\begin{spacing}{0}

\chapter{Mathematical Derivation for Chapter~\ref{chap:mhd_math}}
\label{app:append_math}
\end{spacing}
%\baselineskip=12pt
\setlength{\baselineskip}{11.5pt}
\ \\
\ \\
\section{Derivation of Rayleigh's Instability at an Interface Separating Two Flows in a Magnetic Field}
\label{sec:app:append_math:instability}

\subsection{Kinematic boundary condition at interface}

We consider the $(x,y,z)$ coordinate system in Fig. \ref{fig:jet_surface}.
A particle of fluid that is at some time on the free surface will always remain on the free surface.
Then, since the equation of the free surface is $ y - (\xi + a) $= 0, it follows that

\begin{equation}
 \label{e:eq272}
\frac{D}{D t} (y - (\xi + a)) = 0 \ .
\end{equation}

Neglecting quadratically small terms, Eqn.~(\ref{e:eq272}) yields at the interface($ y = \pm a$):

\begin{equation}
 \label{e:eq273}
\frac{\partial \xi}{\partial t} + U_{i} \frac{\partial \xi}{\partial x} = \frac{\partial \phi_{i}}{\partial y}
\ .
\end{equation}

In the region ($ -a < y < a $), the velocity potential $ \phi _{i} $ must satisfy
$ \frac{\partial^{2} \phi_{1}}{\partial x^{2}} + \frac{\partial^{2} \phi_{1}}{\partial y^{2}} = 0 $,
$ |\nabla \phi_{1} | $=finite. In the region $ y > a $, $ y < -a $, the velocity potential must satisfy
$ \frac{\partial^{2} \phi_{2}}{\partial x^{2}} + \frac{\partial^{2} \phi_{2}}{\partial y^{2}} = 0 $,
$ |\nabla \phi_{2} | $=finite. In view of the shape of the interface,
the solutions should be trigonometric in x, then the y dependence will be exponential. In view of the
finite conditions of velocity potentials, the negative exponential should be rejected for $ \phi_{1} $
and the positive exponential should be rejected for $ \phi_{2} $. Therefore, the general solutions are

\begin{equation}
 \label{e:eq274}
 \begin{split}
&\phi_{1} (x,y,t) = A_{1} e^{(2 \pi / \lambda)y} e^{i (2 \pi / \lambda)(x-\textit{c}t)} \ , \\
&\phi_{2} (x,y,t) = A_{2} e^{-(2 \pi / \lambda)y} e^{i (2 \pi / \lambda)(x-\textit{c}t)}  \ .
\end{split}
\end{equation}

Imposing the kinematic conditions on these solutions, the coefficients are determined
at $ y = a$ and $ y = -a$ respectively as follow:

\begin{equation}
 \label{e:eq275}
 \begin{split}
&\phi_{1} (x,y,t) = - i \epsilon ( \textit{c} - U_{1})  e^{i (2 \pi / \lambda)(x-\textit{c}t)} \ , \\
&\phi_{2} (x,y,t) = i \epsilon (\textit{c} - U_{2})  e^{i (2 \pi / \lambda)(x-\textit{c}t)} \ ,
\end{split}
\end{equation}

where $ U_{1} = U_{1}(a)$, $ U_{2} = U_{2}(a) $, and

\begin{equation}
 \label{e:eq276}
 \begin{split}
&\phi_{1} (x,y,t) =  i \epsilon ( \textit{c} - U_{1})  e^{i (2 \pi / \lambda)(x-\textit{c}t)} \ , \\
&\phi_{2} (x,y,t) = -i \epsilon (\textit{c} - U_{2})  e^{i (2 \pi / \lambda)(x-\textit{c}t)} \ ,
\end{split}
\end{equation}

where $ U_{1} = U_{1}(-a)$ and $ U_{2} = U_{2}(-a) $.
Since the perturbed surface at $ y = a$ and $ y = -a $ are supposed to be symmetric,
half of the jet section for the surface stability is considered in the following work.

\subsection{Hydrodynamic stability in a magnetic field}

Substituting the perturbed expressions into the equations of motion, neglecting second order terms in the
perturbed quantities, and making use of the fact that U, P satisfy
the flow equations and the current density in Lorentz force term can be represented using Ohm's law,
one will have the linearized equations governing the motion of disturbance as follows:

\begin{equation}
 \label{e:eq277}
\begin{split}
&\frac{\partial \textit{v}'_{xi}}{\partial t} + U_{i} \frac{\partial \textit{v}'_{xi}}{\partial x} +
\textit{v}'_{xi} \frac{d U_{i}}{d y}    \\
&= - \frac{1}{\rho_{i}} \frac{\partial
p'_{i}}{\partial x} - \frac{\sigma_{i}}{\rho_{i}} \textrm{B}_{y}^{2} \textit{v}'_{xi} + \frac{\sigma_{i}}{\rho_{i}} \textrm{B}_{x} \textrm{B}_{y}  \textit{v}'_{yi}
\end{split}
\end{equation}

and

\begin{equation}
 \label{e:eq278}
 \begin{split}
&\frac{\partial \textit{v}'_{yi}}{\partial t} + U_{i} \frac{\partial \textit{v}'_{yi}}{\partial x}  \\
&= - \frac{1}{\rho_{i}} \frac{\partial p'_{i}}{\partial y} - \frac{\sigma_{i}}{\rho_{i}} \textrm{B}_{x}^{2} \textit{v}'_{yi} + \frac{\sigma_{i}}{\rho_{i}} \textrm{B}_{x} \textrm{B}_{y}  \textit{v}'_{xi} \ ,
\end{split}
\end{equation}

where $ p'_{i} = f_{i}(c, \lambda, y) e^{i (2 \pi/\lambda) ( x - \textit{c} t )}$.
The perturbed velocity $ \textit{v}'_{x}, \ \textit{v}'_{y} $ are given as follow:

\begin{equation}
 \label{e:eq279}
 \begin{split}
&\textit{v}'_{x} = \frac{\partial \phi_{1}}{\partial x} = i (\frac{2 \pi}{\lambda}) A_{1} e^{ (2 \pi /\lambda) y } e^{i(2 \pi/\lambda)(x-\textit{c}t)}  \\
&\textit{v}'_{y} = \frac{\partial \phi_{1}}{\partial y} =  (\frac{2 \pi}{\lambda}) A_{1} e^{ (2 \pi /\lambda) y } e^{i(2 \pi/\lambda)(x-\textit{c}t)} \ .
\end{split}
\end{equation}

Putting Eqn.~(\ref{e:eq279}) into Eqn.~(\ref{e:eq277}) and Eqn.~(\ref{e:eq278}), equate the hydrodynamic
pressures since it is isotropic, which leads to Rayleigh's stability equation for the flow in a magnetic field 
as follow:

\begin{equation}
 \label{e:eq280}
\sigma_{1} B_{x} B_{y} + i \sigma_{1} B_{x}^{2} = \sigma_{1} B_{y}^{2} i - \sigma_{1} B_{x} B_{y}
+ \rho_{1} (\frac{\lambda}{2 \pi} ) \frac{d^{2} U_{1}}{d y^{2}} \ ,
\end{equation}

where $ U_{1} = U_{1}(y)$.
In the same manner, the Rayleigh's stability equation for the upper flow in magnetic field is derived as follow:

\begin{equation}
 \label{e:eq281}
\sigma_{2} B_{x}^{2} + \sigma_{2} B_{x} B_{y} i = \sigma_{2} B_{y}^{2} - i \sigma_{2} B_{x} B_{y}
- \rho_{2} i (\frac{\lambda}{2 \pi}) \frac{d^2 U_{2}}{d y^{2}} \ ,
\end{equation}

where $ U_{2} = U_{2}(y)$.

\subsection{Dynamic boundary condition at interface}

The difference of the normal stress must be balanced by the normal stress induced by
surface tension at the interface, which is expresses as follow:

\begin{equation}
 \label{e:eq282}
(P_{1} + \frac{\partial P_{1}}{\partial y} \xi + \frac{\partial^{2} P_{1}}{\partial y^{2}} \xi + \ldots + p'_{1} ) - (P_{2} + \frac{\partial P_{2}}{\partial y} \xi + \frac{\partial^{2} P_{2}}{\partial y^{2}} \xi + \ldots + p'_{2} ) + \Gamma \frac{\partial^{2} \xi}{\partial x^{2}} = 0  \ ,
\end{equation}

where $ \Gamma $ is surface tension.
Considering the gravity force in the free surface waves, Eqn.~(\ref{e:eq282}) can be rewritten as follow:

\begin{equation}
 \label{e:eq283}
 \begin{split}
&(\rho_{2} - \rho_{1}) g \cos \theta + \rho_{1} (\textit{c}-U_{1})^{2}(\frac{2 \pi}{\lambda}) + \rho_{2}
(\textit{c}-U_{2})^{2} (\frac{2 \pi}{\lambda}) \\
&+ \rho_{1} (\textit{c}-U_{1})\frac{d U_{1}}{d y} - \rho_{2} (\textit{c}-U_{2})\frac{d U_{2}}{d y} + i B_{y}^{2} ( \sigma_{1} (\textit{c}-U_{1})+\sigma_{2}
(\textit{c}-U_{2})) \\
&+ B_{x} B_{y}(\sigma_{2}(\textit{c}-U_{2})-\sigma_{1}(\textit{c}-U_{1})) - \Gamma (\frac{2 \pi}{\lambda})^{2} = 0 \ ,
\end{split}
\end{equation}

where $ U_{1} = U_{1}(a), \ U_{2} = U_{2}(a) $.
Consider the case that $ U_{2} =0, \frac{d U_{2}}{d y} =0, \rho_{2} =0, \sigma_{2} = 0$. This would
correspond to the stationary fluid on the upper and the density and conductivity of the upper fluid are
very small compared with these of the lower fluid. The wave velocity is represented as follow:

\begin{equation}
 \label{e:eq284}
  \begin{split}
&\textit{c} = [ \ \ - \rho_{1} \frac{d U_{1}}{d y} + B_{x} B_{y} \sigma_{1} - i B_{y}^{2} \sigma_{1}
+ 2 (\frac{2\pi}{\lambda}) \rho_{1} U_{1}  \\
&\pm \sqrt{\rho_{1}^{2} (\frac{d U_{1}^{2}}{d y}) +
4(\frac{2\pi}{\lambda}) \cos \theta g \rho_{1}^{2} - 2 B_{x} B_{y} \rho_{1} \sigma_{1} \frac{d U_{1}}{d y}
+ 2 i B_{y}^{2} \rho_{1} \sigma_{1} \frac{d U_{1}}{d y}} \\
&\overline{+ B_{x}^{2} B_{y}^{2} \sigma_{1}^{2}
- 2 i B_{x} B_{y}^{3} \sigma_{1}^{2}
- B_{y}^{4} \sigma_{1}^{2} + 4 (\frac{2\pi}{\lambda})^{3} \rho_{1} \Gamma } \ \ \ ] \times \frac{1}{2(\frac{2\pi}{\lambda}) \rho_{1}} \ .
\end{split}
\end{equation}

\section{The Governing Equations of MHD Flow in Cylindrical Coordinates}
\label{sec:app:append_math:mhd_cylind}

The momentum equations in the $(r,\theta, z)$ coordinates can be written as follows:

\begin{equation}
 \label{e:eq219}
 \begin{split}
&-\rho (v_{r} \frac{\partial v_{r}}{\partial r} + \frac{v_{\theta}}{r} \frac{\partial v_{r}}{\partial \theta} + v_{z} \frac{\partial v_{r}}{\partial z} ) -\frac{\partial p_{t}}{\partial r} - \rho g \cos \theta + \eta (\frac{\partial^{2} v_{r}}{\partial r^{2}} + \frac{1}{r} \frac{\partial v_{r}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} \textit{v}_{r}}{\partial \theta^{2}} + \frac{\partial^{2} v_{r}}{\partial z^{2}})  \\
 &+ \frac{1}{\mu} ( B_{r} \frac{\partial B_{r}}{\partial r} + \frac{B_{\theta}}{r} \frac{\partial B_{r}}{\partial \theta} + B_{z} \frac{\partial B_{r}}{\partial z} ) = \rho \frac{\partial v_{r}}{\partial t} \ ,
 \end{split}
\end{equation}

\begin{equation}
 \label{e:eq220}
 \begin{split}
&-\rho (v_{r} \frac{\partial v_{\theta}}{\partial r} + \frac{v_{\theta}}{r} \frac{\partial v_{\theta}}{\partial \theta} + v_{z} \frac{\partial v_{\theta}}{\partial z} ) -\frac{1}{r}\frac{\partial p_{t}}{\partial \theta} + \rho g \sin \theta + \eta (\frac{\partial^{2} v_{\theta}}{\partial r^{2}} + \frac{1}{r} \frac{\partial v_{\theta}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} v_{\theta}}{\partial \theta^{2}} + \frac{\partial^{2} v_{\theta}}{\partial z^{2}})  \\
 &+ \frac{1}{\mu} ( B_{r} \frac{\partial B_{\theta}}{\partial r} + \frac{B_{\theta}}{r} \frac{\partial B_{\theta}}{\partial \theta} + B_{z} \frac{\partial B_{\theta}}{\partial z} ) = \rho \frac{\partial v_{\theta}}{\partial t} \ ,
 \end{split}
\end{equation}

and

\begin{equation}
 \label{e:eq221}
 \begin{split}
&-\rho (v_{r} \frac{\partial v_{z}}{\partial r} + \frac{v_{\theta}}{r} \frac{\partial v_{z}}{\partial \theta} + v_{z} \frac{\partial v_{z}}{\partial z} ) -\frac{\partial p_{t}}{\partial z} + \eta (\frac{\partial^{2} v_{z}}{\partial r^{2}} + \frac{1}{r} \frac{\partial v_{z}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} v_{z}}{\partial \theta^{2}} + \frac{\partial^{2} v_{z}}{\partial z^{2}}) \\
 &+ \frac{1}{\mu} ( B_{r} \frac{\partial B_{z}}{\partial r} + \frac{B_{\theta}}{r} \frac{\partial B_{z}}{\partial \theta} + B_{z} \frac{\partial B_{z}}{\partial z} ) = \rho \frac{\partial v_{z}}{\partial t} \ ,
 \end{split}
\end{equation}

where $\textit{p}_{t}$ = $ \textit{p} + \frac{\textbf{B}^{2}}{2 \mu } $.
The magnetic induction equation in the $(r, \theta, z)$ coordinates can be written as follows: 

\begin{equation}
 \label{e:eq222}
 \begin{split}
&\frac{1}{\mu \sigma} [ \frac{\partial^{2} B_{r}}{\partial r^{2}} + \frac{1}{r} \frac{\partial B_{r}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} B_{r}}{\partial \theta^{2}} + \frac{\partial^{2} B_{r} }{\partial z^{2}} ] + \frac{1}{r} B_{r} (r \frac{\partial v_{r}}{\partial r}) + \frac{1}{r} B_{\theta} \frac{\partial v_{r}}{\partial \theta} + B_{z} \frac{\partial v_{r}}{\partial z }  \\
&- \frac{1}{r} v_{r} (r \frac{\partial B_{r}}{\partial r} ) - \frac{1}{r} v_{\theta} \frac{\partial B_{r}}{\partial \theta} -v_{z} \frac{\partial B_{r}}{\partial z} = \frac{\partial B_{r}}{\partial t} \ ,
\end{split}
\end{equation}

\begin{equation}
 \label{e:eq223}
 \begin{split}
&\frac{1}{\mu \sigma} [ \frac{\partial^{2} B_{\theta}}{\partial r^{2}} + \frac{1}{r} \frac{\partial B_{\theta}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} B_{\theta}}{\partial \theta^{2}} + \frac{\partial^{2} B_{\theta} }{\partial z^{2}} ] + \frac{1}{r} B_{r} (r \frac{\partial v_{\theta}}{\partial r}) + \frac{1}{r} B_{\theta} \frac{\partial v_{\theta}}{\partial \theta} + B_{z} \frac{\partial v_{\theta}}{\partial z }  \\
&- \frac{1}{r} v_{r} (r \frac{\partial B_{\theta}}{\partial r} ) - \frac{1}{r} v_{\theta} \frac{\partial B_{\theta}}{\partial \theta} -v_{z} \frac{\partial B_{\theta}}{\partial z} = \frac{\partial B_{\theta}}{\partial t} \ ,
\end{split}
\end{equation}

and

\begin{equation}
 \label{e:eq224}
 \begin{split}
&\frac{1}{\mu \sigma} [ \frac{\partial^{2} B_{z}}{\partial r^{2}} + \frac{1}{r} \frac{\partial B_{z}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} B_{z}}{\partial \theta^{2}} + \frac{\partial^{2} B_{z} }{\partial z^{2}} ] + \frac{1}{r} B_{r} (r \frac{\partial v_{z}}{\partial r}) + \frac{1}{r} B_{\theta} \frac{\partial v_{z}}{\partial \theta} + B_{z} \frac{\partial v_{z}}{\partial z } \\
&- \frac{1}{r} v_{r} (r \frac{\partial B_{z}}{\partial r} ) - \frac{1}{r} v_{\theta} \frac{\partial B_{z}}{\partial \theta} -v_{z} \frac{\partial B_{z}}{\partial z} = \frac{\partial B_{z}}{\partial t} \ .
\end{split}
\end{equation}

The Amp\`{e}re's law can be written as

\begin{equation}
 \label{e:eq225}
 \begin{split}
&j_{r} = \frac{1}{\mu} (\frac{1}{r} \frac{\partial B_{z}}{\partial \theta} - \frac{\partial B_{\theta}}{\partial z}) \ , \\
&j_{\theta} = \frac{1}{\mu} (- \frac{\partial B_{z}}{\partial r} + \frac{\partial B_{r}}{\partial z}) \ , \\
&j_{z} = \frac{1}{\mu} (\frac{\partial B_{\theta}}{\partial r} -\frac{1}{r} \frac{\partial B_{r}}{\partial \theta}) \ .
\end{split}
\end{equation}

The equation of continuity and the solenoidal condition for the magnetic field are

\begin{equation}
 \label{e:eq226}
\frac{1}{r} \frac{\partial}{\partial r} (r v_{r}) + \frac{1}{r} \frac{\partial v_{\theta}}{\partial \theta} + \frac{\partial v_{z}}{\partial z} = 0 \
\end{equation}

and

\begin{equation}
 \label{e:eq227}
\frac{1}{r} \frac{\partial}{\partial r} (r B_{r}) + \frac{1}{r} \frac{\partial B_{\theta}}{\partial \theta} + \frac{\partial B_{z}}{\partial z} = 0 \ .
\end{equation}

