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\documentclass{article}
\usepackage{graphicx}
\usepackage{color}


\begin{document}

%\end{document}
%\pagestyle{empty}
%\setlength{\oddsidemargin}{.2in}
%\Large
%\begin{centering}
\title{\bf\rr\Large First order perturbative calculations for a conducting liquid jet in a solenoid\\
  \bb DRAFT 3}
%\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}

\newpage
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.in\hskip1in
\special{bmp:jet5.bmp y=3in }
\vskip.2in

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~
=~x~v~\kappa~B_y~ {dB_y \over dz}
\end{equation}

This force will distrot the cross section. Assuming that the liquid in incompresible, we must find the induced pressures and motions 
$\Delta \vec{r}$
within the cross section that are driven by this force, with the constraint that the diverence of these motions
is zero:
$$ Div (\Delta \vec{r})~=~0 $$

Defining 
$$ F_o~=~v~\kappa~B_y~ {dB_y \over dz}$$
so that the magnet force per unit volume:
$$ f_x({\rm magnetic})~=~ x~F_o$$

\subsubsection{Magnetic Forces}
\vskip2.in
\special{bmp:dist1.bmp y=2in }
\vskip.2in

The pressure on the surface of the jet will be independent of azymuth $\phi$:
$$p(r_o)_{\rm circular}=p_{\rm atm}~+~T/r_o ~,$$
 T being the surface tension.
If the initial cross section is circular, we can consider pressures within the cross section:
$$ p~=~p_o~-~r^2~{F_o \over 4}$$
where $p_o$ is set by the constraint the above surface pressure at $r=r_o$. This bulk pressure will induce radial pressure forces:
$$f_r({\rm pressure})~=~{dp \over dr}~=~-~{r~F_o \over 2}$$
so
$$f_x({\rm total})~=~F_o~\(x~-~{r~\cos(\phi) \over 2 }\)~=~ {x~F_o \over 2}$$

$$f_y({\rm total})~=~-~{F_o~r~\sin(\phi) \over 2 }~=~ -{y~F_o \over 2}$$
and the accelerations:
$${d^2x,y\over dz^2}~=~{f_{x,y} \over v^2~\rho}$$
will give dispacements:
$$\Delta x~=~ \int \int {f_{x,y} \over v^2~\rho}~dz^2 ~=~x~{\kappa \over 2~v~\rho}~\int \int~B_y~{dB_y\over dz}~dz^2 $$
$$\Delta y~=~ \int \int {f_{x,y} \over v^2~\rho}~dz^2 ~=~-y~{\kappa \over 2~v~\rho}~\int \int~B_y~{dB_y\over dz}~dz^2 $$

Such motions are quadrupole ( see fig ***) and, as required, give
$Div~(dx,dy)~=~0$. The resulting ellipticity $\epsilon = {\Delta x_o\over r_o}~=~-{\Delta y_o\over r_o}$:

{\rr
\begin{equation} \epsilon(z)~=~~{r_o~\kappa \over 2~v~ \rho}~\int \int B_y~ {dB_y \over dz}~dz^2  \end{equation}
} 

\subsubsection{Surface Tension Forces}
\vskip2.in
\special{bmp:dist2.bmp y=2in }
\vskip.2in


Once the cross section becomes somewhat elliptical (we consider only a small ellipticity), then the pressure at the surface is no longer independent of the asymuthal angle $\phi$, but is given by:

$$p(r=r_o)~=~ p_{\rm atm}~ +~ {T \over r_0}~(1~ -~ \epsilon)~\cos(2~\phi)$$

Consider, in addition to those given above for the circular case, pressures within the cross section:

$$p(x,y)_{\rm elliptical}~=~p_o~+~{T~\epsilon \over r_o}~\(\cos^2(2~\phi)~-~\sin^2(2~\phi)\)
~=~p_o~+~{T~\epsilon \over r_o^3}~( x^2~-~y^2)$$

which has the correct values at $r=r_o$.

Defining 
$$S_o~=~{2~T~\epsilon \over r_o^3},$$
the resulting bulk forces are:

$$f_x({\rm elliptical})~=~{dp\over dx}~=~x~S_o$$
$$f_y({\rm elliptical})~=~{dp\over dy}~=~ -y~S_o$$
which are, once again, quadrupole forces that will generate quadrupole motions with Div($\vec{dr}$)=0.

$$\Delta x,\Delta y~=~\int\int {(x,~-y)\over v~\rho}~S_o~dz^2$$

\subsubsection{Magnetic and Surface Tension Forces}

Adding these surface tension dispalcements to the forces derived for the circular case:

$$\Delta x,\Delta y~=~(x,~-y)~\int\int \( {\kappa \over 2~v~\rho}~B_y~{dB_y\over dz}~+~{2~T~\epsilon \over v~\rho~ r_o^3} \) 
~dz^2 $$

and the resulting ellipticity $\epsilon = {\Delta x_o\over r_o}~=~-{\Delta y_o\over r_o}$:

{\rr
\begin{equation} \epsilon(z)~=~\int\int \( {r_o~\kappa \over 2~v~\rho}~B_y~ {dB_y \over dz}~+~{2~T~\epsilon \over v~\rho~r_o^2}   \)
~dz^2  \end{equation}
} 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 Early Example}

In our earlier studies we had considered a jet entering from outside the field, with the following parameters:

\vskip.2in
\bt{|l|cc|}
\hline
$\sigma_r$ & 3&mm\\
$r_o$& 10 & mm\\
$v_o$ & 20 & m/sec\\
$\theta$&100&mrad\\
$\sigma_z$&0.6 &m\\
\hline
\et
\vskip.2in

Using the above fomulae we obtain the results plotted in fig***.

\input{jetold.fig}

It is seen that although the deflection of the jet is small (3mm) and the average decelleration is reasonable (0.5 m/sec, yet there are several unacceptable results:
\bi
\item The hydrostatic pressure falls to -1.5 atmosphers, and would require a high pressure environment to stop the jet breaking up.
\item There are shear accelerations of $+/-$ 11 m/sec. leading to a 3:1 ratio of velocity across the jet.
\item The calculated distortion with surface tension included is 340\% (without the surface tension it is 700\%), indicating that the calculation is beyond its valid region. But indicating that the jet will be badly disrupted.
\ei
Clearly these parameters are unacceptable.

\section{\rr Study 2 Example}

For Study 2, several parameters were changed from the above example. The jet radius was halved, the jet velocity increased, the magnetic field was kept flatter, and the nozzle introduced inside the magnetic field.

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{tgta.fig}
%\input{gen.fig}
Fig***

\vskip.2in
The geometry is shown in fig***, with 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 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\\
$\theta_{jet}$ & -100 & mrad\\
$\kappa$& $10^6$ &$\Omega$ 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

The following figures shown here use a horizontal scale with $z=0$ at the magnetic center.
Plots are shown for
\begin{enumerate}
\item[a)] The axial magnetic field $B_z$
\item[b)] The hydrostatic pressure on the jet axis with respect to the environment outside the jet ($p_{\rm axis}-p_{\rm gas}$) 
\item[c)] The average deceleration of the jet $\Delta_v$(Ave)
\item[d)] The maximum shear acceleration/deceleration of the upper/lower limits of the jet $\Delta_v$(shear)
\item[e)] The vertical displacement of the jet due to deflecting forces $y$
\item[f)] The total transverse distorting forces $F_{\rm distort}$
\item   The resulting elliptical distortion ($\delta$x/r=$-\delta$y/r), without and without surface tension

\end{enumerate}


\input{jet30.fig}
\vskip.3in
 We see that over the extent of the new jet (from - .3 to .3 m):

\bi
\item The maximum axial field deviations are $+/-$ 1.1 T = 5\%


\item The axial pressure difference has a minimum of - 0.25 atmospheres. Thus if the jet is operating in a gas (He or Argon) at a pressure greater than or equal to .25 atmosphere, then the negative  pressures will be avoided, and there will be no tendency to cavitate prior to the arrival of the beam.

\item The maximum average deceleration of the jet is very small compared to the average jet velocity: 0.06/30$\approx$0.2\%.

\item The maximum decelerations (from shear forces) are also small compared to the average jet velocity: 0.4/30$\approx$1.3\%.

\item The deflections of the jet are very small: 5 $\mu$m.

\item The jet distortions ($\Delta$ width / ave width)  are approximately 0.4\% without surface tension, and less than 0.2% with surface tension.
\ei

Beyond the target region (z=.3 to 1.5 m), the effects are larger, but still not sufficient to break up the jet. The maximum shear is about 5 m/sec, and the distortion 20 \%. But these numbers are probably meaningless, since the jet will have been disrupted by the beam.
These results are much better than in the earlier example and are considered to be acceptable.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\section{\rr Coil Design}

The coil dimensions are given in the following table and plotted in figure a.
\vskip.2in

\begin{tabular}{|c|ccccc|cc|}
\hline
 len1&gap &  dl  &  rad  &  dr   &  I/A & n I & n I l\\ 
  m  &    m  &   m   &   m  & m &  A/mm$^2$ & A & A m \\ 
\hline
Fe &&&&&&&\\
 0.980 & 0.980 &  0.108 &  0.000 &  0.313 &   0.00 &  0.00 &  0.00  \\
 1.088 & -.000 &  0.312 &  0.000 &  0.168 &   0.00 &  0.00 &  0.00  \\
\hline
Hollow &&&&&&&\\
 1.288 & -.112 &  0.749 &  0.178 &  0.054 & -24.37 &  0.98 &  1.26  \\
 1.288 & -.749 &  0.877 &  0.231 &  0.122 & -19.07 &  2.04 &  3.74  \\
 1.288 & -.877 &  1.073 &  0.353 &  0.137 & -14.87 &  2.18 &  5.78  \\
\hline
SC &&&&&&&\\
 0.747 & -1.614 &  1.781 &  0.636 &  0.642 & -23.39 & 26.77 & 160.95  \\
 2.628 & 0.100 &  0.729 &  0.686 &  0.325 & -25.48 &  6.04 & 32.23  \\
 3.457 & 0.100 &  0.999 &  0.776 &  0.212 & -29.73 &  6.29 & 34.86  \\
 4.556 & 0.100 &  1.550 &  0.776 &  0.107 & -38.26 &  6.36 & 33.15  \\
 6.206 & 0.100 &  1.859 &  0.776 &  0.066 & -49.39 &  6.02 & 30.59  \\
 8.000 & -.065 &  0.103 &  0.416 &  0.051 & -68.32 &  0.36 &  1.00  \\
 8.275 & 0.172 &  2.728 &  0.422 &  0.029 & -69.27 &  5.42 & 14.88  \\
11.053 & 0.050 &  1.749 &  0.422 &  0.023 & -75.62 &  3.00 &  8.18  \\
12.852 & 0.050 &  1.750 &  0.422 &  0.019 & -77.37 &  2.61 &  7.09  \\
14.652 & 0.050 &  1.749 &  0.422 &  0.017 & -78.78 &  2.30 &  6.22  \\
16.451 & 0.050 &  1.750 &  0.422 &  0.015 & -79.90 &  2.07 &  5.59  \\
18.251 & 0.050 &  2.366 &  0.422 &  0.013 & -80.85 &  2.53 &  6.80  \\
\hline
\end{tabular}
\vskip.2in

%\section{\rr Using Calculated Fields}
The axial fields are shown in figures b and c. The components are shown from: the use of iron (green), superconductors (blue), and the total (red). The Gaussian distribution used in the above calculations is also given in figfure c (black), and is seen to be a good match to the total field over thetarget region (-.6 m to 0).

\input{wegf.fig}
%\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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\sectio


\section{\rr Conclusion}

\bi
\item It is not acceptable to have the jet nozzle outside the magnetic field.
\item With the study 2 parameters, all disruptive effects are negligible up to the distance traveled by the jet since the last pulse.
\item Even beyond this location, the disruptions are not unreasonable, and would not, of themselves disrupt the jet.
\item It would probably be acceptable to shorten the high field region, if this were desired for cost reasons.
\ei




\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}