation}
 \label{e:eq24}
\textbf{B} = \mu_{o} ( \textbf{H} + \textbf{M} ) = \mu \textbf{H} \ ,
\end{equation}

where $\varepsilon$ is the electrical permittivity and $\mu$ is the magnetic permeability of the material.

\subsubsection{\textit{Maxwell's equations}}

The solenoidal condition for the magnetic induction, indicating that there are no magnetic
monopoles, is given by

\begin{equation}
 \label{e:eq25}
 \nabla \cdot \textbf{B}=0 \ .
\end{equation}

This shows that there are no sources and sinks for magnetic field lines.
Faraday's law of magnetic induction is given by

\begin{equation}
 \label{e:eq26}
\nabla \times \textbf{E} =  - \partial \textbf{B} / \partial t   \ ,
\end{equation}

showing that a spatially varying electric field can induce a magnetic field.
Charge conservation gives

\begin{equation}
 \label{e:eq27}
\nabla \cdot \textbf{E} =  \rho^{*} / \varepsilon_{o} \ ,
\end{equation}

where $ \rho^{*} = \varepsilon_{o}$ (n$^{+}$ - n$^{-}$)  is the charge density,  n$^{+} $
is the number of ions, and  n$^{-} $ is the number of electrons. Amp\`{e}re's law is given by

\begin{equation}
 \label{e:eq28}
\nabla \times \textbf{B} = \mu \textbf{j} + \mu \varepsilon \partial \textbf{E} / \partial t \ ,
\end{equation}

where the last term on the right hand side is the displacement current.
Introducing the fundamental units of mass M, length L, velocity \textit{v}, and time t,
we consider the dimensions of the displacement current in Eqn.~(\ref{e:eq28}).
The dimensions of the magnetic field B, electric field E, and the speed of light c are considered for simplicity.

$ \nabla \times \textbf{E} \sim \frac{E}{L}, \frac{\partial \textbf{B}}{\partial t} \sim \frac{B}{t}$ gives  E = \textit{v} B . From the speed of light,  c = $\frac{1}{\sqrt{\mu \varepsilon}}$,
$ \mu \varepsilon \partial \textbf{E} / \partial t
= \frac{1}{c^{2}} \partial \textbf{E} / \partial t \sim \frac{1}{c^{2}} \frac{E}{t} = \frac{\textit{v}}{c^{2}} \frac{B}{t} = \frac{B}{L} \frac{\textit{v}^{2}}{c^{2}}$.

Therefore, the displacement current in Amp\`{e}re's law can be neglected if the flow velocity is much
less than the speed of light.
By assuming the flow obeys charge neutrality,  n$^{+}$ - n$^{-}$ $\ll$ n, where n is the total number
density, the charge density in Eqn.~(\ref{e:eq27}) can be neglected.
Note that n includes positive, negative, and neutral charge. Thus, n$^{+}$ - n$^{-}$ $\ll$ n indicates that
most of charges are assumed to obey neutrality. Finally, Ohm's law without Hall effect is given by

\begin{equation}
 \label{e:eq29}
\textbf{j} = \sigma ( \textbf{E} + \textbf{v} \times \textbf{B} ) \ .
\end{equation}

This is the generalization of the relation between voltage and current in a moving conductor.
It provides the link between the electromagnetic equations and the fluid equations.
The electric charge is conserved, which is given by Kirchhoff's law:

\begin{equation}
 \label{e:eq210}
\nabla \cdot \textbf{j} = 0 .
\end{equation}


\subsection{The Navier Stokes and magnetic induction equations in a conducting liquid flow}

The motion of an electrically conducting fluid in the presence of
a magnetic field obeys the equations of MHD. The
fluid is treated as a continuum, and the classical results of fluid
dynamics and electro-dynamics are combined in the derivation of the equations.
The first equation is from mass conservation:

\begin{equation}
 \label{e:eq211}
 \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \textbf{v}) = 0 \ .
\end{equation}

Next, Newton's second law of motion gives

\begin{equation}
 \label{e:eq212}
 \rho \frac{D \textbf{v} }{D t} = - \nabla p + \textbf{F} \ ,
\end{equation}

where the external force  \textbf{F}  consists of several terms, such as the Lorentz force,
given by  \textbf{j} $\times$ \textbf{B} , the gravitational force $\rho$ \textbf{g} , and
the viscous force. The viscous term is given by a kinematic viscosity of the form $ \rho \nu \nabla^{2}$ \textbf{v} for an incompressible flow. Thus, Equation~(\ref{e:eq212}) becomes

\begin{equation}
 \label{e:eq213}
 \rho \frac{\partial \textbf{v}}{\partial t} + \rho (\textbf{v}\cdot \nabla) \textbf{v}
  = - \nabla p + \rho \textbf{g} + \eta \nabla^{2} \textbf{v} + \textbf{j} \times \textbf{B} \ .
\end{equation}

Note that the Lorentz force couples the fluid equations to the electromagnetic equations.
Equation~(\ref{e:eq213}) can be reduced to a dimensionless form.

\begin{equation}
 \label{e:eq21a}
\frac{\partial \textbf{v}}{\partial t} + \textbf{v} \cdot \nabla
\textbf{v} = -\nabla p + \frac{\textbf{g}}{\textrm{Fr}^{2}} + \frac{1}{\textrm{Re}} \nabla^{2} \textbf{v}
+ \textrm{Al} ( \textbf{j} \times \textbf{B}) \ ,
\end{equation}

where \textrm{Fr} = $\textit{v}/\sqrt{g L}$, \textrm{Re}= $\rho \textit{v}L/\eta$, \textrm{Re}$_{m}$= $\mu \sigma \textit{v}L$, and \textrm{Al} = $B^{2}_{o}/\mu \rho \textit{v}^{2}$ denote the Froude, Reynolds, magnetic
Reynolds, and Alfv\'{e}n numbers, respectively. The Hartmann number, \textrm{Ha} = $B_{o}L(\sigma / \eta)^{\frac{1}{2}}$, gives the ratio of magnetic forces to viscous forces. Thus, this number is the important
parameter in cases where the inertial effects are small. On the other hand,
the Stuart number, \textrm{N}, gives the ratio of magnetic forces to inertial forces, Thus, this number is
the important parameter where dealing with inviscid or turbulent flows. \textrm{Ha} and \textrm{N} are related
through $\textrm{Ha}^{2} = \textrm{Re} \textrm{Re}_{m} \textrm{Al}$ and
$\textrm{N}=\textrm{Re}_{m}\textrm{Al}$. Note that the ratio of Hartmann number and
Reynolds number involves viscous, magnetic, and inertial
forces and can be thought as the square root of the product of the viscous and magnetic forces
divided by the inertial forces.

We consider components of the magnetic induction field $B_{x}, B_{y}, B_{z}$.
Note that the longitudinal magnetic field along the jet axis $x$ and the transverse magnetic
field normal to the jet axis are given by $ B_{x} = B_{X} cos \theta - B_{Y} sin \theta ,
B_{y} = - B_{X} sin \theta + B_{Y} cos \theta  $ respectively, where $ B_{X} $ is axial magnetic field
and $ B_{Y} $ is radial magnetic field. Also note that the jet flow uses $(x,y,z)$ coordinate system and
magnetic field uses $(X,Y,Z)$ coordinate system in Fig.~\ref{fig:jet_surface}.
The nondimensionalized momentum equations in the $(x,y,z)$ coordinate system
in Fig.~\ref{fig:jet_surface} is represented as Eqn.~(\ref{e:eq22a}) using Ohm's equation:

\begin{equation}
 \label{e:eq22a}
\begin{split}
&\frac{\partial \textit{v}_{x}}{\partial t} + \textit{v}_{x} \cdot \nabla \textit{v}_{x} = -\nabla p +
\frac{1}{\textrm{Re}} \nabla^{2} \textit{v}_{x} -
\frac{\textrm{Ha}_{y}^{2}}{\textrm{Re}}\textit{v}_{x} +
\frac{\textrm{Ha}_{x}\textrm{Ha}_{y}}{\textrm{Re}} \textit{v}_{y} \ , \\
&\frac{\partial \textit{v}_{y}}{\partial t} + \textit{v}_{y} \cdot \nabla \textit{v}_{y} = -\nabla p +
\frac{1}{\textrm{Re}} \nabla^{2} \textit{v}_{y} -
\frac{\textrm{Ha}_{x}^{2}}{\textrm{Re}} \textit{v}_{y} +
\frac{\textrm{Ha}_{x}\textrm{Ha}_{y}}{\textrm{Re}} \textit{v}_{x} \ ,  \\
&\frac{\partial \textit{v}_{z}}{\partial t} + \textit{v}_{z} \cdot \nabla \textit{v}_{z} = -\nabla p +
\frac{1}{\textrm{Re}} \nabla^{2} \textit{v}_{z} -
\frac{\textrm{Ha}_{x}^{2}}{\textrm{Re}} \textit{v}_{z} -
\frac{\textrm{Ha}_{y}^{2}}{\textrm{Re}} \textit{v}_{z} \ ,
\end{split}
\end{equation}

where $\textrm{Ha}_{x} = B_{x}L(\sigma / \eta)^{\frac{1}{2}}$, $\textrm{Ha}_{y} = B_{y}L(\sigma / \eta)^{\frac{1}{2}}$, and $\textrm{Ha}_{z} = B_{z}L(\sigma / \eta)^{\frac{1}{2}}$.
In MHD, to eliminate the electric field \textbf{E} and the electric current density  \textbf{j},
we use the Ampere's law and Ohm's law. Then, Faraday's law gives the magnetic induction equation:

\begin{equation}
 \label{e:eq214}
 \frac{\partial \textbf{B}}{\partial t} = \nabla \times ( \textbf{v} \times \textbf{B} )
 - \frac{1}{\mu \sigma } \nabla \times \nabla \times \textbf{B}
 = \nabla \times ( \textbf{v} \times \textbf{B} ) + \frac{1}{ \mu \sigma } \nabla^{2} \textbf{B} \ .
\end{equation}


\subsubsection{\textit{magnetic Reynolds number}}

In Eqn.~(\ref{e:eq214}), the dimension of the term on the left hand side is $ \frac{B}{t}$
and that of the second term on the right hand side is $ \frac{B}{\sigma \mu L^{2} } $.
Therefore, $ \sigma \mu \sim \frac{t}{L^{2}} $. The magnetic induction equation can be written
in a dimensionless form:

\begin{equation}
 \label{e:eq215}
 \frac{\partial \textbf{B}}{\partial t}
 = \nabla \times ( \textbf{v} \times \textbf{B} ) + \frac{1}{\sigma \mu L \textit{v}} \nabla^{2} \textbf{B} \ ,
\end{equation}

where the quantity $ \sigma \mu L \textit{v }$ is the magnetic Reynolds number, $ \textrm{Re}_{m} $,
which is a measure of the size of the advection term, $ \nabla \times ( \textbf{v} \times \textbf{B} ) $,
relative to the diffusion term, $\frac{1}{\sigma \mu L \textit{v}} \nabla^{2} \textbf{B} $.
Reynolds number \textrm{Re} measures the extent to which
a convective process prevails over a diffusive one. In viscous flow, the viscosity causes
vorticity to diffuse in the face of convection and the Reynolds number measures the power of
convection over diffusion of vorticity. In MHD, the conductivity causes convection to
overcome diffusion of the magnetic field to a degree
measured by the magnetic Reynolds number $ \textrm{Re}_{m} $. If $ \textrm{Re}_{m} $ is large,
convection dominates over diffusion and magnetic boundary layer near the magnetic field are to be expected.
The magnetic Prandtl number, $\textrm{Pr}_{m}$, measures the ratio of viscous diffusivity and magnetic
diffusivity and is defined as $\textrm{Re}_{m} / \textrm{Re} $. When it is small, magnetic fields diffuse much
more rapidly than vorticity and magnetic boundary layers are much thicker than viscous layers.
From Eqn.~(\ref{e:eq216}) and (\ref{e:eq217}), the time scales with changes due to fluid motions and field diffusion
can be thought of $t_{convection} = \frac{v}{L}$ and $t_{diffusion} = \sigma \mu L^{2}$.
Thus, the time scales for magnetic Prandtl number can be $\mu \sigma v L $, which corresponds to $\textrm{Re}_{m}$.
Therefore, if $\textrm{Re}_{m} \ll 1$, the dissipation time is much smaller than characteristic time in the flow.
During the characteristic time in the flow, a field disturbance diffuses $ (L/\mu\sigma \textit{v})^{1/2} $.
This is much larger than $L$ if $\textrm{Re}_{m} \ll 1$. This makes for simplifications such as the neglect of
viscosity in the magnetic boundary layer.
In any region of length scale $\delta$ where magnetic convection and diffusion are equally important, $\delta$ must be of order $1/\mu \sigma \textit{v}$. Only within limited regions where B changes significantly in a distance $\delta $ can the gradients be high enough for diffusion and dissipation to matter. The characteristic time in the flow is the transit time $L/\textit{v}$, during which a field disturbance diffuses a distance of order $ (L/\mu\sigma \textit{v})^{1/2} $. This is much less than $L$ if $\textrm{Re}_{m} \gg$ 1, in which case diffusion is negligible
and diffuses a distance of order $ (t/\mu\sigma)^{1/2} $. This is the required criterion for the perfect conductivity approximation to be valid. At the other extreme case where diffusion is dominant is that the medium diffuses to the form it would be in stationary fluid, where no induced magnetic field would occur. The ratio of the induced magnetic field and the imposed magnetic field is of order $ \mu \sigma \textit{v} L $, which is $\textrm{Re}_{m}$. The low  $\textrm{Re}_{m}$ approximation is to ignore the induced field, to replace \textbf{B} by the known field $\textbf{B}_{o}$ in all MHD equations.

\subsubsection{\textit{frozen-in theorem in magnetic induction equation}}

If $ \textrm{Re}_{m} \gg $ 1 , the induction equation Eqn.~(\ref{e:eq214}) is approximated by

\begin{equation}
 \label{e:eq216}
 \frac{\partial \textbf{B}}{\partial t}
 = \nabla \times ( \textbf{v} \times \textbf{B} ) .
\end{equation}

The timescale with changes due to the fluid motion from Eqn.~(\ref{e:eq216}) is given by
$ t_{motion} \sim \frac{L}{\textit{v}} $. In the case $ t_{motion} \ll t_{diffusion} $, which corresponds to
$ \textrm{R}_{m} \gg $ 1, the diffusion term is negligible. According to the frozen-flux theorem of
Alfv\'{e}n, in a perfectly conducting fluid, where $ \textrm{Re}_{m} \rightarrow \infty $, the
magnetic field lines move with the fluid: the field lines are `frozen' into the fluid. This theorem
states that motions along the field lines do not change the field but motions transverse to the field
carry the field with them. 

\subsubsection{\textit{diffusion limit in induction equation}}

If $ \textrm{Re}_{m} \ll $ 1, the induction equation Eqn.~(\ref{e:eq214}) is approximated by

\begin{equation}
 \label{e:eq217}
 \frac{\partial \textbf{B}}{\partial t}
 =  \frac{1}{ \mu \sigma } \nabla^{2} \textbf{B} .
\end{equation}

The timescale with changes due to field diffusion from Eqn.~(\ref{e:eq217}) is given by
$ t_{diffusion} \approx \sigma \mu L^{2} $. The diffusion equation indicates that any irregularities
in an initial magnetic field will diffuse away and be smoothed out. The field will tend to be a simpler
uniform field. This process of smoothing out will occur on the given diffusion timescale.

\section{The Energy Equation in MHD}

\subsection{Energetics and effects of Lorentz force}

The energy equation that contains all the various types of energy, including kinetic energy,
gravitational energy, the internal energy, and the magnetic energy is obtained using the
MHD governing equations. The gravitational potential $\Phi $ is defined by $ - \nabla \Phi = \textbf{g} $.
By multiplying Eqn.~(\ref{e:eq211}) by $ v^{2}/2 $ and dotting Eqn.~(\ref{e:eq212}) with  \textbf{v},
the energy equation can be given by

\begin{equation}
 \label{e:eq24b}
\frac{\partial }{\partial t} ( \frac{1}{2} \rho v^{2}) + \nabla \cdot ( \frac{1}{2} \rho v^{2} \textbf{v}) = - \textbf{v} \cdot \nabla p + \textbf{v} \cdot ( \textbf{j} \times \textbf{B})
- \textbf{v} \cdot \rho \nabla \Phi + \textbf{v}\cdot \eta \nabla^{2} \textbf{v} \ .
\end{equation}

The gravitational term is given using Eqn.~(\ref{e:eq211}) and the fact that
$ \partial \Phi / \partial t = 0 $ as follow:

\begin{equation}
 \label{e:eq25b}
\textbf{v}\cdot \rho \nabla \Phi = \nabla \cdot (\rho \Phi \textbf{v}) + \frac{\partial }{\partial t} ( \rho \Phi ) \ .
\end{equation}

The Lorentz force term is given using Eqn.~(\ref{e:eq29}) as follow:

\begin{equation}
 \label{e:eq26b}
\textbf{v}\cdot(\textbf{j}\times \textbf{B}) = -\textbf{j}\cdot(\textbf{v}\times \textbf{B})
= - \frac{j^{2}}{\sigma} + \textbf{j} \cdot \textbf{E} \ .
\end{equation}

Equation~(\ref{e:eq26b}) is rearranged using Eqn.~(\ref{e:eq26}) as follow:

\begin{equation}
 \label{e:eq27b}
\textbf{v}\cdot(\textbf{j}\times \textbf{B}) = - \frac{j^{2}}{\sigma} - \nabla \cdot
(\frac{\textbf{E} \times \textbf{B}}{\mu}) - \frac{\partial}{\partial t} (\frac{B^{2}}{2\mu}) \ .
\end{equation}

The pressure gradient term gives

\begin{equation}
 \label{e:eq28b}
- \textbf{v}\cdot \nabla p = - \nabla \cdot (p \textbf{v}) + p \nabla \cdot \textbf{v} \ .
\end{equation}

Equation~(\ref{e:eq28b}) can also be expressed using Eqn.~(\ref{e:eq211}) as follow:

\begin{equation}
 \label{e:eq29b}
p \nabla \cdot \textbf{v} = - \frac{\partial p}{\partial t} \ .
\end{equation}

Substituting the foregoing relations, the full energy equation is as follow:

\begin{equation}
 \label{e:eq30z}
\frac{\partial }{\partial t} [ \frac{1}{2} \rho v^{2} + \rho \Phi + p + \frac{B^{2}}{2\mu}] + \nabla \cdot \{ [ \frac{1}{2} \rho v^{2} + \rho \Phi + p ] \textbf{v} + \frac{\textbf{E} \times \textbf{B}}{\mu} \} = - \frac{j^{2}}{\sigma} - D \ ,
\end{equation}

where D is viscous dissipation term.

\subsection{Proton beam induced energy deposition and equation of state}

Due to the sudden energy deposition by proton beam, it is worthy to consider the components of added energy and
the state of energy from compressible density variation as well as ionization to the right hand side of the full
energy equation Eqn.~(\ref{e:eq30z}). It is reported \cite{Hassanein2000} that the instantaneous beam energy
deposition is

\begin{equation}
 \label{e:eq36z}
E_{beam} (r) = E_{p} (r) \cdot \delta ( t - t_{beam} ) ,
\end{equation}

where

\begin{equation}
 \label{e:eq37z}
E_{p} (r) = E_{o} exp [ - \frac{r}{a}  ] \ .
\end{equation}


$E_{p} \ (r)$ is radial energy density distribution and the proton beam energy is
assumed to be deposited as a $\delta$ function at time $t = t_{beam}$.
The equation of state (EOS) is considered as the sum of compression, ion thermal, and electron thermal terms.
The compressible pressure $P_{c}$ and energy $E_{c}$ are

\begin{equation}
 \label{e:eq39z}
P_{c} = P_{co} [ (\frac{\rho}{\rho_{o}})^{\gamma} - 1 ]
\end{equation}

and

\begin{equation}
 \label{e:eq40z}
E_{c} = E_{co} [ (\frac{\rho}{\rho_{o}})^{\gamma - 1} - 1 ] \frac{\rho}{\rho_{o}} +  P_{co} (1 - \frac{\rho}{\rho_{o}})   ,
\end{equation}

where

\begin{equation}
 \label{e:eq41z}
P_{co} = \frac{\rho  c^{2}}{\gamma} \ \ \ \ , \ \ \ \
E_{co} = \frac{P_{co}}{\gamma - 1}   .
\end{equation}

Ion and electron thermal pressure and energy are

\begin{equation}
 \label{e:eq42z}
E_{I} = 3 n k (T - T_{o} ) \ \ \ , \ \ \ P_{I} = G_{I} E_{I} \ \ \
\end{equation}

and

\begin{equation}
 \label{e:eq43z}
E_{e} = \frac{1}{2} \beta ( T - T_{o} )^{2} \ \ \ , \ \ \  \beta = \beta_{o}(\frac{\rho_{o}}{\rho})^{2/3} \ \ \ , \ \ \ \ P_{e} = G_{e} E_{e}  ,
\end{equation}

where $n = \frac{\rho}{M}$ and $k$ is Boltzmann's constant. Thus, the total energy deposition and
pressure are

\begin{equation}
 \label{e:eq44z}
E_{p} = E_{c} + E_{I} + E_{e} \ \ \ , \ \ \ P_{p} = P_{c} + P_{I} + P_{e} \ \ ,
\end{equation}

where subscripts c, I, and e correspond to compression, ion thermal, and electron thermal components, respectively.
$G_{I}$ and $G_{e}$ are the Gruneisen coefficients for the ion and electron. $c$ is the speed of sound in
the material. Initial Hg pressure $P$ is 0 at $T = T_{o}$ and normal density $\rho = \rho_{o}$. At higher temperatures, the Hg can be ionized and the resulting energy and pressure by free-electron
component is added to the EOS.

It is reported \cite{Samulyak2007} that a single phase EOS for liquid Hg showed that the strength of rarefaction
waves in the jet significantly exceeds the Hg cavitation threshold. The formation of cavities is believed
to take place in strong rarefaction waves, and cavitation bubbles influence the wave dynamics
in Hg \cite{Samulyak2007}. In order to capture the formation and dynamics of the two-phase domain of bubbly
Hg, a homogenized two-phase equation of state is reported \cite{Samulyak2004}. EOS consists of three branches.
The pure vapor and liquid branches are described by the polytropic and stiffened polytropic EOS model
respectively \cite{Menikoff}. The two branches are connected by a model for the liquid-vapor mixture.
The speed of sound in an equilibrium homogeneous mixture of liquid and vapor are given:

\begin{equation}
 \label{e:eq45z}
\frac{1}{c^{2}} = (\alpha \rho_{sat, v} + (1 - \alpha) \rho_{sat, l} ) ( \frac{\alpha}{\rho_{sat, v} c^{2}_{sat, v}} + \frac{1-\alpha}{\rho_{sat, l} c^{2}_{sat, l}}) \ ,
\end{equation}

where $\rho_{sat, v} , \rho_{sat, l}, a_{sat, v}, a_{sat, l}$ are the density and the speed of sound
of vapor and liquid in saturation point, and $\alpha$ is the void fraction:

\begin{equation}
 \label{e:eq46z}
\alpha = \frac{\rho - \rho_{sat, l}}{\rho_{sat, v} - \rho_{sat, l}} \ .
\end{equation}

Integrating the sound speed with respect to the density, the pressure-density relation is given as follow:

\begin{equation}
 \label{e:eq47z}
P = P_{sat, l} + P_{vl} log [ \frac{\rho_{sat, v} c^{2}_{sat, v} (\rho_{sat, l} + \alpha (\rho_{sat, v}-\rho_{sat, l}))}{\rho_{sat, l}(\rho_{sat, v} c^{2}_{sat, v} - \alpha (\rho_{sat, v} c^{2}_{sat, v} - \rho_{sat, l} c^{2}_{sat, l}))}  ]  \ ,
\end{equation}

where $P_{sat, l}$ is the liquid pressure in the saturation point and

\begin{equation}
 \label{e:eq48z}
P_{vl} = \frac{\rho_{sat, v} c^{2}_{sat, v} \rho_{sat, l} c^{2}_{sat, l} (\rho_{sat, v} - \rho_{sat, l})}{\rho^{2}_{sat, v} c^{2}_{sat, v} - \rho^{2}_{sat, l} c^{2}_{sat, l}}  \ .
\end{equation}

Using the second law of thermodynamic at constant entropy and Eqn.~(\ref{e:eq47z}), the specific internal
energy of the vapor-liquid mixture is calculated \cite{Samulyak2004}.


\subsection{Magnetic damping with joule dissipation}

It is known that a static magnetic field can suppress motion of an electrically conducting liquid.
If a conducting liquid moves through an imposed static magnetic field, electric currents are
generated. These, in turn, lead to ohmic heating such as Joule dissipation.  As the thermal energy of the fluid rises, there is a corresponding drop in its kinetic energy, and so the fluid decelerates. This is to suppress the motion of liquid jets. In many applications, it is believed
that the imposition of a static magnetic field is used as one means of suppressing unwanted
motion. Considering the uniform perpendicularly imposed magnetic field to the flow direction for
simplicity, the damping effect of Lorentz force can be quantified. If the magnetic field
is uniform, the Faraday' law requires that $ \nabla \times \textbf{E} = 0 $. Using Ohm's law and the
fact that the current density is solenoidal, the current relationship is given by

\begin{equation}
 \label{e:eq311z}
\nabla \cdot \textbf{j} = 0 \ \ \ , \ \ \ \nabla \times \textbf{j} = \sigma \textbf{B} \cdot \nabla \textbf{v} \ .
\end{equation}

Thus, \textbf{j}  is zero if \textbf{v} is independent of the magnetic field direction.
By doing cross product of \textbf{j} and \textbf{B} and using the vector identity,
Lorentz force per unit mass is given by

\begin{equation}
 \label{e:eq312z}
\textbf{F} = - \frac{\textbf{v}_{\perp}}{\tau} + \frac{\sigma (\textbf{B} \times \nabla \phi_{E})}{\rho} \ ,
\end{equation}

where $ \tau = \rho / \sigma B^{2} $ is Joule damping term and $\phi_{E}$ is electrical potential,
which is given by the divergence of Ohm's law: $\phi_{E} = \nabla^{-2} (B \cdot \omega)$. The
Lorentz force then simplifies to $ - \textbf{v}/\tau $ when the magnetic field and the vorticity field
are mutually perpendicular. Thus, the $\textbf{v}_{\perp}$ to magnetic field declines on a
time scale of $\tau$, which is the phenomenological basis of magnetic damping. The ratio of the
damping time $\tau$ to the characteristic time $L/\textit{v}$ gives the interaction
parameter $ \textrm{N} = \sigma B^{2} L/\rho \textit{v}$, which is also used for the indication of
the ratio of the magnetic and inertial forces.

To investigate the role of Joule dissipation, consider the fully derived energy equation in
inviscid flow from Eqn.~(\ref{e:eq30z}) is given:

\begin{equation}
 \label{e:eq313z}
\frac{d E}{d t} = -\frac{1}{\sigma \rho} \int j^{2} dV  \ ,
\end{equation}

where $j$ is current density, $E$ is global kinetic energy, and the Joule dissipation is shown as
per density. $ \textbf{j}^{2} $ from Eqn.~(\ref{e:eq311z}) was estimated \cite{Davidson} and is given:

\begin{equation}
 \label{e:eq314z}
\frac{d E}{d t} \sim - (\frac{L_{min}}{L_{\parallel}})^{2} \frac{E}{\tau} \ ,
\end{equation}

from which

\begin{equation}
 \label{e:eq315z}
E \sim E_{o} \ \ exp \  ( - \tau^{-1} \int_{0}^{t} (L_{min}/L_{\parallel})^{2} dt ) \ ,
\end{equation}

where $L_{\parallel}$ and $L_{min}$ are the characteristic length scales for the flow, parallel to the
magnetic field.
The time constant required for energy dissipation is getting smaller as the magnetic field strength increases.
Figure~\ref{fig:energy_decay} (a) and (b) show the decay of energy depending on the Joule damping term with
various magnetic fields and its integration with respect to time, provided  $L_{\parallel}$ and $L_{min}$ remain of
the same order. It indicates that the flow decays on a time scale of $\tau $.
However, the Lorentz force cannot create or destroy linear momentum, nor angular momentum \cite{Davidson}.
As long as momentum is conserved, despite the Joule dissipation, the flow will not be annihilated on a
time scale of $\tau$, and one may infer from Eqn.~(\ref{e:eq314z}) that $\frac{L_{\parallel}}{L_{min}}$ must
increase with time \cite{Davidson}. Therefore, it is expected that the flow will experience anisotropy,
with $ L_{\parallel}$ increasing as the flow evolves.

\subsection{Discussion}

The purpose of this Chapter is to review governing equations of conducting flow in a magnetic field and
the effect of Lorentz force on flow. Important equations are as follows: Maxwell equations
(Eqn.~(\ref{e:eq25}) $\sim$ Eqn.~(\ref{e:eq210})), Navier-Stokes equation (Eqn.~(\ref{e:eq211}),
Eqn.~(\ref{e:eq21a})),  Induction equation (Eqn.~(\ref{e:eq215})), Energy equation (Eqn.~(\ref{e:eq30z})),
Energy deposition (Eqn.~(\ref{e:eq36z}), Eqn.~(\ref{e:eq44z})), EOS equation (Eqn.~(\ref{e:eq47z})),
and Magnetic damping equation (Eqn.~(\ref{e:eq315z})). These equations are employed as governing equations
in Frontier MHD code using front tracking method \cite{Samulyak2004, Samulyak2007}, whose calculated results
are referenced for comparison of jet size and filament velocity with experimental measurement.

\setlength{\baselineskip}{12pt}

\newpage
\clearpage
\input{epsf}
\begin{landscape}
\begin{figure}
\centering
\leavevmode \epsfxsize=18 cm \epsfysize=10 cm
\epsfbox{jet_surface3.eps} \caption{Wave-shaped interface separating two different fluids traveling
at different average speeds.}
\label{fig:jet_surface}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{energy4}
\caption{Energy decay in a magnetic field. a.) Normalized energy decay. b.) Integration of
normalized energy with respect to time.}
\label{fig:energy_decay}
\end{figure}
\end{landscape}
\clearpage

