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\begin{document}

%\end{document}
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%\Large
%\begin{centering}
\title{\bf\rr\Large First order perturbative calculations for a conducting liquid jet in a solenoid\\
  \bb DRAFT 2}
%\vskip.3in
\author{J. Gallardo, S. Kahn, R. B. Palmer, P. Thieberger, R Weggel\\
  Brookhaven National Laboratory\\\\
 K. McDonald\\
Princeton}

\maketitle
\begin{abstract}
  A perturbative calculation is given of the behavior of a continuous
  jet of conducting fluid as it enters and leaves a solenoidal
  magnetic field. It is assumed that the changes in direction, jet
  cross section and velocity are small.
  
  If the jet enters the field along, or close to the axis, then the
  induced forces are compressive and retarding. The jet slows, suffers
  an increase in hydrostatic pressure, and increases in diameter;
  later, the jet re accelerates and shrinks. As the jet leaves the
  field, the hydrostatic pressure becomes negative and cavitation may
  occur.
  
  If the jet enters at an angle to the axis, there are, in addition,
  deflections and elliptical deformations of the jet.
  
  Formulae are given for these effects and numerical values given for
  the example of a solenoidal field with a Gaussian axial profile.
\end{abstract}

\tableofcontents
\vskip.5in

\section{\rr Introduction}
A mercury jet, injected at an angle respect to the axis of the solenoidal field, is the current baseline solution for the Feasibility Study II\cite{ref1}. The interaction of the liquid-metal jet with the strong $20$~T target solenoid has as result a number of forces on the jet which potentially may affect the viability of this target.\cite{ref2},\cite{ref3},\cite{ref4} We present here perturbative calculations which confirm the findings of previous authors.  
\section{\rr Formulae}

\subsection{Introduction}

The jet is assumed to have an initial radius $r$, and be traveling at
a velocity $v$. Changes in radius, shape, direction and velocity are
all assumed to be small. The angle between the jet and solenoid axes
is also assumed to be small.

No viscosity

$r<<L$ 

In the following formulae, the coordinate system is defined by the
jet; $z$ is along the direction of motion and $r$ is perpendicular to
$z$. 


If the jet is not directed along the solenoid axis, then we also define
 $y$ in a direction perpendicular to $z$ (the jet axis) and away
from the solenoid axis; and $x$ perpendicular to $y$ and $z$; and we define
a second coordinate system $x$~$y'$~ $z'$, where $z'$ is aligned along the magnet axis.
Assuming a small angle $\theta^2<<1$ then (See fig.~\ref{fig1}):


$$x'~\approx~x$$,
 $$y'~\approx~y_o~+~z~\theta$$,
 $$r'^2~\approx~r^2~+~2z~y_o~\theta$$,
 $$z'~\approx~z-y_o$$

%In this case, $B_x=0$.
\begin{figure}
\vskip1.5in
\hskip1in\special{bmp:jet0.bmp y=1.5in }
%\vskip.4in
%  \centering \includegraphics[width=3.5in,angle=-90]{jet_fig1.ps}
%\includegraphics[bb= 45 90 480 700,angle=-90]{jet_fig1.ps}
\label{fig1}
\caption{Schematic of the geometrical arrangement of solenoid and jet}
\end{figure}


\subsection{Induced azimuthal current}


The magnetic flux through a circle of radius r perpendicular to the
jet axis is
\begin{equation}
\Phi = \int_S dS\vec{n}\cdot \vec{B}\approx \pi r^2 B_z(x,y,z)
\end{equation}
As a liquid metal jet passes axially down such a field at a velocity
$v={dz / dt}$, a circumferential potential will be generated\cite{ref6}
\begin{equation}
V \equiv \oint\vec{E}\cdot \vec{dl}=-{d\Phi \over dt}= \pi r^2v {dB_z(x,y,z) \over dz} 
\end{equation}
%\marginpar{this equation is not correct!!! additional term proportional to $\dot{r}$. See Ref.~\cite{ref6}}
If the metal electrical conductivity $\sigma$ is low enough so that
the resulting current has a negligible effect on the field, then the
azimuthal current density $i_{\phi}$ will be
\begin{equation}
i_{\phi}\approx {V \over 2\pi r}\sigma
\end{equation}
\begin{equation}
{\rr i_{\phi}\approx {rv\kappa\over 2} {dB_z(0,0,z) \over dz}} 
\end{equation}

\subsection{Radial forces and Hydrostatic pressure}
\vskip2.3in
\special{bmp:jet2.bmp y=2in }
\vskip.4in


The induced radial force per unit volume $(dr~r~d\phi dz)$ is
\begin{equation}
f_r=B_z i_{\phi} \approx  {r \over 2 }~v \kappa ~B_z {dB_z \over dz} 
\end{equation}
If we assume the effects of the fields are small so that the jet
radius and liquid velocities do not vary by large fractions, and if we
ignore radial inertia, then the hydrostatic pressures in a jet of
outside radius $r_o$, at radius r, will be given by
\begin{equation}
p(r,z)=\int_{r_o}^r -{f_r}dr\approx \({r_o^2-r^2 \over 4}\)~v \kappa~ B_z {dB_z \over dz}
\end{equation}

\subsection{Axial force}

The above hydrostatic pressure is a function of $z$, and gradients in it will exert
axial pressures $f_p$ on the liquid that must be added to the magnetic
term $f_z({\rm hydrostatic})$.
\begin{equation}
f_p({\rm hydrostatic})= {dp(r,z) \over dz} \approx~-~ \({r_o^2-r^2 \over 4}\)~v \kappa~
 {d \over dz}\( B_z {dB_z \over dz}\)
\end{equation}

To this must be added the axial forces induced directly by the fields acting on the asymuthal currents:

\vskip2.3in
\special{bmp:jet1.bmp y=2in }
\vskip.4in

If the jet is aligned with the field axis ($\theta=0$, the radial field is given by
\begin{equation}
B_r(\theta=0)\approx -{r \over 2}{\partial B_z(0,0,z) \over \partial z} 
\end{equation} 
%\marginpar{ CHECK THIS EQUATION: $B_z$ is mainly $\theta B_z$}
The induced axial force per unit volume $(dr~ d\phi~ dz)$ is
$$
f_z(\theta=0)~=~f_p(r,z)~-~B_r i_{\phi}
$$

If the jet is at an able to the magnetic axis, then there is an additional shear force:

%\vskip2.3in
%\special{bmp:jet1.bmp y=2in }
%\vskip.4in

$$
f_z(\theta)~=~f_p(r,z)~-~B_y i_{\phi}~\sin(\phi)
$$

giving, in all:

\begin{equation}
f_z~ \approx~-~ \({r_o^2-r^2 \over 4}\)~v \kappa~
 {d \over dz}\( B_z {dB_z \over dz}\)
~+~{r^2 \over 4}v \kappa\({dB_z \over dz} \)^2 
~+~{rv\kappa\over 2}~B_y~{dB_z\over dz} \sin (\phi)
\end{equation} 
On the jet axis:
\begin{equation}
f_z~ \approx~-~ \({r_o^2 \over 4}\)~v \kappa~
 {d \over dz}\( B_z {dB_z \over dz}\)
\end{equation} 
On the outer surface, averaged over the azymuthal angle $\phi$, or in the absence of a $B_y$:
\begin{equation}
f_z~ \approx~
 {d \over dz}\( B_z {dB_z \over dz}\)
~+~{r^2 \over 4}v \kappa\({dB_z \over dz} \)^2 
\end{equation} 
On the outer surface, with a finite $B_y$, as in the case of a jet at an angle to the magnetic axis:
\begin{equation}
f_z~ \approx~
 {d \over dz}\( B_z {dB_z \over dz}\)
~+~{r_o^2 \over 4}v \kappa\({dB_z \over dz} \)^2 
~+~{yv\kappa\over 2}~B_y~{dB_z\over dz}
\end{equation} 

and the average force of the disk of radius $r_o$ is given by integrating the terms
\begin{equation}
{\rr <{f_z}> \approx{r_o^2 \over 8}v \kappa 
\(   \({dB_z \over dz} \)^2    ~+~   {d \over dz} \(B_z~{dB_z \over dz} \)\)}
\end{equation} 

 
\subsection{Axial accelerations}


These forces will then decelerate, or accelerate layers of the fluid,
thus inducing differences of liquid velocity as a function of radius
\begin{equation}
{dv \over dz}~=~{f \over \rho~v}
\end{equation}
\begin{equation}
\Delta~ v(r,z)~=~ \int_{z_o}^z{(f_z+f_p)~{1 \over v \rho}~ dz}
\end{equation}
The average change in velocity is then
\begin{equation}
{\rr <\Delta v>(z)~=~{\kappa \over \rho}~{r_o^2 \over 8}~   \(\int_{z_o}^z{\({dB_z \over dz} \)^2 
~+~{d \over dz}\( B_z {dB_z \over dz}\)~dz} \)}
\end{equation}
and the radius as a function of z is
\begin{equation}
r(z)~=~r_o~\(1~-~{<\Delta v>(z) \over v} \)
\end{equation}
%\marginpar{WRONG DIMENSIONS!!!}

\subsection{Transverse forces and deflections}
\vskip2.3in
\special{bmp:jet3.bmp y=2in }
\vskip.4in


>From above, the radial force per unit volume $(dr~r~d\phi dz)$ is
\begin{equation}
f_r=B_z i_{\phi} \approx  {r \over 2 }~v \kappa ~B_z {dB_z \over dz} 
\end{equation}
If $B_z$ varies with a transverse distance $y$, then the component of
this radial force in the $y$ direction is
\begin{equation}
f_y = f_r~\sin{\phi}
\end{equation}
and the net deflective force $dF_y$ per unit length $dz$ is
\begin{equation}
{dF_y \over dz}~ =~ \int_o^r ~\int_o^{2\pi} 
   {r \over 2 }~v \kappa ~{dB_z \over dy}~r \sin{\phi}^2 {dB_z \over dz} ~r~ dr d\phi 
\end{equation}
\begin{equation}
= ~{v \kappa \over 2} ~{dB_z \over dy}~ {dB_z \over dz} ~\int_o^{2\pi}{ \sin{\phi}^2 d \phi}
\int_o^r {r^3~dr }
 \end{equation}
\begin{equation}
= ~{\pi \over 8}v \kappa ~r^4 ~{dB_z \over dy}~ {dB_z \over dz} 
\end{equation}
and the change in transverse velocity
\begin{eqnarray}
{d v_y \over dz} & = & {1 \over v}{d v_y \over dt}\nonumber \cr\cr
&=& {{dF_y \over dz}dz\over v \rho \pi  r^2  dz}\nonumber \cr\cr
&=&{\kappa r^2 \over 8 \rho}  {dB_z \over dy} {dB_z \over dz}\cr 
\end{eqnarray}
and the inverse radius of bend is {\rr 
\begin{equation} {d^2y\over dz^2}~=~{d \theta
      \over dz}~=~{\kappa~r^2 \over 8~v~\rho} ~{dB_z \over dy}~ {dB_z
      \over dz}
\end{equation}}


\subsection{Induced axial current}
\vskip3.3in
\special{bmp:jet5.bmp y=3in }
\vskip.4in

Consider a transverse field component $B_y$

The magnetic flux between transverse positions $-x$ to $x$ and $dz$ is
\begin{equation}
d \Phi_y = 2x dz B_y(z)
\end{equation}
As a liquid metal jet passes axially down such a field at a velocity
$v={dz / dt}$, axial voltage gradients will be generated
\begin{equation}
G=x {dB_y \over dz} v
\end{equation}
If the metal electrical conductivity $\kappa$ is low enough so that
the resulting current has a negligible effect on the field, then the
axial current density $i_z$ will be
\begin{equation}
i_z= G\sigma
\end{equation}
\begin{equation}
{\rr i_z~=xv\kappa{dB_y(0,0,z) \over dz}}
\end{equation}


\subsection{Transverse elliptical distortion}
\vskip3.3in
\special{bmp:jet6.bmp y=3in }
\vskip.4in

If the jet is not on the solenoid axis, the axial induced currents
interacting with the transverse fields will generate distorting forces
on the jet. These transverse forces per unit volume $dx~ dy~ dz$ are
\begin{equation}
f_x~=~ i_z~B_y~
\end{equation}
\begin{equation}
=~x~v~\kappa~B_y~ {dB_y \over dz}
\end{equation}
Ignoring surface tension, there is no restoring force, so the shape
will change. The sides will go in and the height increase. A
conservative estimate of the rate of width reduction can be obtained
assuming that the only inertial resistance comes from the elements
themselves. Then

\begin{equation}
 {d^2x \over dz^2}~<~{ f_x \over v~\rho } =~~{x~\kappa \over \rho}~B_y~ {dB_y \over dz}\end{equation}

In reality the inertial constraint will be greater, because the inward
motion of elements on the sides must generate equal outward motions at
the top and bottom, and azymuthal motions between the two. Including
these effects we can guess {\rr
\begin{equation} {d^2y \over dz^2}~\approx~{1\over3}~{x~\kappa \over \rho}~B_y~ {dB_y \over dz}\end{equation}
} If this velocity is sufficiently small, then surface tension will
restrain it. The total inward force on each side is
\begin{eqnarray}
{dF_B \over dz}&\approx &{1\over3}~v~\kappa ~B_y~{dB_y \over dz}~~ \int_{-\pi}^\pi \int_o^r  r ~\cos{\phi} dr~ r~ d\phi \nonumber \cr 
%\end{equation}
%\begin{equation}
&\approx &{2\over 9}~r^3~v~\kappa B_y {dB_y \over dz}
\end{eqnarray}
and this is opposed by the surface tension difference for a finite
ellipticity $\epsilon=y/x$
\begin{equation}
{dF_T \over dz}~\approx~2~T~\epsilon
\end{equation}
where $T$ is the surface tension.

So the equilibrium ellipticity is {\rr \begin{equation}
    \epsilon_{equilib}~\approx~{1\over3}~{r^3\over 3~T}~\kappa~B_y~
    {dB_y \over dz}
\end{equation}}

Again, this is only an order of magnitude calculation since the factor
1/3 is only an estimate for the inertial effects in distorting the
ellipse. With the factor removed, however, the equation gives a harder
maximum value.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\rr Gaussian Case}

We can consider a field that varies as a Gaussian in $z$
The fields of the solenoid and coordinates are denoted with primes $({}^{\prime})$.

The Fields in the magnet system are:

\begin{eqnarray}
B_z'(r',z') &\approx &B_o e^{-{z'^2\over 2\sigma_z^2}}-{1\over 4}r'^2{\partial^2B_z'(0,z')\over \partial z'^2}\\
B'_r(r',z') &\approx &-{1\over 2}r' {\partial B_z'(0,z')\over \partial z'} 
\end{eqnarray}

In the coordinate system of the jet, see Fig.~\ref{fig1}, assuming a very small angle $\theta$ then $x'=x$, $y'=y_o+z\theta$, $r'^2=r^2+2zy_o\theta$ and $z'=z-y_o\theta$ and the fields are
\begin{eqnarray}
B_z(x,y,z) &\approx &\left[B_z'(r',z')-{1\over 2}\theta ~y_o~{\partial B_z'(0,z')\over \partial z'}\right]
%\Bigg|_{r'=r+{~y_o~\theta z\over r}}
%, s'=s-~y_o~\theta} 
\nonumber \\
B_x(x,y,z) &\approx &-{1\over 2}x{\partial B_z'(0,z')\over \partial z'}
%\Bigg|_{s'=s-~y_o~\theta} 
\nonumber \\
B_y(x,y,z) &\approx & -\left[{1\over 2} (~y_o~+z\theta){\partial B_z'(0,z')\over \partial z'}-\theta B_z'(r',z')\right]
%\Bigg|_{r'=r+{~y_o~\theta z\over r}, s'=s-~y_o~\theta}
\nonumber \\
{\partial B_z(x,y,z)\over \partial z}&\approx &\left[{\partial B_z'(0,z')\over \partial z'} -{1\over 2}~y_o~\theta {\partial^2 B_z'(0,z')\over \partial z'^2} 
%-{1\over 4} (r^2+2z~y_o~\theta){\partial^3B_z'(0,z') \over \partial z'^3}
\right]
%\Bigg|_{s'=s-~y_o~\theta} 
\nonumber \\
%{dB_z(x,y,z)\over dy}&\approx &-\left[\theta {\partial B_z'(0,z')\over \partial z'}+{1\over 2}(z\theta+~y_o~){\partial^2 $B_z'(0,z')\over \partial z'^2}
%%-{1\over 4}r^2\theta {\partial^3B_z'(0,z') \over \partial z'^3}   
%\right]
%\Bigg|_{s'=s-~y_o~\theta} 
%\nonumber  \\
{dB_y(x,y,z)\over dz}&\approx & \left[{1\over 2}\theta{\partial B_z'(0,z')\over \partial z'}~-~
{1\over 2}(~y_o~+z\theta){\partial^2 B_z'(0,z')\over \partial z'^2}\right]
%\Bigg|_{s'=s-~y_o~\theta} 
\end{eqnarray}
%\marginpar{ CHECK THIS EQUATION: I corrected it}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\rr Study 2 Example}

In the Study 2 target system, the beam with rms radius $\sigma_r$ intersects a mercury jet of radius $r_o$ at an angle $\theta_{crossing}$. The forward velocity of the jet is $v_o$. The intervals between pulses is $t$, and it will be assumed here that after a pulse, all the mercury outside of the nozzle is dispersed. The nozzle is at $z_{nozzel}$ with respect to the intersection of the beam and jet center lines. Consider the following parameters:

\vskip.2in
\bt{|l|cc|}
\hline
$\sigma_r$ & 1.5&mm\\
$r_o$& 5 & mm\\
$\theta_{crossing}$&33&mrad\\
$v_o$ & 30 & m/sec\\
$t$& 20& ms\\
$z_{nozzel}$ & -.375 & m\\
\hline
\et
\vskip.2in


\input{gen.fig}

\vskip.2in
The distribution of resulting interactions as a function of $z$ is shown above. At the time of a second, or subsequent bunch, the newly established jet will extend a distance $z_{jet}=v_o~t=0.6$ m from the nozzle.
It is seen that only 2.5 \% of the interactions would occur after this location, had the beam extended indefinitely.
Thus there is a negligible loss from this limited jet extent.

Thus the total length over which the jet must propagate without serious magnetic disruption is from the nozzle to a point 0.6 m downstream. In order to minimize the field non uniformity over this length, the magnetic center (approximate point of maximum $B_z$ is placed at the center of this length. i.e. the magnetic center is set at a distance 
$z_{magnet}=z_{jet}/2-z_{nozzel}=~-.15$ m with respect to the jet-beam intersection. the figures shown here use a horizontal scale with $z=0$ at the magnetic center.


\vskip3.3in
\special{bmp:jet8.bmp y=3in }
\vskip.4in

A sketch of the layout is shown above. The proton beam enters at an angle $\theta_{beam}$ with respect to the magnet axis. The jet is at an angle $\theta_{jet}=\theta_{beam}-\theta_{crossing}$. The vertical distance $y_o$ from the magnet center ($z=0, r=0$) to the jet axis at $z=0$ can be chosen to minimize beam disruption.
We assuming a Gaussian distribution of $B_z'$ vs $z'$, with a maximum value of $B_o$,
The jet conductivity $\kappa$, density $\rho$, and surface tension $T_{surface}$, and the other parameters are given below: 

\vskip.2in

\bt{|l|cc|}
\hline
$B_o$ & 20 & T \\
$\sigma_z'$ & .8 & m\\
$\theta_{jet}$ & -67 & mrad\\
$y_o$ (without surface tension)& 2 & mm\\
$y_o$ (with surface tension)& 1 & mm\\
$\kappa$& $10^6$ &$\Omega$ m\\
$\rho$ & $13.5~10^4$ & $kg/m^3$\\
$T_{\rm surface}$&$ .456 $&$ N/m $\\
$p_{\rm gas}=p_{\rm atmospheric}$&$10^5$&$N/m^2$\\
\hline
\et
\vskip.2in

Plots are shown for
\begin{enumerate}
\item[a)] The axial magnetic field $B_z$
\item[b)] The vertical magnetic field $B_y$
\item[c)] The rate of change of the transverse field $B_y$
\item[c)] The hydrostatic pressure on the jet axis with respect to the environment outside the jet ($p_{\rm axis}-p_{\rm gas}$) 
\item[d)] The average deceleration of the jet $\Delta_v$(Ave)
\item[e)] The maximum shear acceleration/deceleration of the upper/lower limits of the jet $\Delta_v$(shear)
\item[f)] The vertical displacement of the jet due to deflecting forces $y$
\item[g)] The total transverse distorting forces $F_{\rm distort}$
\item[h)] The resulting elliptical distortion ($\delta$x/r=$-\delta$y/r), without surface tension
\item   The resulting elliptical distortion ($\delta$x/r=$-\delta$y/r), without surface tension

\end{enumerate}


\input{jet2.fig}

 We see that over the extent of the new jet (from - .3 to .3 m):

\bi
\item The maximum axial field deviations are $+/-$ 1.25 T = 6\%

\item The transverse fields seen by the jet are approximately 1.4 T, and vary at the ends by only up to 
$+/-$ 0.01 T = $+/-$ 0.7 \%. But it is this small variation of transverse field that generate axial current in the jet, leading to jet shape distortions that are the most serious constraint on the magnet design.

\item The axial pressure difference has a minimum of - 0.5 atmospheres. Thus if the jet is operating in a gas (He or Argon) at a pressure of 1 atmosphere, then the minimum  pressure will be + 0.5 atmospheres, and there will be no tendency to cavitate prior to the arrival of the beam.

\item The maximum average deceleration of the jet is small compared to the average jet velocity: 0.11/30$\approx$0.3\%.

\item The maximum decelerations (from shear forces) are also small compared to the average jet velocity: 0.7/30$\approx$2.3\%.

\item The deflections of the jet are very small: 20 $\mu$m.

\item The distorting Forces, in the absence of surface tension, are of the same order of magnitude as the surface tension: .15 N/m compared with the surface tension of .5 N/m.

\item The resulting calculated jet distortions ($\Delta$ width / ave width)  with no jet axis displacement ($y_o=0$) are 18\% without surface tension, and 7\%, with surface tension. Both can be reduced by  small jet axis displacements: to 6\% without surface tension, and 4\% with surface tension.
\ei

These disruptions  are all relatively small, and should, with the fields assumed, cause no problems. But relatively small field changes can make the beam distortions uncomfortably large. It thus appears that considerations of these distortions will set the tolerances on the magnetic field shape. Tuning of these distortions can be achieved by vertical displacements of the jet, and provision for mechanically adjusting this should be considered. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage

\section{\rr Using Calculated Fields}


%\newpage
Plots of the magnetic fields and derivatives needed in this calculations including the relative tilded of $\theta=0.1$ between  jet and solenoid axis are shown in Figs.\ref{fig2} and \ref{fig3}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip.5in

*************
\vskip.5in

Compared with a jet velocity of 20 m/sec, these are all relatively
low. Earlier optimizations had called for a jet radius of 1 cm, which
gave values of $\Delta v$ 4 times higher, and thus more significant.
%\sectio


\section{\rr Conclusion}


************



\begin{thebibliography}{}
\bibitem{ref1}\begin{verbatim}http://www.cap.bnl.gov/mumu/studyii/\end{verbatim}
\bibitem{ref2}K.~McDonald (Ed), \textsl{An R\&D Program for Targetry and Capture at a Muon-Collider Source}, Proposal to the BNL AGS Division, p.23
\bibitem{ref3}R.~Weggel, \textsl{Behavior of conducting solid or liquid jet moving in magnetic field: 1) Paraxial; 2) Transverse; 3) Oblique}, BNL report BNL-65611/CAP-220-Muon-98R
\bibitem{ref4}\textbf{Phys. Today}, Feb 2000, p.29
\bibitem{ref5}R.~Weggel, \textsl{ 20~T Hybrid magnet for Study II}
\bibitem{ref6} K.~McDonald, \textsl{Damping and Radial Pinching Effects},
\begin{verbatim}
http://www.hep.princeton.edu/~mcdonald/mumu/target/radialpinch.ps\end{verbatim}  
\end{thebibliography}
\end{document}



====================================================================


\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip.5in
n{References}

%%%%%%%%%%%%%%%%%%%%%%
Elliptical distortion can result from a second order variation of the $B_z$ field.
\vskip.2in

The hydrostatic pressure will restrain the average inward radial forces, and the net transverse force results in a deflection. Any remaining difference in the  inward forces at $\phi=0$ and $\phi=\pi$ will result in inward motion of one and  equal outward motion of the other. The driving force for these asymmetric motions, per unit of volume, is 

$$\Delta f_r~= ~ {f_r(\phi=0)~-~f_r(\phi=\pi) \over 2}    $$

$$\Delta f_r~=~  {r^3 \over 4 }~v \kappa~{d_2 B_z \over dy^2} {dB_z \over dz} $$

Ignoring surface tension, there is no restoring force, so the shape will change at a rate

$$ {dy \over dz}~=~-{dx \over dz}~=~{\Delta f_r \over v~\rho } $$

$$ =~{r^3 \over 4~\rho }~ \kappa~{d_2 B_z \over dy^2} {dB_z \over dz} $$



\begin{eqnarray}
B_z'(r',z') &\approx &B_o e^{-{z'^2\over 2\sigma_z^2}}-{1\over 4}r'^2{\partial^2B_z'(0,z')\over \partial z'^2}\\
B'_r(r',z') &\approx &-{1\over 2}r' {\partial B_z'(0,z')\over \partial z'} 
\end{eqnarray}
In the coordinate system of the jet, see Fig.~\ref{fig1}, assuming a very small angle $\theta$ then $x'=x$, $y'=y+z\theta$, $r'^2=r^2+2zy\theta$ and $s'=s-y\theta$ and the fields are
\begin{eqnarray}
B_z(x,y,z) &\approx &B_o e^{-{z^2\over 2\sigma_z^2} (1-{2y\theta \over z})}\left[1-{r^2\over 4\sigma_z^2}(1+{z^2\over \sigma_z^2}) +{\theta y z\over 2\sigma_z^2} \{1+{1\over \sigma_z^2}(r^2-z^2)\}\right] \nonumber \\
B_x(x,y,z) &\approx &B_o e^{-{z^2\over 2\sigma_z^2}(1-{2y\theta \over z})} {x(z-y\theta)\over 2 \sigma_z^2}\nonumber \\
B_y(x,y,z) &\approx &B_o e^{-{z^2\over 2\sigma_z^2}}\left[ {yz\over 2 \sigma_z^2} -\theta \{1 -{r^2\over 4\sigma_z^2}(1-{z^2\over \sigma_z^2}) +{(y^2-z^2)\over 2 \sigma_z^2}\}\right]\nonumber \\
{dB_z\over dz}(0,0,z)&\approx &-B_o {z\over \sigma_z^2} e^{-{z^2\over 2\sigma_z^2}}\nonumber \\
{dB_z\over dy}(0,0,z)&\approx & B_o {\theta z \over \sigma_z^2}({7\over 2}-{2z^2\over \sigma_z^2})e^{-{z^2\over 2\sigma_z^2}}\nonumber \cr
{dB_y\over dz}(0,0,z)&\approx & B_o {\theta z \over \sigma_z^2}\left[ 2-{z^2\over 2 \sigma_z^2}\right]e^{-{z^2\over 2\sigma_z^2}}
\end{eqnarray}



For three possible liquids, we obtain

\begin{center}
\begin{tabular}{|l|cccc|}
\hline
material & conductivity & density & $\Delta v$ (o)& $\Delta v$ (tot)\\
      & ohm m & kg/m$^3$ & m/s & m/s \\
\hline
%gallium & 3.8 $10^6$ & 6~$10^3$ & 2.15 & 4.3 \\ 
mercury & $10^6$ & 13.5~$10^3$& .95 & 1.9\\
Pb(81) In(19) & 2.6~$10^6$ & 10.3~$10^3$ & .8 & 1.6 \\
\hline
\end{tabular}
\end{center}



\begin{figure}
%  \centering \includegraphics[width=4in,angle=90]{fields_vs_s.ps}
\label{fig2}
\caption{$B_z$ and $B_y$ vs z for a gaussian target magnet at $y=-4,-2,0,2,4$ cm }
\end{figure}
\begin{figure}
%  \centering \includegraphics[width=4in,angle=90]{deriv_vs_s.ps}
\label{fig3}
\caption{${dB_z\over dz}~,{dB_y\over dz}\textrm{and} {dB_z\over dy}$ vs z for a gaussian target magnet at $y=-4,-2,0,2,4$ cm}
\end{figure} 
Using a more realistic hibrid target magnet as suggested by 
B.~Weggel\cite{ref5} yields fields and relevant derivatives shown in Figs.\ref{fig4},\ref{fig7}
\begin{figure}
% \centering \includegraphics[width=4.5in,angle=0]{steve_fig1.eps}
%  \centering \includegraphics{steve_fig1.eps}
\label{fig4}
\caption{${dB_y\over dz}$ vs z for the hybrid 20~T target magnet}
\end{figure}
\begin{figure}
%  \centering \includegraphics[width=4.5in,angle=0]{steve_fig2.eps}
\label{fig5}
\caption{${dB_y\over dz}$ vs z for the hybrid 20~T target magnet}
\end{figure}
\begin{figure}
%  \centering \includegraphics[width=4.5in,angle=0]{steve_fig3.eps}
\label{fig6}
\caption{${dB_y\over dz}$ vs z for the hybrid 20~T target magnet}
\end{figure}
\begin{figure}
%  \centering \includegraphics[width=4.5in,angle=0]{steve_fig4.eps}
\label{fig7}
\caption{${dB_y\over dz}$ vs z for the hybrid 20~T target magnet}
\end{figure}
\begin{figure}
%  \centering \includegraphics[width=4.5in,angle=0]{steve_fig5.eps}
\label{fig8}
\caption{${dB_y\over dz}$ vs z for the hybrid 20~T target magnet}
\end{figure}
\begin{figure}
%  \centering \includegraphics[width=4.5in,angle=0]{steve_fig6.eps}
\label{fig9}
\caption{${dB_y\over dz}$ vs z for the hybrid 20~T target magnet}
\end{figure}
\begin{figure}
%  \centering \includegraphics[width=4.5in,angle=0]{steve_fig7.eps}
\label{fig10}
\caption{${dB_y\over dz}$ vs z for the hybrid 20~T target magnet}
\end{figure}
\begin{figure}
%  \centering \includegraphics[width=4.5in,angle=0]{steve_fig9.eps}
\label{fig11}
\caption{${dB_y\over dz}$ vs z for the hybrid 20~T target magnet}
\end{figure}