\def\rr{\color{red}}
\def\bw{\color{black}}
\def\gr{\color{green}}
\def\mm{\color{magenta}}
\def\bb{\color{blue}}
\def\bw{\color{black}}

\def\({ \left( }
\def\){ \right) }
\def\b{ \color{red} \begin{equation}}
\def\e{\end{equation} \color{black}}
%\def\b{\begin{equation}}
%\def\e{\end{equation}}
\def\={\ =\ }
\def\ap{\ \approx\ }
\def\+{\ +\ }
\def\-{\ -\ }
\def\lc{\ell_c}
\def\Ls{\cal L \rm}
\def\ls{\cal l \rm}
\def\mumu{$\mu^+\mu^-$}
\def\ee{$e^+e^-$}
\def\bt{\begin{tabular}}
\def\et{\end{tabular}}
\def\bi{\begin{itemize}}
\def\ei{\end{itemize}}

\documentclass{article}
\usepackage{graphicx}
\usepackage{color}
%\usepackage{graphicx}
\usepackage{pstcol}
%\usepackage{pstricks}
\begin{document}

%\end{document}
%\pagestyle{empty}
%\setlength{\oddsidemargin}{.2in}
%\Large
%\begin{centering}
%\vskip-2.5in
%\hskip 2.5in MUC-NOTE-COOL\_THEORY-233
%\vskip.05in
%\hskip 2.5in Muon Tech. Note MU-046 
%\vskip 1in
\title{\bf\rr\Large First order perturbative calculations for a conducting liquid jet in a solenoid}
%\vskip.3in
\author{J. Gallardo, S. Kahn, R. B. Palmer, P. Thieberger, R. Weggel\\
  Brookhaven National Laboratory\\
Upton, NY 11973\\
\\
 K. McDonald\\
Princeton University\\
Princeton, NJ 08544}

\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 elongates. 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 with 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{ref6} We present here perturbative calculations which confirm and extend the findings of previous authors.  

\section{\rr Analytic Treatment}

\subsection{Introduction}

The jet is assumed to have an initial radius $r_o$, and initial 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. Furthermore, we assume the jet to be very long $r_o\ll L,$ and with zero viscosity.

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


If the jet is not directed along the solenoid axis, then we also define
 $y$ in the plane of that axis and the jet and in a direction perpendicular to $z$ (the jet axis) and away
from the solenoid axis; and $x$ perpendicular to $y$ and $z.$ We also 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}):

\begin{eqnarray}
x'&=&x,\nonumber \\
y'&\approx&y_o+y+z\theta,\\
z'&\approx&z -(y+y_o)\theta \nonumber \\
r'^2&\approx&r^2+y_o^2+2yy_o+2z\theta(y+y_o)
\label{eq1}
\end{eqnarray}

%In this case, $B_x=0$.
\begin{figure}[htb!]
%\vskip1.5in
%\hskip1in\special{bmp:jet0.bmp y=1.5in }
%\vskip.4in
%  \centering \includegraphics[bb=105 520 328 730,width=3.5in]{jet_fig1.ps}
\centering \includegraphics*[bb=105 520 328 730]{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).
\label{eq2}
\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
\begin{equation}
V \equiv \oint\vec{E}\cdot \vec{dl}=-{d\Phi \over dt}= \pi r^2v \frac{\partial B_z(x,y,z)}{\partial z}. \label{eq3}
\end{equation}
%\marginpar{this equation is not correct!!! additional term proportional to $\dot{r}$. See Ref.~\cite{ref6}}
If the metal electrical conductivity $\kappa $ 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 \frac{V}{2\pi r}\kappa  \approx -\frac{rv\kappa}{2} \frac{\partial B_z(0,0,z)} {\partial z}.
\label{eq4}
\end{equation}

\subsection{Radial forces and hydrostatic pressure}
The induced radial force per unit volume $(rdrd\phi dz)$ is
\begin{equation}
f_r=B_z i_{\phi} \approx \frac{r}{2}v \kappa B_z \frac{\partial B_z}{\partial z}.
\label{eq6} 
\end{equation}

\begin{figure}[htb!]
\includegraphics*[bb= 120 480 435 670]{jet_fig2.ps}
\label{fig2}
\caption{Radial pressure force produced by the axial magnetic field and the induced azimuthal current.}
\end{figure}

If we assume that 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 \(\frac{r_o^2-r^2}{4}\)~v \kappa~ B_z \frac{dB_z}{dz}.
\label{eq7}
\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 $f_z.$
\begin{equation}
f_p({\rm hydrostatic})= {\partial p(r,z) \over \partial z} \approx~-~ \({r_o^2-r^2 \over 4}\)~v \kappa~
 \frac{\partial} {\partial z}\( B_z \frac{\partial B_z}{\partial z}\).
\label{eq8}
\end{equation}

%\newpage
To this must be added the axial forces induced directly by the fields acting on the asimuthal currents:
\begin{figure}[htb!]
%\vskip2.3in
%\special{bmp:jet1.bmp y=2in }
\includegraphics*[bb=120 470 425 660]{jet_fig3.ps}
\caption{ Retarding force in the axial direction}
\label{fig3}
\end{figure}
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}.
\label{eq9} 
\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
\begin{equation}
f_z(\theta=0)=f_p(r,z)-B_r i_{\phi}.
\label{eq10} 
\end{equation} 

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

\begin{figure}[htb!]

%\vskip2.3in
%\special{bmp:jet1.bmp y=2in }
%\vskip.4in
\includegraphics*[bb=90 450 440 675]{jet_fig4.ps}
\caption{Deflection in the tilded case}
\label{fig4}
\end{figure}

\begin{equation}
f_z(\theta)=f_p(r,z)-B_y i_{\phi}~\sin(\phi)\quad \textrm{with} \quad  B_y\approx \theta B_z(0,0,z),
\label{eq11} 
\end{equation} 

giving, in all:

\begin{equation}
f_z \approx-\({r_o^2-r^2 \over 4}\)v \kappa
 \frac{\partial }{\partial z}\( B_z \frac{\partial B_z}{\partial z}\)
+\frac{r^2}{4}v \kappa\(\frac{\partial B_z}{\partial z} \)^2 
+\frac{rv\kappa}{2}B_y\frac{\partial B_z}{\partial z}\sin{\phi}
\label{eq12}
\end{equation} 
On the jet axis:
\begin{equation}
f_z~ \approx~-~ \({r_o^2 \over 4}\)~v \kappa~
 \frac{\partial } {\partial z}\( B_z \frac{\partial B_z}{\partial z}\).
\label{eq13}
\end{equation} 
On the outer surface, averaged over the azimuthal angle $\phi$, or in the absence of $B_y$:
\begin{equation}
f_z~ \approx~
% {\partial  \over \partial z}\( B_z {\partial B_z \over \partial z}\)
%~+~
\frac{r_o^2}{4}v \kappa\(\frac{\partial B_z}{\partial z} \)^2. 
\end{equation}
% \marginpar{the first term seems out of place WRONG DIMENSIONS}
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~
% {\partial  \over \partial z}\( B_z {\partial B_z \over \partial z}\)
%~+~
\frac{r_o^2}{4}v \kappa\(\frac {\partial B_z}{\partial z} \)^2 
~+~\frac{y}{2}v\kappa~B_y~{\partial B_z\over \partial z}
\end{equation}
%\marginpar{the first term seems out of place WRONG DIMENSIONS} 
and the average force of the disk of radius $r_o$ is given by integrating the terms
\begin{equation}
<f_z> \approx{r_o^2 \over 8}v \kappa 
\left[ \left(\frac{\partial B_z}{\partial z} \right)^2+\frac{\partial }{\partial z} \left( B_z\frac{\partial B_z}{\partial z}\right) \right]  
\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}
where $\rho$ is the fluid density.
\begin{equation}
\Delta~ v(r,z)~=~ \int_{z_o}^z{(f_z+f_p)~{1 \over v \rho}~ dz}
\end{equation}
%\marginpar{f$_p$ do not belong there}
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 Eq.~\ref{eq6}, the radial force per unit volume $(dr~r~d\phi dz)$ is
 $f_r=B_z i_{\phi} \approx  {r \over 2 }~v \kappa ~B_z {\partial B_z \over \partial z};$  
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}
{\partial F_y \over \partial z}~ =~ \int_0^r ~\int_0^{2\pi} 
   {r \over 2 }~v \kappa ~{\partial B_z \over \partial y}~r \sin^2{\phi} {\partial B_z \over \partial z} ~r~ dr d\phi 
\end{equation}
\begin{equation}
= ~{v \kappa \over 2} ~{\partial B_z \over \partial y}~ {\partial B_z \over \partial z} ~\int_0^{2\pi}{ \sin{\phi}^2 d \phi}
\int_0^r {r^3~dr }
 \end{equation}
\begin{equation}
= ~{\pi \over 8}v \kappa ~r^4 ~{\partial B_z \over \partial y}~ {\partial B_z \over \partial z}. 
\end{equation}
The change in transverse velocity is 
\begin{eqnarray}
{\partial  v_y \over \partial z} & = & {1 \over v}{\partial  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}  {\partial B_z \over \partial y} {\partial B_z \over \partial z}. 
\end{eqnarray}
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} ~{\partial B_z \over \partial y}~ {\partial B_z
      \over \partial z}.
\end{equation}}

\subsection{Induced axial current}
%\vskip3.in\hskip1in
%\special{bmp:jet5.bmp y=3in }
%\vskip.2in
\begin{figure}[hbt!]
\includegraphics*[bb=90 350 380 675]{jet_fig5.ps}
\caption{Induced axial current due to gradients in the radial field}
\label{fig5}
\end{figure}

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 {\partial B_y \over \partial z} 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\kappa ={\rr i_z~=xv\kappa{\partial B_y(0,0,z) \over \partial z}}
\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~ {\partial B_y \over \partial z}
\end{equation}

This force will distort the cross section. Assuming that the liquid is 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 divergence of these motions
is zero:
\begin{equation}
 Div (\Delta \vec{r})~=~0
\end{equation}

Defining 
\begin{equation}
F_o~=~v~\kappa~B_y~ {\partial B_y \over \partial z},
\end{equation}
so that the magnet force per unit volume is
\begin{equation}
f_x({\rm magnetic})~=~ x~F_o.
\end{equation}

\subsubsection{Magnetic Forces}
%\vskip2.in
%%\special{bmp:dist1.bmp y=2in }
\vskip.2in
\begin{figure}[htb!]
\centering
\includegraphics[angle=-90,width=4in]{jet_fig6.ps}
\caption{Distortion produced by the magnetic forces}
\label{fig6}
\end{figure}
The pressure on the surface of the jet will be independent of azimuthal angle $\phi$
\begin{equation}p(r_o)_{\rm circular}=p_{\rm atm}~+~T/r_o ~,\end{equation}
 T being the surface tension.
If the initial cross section is circular, we can consider pressures within the cross section:
\begin{equation} 
p=p_o+r^2\frac{F_o}{4},
\end{equation}
where $p_o$ is set by the constraint on the above surface pressure at $r=r_o$. This bulk pressure will induce radial pressure forces:
\begin{equation}f_r({\rm pressure})={\partial p \over \partial r}=-{rF_o \over 2};\end{equation}
so
\begin{equation}
f_x({\rm total})=-{F_o~r~\cos(\phi) \over 2 }~=~ {x~F_o \over 2};\end{equation}

\begin{equation}f_y({\rm total})~=~-~{F_o~r~\sin(\phi) \over 2 }~=~ -{y~F_o \over 2}.\end{equation}
The accelerations
\begin{equation}{d^2x,y\over dz^2}~=~{f_{x,y} \over v^2~\rho}\end{equation}
will give dispacements:
\begin{equation}\Delta x~=~ \int \int {f_{x,y} \over v^2~\rho}~dz^2 ~=~x~{\kappa \over 2~v~\rho}~\int \int~B_y~{\partial B_y\over \partial z}~dz^2, \end{equation}
\begin{equation}\Delta y~=~ \int \int {f_{x,y} \over v^2~\rho}~dz^2 ~=~-y~{\kappa \over 2~v~\rho}~\int \int~B_y~{\partial B_y\over \partial z}~dz^2. \end{equation}

Such motions are quadrupole ( see Fig.~\ref{fig6}) and, as required, give
$Div~(\Delta x,\Delta y)=0$. The resulting ellipticity $\epsilon = {\Delta x_o\over r_o}=-{\Delta y_o\over r_o}$

\begin{equation} 
\epsilon(z)~=\frac{\kappa }{2v\rho}~\int \int B_y~ {\partial B_y \over \partial z}~dz^2.  
\end{equation}
 
\subsubsection{Surface tension forces}
%\vskip2.in
%\special{bmp:dist2.bmp y=2in }
%\vskip.2in
\begin{figure}[htb!]
\includegraphics[angle=-90,width=4in]{jet_fig6.ps}
\caption{Restoring surface tension forces}
\label{fig7}
\end{figure}


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 asimuthal angle $\phi$, but is given by:

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

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

%\begin{equation}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),\end{equation}
\begin{equation}
p(x,y)_{\rm elliptical}=p_o-\frac{T\epsilon}{r_o}\(\cos^2(\phi)-\sin^2(\phi)\)=p_o-\frac{T\epsilon}{r_o^3}( x^2-y^2),
\end{equation}
which has the correct values at $r=r_o$.

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

\begin{equation}f_x({\rm elliptical})~=~{\partial p\over \partial x}~=~x~S_o,\end{equation}
\begin{equation}f_y({\rm elliptical})~=~{\partial p\over \partial y}~=~ -y~S_o.\end{equation}
Once again, these are quadrupole forces that will generate quadrupole motions with Div($\Delta\vec{r}$)=0:

\begin{equation}(\Delta x,\Delta y)=\int\int {(x,~-y)\over v^2 \rho}S_odz^2,\end{equation}

\subsubsection{Magnetic and surface tension forces}

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

\begin{equation}
(\Delta x,\Delta y)=(x,-y)~\int\int \( {\kappa \over 2~v~\rho}~B_y~{\partial B_y\over \partial z}~+~{2~T~\epsilon \over v^2~\rho~ r_o^3} \) 
~dz^2, 
\end{equation}
and the resulting ellipticity $\epsilon = {\Delta x_o\over r_o}~=~-{\Delta y_o\over r_o}$:
{\rr
\begin{equation} 
\epsilon(z)~=~\int\int \( {r_o~\kappa \over 2~v~\rho}~B_y~ {\partial B_y \over \partial z}~+~{2~T~\epsilon \over v^2~\rho~r_o^2}   \)
~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'(0,z') &\approx &B_o e^{-{z'^2\over 2\sigma_z^2}}\\
B_z'(r',z') &\approx &B_z'(0,z')-{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 $\theta,$ 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  \\
{\partial B_y(x,y,z)\over \partial z}&\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:

\begin{table}[tbh!]
\centering
\caption{Jet parameters for our earlier studies.}
\label{tb1}
\begin{tabular}{|lc|}
\hline
%$\sigma_r$ (mm) & 3\\
$r_o$ (mm)& 5\\
$v_o$ (m/s) & 20\\
$\theta$ (mrad)&100\\
$\sigma_z$ (m)&0.6\\
\hline
\end{tabular}
\end{table}

Using the above fomulae we obtain the results plotted in Figs.~\ref{fig8a} and summarized in Tb.~\ref{tb10},

\begin{figure}[thb!]
\input{jetk1.fig}
%{\includegraphics*[bb=175 480 400 675,width=0.5\textwidth]{jet_fig8a.ps}
%\quad
%{\includegraphics*[bb=175 480 400 675,width=0.5\textwidth]{jet_fig8aa.ps}}
%\end{figure}
%\begin{figure}[bth!]
%{\includegraphics[bb=175 480 400 675,width=0.5\textwidth]{jet_fig8b.ps}}
%\quad
%{\includegraphics[bb=175 480 400 675,width=0.5\textwidth]{jet_fig8c.ps}}
\caption{Results for the case of a jet entering a magnetic field with parameters listed in Tb.~\ref{tb1}.}
\label{fig8a}
\end{figure}
%\begin{figure}
%{\includegraphics[bb=175 480 400 675,width=0.5\textwidth]{jet_fig8d.ps}}
%\quad
%{\includegraphics[bb=175 480 400 675,width=0.5\textwidth]{jet_fig8e.ps}}
%\caption{Distortion due to the magnetic field jetold.fig}
%\label{fig8b}
%\end{figure}
\begin{table}[hbt!]
\centering
\caption{Summary of maximum and minimum values of the quantities shown in Fig.~\ref{fig8a}  }
\label{tb10}
 \begin{tabular}{|ccccccc|}
 \hline
 $B_z$ & $\Delta p$ & $< \Delta v>$ & $\Delta v$ & $\Delta y$ & $\Delta x$/r&$\Delta x$/r\\
&&&&&No ST&\\
 T     &  Atm.   &      m/s    &          m/s           &   mm        &    \%  &\%\\
 \hline
 0.077 &-0.245 &-0.125 & -5.542 &-0.060 &0.0 &26\\
  20.0 &0.351 &0. &0 &0.045 & 423 &-21\\
 %0.125 &5.542 &0.060 & 423 &\\
 \hline
 \end{tabular}
\end{table}

It is seen that although the deflection of the jet is small (0.04~mm) and the average deceleration is reasonable (0.125~m/s), yet there are several unacceptable results:
\bi
\item The hydrostatic pressure falls to -0.24~atmosphers, and would require a high-pressure environment to stop the jet breaking up.
\item There are shear accelerations of $\pm 5.5$~m/s.
\item The calculated distortion with surface tension included is 21\% (without the surface tension it is 423\%), indicating that the calculation is beyond its valid region, but suggesting that the jet will be badly disrupted.
\ei
Clearly these parameters are unacceptable.

\section{Study~II Example with $v=30$~m/s }

For Study~II, several parameters were changed from the above example. The  jet  velocity increased, the magnetic field was kept more uniform, and the nozzle introduced inside the magnetic field.

The beam with rms radius $\sigma_r=3$~mm 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 $z_{nozzle}$ with respect to the intersection of the beam and jet center lines. Consider the following parameters:
\begin{table}[htb!]
\centering
\caption{Proton beam and jet parameters for Study~II example.}
\label{tb2}
\bt{|lc|}
\hline
$\sigma_r$ (mm) & 1.5\\
$r_o$ (mm)& 5\\
$\theta_{crossing}$ (mrad)&33\\
$v_o$ (m/s)& 30\\
$t$ (ms)& 20\\
$z_{nozzle}$ (m) & -0.375\\
\hline
\et
\end{table}

\begin{figure}[bht!]
\begin{center}
\input{tgta.fig}
%\input{gen.fig}
\caption{Geometry of the Hg jet, proton beam and target magnet; on top is the distribution of interactions as a function of $z.$}
\label{fig30}
\end{center}
\end{figure}
The geometry is shown in Fig.~\ref{fig30}, 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. \textsl{i.e.} the magnetic center is set at a distance 
$z_{magnet}=z_{jet}/2-z_{nozzel}=~-0.15$ 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 assume 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: 

\begin{table}
\centering
\caption{Hg jet parameters.}
\label{tb3}
\bt{|l|c|}
\hline
$B_o$ (T) & 20 \\
$\sigma_z'$ (m) & 0.8\\
$\theta_{jet}$ (mrad) & -100 \\
$\kappa$ ($\Omega$ m) & $10^6$ \\
$\rho$ ($kg/m^3$) & $13.5~10^4$  \\
$T_{\rm surface}$ ($ N/m $)&$ 0.456 $\\
$p_{\rm gas}=p_{\rm atmospheric}$ ($N/m^2$)&$10^5$\\
\hline
\et
\end{table}

Figures~\ref{fig20} 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[g)]   The resulting elliptical distortion $(\Delta x/r=-\Delta y/r),$ with and without surface tension.

\end{enumerate}

\begin{figure}[tphb!]
\psset{unit=.2in}
\hskip-.4in
\begin{pspicture}(13,11)
\thicklines
\color{black}
\put( 1.0000, 9.8000){\bf { \normalsize \rr~~{\Large~a)}~~~~}}
\put( 6.250, 12.0000){\bf { \large ~$\sigma_z$=~0.9~(m)~~$\theta$=-0.1~~~$v_o$=~30~(m/s)~~$r$=~5~(mm)}}
%\put( 6.2500,12.0000){\bf { \large $\sigma_z =~0.9$~(m)~~$\theta$=~-0.1~~~$v_o$=~30~(m/s)~~$r$=~5~(mm)}}
\put(-1.5 , 4.2  ){\rotatebox{ 90 }{\bf { \normalsize ~B$_z$~~~(T)}}}
%\put( 2.0733, 0.2000){\bf { \normalsize z~~~(m)~}}
\color{red}
\put( 7.0000, 8.0000){\bf { \normalsize max~20.0~~T~}}
\put( 7.0000, 7.4000){\bf { \normalsize min~18.8~~T~}}
\color{black}
\put( 1 , 1.5 ){\line(1,0){ 10.73333 }}
\put(  2.79,  1.50){\line(0,1){.25}}
\put(  2.79,  1.00){\makebox(0,0){\normalsize  0.0}}
\put(  5.77,  1.50){\line(0,1){.25}}
\put(  5.77,  1.00){\makebox(0,0){\normalsize  0.5}}
\put(  8.75,  1.50){\line(0,1){.25}}
\put(  8.75,  1.00){\makebox(0,0){\normalsize  1.0}}
\put( 11.73,  1.00){\makebox(0,0){\normalsize  1.5}}
\put( 1 , 1.5 ){\line(0,1){ 7 }}
\put(  1.00,  2.57){\line(1,0){.25}}
\put(  0.00,  2.57){\makebox(0,0){\normalsize     5}}
\put(  1.00,  4.35){\line(1,0){.25}}
\put(  0.00,  4.35){\makebox(0,0){\normalsize    10}}
\put(  1.00,  6.12){\line(1,0){.25}}
\put(  0.00,  6.12){\makebox(0,0){\normalsize    15}}
\put(  1.00,  7.90){\line(1,0){.25}}
\put(  0.00,  7.90){\makebox(0,0){\normalsize    20}}
\color{red}
\color{red}
\psset{linewidth= 1.2 pt,linecolor=red}
\psset{linestyle=solid}
\pscurve
(  1.008937  ,  7.48186  )
(  1.053435  ,  7.500217  )
(  1.097935  ,  7.518159  )
(  1.142433  ,  7.535678  )
(  1.186932  ,  7.552775  )
(  1.231431  ,  7.569446  )
(  1.27593  ,  7.585684  )
(  1.320429  ,  7.601492  )
(  1.364927  ,  7.616864  )
(  1.409426  ,  7.631792  )
(  1.453925  ,  7.64628  )
(  1.498424  ,  7.660323  )
(  1.542922  ,  7.673916  )
(  1.587422  ,  7.687055  )
(  1.63192  ,  7.699743  )
(  1.676419  ,  7.711977  )
(  1.720918  ,  7.723745  )
(  1.765417  ,  7.735055  )
(  1.809915  ,  7.745905  )
(  1.854414  ,  7.756281  )
(  1.898913  ,  7.766193  )
(  1.943412  ,  7.775636  )
(  1.987911  ,  7.784603  )
(  2.03241  ,  7.793098  )
(  2.076908  ,  7.801118  )
(  2.121407  ,  7.808659  )
(  2.165906  ,  7.81572  )
(  2.210405  ,  7.822301  )
(  2.254903  ,  7.828401  )
(  2.299402  ,  7.834016  )
(  2.343901  ,  7.839146  )
(  2.3884  ,  7.843793  )
(  2.432899  ,  7.847953  )
(  2.477397  ,  7.851624  )
(  2.521896  ,  7.854809  )
(  2.566395  ,  7.857507  )
(  2.610894  ,  7.859712  )
(  2.655393  ,  7.86143  )
(  2.699892  ,  7.862654  )
(  2.74439  ,  7.863389  )
(  2.788889  ,  7.863634  )
(  2.833388  ,  7.863389  )
(  2.877887  ,  7.862654  )
(  2.922385  ,  7.86143  )
(  2.966884  ,  7.859712  )
(  3.011383  ,  7.857507  )
(  3.055882  ,  7.854809  )
(  3.100381  ,  7.851624  )
(  3.144879  ,  7.847953  )
(  3.189378  ,  7.843793  )
(  3.233877  ,  7.839146  )
(  3.278376  ,  7.834016  )
(  3.322875  ,  7.828401  )
(  3.367373  ,  7.822301  )
(  3.411872  ,  7.81572  )
(  3.456371  ,  7.808659  )
(  3.50087  ,  7.801118  )
(  3.545368  ,  7.793098  )
(  3.589867  ,  7.784603  )
(  3.634366  ,  7.775636  )
(  3.678865  ,  7.766193  )
(  3.723364  ,  7.756281  )
(  3.767863  ,  7.745905  )
(  3.812361  ,  7.735055  )
(  3.856861  ,  7.723745  )
(  3.901359  ,  7.711977  )
(  3.945858  ,  7.699743  )
(  3.990356  ,  7.687055  )
(  4.034856  ,  7.673916  )
(  4.079354  ,  7.660323  )
(  4.123853  ,  7.64628  )
(  4.168352  ,  7.631792  )
(  4.212851  ,  7.616864  )
(  4.257349  ,  7.601492  )
(  4.301848  ,  7.585684  )
(  4.346347  ,  7.569446  )
(  4.390846  ,  7.552775  )
(  4.435345  ,  7.535678  )
(  4.479844  ,  7.518159  )
(  4.524343  ,  7.500217  )
(  4.568841  ,  7.48186  )
\psset{linewidth= 1.2 pt,linecolor=red}
\psset{linestyle=dashed,linewidth=1.6pt,dash=1pt 4pt}
\pscurve
(  4.61334  ,  7.46309  )
(  4.657838  ,  7.443913  )
(  4.702338  ,  7.424332  )
(  4.746837  ,  7.404348  )
(  4.791335  ,  7.383967  )
(  4.835834  ,  7.363193  )
(  4.880333  ,  7.342031  )
(  4.924831  ,  7.320482  )
(  4.96933  ,  7.298553  )
(  5.013829  ,  7.276252  )
(  5.058328  ,  7.253578  )
(  5.102827  ,  7.230532  )
(  5.147326  ,  7.207129  )
(  5.191824  ,  7.183362  )
(  5.236323  ,  7.159247  )
(  5.280821  ,  7.13478  )
(  5.325321  ,  7.109972  )
(  5.369819  ,  7.08482  )
(  5.414318  ,  7.059338  )
(  5.458817  ,  7.033522  )
(  5.503316  ,  7.007383  )
(  5.547814  ,  6.980925  )
(  5.592314  ,  6.954154  )
(  5.636812  ,  6.927071  )
(  5.681311  ,  6.899682  )
(  5.72581  ,  6.871995  )
(  5.770308  ,  6.844018  )
(  5.814807  ,  6.815745  )
(  5.859306  ,  6.787192  )
(  5.903804  ,  6.758362  )
(  5.948304  ,  6.729255  )
(  5.992802  ,  6.699886  )
(  6.037302  ,  6.67025  )
(  6.0818  ,  6.640358  )
(  6.126299  ,  6.610216  )
(  6.170798  ,  6.579827  )
(  6.215296  ,  6.549197  )
(  6.259796  ,  6.518333  )
(  6.304295  ,  6.487242  )
(  6.348793  ,  6.455924  )
(  6.393292  ,  6.424389  )
(  6.437791  ,  6.392641  )
(  6.482289  ,  6.360687  )
(  6.526788  ,  6.32853  )
(  6.571287  ,  6.296175  )
(  6.615787  ,  6.263632  )
(  6.660285  ,  6.230908  )
(  6.704784  ,  6.197999  )
(  6.749282  ,  6.16492  )
(  6.793781  ,  6.131674  )
(  6.83828  ,  6.098265  )
(  6.882779  ,  6.064699  )
(  6.927277  ,  6.030984  )
(  6.971776  ,  5.99712  )
(  7.016274  ,  5.963122  )
(  7.060774  ,  5.928988  )
(  7.105272  ,  5.894723  )
(  7.149772  ,  5.860337  )
(  7.19427  ,  5.825835  )
(  7.238769  ,  5.791222  )
(  7.283268  ,  5.756502  )
(  7.327767  ,  5.72168  )
(  7.372266  ,  5.686765  )
(  7.416764  ,  5.651762  )
(  7.461263  ,  5.616673  )
(  7.505762  ,  5.581507  )
(  7.550261  ,  5.546266  )
(  7.594759  ,  5.510957  )
(  7.639258  ,  5.475588  )
(  7.683757  ,  5.440162  )
(  7.728256  ,  5.404683  )
(  7.772754  ,  5.369162  )
(  7.817254  ,  5.333593  )
(  7.861752  ,  5.297994  )
(  7.906251  ,  5.262362  )
(  7.950749  ,  5.226706  )
(  7.995248  ,  5.191031  )
(  8.039747  ,  5.155339  )
(  8.084246  ,  5.119637  )
(  8.128745  ,  5.08393  )
(  8.173244  ,  5.048224  )
(  8.217743  ,  5.012521  )
(  8.262241  ,  4.976829  )
(  8.30674  ,  4.941151  )
(  8.351239  ,  4.905495  )
(  8.395739  ,  4.869859  )
(  8.440236  ,  4.834256  )
(  8.484735  ,  4.798685  )
(  8.529234  ,  4.763152  )
(  8.573733  ,  4.727663  )
(  8.618232  ,  4.692223  )
(  8.66273  ,  4.656832  )
(  8.707229  ,  4.621495  )
(  8.751728  ,  4.586222  )
(  8.796228  ,  4.551013  )
(  8.840724  ,  4.515874  )
(  8.885226  ,  4.480807  )
(  8.929722  ,  4.445821  )
(  8.974223  ,  4.410914  )
(  9.018719  ,  4.376092  )
(  9.06322  ,  4.34136  )
(  9.107717  ,  4.306723  )
(  9.152218  ,  4.272182  )
(  9.196713  ,  4.237743  )
(  9.241215  ,  4.203409  )
(  9.285716  ,  4.169184  )
(  9.330213  ,  4.135071  )
(  9.374714  ,  4.101074  )
(  9.419209  ,  4.067197  )
(  9.463711  ,  4.033442  )
(  9.508206  ,  3.999814  )
(  9.552708  ,  3.966316  )
(  9.597204  ,  3.93295  )
(  9.641706  ,  3.899722  )
(  9.686201  ,  3.866632  )
(  9.730702  ,  3.833683  )
(  9.775198  ,  3.800882  )
(  9.8197  ,  3.768228  )
(  9.864202  ,  3.735725  )
(  9.908697  ,  3.703377  )
(  9.953198  ,  3.671185  )
(  9.997694  ,  3.639153  )
(  10.0422  ,  3.607284  )
(  10.08669  ,  3.575578  )
(  10.13119  ,  3.54404  )
(  10.17569  ,  3.512673  )
(  10.22019  ,  3.481477  )
(  10.26469  ,  3.450455  )
(  10.30919  ,  3.419611  )
(  10.35368  ,  3.388945  )
(  10.39818  ,  3.358461  )
(  10.44269  ,  3.328159  )
(  10.48718  ,  3.298044  )
(  10.53168  ,  3.268115  )
(  10.57618  ,  3.238374  )
(  10.62068  ,  3.208826  )
(  10.66518  ,  3.179469  )
(  10.70968  ,  3.150307  )
(  10.75417  ,  3.121341  )
(  10.79868  ,  3.092572  )
(  10.84317  ,  3.064003  )
(  10.88767  ,  3.035634  )
(  10.93217  ,  3.007467  )
(  10.97667  ,  2.979503  )
(  11.02117  ,  2.951744  )
(  11.06567  ,  2.924191  )
(  11.11016  ,  2.896845  )
(  11.15466  ,  2.869707  )
(  11.19916  ,  2.842777  )
(  11.24366  ,  2.816058  )
(  11.28816  ,  2.789551  )
(  11.33266  ,  2.763255  )
(  11.37716  ,  2.737172  )
(  11.42166  ,  2.711302  )
(  11.46616  ,  2.685647  )
(  11.51065  ,  2.660206  )
(  11.55515  ,  2.634981  )
(  11.59965  ,  2.609973  )
(  11.64415  ,  2.585181  )
(  11.68865  ,  2.560606  )
\color{green}
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  1.894444  ,  1.85  )
(  3.683334  ,  2.06  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  3.683334  ,  2.06  )
(  5.472222  ,  1.85  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  5.472222  ,  1.85  )
(  3.683334  ,  1.64  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  3.683334  ,  1.64  )
(  1.894444  ,  1.85  )
\color{magenta}
\psset{linewidth= 1.2 pt,linecolor=magenta}
\psset{linestyle=solid}
\psline
(  1.894444  ,  1.5  )
(  1.894444  ,  8.5  )
\psset{linewidth= 1.2 pt,linecolor=magenta}
\psset{linestyle=solid}
\psline
(  4.577778  ,  1.5  )
(  4.577778  ,  8.5  )
\end{pspicture}
\hskip.5in\begin{pspicture}(13,11)
\thicklines
\bw
\color{black}
\put( 1.0000, 9.8000){\bf { \normalsize \rr~~{\Large~b)}~~~~}}
\put(-1.5 , 2.2  ){\rotatebox{ 90 }{\bf { \normalsize ~$\Delta$~Pressure~(atm)}}}
%\put( 2.0733, 0.2000){\bf { \normalsize z~~~(m)~}}
\color{red}
\put( 7.0000, 8.0000){\bf { \normalsize max~~0.245~~Atm.~}}
\put( 7.0000, 7.4000){\bf { \normalsize min~-0.240~~Atm.~}}
\color{black}
\put( 1 , 1.5 ){\line(1,0){ 10.73333 }}
\put(  2.79,  1.50){\line(0,1){.25}}
\put(  2.79,  1.00){\makebox(0,0){\normalsize  0.0}}
\put(  5.77,  1.50){\line(0,1){.25}}
\put(  5.77,  1.00){\makebox(0,0){\normalsize  0.5}}
\put(  8.75,  1.50){\line(0,1){.25}}
\put(  8.75,  1.00){\makebox(0,0){\normalsize  1.0}}
\put( 11.73,  1.00){\makebox(0,0){\normalsize  1.5}}
\put( 1 , 1.5 ){\line(0,1){ 7 }}
\put(  1.00,  3.47){\line(1,0){.25}}
\put(  0.00,  3.47){\makebox(0,0){\normalsize -0.25}}
\put(  1.00,  5.69){\line(1,0){.25}}
\put(  0.00,  5.69){\makebox(0,0){\normalsize  0.00}}
\put(  1.00,  7.91){\line(1,0){.25}}
\put(  0.00,  7.91){\makebox(0,0){\normalsize  0.25}}
\color{red}
\color{red}
\psset{linewidth= 1.2 pt,linecolor=red}
\psset{linestyle=solid}
\pscurve
(  1.008937  ,  7.863636  )
(  1.053435  ,  7.820922  )
(  1.097935  ,  7.777381  )
(  1.142433  ,  7.733028  )
(  1.186932  ,  7.687877  )
(  1.231431  ,  7.641943  )
(  1.27593  ,  7.595242  )
(  1.320429  ,  7.547789  )
(  1.364927  ,  7.499599  )
(  1.409426  ,  7.45069  )
(  1.453925  ,  7.401081  )
(  1.498424  ,  7.350787  )
(  1.542922  ,  7.299828  )
(  1.587422  ,  7.24822  )
(  1.63192  ,  7.195986  )
(  1.676419  ,  7.143143  )
(  1.720918  ,  7.089712  )
(  1.765417  ,  7.035712  )
(  1.809915  ,  6.981164  )
(  1.854414  ,  6.92609  )
(  1.898913  ,  6.870512  )
(  1.943412  ,  6.814448  )
(  1.987911  ,  6.757925  )
(  2.03241  ,  6.700962  )
(  2.076908  ,  6.643582  )
(  2.121407  ,  6.58581  )
(  2.165906  ,  6.527668  )
(  2.210405  ,  6.469179  )
(  2.254903  ,  6.410367  )
(  2.299402  ,  6.351256  )
(  2.343901  ,  6.291871  )
(  2.3884  ,  6.232235  )
(  2.432899  ,  6.172374  )
(  2.477397  ,  6.112312  )
(  2.521896  ,  6.052073  )
(  2.566395  ,  5.991683  )
(  2.610894  ,  5.931167  )
(  2.655393  ,  5.87055  )
(  2.699892  ,  5.809857  )
(  2.74439  ,  5.749113  )
(  2.788889  ,  5.688344  )
(  2.833388  ,  5.627575  )
(  2.877887  ,  5.566831  )
(  2.922385  ,  5.506138  )
(  2.966884  ,  5.445521  )
(  3.011383  ,  5.385005  )
(  3.055882  ,  5.324615  )
(  3.100381  ,  5.264376  )
(  3.144879  ,  5.204314  )
(  3.189378  ,  5.144453  )
(  3.233877  ,  5.084817  )
(  3.278376  ,  5.025432  )
(  3.322875  ,  4.966321  )
(  3.367373  ,  4.907509  )
(  3.411872  ,  4.84902  )
(  3.456371  ,  4.790878  )
(  3.50087  ,  4.733106  )
(  3.545368  ,  4.675726  )
(  3.589867  ,  4.618763  )
(  3.634366  ,  4.56224  )
(  3.678865  ,  4.506177  )
(  3.723364  ,  4.450598  )
(  3.767863  ,  4.395524  )
(  3.812361  ,  4.340976  )
(  3.856861  ,  4.286977  )
(  3.901359  ,  4.233545  )
(  3.945858  ,  4.180702  )
(  3.990356  ,  4.128468  )
(  4.034856  ,  4.07686  )
(  4.079354  ,  4.025902  )
(  4.123853  ,  3.975607  )
(  4.168352  ,  3.925998  )
(  4.212851  ,  3.87709  )
(  4.257349  ,  3.8289  )
(  4.301848  ,  3.781446  )
(  4.346347  ,  3.734745  )
(  4.390846  ,  3.688811  )
(  4.435345  ,  3.64366  )
(  4.479844  ,  3.599307  )
(  4.524343  ,  3.555766  )
(  4.568841  ,  3.513052  )
\psset{linewidth= 1.2 pt,linecolor=red}
\psset{linestyle=dashed,linewidth=1.6pt,dash=1pt 4pt}
\pscurve
(  4.61334  ,  3.471176  )
(  4.657838  ,  3.430152  )
(  4.702338  ,  3.389992  )
(  4.746837  ,  3.350708  )
(  4.791335  ,  3.312311  )
(  4.835834  ,  3.274811  )
(  4.880333  ,  3.238217  )
(  4.924831  ,  3.202541  )
(  4.96933  ,  3.167789  )
(  5.013829  ,  3.133971  )
(  5.058328  ,  3.101094  )
(  5.102827  ,  3.069165  )
(  5.147326  ,  3.038192  )
(  5.191824  ,  3.008178  )
(  5.236323  ,  2.979131  )
(  5.280821  ,  2.951053  )
(  5.325321  ,  2.92395  )
(  5.369819  ,  2.897827  )
(  5.414318  ,  2.872686  )
(  5.458817  ,  2.848527  )
(  5.503316  ,  2.825356  )
(  5.547814  ,  2.803171  )
(  5.592314  ,  2.781975  )
(  5.636812  ,  2.761766  )
(  5.681311  ,  2.742547  )
(  5.72581  ,  2.724314  )
(  5.770308  ,  2.707068  )
(  5.814807  ,  2.690805  )
(  5.859306  ,  2.675524  )
(  5.903804  ,  2.661222  )
(  5.948304  ,  2.647895  )
(  5.992802  ,  2.635539  )
(  6.037302  ,  2.62415  )
(  6.0818  ,  2.613721  )
(  6.126299  ,  2.604249  )
(  6.170798  ,  2.595729  )
(  6.215296  ,  2.588152  )
(  6.259796  ,  2.581512  )
(  6.304295  ,  2.575802  )
(  6.348793  ,  2.571014  )
(  6.393292  ,  2.567139  )
(  6.437791  ,  2.564171  )
(  6.482289  ,  2.562099  )
(  6.526788  ,  2.560913  )
(  6.571287  ,  2.560606  )
(  6.615787  ,  2.561166  )
(  6.660285  ,  2.562582  )
(  6.704784  ,  2.564845  )
(  6.749282  ,  2.567942  )
(  6.793781  ,  2.571863  )
(  6.83828  ,  2.576595  )
(  6.882779  ,  2.582126  )
(  6.927277  ,  2.588445  )
(  6.971776  ,  2.595539  )
(  7.016274  ,  2.603393  )
(  7.060774  ,  2.611996  )
(  7.105272  ,  2.621335  )
(  7.149772  ,  2.631394  )
(  7.19427  ,  2.642159  )
(  7.238769  ,  2.65362  )
(  7.283268  ,  2.665759  )
(  7.327767  ,  2.678562  )
(  7.372266  ,  2.692016  )
(  7.416764  ,  2.706104  )
(  7.461263  ,  2.720814  )
(  7.505762  ,  2.736128  )
(  7.550261  ,  2.752033  )
(  7.594759  ,  2.768512  )
(  7.639258  ,  2.785552  )
(  7.683757  ,  2.803135  )
(  7.728256  ,  2.821247  )
(  7.772754  ,  2.839872  )
(  7.817254  ,  2.858995  )
(  7.861752  ,  2.878601  )
(  7.906251  ,  2.898672  )
(  7.950749  ,  2.919193  )
(  7.995248  ,  2.940151  )
(  8.039747  ,  2.961526  )
(  8.084246  ,  2.983307  )
(  8.128745  ,  3.005475  )
(  8.173244  ,  3.028017  )
(  8.217743  ,  3.050914  )
(  8.262241  ,  3.074154  )
(  8.30674  ,  3.097721  )
(  8.351239  ,  3.121598  )
(  8.395739  ,  3.145771  )
(  8.440236  ,  3.170224  )
(  8.484735  ,  3.194943  )
(  8.529234  ,  3.219914  )
(  8.573733  ,  3.245121  )
(  8.618232  ,  3.27055  )
(  8.66273  ,  3.296184  )
(  8.707229  ,  3.322012  )
(  8.751728  ,  3.348019  )
(  8.796228  ,  3.37419  )
(  8.840724  ,  3.400511  )
(  8.885226  ,  3.426969  )
(  8.929722  ,  3.453551  )
(  8.974223  ,  3.480243  )
(  9.018719  ,  3.507033  )
(  9.06322  ,  3.533907  )
(  9.107717  ,  3.560852  )
(  9.152218  ,  3.587856  )
(  9.196713  ,  3.614906  )
(  9.241215  ,  3.641991  )
(  9.285716  ,  3.669099  )
(  9.330213  ,  3.696218  )
(  9.374714  ,  3.723336  )
(  9.419209  ,  3.750444  )
(  9.463711  ,  3.777529  )
(  9.508206  ,  3.80458  )
(  9.552708  ,  3.831588  )
(  9.597204  ,  3.858542  )
(  9.641706  ,  3.885432  )
(  9.686201  ,  3.91225  )
(  9.730702  ,  3.938983  )
(  9.775198  ,  3.965624  )
(  9.8197  ,  3.992164  )
(  9.864202  ,  4.018593  )
(  9.908697  ,  4.044905  )
(  9.953198  ,  4.07109  )
(  9.997694  ,  4.09714  )
(  10.0422  ,  4.123047  )
(  10.08669  ,  4.148805  )
(  10.13119  ,  4.174405  )
(  10.17569  ,  4.19984  )
(  10.22019  ,  4.225105  )
(  10.26469  ,  4.250192  )
(  10.30919  ,  4.275095  )
(  10.35368  ,  4.299809  )
(  10.39818  ,  4.324328  )
(  10.44269  ,  4.348645  )
(  10.48718  ,  4.372756  )
(  10.53168  ,  4.396656  )
(  10.57618  ,  4.420339  )
(  10.62068  ,  4.443802  )
(  10.66518  ,  4.46704  )
(  10.70968  ,  4.490048  )
(  10.75417  ,  4.512824  )
(  10.79868  ,  4.535362  )
(  10.84317  ,  4.557661  )
(  10.88767  ,  4.579716  )
(  10.93217  ,  4.601523  )
(  10.97667  ,  4.623083  )
(  11.02117  ,  4.644392  )
(  11.06567  ,  4.665445  )
(  11.11016  ,  4.686241  )
(  11.15466  ,  4.706779  )
(  11.19916  ,  4.727058  )
(  11.24366  ,  4.747075  )
(  11.28816  ,  4.766828  )
(  11.33266  ,  4.786316  )
(  11.37716  ,  4.805539  )
(  11.42166  ,  4.824496  )
(  11.46616  ,  4.843185  )
(  11.51065  ,  4.861607  )
(  11.55515  ,  4.87976  )
(  11.59965  ,  4.897645  )
(  11.64415  ,  4.915262  )
(  11.68865  ,  4.93261  )
\color{green}
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  1.894444  ,  1.85  )
(  3.683334  ,  2.06  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  3.683334  ,  2.06  )
(  5.472222  ,  1.85  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  5.472222  ,  1.85  )
(  3.683334  ,  1.64  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  3.683334  ,  1.64  )
(  1.894444  ,  1.85  )
\color{magenta}
\psset{linewidth= 1.2 pt,linecolor=magenta}
\psset{linestyle=solid}
\psline
(  1.894444  ,  1.5  )
(  1.894444  ,  8.5  )
\psset{linewidth= 1.2 pt,linecolor=magenta}
\psset{linestyle=solid}
\psline
(  4.577778  ,  1.5  )
(  4.577778  ,  8.5  )
\color{black}
\psset{linewidth= 1.2 pt,linecolor=black}
\psset{linestyle=solid}
\psline
(  1  ,  5.688344  )
(  11.73333  ,  5.688344  )
\end{pspicture}

\hskip-.4in\begin{pspicture}(13,11)
\thicklines
\bw
\put( 1.0000, 9.8000){\bf { \normalsize \rr~~{\Large~c)}~~~~}}
\put(-1.5 , 2.2  ){\rotatebox{ 90 }{\bf { \normalsize ~Ave~$\Delta$v~~(m/s)}}}
%\put( 2.0733, 0.2000){\bf { \normalsize z~~~(m)~}}
\color{red}
\put( 7.0000, 8.0000){\bf { \normalsize max~~0.058~~m/s~}}
\color{black}
\put( 1 , 1.5 ){\line(1,0){ 10.73333 }}
\put(  2.79,  1.50){\line(0,1){.25}}
\put(  2.79,  1.00){\makebox(0,0){\normalsize  0.0}}
\put(  5.77,  1.50){\line(0,1){.25}}
\put(  5.77,  1.00){\makebox(0,0){\normalsize  0.5}}
\put(  8.75,  1.50){\line(0,1){.25}}
\put(  8.75,  1.00){\makebox(0,0){\normalsize  1.0}}
\put( 11.73,  1.00){\makebox(0,0){\normalsize  1.5}}
\put( 1 , 1.5 ){\line(0,1){ 7 }}
\put(  1.00,  2.57){\line(1,0){.25}}
\put(  0.00,  2.57){\makebox(0,0){\normalsize  0.000}}
\put(  1.00,  4.57){\line(1,0){.25}}
\put(  0.00,  4.57){\makebox(0,0){\normalsize  0.025}}
\put(  1.00,  6.57){\line(1,0){.25}}
\put(  0.00,  6.57){\makebox(0,0){\normalsize  0.050}}
\color{red}
\color{red}
\psset{linewidth= 1.2 pt,linecolor=red}
\psset{linestyle=solid}
\pscurve
(  1.008937  ,  2.560606  )
(  1.053435  ,  2.602103  )
(  1.097935  ,  2.644827  )
(  1.142433  ,  2.688756  )
(  1.186932  ,  2.73387  )
(  1.231431  ,  2.780147  )
(  1.27593  ,  2.827565  )
(  1.320429  ,  2.876101  )
(  1.364927  ,  2.925732  )
(  1.409426  ,  2.976431  )
(  1.453925  ,  3.028175  )
(  1.498424  ,  3.080936  )
(  1.542922  ,  3.134688  )
(  1.587422  ,  3.189404  )
(  1.63192  ,  3.245054  )
(  1.676419  ,  3.30161  )
(  1.720918  ,  3.359043  )
(  1.765417  ,  3.417322  )
(  1.809915  ,  3.476417  )
(  1.854414  ,  3.536295  )
(  1.898913  ,  3.596927  )
(  1.943412  ,  3.658279  )
(  1.987911  ,  3.720316  )
(  2.03241  ,  3.783008  )
(  2.076908  ,  3.846319  )
(  2.121407  ,  3.910214  )
(  2.165906  ,  3.974661  )
(  2.210405  ,  4.039622  )
(  2.254903  ,  4.105063  )
(  2.299402  ,  4.170948  )
(  2.343901  ,  4.23724  )
(  2.3884  ,  4.303903  )
(  2.432899  ,  4.3709  )
(  2.477397  ,  4.438194  )
(  2.521896  ,  4.505747  )
(  2.566395  ,  4.573524  )
(  2.610894  ,  4.641484  )
(  2.655393  ,  4.709592  )
(  2.699892  ,  4.777808  )
(  2.74439  ,  4.846097  )
(  2.788889  ,  4.914418  )
(  2.833388  ,  4.982735  )
(  2.877887  ,  5.051009  )
(  2.922385  ,  5.119201  )
(  2.966884  ,  5.187276  )
(  3.011383  ,  5.255194  )
(  3.055882  ,  5.322918  )
(  3.100381  ,  5.39041  )
(  3.144879  ,  5.457633  )
(  3.189378  ,  5.524549  )
(  3.233877  ,  5.591124  )
(  3.278376  ,  5.657317  )
(  3.322875  ,  5.723093  )
(  3.367373  ,  5.788417  )
(  3.411872  ,  5.853253  )
(  3.456371  ,  5.917565  )
(  3.50087  ,  5.981318  )
(  3.545368  ,  6.044478  )
(  3.589867  ,  6.107009  )
(  3.634366  ,  6.16888  )
(  3.678865  ,  6.230054  )
(  3.723364  ,  6.290501  )
(  3.767863  ,  6.350188  )
(  3.812361  ,  6.409082  )
(  3.856861  ,  6.467154  )
(  3.901359  ,  6.524372  )
(  3.945858  ,  6.580707  )
(  3.990356  ,  6.636128  )
(  4.034856  ,  6.690608  )
(  4.079354  ,  6.744117  )
(  4.123853  ,  6.796629  )
(  4.168352  ,  6.848118  )
(  4.212851  ,  6.898556  )
(  4.257349  ,  6.94792  )
(  4.301848  ,  6.996183  )
(  4.346347  ,  7.043323  )
(  4.390846  ,  7.089318  )
(  4.435345  ,  7.134144  )
(  4.479844  ,  7.17778  )
(  4.524343  ,  7.220207  )
(  4.568841  ,  7.261403  )
\psset{linewidth= 1.2 pt,linecolor=red}
\psset{linestyle=dashed,linewidth=1.6pt,dash=1pt 4pt}
\pscurve
(  4.61334  ,  7.301349  )
(  4.657838  ,  7.340029  )
(  4.702338  ,  7.377425  )
(  4.746837  ,  7.41352  )
(  4.791335  ,  7.4483  )
(  4.835834  ,  7.481749  )
(  4.880333  ,  7.513854  )
(  4.924831  ,  7.544601  )
(  4.96933  ,  7.573981  )
(  5.013829  ,  7.60198  )
(  5.058328  ,  7.62859  )
(  5.102827  ,  7.653799  )
(  5.147326  ,  7.677602  )
(  5.191824  ,  7.699989  )
(  5.236323  ,  7.720954  )
(  5.280821  ,  7.740493  )
(  5.325321  ,  7.7586  )
(  5.369819  ,  7.77527  )
(  5.414318  ,  7.790501  )
(  5.458817  ,  7.804292  )
(  5.503316  ,  7.816639  )
(  5.547814  ,  7.827545  )
(  5.592314  ,  7.837008  )
(  5.636812  ,  7.845031  )
(  5.681311  ,  7.851614  )
(  5.72581  ,  7.856762  )
(  5.770308  ,  7.860479  )
(  5.814807  ,  7.862769  )
(  5.859306  ,  7.863636  )
(  5.903804  ,  7.86309  )
(  5.948304  ,  7.861135  )
(  5.992802  ,  7.857779  )
(  6.037302  ,  7.853032  )
(  6.0818  ,  7.846903  )
(  6.126299  ,  7.839402  )
(  6.170798  ,  7.830539  )
(  6.215296  ,  7.820326  )
(  6.259796  ,  7.808776  )
(  6.304295  ,  7.795902  )
(  6.348793  ,  7.781716  )
(  6.393292  ,  7.766234  )
(  6.437791  ,  7.749471  )
(  6.482289  ,  7.73144  )
(  6.526788  ,  7.712159  )
(  6.571287  ,  7.691643  )
(  6.615787  ,  7.669911  )
(  6.660285  ,  7.64698  )
(  6.704784  ,  7.622869  )
(  6.749282  ,  7.597595  )
(  6.793781  ,  7.571177  )
(  6.83828  ,  7.543638  )
(  6.882779  ,  7.514995  )
(  6.927277  ,  7.48527  )
(  6.971776  ,  7.454483  )
(  7.016274  ,  7.422656  )
(  7.060774  ,  7.389809  )
(  7.105272  ,  7.355968  )
(  7.149772  ,  7.321151  )
(  7.19427  ,  7.285384  )
(  7.238769  ,  7.248686  )
(  7.283268  ,  7.211084  )
(  7.327767  ,  7.172601  )
(  7.372266  ,  7.133259  )
(  7.416764  ,  7.093083  )
(  7.461263  ,  7.052095  )
(  7.505762  ,  7.010323  )
(  7.550261  ,  6.967788  )
(  7.594759  ,  6.924517  )
(  7.639258  ,  6.880533  )
(  7.683757  ,  6.835861  )
(  7.728256  ,  6.790525  )
(  7.772754  ,  6.744552  )
(  7.817254  ,  6.697965  )
(  7.861752  ,  6.65079  )
(  7.906251  ,  6.603051  )
(  7.950749  ,  6.554774  )
(  7.995248  ,  6.505982  )
(  8.039747  ,  6.456701  )
(  8.084246  ,  6.406956  )
(  8.128745  ,  6.35677  )
(  8.173244  ,  6.30617  )
(  8.217743  ,  6.255177  )
(  8.262241  ,  6.203819  )
(  8.30674  ,  6.152118  )
(  8.351239  ,  6.100097  )
(  8.395739  ,  6.047782  )
(  8.440236  ,  5.995196  )
(  8.484735  ,  5.942361  )
(  8.529234  ,  5.889301  )
(  8.573733  ,  5.836039  )
(  8.618232  ,  5.782598  )
(  8.66273  ,  5.729  )
(  8.707229  ,  5.675267  )
(  8.751728  ,  5.621421  )
(  8.796228  ,  5.567484  )
(  8.840724  ,  5.513476  )
(  8.885226  ,  5.45942  )
(  8.929722  ,  5.405336  )
(  8.974223  ,  5.351243  )
(  9.018719  ,  5.297161  )
(  9.06322  ,  5.243111  )
(  9.107717  ,  5.189113  )
(  9.152218  ,  5.135183  )
(  9.196713  ,  5.081343  )
(  9.241215  ,  5.027608  )
(  9.285716  ,  4.973999  )
(  9.330213  ,  4.920531  )
(  9.374714  ,  4.867222  )
(  9.419209  ,  4.814089  )
(  9.463711  ,  4.761148  )
(  9.508206  ,  4.708415  )
(  9.552708  ,  4.655905  )
(  9.597204  ,  4.603633  )
(  9.641706  ,  4.551616  )
(  9.686201  ,  4.499865  )
(  9.730702  ,  4.448394  )
(  9.775198  ,  4.39722  )
(  9.8197  ,  4.346352  )
(  9.864202  ,  4.295806  )
(  9.908697  ,  4.245592  )
(  9.953198  ,  4.195722  )
(  9.997694  ,  4.146207  )
(  10.0422  ,  4.097061  )
(  10.08669  ,  4.04829  )
(  10.13119  ,  3.999907  )
(  10.17569  ,  3.951922  )
(  10.22019  ,  3.904343  )
(  10.26469  ,  3.85718  )
(  10.30919  ,  3.810441  )
(  10.35368  ,  3.764132  )
(  10.39818  ,  3.718265  )
(  10.44269  ,  3.672845  )
(  10.48718  ,  3.627879  )
(  10.53168  ,  3.583373  )
(  10.57618  ,  3.539334  )
(  10.62068  ,  3.495769  )
(  10.66518  ,  3.45268  )
(  10.70968  ,  3.410076  )
(  10.75417  ,  3.367959  )
(  10.79868  ,  3.326334  )
(  10.84317  ,  3.285206  )
(  10.88767  ,  3.244578  )
(  10.93217  ,  3.204453  )
(  10.97667  ,  3.164834  )
(  11.02117  ,  3.125724  )
(  11.06567  ,  3.087125  )
(  11.11016  ,  3.04904  )
(  11.15466  ,  3.011469  )
(  11.19916  ,  2.974414  )
(  11.24366  ,  2.937877  )
(  11.28816  ,  2.901858  )
(  11.33266  ,  2.866358  )
(  11.37716  ,  2.831377  )
(  11.42166  ,  2.796914  )
(  11.46616  ,  2.76297  )
(  11.51065  ,  2.729544  )
(  11.55515  ,  2.696635  )
(  11.59965  ,  2.664244  )
(  11.64415  ,  2.632366  )
(  11.68865  ,  2.601002  )
\color{green}
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  1.894444  ,  1.85  )
(  3.683334  ,  2.06  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  3.683334  ,  2.06  )
(  5.472222  ,  1.85  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  5.472222  ,  1.85  )
(  3.683334  ,  1.64  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  3.683334  ,  1.64  )
(  1.894444  ,  1.85  )
\color{magenta}
\psset{linewidth= 1.2 pt,linecolor=magenta}
\psset{linestyle=solid}
\psline
(  1.894444  ,  1.5  )
(  1.894444  ,  8.5  )
\psset{linewidth= 1.2 pt,linecolor=magenta}
\psset{linestyle=solid}
\psline
(  4.577778  ,  1.5  )
(  4.577778  ,  8.5  )
\color{black}
\psset{linewidth= 1.2 pt,linecolor=black}
\psset{linestyle=solid}
\psline
(  1  ,  2.567434  )
(  11.73333  ,  2.567434  )
\end{pspicture}
\hskip.5in\begin{pspicture}(13,11)
\thicklines
\bw
\put( 1.0000, 9.8000){\bf { \normalsize \rr~~{\Large~d)}~~~~}}
\put(-1.5 , 2.2  ){\rotatebox{ 90 }{\bf { \normalsize ~Shear~$\Delta$v~~(m/s)}}}
%\put( 2.0733, 0.2000){\bf { \normalsize z~~~(m)~}}
\color{red}
\put( 7.0000, 8.0000){\bf { \normalsize max~-0.019~~m/s~}}
\put( 7.0000, 7.4000){\bf { \normalsize min~-0.409~~m/s~}}
\color{black}
\put( 1 , 1.5 ){\line(1,0){ 10.73333 }}
\put(  2.79,  1.50){\line(0,1){.25}}
\put(  2.79,  1.00){\makebox(0,0){\normalsize  0.0}}
\put(  5.77,  1.50){\line(0,1){.25}}
\put(  5.77,  1.00){\makebox(0,0){\normalsize  0.5}}
\put(  8.75,  1.50){\line(0,1){.25}}
\put(  8.75,  1.00){\makebox(0,0){\normalsize  1.0}}
\put( 11.73,  1.00){\makebox(0,0){\normalsize  1.5}}
\put( 1 , 1.5 ){\line(0,1){ 7 }}
\put(  1.00,  3.00){\line(1,0){.25}}
\put(  0.00,  3.00){\makebox(0,0){\normalsize     0}}
\put(  1.00,  5.17){\line(1,0){.25}}
\put(  0.00,  5.17){\makebox(0,0){\normalsize     2}}
\put(  1.00,  7.34){\line(1,0){.25}}
\put(  0.00,  7.34){\makebox(0,0){\normalsize     4}}
\color{red}
\color{red}
\psset{linewidth= 1.2 pt,linecolor=red}
\psset{linestyle=solid}
\pscurve
(  1.008937  ,  2.983277  )
(  1.053435  ,  2.962701  )
(  1.097935  ,  2.942596  )
(  1.142433  ,  2.922967  )
(  1.186932  ,  2.903817  )
(  1.231431  ,  2.885152  )
(  1.27593  ,  2.866975  )
(  1.320429  ,  2.84929  )
(  1.364927  ,  2.8321  )
(  1.409426  ,  2.81541  )
(  1.453925  ,  2.799224  )
(  1.498424  ,  2.783543  )
(  1.542922  ,  2.768373  )
(  1.587422  ,  2.753716  )
(  1.63192  ,  2.739574  )
(  1.676419  ,  2.725953  )
(  1.720918  ,  2.712853  )
(  1.765417  ,  2.700278  )
(  1.809915  ,  2.688231  )
(  1.854414  ,  2.676714  )
(  1.898913  ,  2.66573  )
(  1.943412  ,  2.65528  )
(  1.987911  ,  2.645367  )
(  2.03241  ,  2.635994  )
(  2.076908  ,  2.627162  )
(  2.121407  ,  2.618873  )
(  2.165906  ,  2.611129  )
(  2.210405  ,  2.603931  )
(  2.254903  ,  2.597281  )
(  2.299402  ,  2.59118  )
(  2.343901  ,  2.58563  )
(  2.3884  ,  2.580631  )
(  2.432899  ,  2.576186  )
(  2.477397  ,  2.572294  )
(  2.521896  ,  2.568956  )
(  2.566395  ,  2.566174  )
(  2.610894  ,  2.563947  )
(  2.655393  ,  2.562277  )
(  2.699892  ,  2.561163  )
(  2.74439  ,  2.560606  )
(  2.788889  ,  2.560606  )
(  2.833388  ,  2.561163  )
(  2.877887  ,  2.562277  )
(  2.922385  ,  2.563947  )
(  2.966884  ,  2.566174  )
(  3.011383  ,  2.568956  )
(  3.055882  ,  2.572294  )
(  3.100381  ,  2.576186  )
(  3.144879  ,  2.580631  )
(  3.189378  ,  2.58563  )
(  3.233877  ,  2.59118  )
(  3.278376  ,  2.597281  )
(  3.322875  ,  2.603931  )
(  3.367373  ,  2.611129  )
(  3.411872  ,  2.618873  )
(  3.456371  ,  2.627162  )
(  3.50087  ,  2.635994  )
(  3.545368  ,  2.645367  )
(  3.589867  ,  2.65528  )
(  3.634366  ,  2.66573  )
(  3.678865  ,  2.676714  )
(  3.723364  ,  2.688231  )
(  3.767863  ,  2.700278  )
(  3.812361  ,  2.712853  )
(  3.856861  ,  2.725953  )
(  3.901359  ,  2.739574  )
(  3.945858  ,  2.753716  )
(  3.990356  ,  2.768373  )
(  4.034856  ,  2.783543  )
(  4.079354  ,  2.799224  )
(  4.123853  ,  2.81541  )
(  4.168352  ,  2.8321  )
(  4.212851  ,  2.84929  )
(  4.257349  ,  2.866975  )
(  4.301848  ,  2.885152  )
(  4.346347  ,  2.903817  )
(  4.390846  ,  2.922967  )
(  4.435345  ,  2.942596  )
(  4.479844  ,  2.962701  )
(  4.524343  ,  2.983277  )
(  4.568841  ,  3.00432  )
\psset{linewidth= 1.2 pt,linecolor=red}
\psset{linestyle=dashed,linewidth=1.6pt,dash=1pt 4pt}
\pscurve
(  4.61334  ,  3.025825  )
(  4.657838  ,  3.047788  )
(  4.702338  ,  3.070204  )
(  4.746837  ,  3.093067  )
(  4.791335  ,  3.116373  )
(  4.835834  ,  3.140117  )
(  4.880333  ,  3.164293  )
(  4.924831  ,  3.188897  )
(  4.96933  ,  3.213922  )
(  5.013829  ,  3.239364  )
(  5.058328  ,  3.265216  )
(  5.102827  ,  3.291473  )
(  5.147326  ,  3.318129  )
(  5.191824  ,  3.345179  )
(  5.236323  ,  3.372615  )
(  5.280821  ,  3.400433  )
(  5.325321  ,  3.428625  )
(  5.369819  ,  3.457186  )
(  5.414318  ,  3.486109  )
(  5.458817  ,  3.515387  )
(  5.503316  ,  3.545015  )
(  5.547814  ,  3.574985  )
(  5.592314  ,  3.605291  )
(  5.636812  ,  3.635926  )
(  5.681311  ,  3.666884  )
(  5.72581  ,  3.698156  )
(  5.770308  ,  3.729737  )
(  5.814807  ,  3.761618  )
(  5.859306  ,  3.793793  )
(  5.903804  ,  3.826256  )
(  5.948304  ,  3.858997  )
(  5.992802  ,  3.89201  )
(  6.037302  ,  3.925288  )
(  6.0818  ,  3.958823  )
(  6.126299  ,  3.992607  )
(  6.170798  ,  4.026634  )
(  6.215296  ,  4.060894  )
(  6.259796  ,  4.095382  )
(  6.304295  ,  4.130088  )
(  6.348793  ,  4.165006  )
(  6.393292  ,  4.200126  )
(  6.437791  ,  4.235443  )
(  6.482289  ,  4.270946  )
(  6.526788  ,  4.306629  )
(  6.571287  ,  4.342485  )
(  6.615787  ,  4.378503  )
(  6.660285  ,  4.414678  )
(  6.704784  ,  4.451  )
(  6.749282  ,  4.487463  )
(  6.793781  ,  4.524055  )
(  6.83828  ,  4.560773  )
(  6.882779  ,  4.597606  )
(  6.927277  ,  4.634546  )
(  6.971776  ,  4.671586  )
(  7.016274  ,  4.708718  )
(  7.060774  ,  4.745933  )
(  7.105272  ,  4.783223  )
(  7.149772  ,  4.820581  )
(  7.19427  ,  4.857998  )
(  7.238769  ,  4.895467  )
(  7.283268  ,  4.932979  )
(  7.327767  ,  4.970527  )
(  7.372266  ,  5.008101  )
(  7.416764  ,  5.045697  )
(  7.461263  ,  5.083304  )
(  7.505762  ,  5.120914  )
(  7.550261  ,  5.158521  )
(  7.594759  ,  5.196117  )
(  7.639258  ,  5.233694  )
(  7.683757  ,  5.271245  )
(  7.728256  ,  5.308762  )
(  7.772754  ,  5.346237  )
(  7.817254  ,  5.383663  )
(  7.861752  ,  5.421033  )
(  7.906251  ,  5.458338  )
(  7.950749  ,  5.495572  )
(  7.995248  ,  5.532729  )
(  8.039747  ,  5.569802  )
(  8.084246  ,  5.606781  )
(  8.128745  ,  5.643661  )
(  8.173244  ,  5.680437  )
(  8.217743  ,  5.717099  )
(  8.262241  ,  5.753642  )
(  8.30674  ,  5.790059  )
(  8.351239  ,  5.826344  )
(  8.395739  ,  5.86249  )
(  8.440236  ,  5.898491  )
(  8.484735  ,  5.934341  )
(  8.529234  ,  5.970036  )
(  8.573733  ,  6.005565  )
(  8.618232  ,  6.040926  )
(  8.66273  ,  6.076112  )
(  8.707229  ,  6.111118  )
(  8.751728  ,  6.145938  )
(  8.796228  ,  6.180568  )
(  8.840724  ,  6.214999  )
(  8.885226  ,  6.24923  )
(  8.929722  ,  6.283254  )
(  8.974223  ,  6.317065  )
(  9.018719  ,  6.350661  )
(  9.06322  ,  6.384035  )
(  9.107717  ,  6.417182  )
(  9.152218  ,  6.450101  )
(  9.196713  ,  6.482785  )
(  9.241215  ,  6.515229  )
(  9.285716  ,  6.547431  )
(  9.330213  ,  6.579386  )
(  9.374714  ,  6.61109  )
(  9.419209  ,  6.642541  )
(  9.463711  ,  6.673733  )
(  9.508206  ,  6.704665  )
(  9.552708  ,  6.735331  )
(  9.597204  ,  6.76573  )
(  9.641706  ,  6.79586  )
(  9.686201  ,  6.825715  )
(  9.730702  ,  6.855294  )
(  9.775198  ,  6.884594  )
(  9.8197  ,  6.913613  )
(  9.864202  ,  6.942348  )
(  9.908697  ,  6.970797  )
(  9.953198  ,  6.998959  )
(  9.997694  ,  7.026829  )
(  10.0422  ,  7.054409  )
(  10.08669  ,  7.081694  )
(  10.13119  ,  7.108684  )
(  10.17569  ,  7.135377  )
(  10.22019  ,  7.161772  )
(  10.26469  ,  7.18787  )
(  10.30919  ,  7.213665  )
(  10.35368  ,  7.23916  )
(  10.39818  ,  7.264353  )
(  10.44269  ,  7.289242  )
(  10.48718  ,  7.313828  )
(  10.53168  ,  7.33811  )
(  10.57618  ,  7.362087  )
(  10.62068  ,  7.38576  )
(  10.66518  ,  7.409129  )
(  10.70968  ,  7.432191  )
(  10.75417  ,  7.45495  )
(  10.79868  ,  7.477404  )
(  10.84317  ,  7.499554  )
(  10.88767  ,  7.521398  )
(  10.93217  ,  7.54294  )
(  10.97667  ,  7.564179  )
(  11.02117  ,  7.585116  )
(  11.06567  ,  7.605752  )
(  11.11016  ,  7.626085  )
(  11.15466  ,  7.646121  )
(  11.19916  ,  7.665857  )
(  11.24366  ,  7.685296  )
(  11.28816  ,  7.704438  )
(  11.33266  ,  7.723286  )
(  11.37716  ,  7.741842  )
(  11.42166  ,  7.760104  )
(  11.46616  ,  7.778076  )
(  11.51065  ,  7.795759  )
(  11.55515  ,  7.813155  )
(  11.59965  ,  7.830265  )
(  11.64415  ,  7.847092  )
(  11.68865  ,  7.863636  )
\color{green}
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  1.894444  ,  1.85  )
(  3.683334  ,  2.06  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  3.683334  ,  2.06  )
(  5.472222  ,  1.85  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  5.472222  ,  1.85  )
(  3.683334  ,  1.64  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  3.683334  ,  1.64  )
(  1.894444  ,  1.85  )
\color{magenta}
\psset{linewidth= 1.2 pt,linecolor=magenta}
\psset{linestyle=solid}
\psline
(  1.894444  ,  1.5  )
(  1.894444  ,  8.5  )
\psset{linewidth= 1.2 pt,linecolor=magenta}
\psset{linestyle=solid}
\psline
(  4.577778  ,  1.5  )
(  4.577778  ,  8.5  )
\color{black}
\psset{linewidth= 1.2 pt,linecolor=black}
\psset{linestyle=solid}
\psline
(  1  ,  3.00432  )
(  11.73333  ,  3.00432  )
\end{pspicture}

\hskip-.4in\begin{pspicture}(13,11)
\thicklines
\bw
\put( 1.0000, 9.8000){\bf { \normalsize \rr~~{\Large~e)}~~~~}}
\put(-1.5 , 2.2  ){\rotatebox{ 90 }{\bf { \normalsize ~y~Deflection~~(mm)}}}
\put( 6.0733, 0.2000){\bf { \normalsize z~~~(m)~}}
\color{red}
\put( 7.0000, 8.0000){\bf { \normalsize max~~0.005~~mm~}}
\color{black}
\put( 1 , 1.5 ){\line(1,0){ 10.73333 }}
\put(  2.79,  1.50){\line(0,1){.25}}
\put(  2.79,  1.00){\makebox(0,0){\normalsize  0.0}}
\put(  5.77,  1.50){\line(0,1){.25}}
\put(  5.77,  1.00){\makebox(0,0){\normalsize  0.5}}
\put(  8.75,  1.50){\line(0,1){.25}}
\put(  8.75,  1.00){\makebox(0,0){\normalsize  1.0}}
\put( 11.73,  1.00){\makebox(0,0){\normalsize  1.5}}
\put( 1 , 1.5 ){\line(0,1){ 7 }}
\put(  1.00,  2.56){\line(1,0){.25}}
\put(  0.00,  2.56){\makebox(0,0){\normalsize  0.000}}
\put(  1.00,  3.73){\line(1,0){.25}}
\put(  0.00,  3.73){\makebox(0,0){\normalsize  0.025}}
\put(  1.00,  4.91){\line(1,0){.25}}
\put(  0.00,  4.91){\makebox(0,0){\normalsize  0.050}}
\put(  1.00,  6.08){\line(1,0){.25}}
\put(  0.00,  6.08){\makebox(0,0){\normalsize  0.075}}
\put(  1.00,  7.26){\line(1,0){.25}}
\put(  0.00,  7.26){\makebox(0,0){\normalsize  0.100}}
\put(  1.00,  8.43){\line(1,0){.25}}
\put(  0.00,  8.43){\makebox(0,0){\normalsize  0.125}}
\color{red}
\color{red}
\psset{linewidth= 1.2 pt,linecolor=red}
\psset{linestyle=solid}
\pscurve
(  1.008937  ,  2.560606  )
(  1.053435  ,  2.560976  )
(  1.097935  ,  2.561518  )
(  1.142433  ,  2.562226  )
(  1.186932  ,  2.563092  )
(  1.231431  ,  2.564108  )
(  1.27593  ,  2.565267  )
(  1.320429  ,  2.566562  )
(  1.364927  ,  2.567985  )
(  1.409426  ,  2.569529  )
(  1.453925  ,  2.571187  )
(  1.498424  ,  2.572953  )
(  1.542922  ,  2.57482  )
(  1.587422  ,  2.576781  )
(  1.63192  ,  2.57883  )
(  1.676419  ,  2.58096  )
(  1.720918  ,  2.583166  )
(  1.765417  ,  2.585442  )
(  1.809915  ,  2.587782  )
(  1.854414  ,  2.590181  )
(  1.898913  ,  2.592633  )
(  1.943412  ,  2.595134  )
(  1.987911  ,  2.597679  )
(  2.03241  ,  2.600263  )
(  2.076908  ,  2.602881  )
(  2.121407  ,  2.605531  )
(  2.165906  ,  2.608207  )
(  2.210405  ,  2.610906  )
(  2.254903  ,  2.613625  )
(  2.299402  ,  2.61636  )
(  2.343901  ,  2.61911  )
(  2.3884  ,  2.62187  )
(  2.432899  ,  2.62464  )
(  2.477397  ,  2.627416  )
(  2.521896  ,  2.630198  )
(  2.566395  ,  2.632982  )
(  2.610894  ,  2.635769  )
(  2.655393  ,  2.638557  )
(  2.699892  ,  2.641346  )
(  2.74439  ,  2.644135  )
(  2.788889  ,  2.646924  )
(  2.833388  ,  2.649713  )
(  2.877887  ,  2.652503  )
(  2.922385  ,  2.655294  )
(  2.966884  ,  2.658087  )
(  3.011383  ,  2.660884  )
(  3.055882  ,  2.663685  )
(  3.100381  ,  2.666494  )
(  3.144879  ,  2.669311  )
(  3.189378  ,  2.67214  )
(  3.233877  ,  2.674982  )
(  3.278376  ,  2.677841  )
(  3.322875  ,  2.68072  )
(  3.367373  ,  2.683622  )
(  3.411872  ,  2.68655  )
(  3.456371  ,  2.68951  )
(  3.50087  ,  2.692504  )
(  3.545368  ,  2.695537  )
(  3.589867  ,  2.698614  )
(  3.634366  ,  2.70174  )
(  3.678865  ,  2.704919  )
(  3.723364  ,  2.708157  )
(  3.767863  ,  2.711458  )
(  3.812361  ,  2.71483  )
(  3.856861  ,  2.718278  )
(  3.901359  ,  2.721807  )
(  3.945858  ,  2.725424  )
(  3.990356  ,  2.729135  )
(  4.034856  ,  2.732947  )
(  4.079354  ,  2.736867  )
(  4.123853  ,  2.740901  )
(  4.168352  ,  2.745056  )
(  4.212851  ,  2.749339  )
(  4.257349  ,  2.753758  )
(  4.301848  ,  2.75832  )
(  4.346347  ,  2.763032  )
(  4.390846  ,  2.767902  )
(  4.435345  ,  2.772937  )
(  4.479844  ,  2.778146  )
(  4.524343  ,  2.783535  )
(  4.568841  ,  2.789113  )
\psset{linewidth= 1.2 pt,linecolor=red}
\psset{linestyle=dashed,linewidth=1.6pt,dash=1pt 4pt}
\pscurve
(  4.61334  ,  2.794887  )
(  4.657838  ,  2.800866  )
(  4.702338  ,  2.807057  )
(  4.746837  ,  2.813468  )
(  4.791335  ,  2.820108  )
(  4.835834  ,  2.826984  )
(  4.880333  ,  2.834104  )
(  4.924831  ,  2.841476  )
(  4.96933  ,  2.849108  )
(  5.013829  ,  2.857007  )
(  5.058328  ,  2.865182  )
(  5.102827  ,  2.873641  )
(  5.147326  ,  2.88239  )
(  5.191824  ,  2.891438  )
(  5.236323  ,  2.900792  )
(  5.280821  ,  2.91046  )
(  5.325321  ,  2.920449  )
(  5.369819  ,  2.930766  )
(  5.414318  ,  2.941418  )
(  5.458817  ,  2.952412  )
(  5.503316  ,  2.963756  )
(  5.547814  ,  2.975455  )
(  5.592314  ,  2.987517  )
(  5.636812  ,  2.999948  )
(  5.681311  ,  3.012754  )
(  5.72581  ,  3.025941  )
(  5.770308  ,  3.039515  )
(  5.814807  ,  3.053482  )
(  5.859306  ,  3.067847  )
(  5.903804  ,  3.082616  )
(  5.948304  ,  3.097793  )
(  5.992802  ,  3.113385  )
(  6.037302  ,  3.129395  )
(  6.0818  ,  3.145828  )
(  6.126299  ,  3.162688  )
(  6.170798  ,  3.17998  )
(  6.215296  ,  3.197707  )
(  6.259796  ,  3.215873  )
(  6.304295  ,  3.234481  )
(  6.348793  ,  3.253534  )
(  6.393292  ,  3.273036  )
(  6.437791  ,  3.292988  )
(  6.482289  ,  3.313394  )
(  6.526788  ,  3.334254  )
(  6.571287  ,  3.355572  )
(  6.615787  ,  3.377349  )
(  6.660285  ,  3.399586  )
(  6.704784  ,  3.422284  )
(  6.749282  ,  3.445445  )
(  6.793781  ,  3.469069  )
(  6.83828  ,  3.493155  )
(  6.882779  ,  3.517705  )
(  6.927277  ,  3.542718  )
(  6.971776  ,  3.568194  )
(  7.016274  ,  3.594131  )
(  7.060774  ,  3.62053  )
(  7.105272  ,  3.647389  )
(  7.149772  ,  3.674706  )
(  7.19427  ,  3.702479  )
(  7.238769  ,  3.730708  )
(  7.283268  ,  3.759389  )
(  7.327767  ,  3.78852  )
(  7.372266  ,  3.818099  )
(  7.416764  ,  3.848123  )
(  7.461263  ,  3.878587  )
(  7.505762  ,  3.90949  )
(  7.550261  ,  3.940828  )
(  7.594759  ,  3.972596  )
(  7.639258  ,  4.004791  )
(  7.683757  ,  4.037408  )
(  7.728256  ,  4.070444  )
(  7.772754  ,  4.103893  )
(  7.817254  ,  4.137751  )
(  7.861752  ,  4.172012  )
(  7.906251  ,  4.206671  )
(  7.950749  ,  4.241724  )
(  7.995248  ,  4.277163  )
(  8.039747  ,  4.312984  )
(  8.084246  ,  4.349181  )
(  8.128745  ,  4.385747  )
(  8.173244  ,  4.422677  )
(  8.217743  ,  4.459963  )
(  8.262241  ,  4.497601  )
(  8.30674  ,  4.535581  )
(  8.351239  ,  4.573899  )
(  8.395739  ,  4.612547  )
(  8.440236  ,  4.651518  )
(  8.484735  ,  4.690804  )
(  8.529234  ,  4.7304  )
(  8.573733  ,  4.770296  )
(  8.618232  ,  4.810486  )
(  8.66273  ,  4.850963  )
(  8.707229  ,  4.891719  )
(  8.751728  ,  4.932745  )
(  8.796228  ,  4.974035  )
(  8.840724  ,  5.015581  )
(  8.885226  ,  5.057374  )
(  8.929722  ,  5.099407  )
(  8.974223  ,  5.141672  )
(  9.018719  ,  5.184161  )
(  9.06322  ,  5.226866  )
(  9.107717  ,  5.269778  )
(  9.152218  ,  5.312891  )
(  9.196713  ,  5.356196  )
(  9.241215  ,  5.399684  )
(  9.285716  ,  5.443349  )
(  9.330213  ,  5.487182  )
(  9.374714  ,  5.531174  )
(  9.419209  ,  5.575318  )
(  9.463711  ,  5.619606  )
(  9.508206  ,  5.664031  )
(  9.552708  ,  5.708582  )
(  9.597204  ,  5.753255  )
(  9.641706  ,  5.79804  )
(  9.686201  ,  5.84293  )
(  9.730702  ,  5.887917  )
(  9.775198  ,  5.932993  )
(  9.8197  ,  5.978152  )
(  9.864202  ,  6.023385  )
(  9.908697  ,  6.068685  )
(  9.953198  ,  6.114045  )
(  9.997694  ,  6.159456  )
(  10.0422  ,  6.204914  )
(  10.08669  ,  6.25041  )
(  10.13119  ,  6.295938  )
(  10.17569  ,  6.34149  )
(  10.22019  ,  6.387059  )
(  10.26469  ,  6.43264  )
(  10.30919  ,  6.478225  )
(  10.35368  ,  6.523808  )
(  10.39818  ,  6.569383  )
(  10.44269  ,  6.614944  )
(  10.48718  ,  6.660485  )
(  10.53168  ,  6.705998  )
(  10.57618  ,  6.75148  )
(  10.62068  ,  6.796923  )
(  10.66518  ,  6.842323  )
(  10.70968  ,  6.887674  )
(  10.75417  ,  6.932971  )
(  10.79868  ,  6.978207  )
(  10.84317  ,  7.02338  )
(  10.88767  ,  7.068482  )
(  10.93217  ,  7.113512  )
(  10.97667  ,  7.15846  )
(  11.02117  ,  7.203327  )
(  11.06567  ,  7.248105  )
(  11.11016  ,  7.292791  )
(  11.15466  ,  7.337381  )
(  11.19916  ,  7.381872  )
(  11.24366  ,  7.426255  )
(  11.28816  ,  7.470534  )
(  11.33266  ,  7.514697  )
(  11.37716  ,  7.558752  )
(  11.42166  ,  7.602684  )
(  11.46616  ,  7.646493  )
(  11.51065  ,  7.690182  )
(  11.55515  ,  7.733743  )
(  11.59965  ,  7.777178  )
(  11.64415  ,  7.820477  )
(  11.68865  ,  7.863639  )
\color{green}
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  1.894444  ,  1.85  )
(  3.683334  ,  2.06  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  3.683334  ,  2.06  )
(  5.472222  ,  1.85  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  5.472222  ,  1.85  )
(  3.683334  ,  1.64  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  3.683334  ,  1.64  )
(  1.894444  ,  1.85  )
\color{magenta}
\psset{linewidth= 1.2 pt,linecolor=magenta}
\psset{linestyle=solid}
\psline
(  1.894444  ,  1.5  )
(  1.894444  ,  8.5  )
\psset{linewidth= 1.2 pt,linecolor=magenta}
\psset{linestyle=solid}
\psline
(  4.577778  ,  1.5  )
(  4.577778  ,  8.5  )
\color{black}
\psset{linewidth= 1.2 pt,linecolor=black}
\psset{linestyle=solid}
\psline
(  1  ,  2.560418  )
(  11.73333  ,  2.560418  )
\end{pspicture}
\hskip.5in\begin{pspicture}(13,11)
\thicklines
\bw
\put( 1.0000, 9.8000){\bf { \normalsize \rr~~{\Large~f)}~~~~}}
\put(-1.5 , 2.2  ){\rotatebox{ 90 }{\bf { \normalsize ~Distortion~~($\Delta r$/r~\%)}}}
\put( 2.0733, 0.2000){\bf { \normalsize z~~~(m)~}}
\color{blue}
\put( 7.0000, 4.0000){\bf { \normalsize max~~0.372~\%~}}
\put( 6.0000, 4.7500){\bf { \normalsize no~Surface~Tension}}
\color{black}
\put( 1 , 1.5 ){\line(1,0){ 10.73333 }}
\put(  2.79,  1.50){\line(0,1){.25}}
\put(  2.79,  1.00){\makebox(0,0){\normalsize  0.0}}
\put(  5.77,  1.50){\line(0,1){.25}}
\put(  5.77,  1.00){\makebox(0,0){\normalsize  0.5}}
\put(  8.75,  1.50){\line(0,1){.25}}
\put(  8.75,  1.00){\makebox(0,0){\normalsize  1.0}}
\put( 11.73,  1.00){\makebox(0,0){\normalsize  1.5}}
\put( 1 , 1.5 ){\line(0,1){ 7 }}
\put(  1.00,  1.50){\line(1,0){.25}}
\put(  0.00,  1.50){\makebox(0,0){\normalsize  -100}}
\put(  1.00,  3.17){\line(1,0){.25}}
\put(  0.00,  3.17){\makebox(0,0){\normalsize   -75}}
\put(  1.00,  4.83){\line(1,0){.25}}
\put(  0.00,  4.83){\makebox(0,0){\normalsize   -50}}
\put(  1.00,  6.50){\line(1,0){.25}}
\put(  0.00,  6.50){\makebox(0,0){\normalsize   -25}}
\put(  1.00,  8.17){\line(1,0){.25}}
\put(  0.00,  8.17){\makebox(0,0){\normalsize     0}}
\color{blue}
\color{blue}
\psset{linewidth= 1.2 pt,linecolor=blue}
\psset{linestyle=solid}
\pscurve
(  1.008937  ,  8.166702  )
(  1.053435  ,  8.166772  )
(  1.097935  ,  8.166872  )
(  1.142433  ,  8.167001  )
(  1.186932  ,  8.167156  )
(  1.231431  ,  8.167336  )
(  1.27593  ,  8.167538  )
(  1.320429  ,  8.16776  )
(  1.364927  ,  8.168001  )
(  1.409426  ,  8.16826  )
(  1.453925  ,  8.168533  )
(  1.498424  ,  8.168821  )
(  1.542922  ,  8.169122  )
(  1.587422  ,  8.169434  )
(  1.63192  ,  8.169756  )
(  1.676419  ,  8.170087  )
(  1.720918  ,  8.170426  )
(  1.765417  ,  8.170773  )
(  1.809915  ,  8.171125  )
(  1.854414  ,  8.171483  )
(  1.898913  ,  8.171845  )
(  1.943412  ,  8.172213  )
(  1.987911  ,  8.172583  )
(  2.03241  ,  8.172956  )
(  2.076908  ,  8.173331  )
(  2.121407  ,  8.173708  )
(  2.165906  ,  8.174088  )
(  2.210405  ,  8.174467  )
(  2.254903  ,  8.174849  )
(  2.299402  ,  8.175231  )
(  2.343901  ,  8.175613  )
(  2.3884  ,  8.175997  )
(  2.432899  ,  8.17638  )
(  2.477397  ,  8.176764  )
(  2.521896  ,  8.177148  )
(  2.566395  ,  8.177531  )
(  2.610894  ,  8.177916  )
(  2.655393  ,  8.178299  )
(  2.699892  ,  8.178683  )
(  2.74439  ,  8.179067  )
(  2.788889  ,  8.179451  )
(  2.833388  ,  8.179834  )
(  2.877887  ,  8.180219  )
(  2.922385  ,  8.180602  )
(  2.966884  ,  8.180986  )
(  3.011383  ,  8.18137  )
(  3.055882  ,  8.181753  )
(  3.100381  ,  8.182137  )
(  3.144879  ,  8.18252  )
(  3.189378  ,  8.182902  )
(  3.233877  ,  8.183285  )
(  3.278376  ,  8.183666  )
(  3.322875  ,  8.184046  )
(  3.367373  ,  8.184425  )
(  3.411872  ,  8.184803  )
(  3.456371  ,  8.185178  )
(  3.50087  ,  8.185551  )
(  3.545368  ,  8.185921  )
(  3.589867  ,  8.186288  )
(  3.634366  ,  8.18665  )
(  3.678865  ,  8.187008  )
(  3.723364  ,  8.187361  )
(  3.767863  ,  8.187707  )
(  3.812361  ,  8.188046  )
(  3.856861  ,  8.188377  )
(  3.901359  ,  8.1887  )
(  3.945858  ,  8.189012  )
(  3.990356  ,  8.189312  )
(  4.034856  ,  8.1896  )
(  4.079354  ,  8.189874  )
(  4.123853  ,  8.190132  )
(  4.168352  ,  8.190373  )
(  4.212851  ,  8.190596  )
(  4.257349  ,  8.190798  )
(  4.301848  ,  8.190977  )
(  4.346347  ,  8.191133  )
(  4.390846  ,  8.191261  )
(  4.435345  ,  8.191361  )
(  4.479844  ,  8.191431  )
(  4.524343  ,  8.191466  )
(  4.568841  ,  8.191466  )
\psset{linewidth= 1.2 pt,linecolor=blue}
\psset{linestyle=dashed,linewidth=1.6pt,dash=1pt 4pt}
\pscurve
(  4.61334  ,  8.191428  )
(  4.657838  ,  8.191348  )
(  4.702338  ,  8.191224  )
(  4.746837  ,  8.191053  )
(  4.791335  ,  8.190832  )
(  4.835834  ,  8.190557  )
(  4.880333  ,  8.190226  )
(  4.924831  ,  8.189833  )
(  4.96933  ,  8.189377  )
(  5.013829  ,  8.188853  )
(  5.058328  ,  8.188257  )
(  5.102827  ,  8.187586  )
(  5.147326  ,  8.186835  )
(  5.191824  ,  8.186  )
(  5.236323  ,  8.185077  )
(  5.280821  ,  8.18406  )
(  5.325321  ,  8.182946  )
(  5.369819  ,  8.181729  )
(  5.414318  ,  8.180406  )
(  5.458817  ,  8.178969  )
(  5.503316  ,  8.177416  )
(  5.547814  ,  8.17574  )
(  5.592314  ,  8.173936  )
(  5.636812  ,  8.171998  )
(  5.681311  ,  8.16992  )
(  5.72581  ,  8.167698  )
(  5.770308  ,  8.165324  )
(  5.814807  ,  8.162794  )
(  5.859306  ,  8.1601  )
(  5.903804  ,  8.157236  )
(  5.948304  ,  8.154198  )
(  5.992802  ,  8.150976  )
(  6.037302  ,  8.147566  )
(  6.0818  ,  8.143959  )
(  6.126299  ,  8.140151  )
(  6.170798  ,  8.136133  )
(  6.215296  ,  8.131899  )
(  6.259796  ,  8.12744  )
(  6.304295  ,  8.122752  )
(  6.348793  ,  8.117826  )
(  6.393292  ,  8.112653  )
(  6.437791  ,  8.107227  )
(  6.482289  ,  8.101542  )
(  6.526788  ,  8.095588  )
(  6.571287  ,  8.089357  )
(  6.615787  ,  8.082844  )
(  6.660285  ,  8.076038  )
(  6.704784  ,  8.068933  )
(  6.749282  ,  8.06152  )
(  6.793781  ,  8.053791  )
(  6.83828  ,  8.045738  )
(  6.882779  ,  8.037354  )
(  6.927277  ,  8.028628  )
(  6.971776  ,  8.019554  )
(  7.016274  ,  8.010123  )
(  7.060774  ,  8.000326  )
(  7.105272  ,  7.990156  )
(  7.149772  ,  7.979604  )
(  7.19427  ,  7.96866  )
(  7.238769  ,  7.957318  )
(  7.283268  ,  7.945568  )
(  7.327767  ,  7.933401  )
(  7.372266  ,  7.92081  )
(  7.416764  ,  7.907786  )
(  7.461263  ,  7.89432  )
(  7.505762  ,  7.880404  )
(  7.550261  ,  7.866028  )
(  7.594759  ,  7.851186  )
(  7.639258  ,  7.835868  )
(  7.683757  ,  7.820066  )
(  7.728256  ,  7.803772  )
(  7.772754  ,  7.786977  )
(  7.817254  ,  7.769672  )
(  7.861752  ,  7.751851  )
(  7.906251  ,  7.733503  )
(  7.950749  ,  7.714622  )
(  7.995248  ,  7.695199  )
(  8.039747  ,  7.675225  )
(  8.084246  ,  7.654694  )
(  8.128745  ,  7.633597  )
(  8.173244  ,  7.611927  )
(  8.217743  ,  7.589674  )
(  8.262241  ,  7.566833  )
(  8.30674  ,  7.543396  )
(  8.351239  ,  7.519354  )
(  8.395739  ,  7.494701  )
(  8.440236  ,  7.469429  )
(  8.484735  ,  7.443532  )
(  8.529234  ,  7.417002  )
(  8.573733  ,  7.389832  )
(  8.618232  ,  7.362015  )
(  8.66273  ,  7.333545  )
(  8.707229  ,  7.304416  )
(  8.751728  ,  7.27462  )
(  8.796228  ,  7.244152  )
(  8.840724  ,  7.213006  )
(  8.885226  ,  7.181175  )
(  8.929722  ,  7.148654  )
(  8.974223  ,  7.115436  )
(  9.018719  ,  7.081517  )
(  9.06322  ,  7.046891  )
(  9.107717  ,  7.011553  )
(  9.152218  ,  6.975498  )
(  9.196713  ,  6.938721  )
(  9.241215  ,  6.901217  )
(  9.285716  ,  6.862981  )
(  9.330213  ,  6.82401  )
(  9.374714  ,  6.784299  )
(  9.419209  ,  6.743845  )
(  9.463711  ,  6.702642  )
(  9.508206  ,  6.660688  )
(  9.552708  ,  6.61798  )
(  9.597204  ,  6.574512  )
(  9.641706  ,  6.530283  )
(  9.686201  ,  6.485291  )
(  9.730702  ,  6.439532  )
(  9.775198  ,  6.393003  )
(  9.8197  ,  6.345702  )
(  9.864202  ,  6.297627  )
(  9.908697  ,  6.248776  )
(  9.953198  ,  6.199148  )
(  9.997694  ,  6.148741  )
(  10.0422  ,  6.097553  )
(  10.08669  ,  6.045582  )
(  10.13119  ,  5.992829  )
(  10.17569  ,  5.939292  )
(  10.22019  ,  5.884971  )
(  10.26469  ,  5.829865  )
(  10.30919  ,  5.773974  )
(  10.35368  ,  5.717299  )
(  10.39818  ,  5.659838  )
(  10.44269  ,  5.601593  )
(  10.48718  ,  5.542563  )
(  10.53168  ,  5.482751  )
(  10.57618  ,  5.422157  )
(  10.62068  ,  5.360781  )
(  10.66518  ,  5.298625  )
(  10.70968  ,  5.235691  )
(  10.75417  ,  5.171979  )
(  10.79868  ,  5.107492  )
(  10.84317  ,  5.042233  )
(  10.88767  ,  4.976202  )
(  10.93217  ,  4.909403  )
(  10.97667  ,  4.841838  )
(  11.02117  ,  4.773509  )
(  11.06567  ,  4.70442  )
(  11.11016  ,  4.634573  )
(  11.15466  ,  4.563972  )
(  11.19916  ,  4.49262  )
(  11.24366  ,  4.420519  )
(  11.28816  ,  4.347674  )
(  11.33266  ,  4.27409  )
(  11.37716  ,  4.199769  )
(  11.42166  ,  4.124717  )
(  11.46616  ,  4.048935  )
(  11.51065  ,  3.97243  )
(  11.55515  ,  3.895206  )
(  11.59965  ,  3.817267  )
(  11.64415  ,  3.738618  )
(  11.68865  ,  3.659264  )
\color{green}
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  1.894444  ,  1.85  )
(  3.683334  ,  2.06  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  3.683334  ,  2.06  )
(  5.472222  ,  1.85  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  5.472222  ,  1.85  )
(  3.683334  ,  1.64  )
\psset{linewidth= 1.2 pt,linecolor=green}
\psset{linestyle=solid}
\psline
(  3.683334  ,  1.64  )
(  1.894444  ,  1.85  )
\color{magenta}
\psset{linewidth= 1.2 pt,linecolor=magenta}
\psset{linestyle=solid}
\psline
(  1.894444  ,  1.5  )
(  1.894444  ,  8.5  )
\psset{linewidth= 1.2 pt,linecolor=magenta}
\psset{linestyle=solid}
\psline
(  4.577778  ,  1.5  )
(  4.577778  ,  8.5  )
\color{black}
\psset{linewidth= 1.2 pt,linecolor=black}
\psset{linestyle=solid}
\psline
(  1  ,  8.166667  )
(  11.73333  ,  8.166667  )
\color{red}
\put( 5.0000, 9.2500){\bf { \normalsize With~Surface~Tension}}
\put( 7.0000, 8.5000){\bf { \normalsize max~~0.179~~\%~}}
\psset{linewidth= 1.2 pt,linecolor=red}
\psset{linestyle=solid}
\pscurve
(  1.008937  ,  8.166702  )
(  1.053435  ,  8.166772  )
(  1.097935  ,  8.166872  )
(  1.142433  ,  8.167  )
(  1.186932  ,  8.167154  )
(  1.231431  ,  8.167333  )
(  1.27593  ,  8.167533  )
(  1.320429  ,  8.167753  )
(  1.364927  ,  8.167991  )
(  1.409426  ,  8.168243  )
(  1.453925  ,  8.16851  )
(  1.498424  ,  8.16879  )
(  1.542922  ,  8.16908  )
(  1.587422  ,  8.169378  )
(  1.63192  ,  8.169684  )
(  1.676419  ,  8.169996  )
(  1.720918  ,  8.170314  )
(  1.765417  ,  8.170633  )
(  1.809915  ,  8.170956  )
(  1.854414  ,  8.17128  )
(  1.898913  ,  8.171603  )
(  1.943412  ,  8.171926  )
(  1.987911  ,  8.172246  )
(  2.03241  ,  8.172564  )
(  2.076908  ,  8.172878  )
(  2.121407  ,  8.173188  )
(  2.165906  ,  8.173493  )
(  2.210405  ,  8.173793  )
(  2.254903  ,  8.174086  )
(  2.299402  ,  8.174372  )
(  2.343901  ,  8.174651  )
(  2.3884  ,  8.174923  )
(  2.432899  ,  8.175186  )
(  2.477397  ,  8.175441  )
(  2.521896  ,  8.175688  )
(  2.566395  ,  8.175924  )
(  2.610894  ,  8.176152  )
(  2.655393  ,  8.176371  )
(  2.699892  ,  8.176579  )
(  2.74439  ,  8.176778  )
(  2.788889  ,  8.176976  )
(  2.833388  ,  8.177164  )
(  2.877887  ,  8.177341  )
(  2.922385  ,  8.177507  )
(  2.966884  ,  8.177663  )
(  3.011383  ,  8.177808  )
(  3.055882  ,  8.17794  )
(  3.100381  ,  8.178061  )
(  3.144879  ,  8.178171  )
(  3.189378  ,  8.178268  )
(  3.233877  ,  8.178353  )
(  3.278376  ,  8.178425  )
(  3.322875  ,  8.178485  )
(  3.367373  ,  8.178531  )
(  3.411872  ,  8.178563  )
(  3.456371  ,  8.178581  )
(  3.50087  ,  8.178585  )
(  3.545368  ,  8.178575  )
(  3.589867  ,  8.178548  )
(  3.634366  ,  8.178505  )
(  3.678865  ,  8.178446  )
(  3.723364  ,  8.17837  )
(  3.767863  ,  8.178274  )
(  3.812361  ,  8.178161  )
(  3.856861  ,  8.178028  )
(  3.901359  ,  8.177874  )
(  3.945858  ,  8.177698  )
(  3.990356  ,  8.177501  )
(  4.034856  ,  8.177279  )
(  4.079354  ,  8.177032  )
(  4.123853  ,  8.17676  )
(  4.168352  ,  8.17646  )
(  4.212851  ,  8.176132  )
(  4.257349  ,  8.175774  )
(  4.301848  ,  8.175384  )
(  4.346347  ,  8.17496  )
(  4.390846  ,  8.174501  )
(  4.435345  ,  8.174007  )
(  4.479844  ,  8.173474  )
(  4.524343  ,  8.172901  )
(  4.568841  ,  8.172285  )
\psset{linewidth= 1.2 pt,linecolor=red}
\psset{linestyle=dashed,linewidth=1.6pt,dash=1pt 4pt}
\pscurve
(  4.61334  ,  8.171625  )
(  4.657838  ,  8.170919  )
(  4.702338  ,  8.170165  )
(  4.746837  ,  8.169359  )
(  4.791335  ,  8.168502  )
(  4.835834  ,  8.167588  )
(  4.880333  ,  8.166616  )
(  4.924831  ,  8.165585  )
(  4.96933  ,  8.16449  )
(  5.013829  ,  8.163329  )
(  5.058328  ,  8.162101  )
(  5.102827  ,  8.160802  )
(  5.147326  ,  8.159428  )
(  5.191824  ,  8.157978  )
(  5.236323  ,  8.156448  )
(  5.280821  ,  8.154836  )
(  5.325321  ,  8.153138  )
(  5.369819  ,  8.151352  )
(  5.414318  ,  8.149473  )
(  5.458817  ,  8.1475  )
(  5.503316  ,  8.145429  )
(  5.547814  ,  8.143256  )
(  5.592314  ,  8.14098  )
(  5.636812  ,  8.138595  )
(  5.681311  ,  8.1361  )
(  5.72581  ,  8.13349  )
(  5.770308  ,  8.130763  )
(  5.814807  ,  8.127915  )
(  5.859306  ,  8.124944  )
(  5.903804  ,  8.121845  )
(  5.948304  ,  8.118616  )
(  5.992802  ,  8.115252  )
(  6.037302  ,  8.111753  )
(  6.0818  ,  8.108113  )
(  6.126299  ,  8.10433  )
(  6.170798  ,  8.100401  )
(  6.215296  ,  8.096323  )
(  6.259796  ,  8.092093  )
(  6.304295  ,  8.087707  )
(  6.348793  ,  8.083162  )
(  6.393292  ,  8.078458  )
(  6.437791  ,  8.073589  )
(  6.482289  ,  8.068555  )
(  6.526788  ,  8.063351  )
(  6.571287  ,  8.057977  )
(  6.615787  ,  8.052428  )
(  6.660285  ,  8.046703  )
(  6.704784  ,  8.040801  )
(  6.749282  ,  8.034718  )
(  6.793781  ,  8.028453  )
(  6.83828  ,  8.022004  )
(  6.882779  ,  8.01537  )
(  6.927277  ,  8.008549  )
(  6.971776  ,  8.001539  )
(  7.016274  ,  7.994339  )
(  7.060774  ,  7.986949  )
(  7.105272  ,  7.979367  )
(  7.149772  ,  7.971592  )
(  7.19427  ,  7.963625  )
(  7.238769  ,  7.955464  )
(  7.283268  ,  7.947109  )
(  7.327767  ,  7.938561  )
(  7.372266  ,  7.929819  )
(  7.416764  ,  7.920884  )
(  7.461263  ,  7.911756  )
(  7.505762  ,  7.902436  )
(  7.550261  ,  7.892925  )
(  7.594759  ,  7.883225  )
(  7.639258  ,  7.873336  )
(  7.683757  ,  7.86326  )
(  7.728256  ,  7.852999  )
(  7.772754  ,  7.842555  )
(  7.817254  ,  7.831931  )
(  7.861752  ,  7.821128  )
(  7.906251  ,  7.810149  )
(  7.950749  ,  7.798998  )
(  7.995248  ,  7.787678  )
(  8.039747  ,  7.776192  )
(  8.084246  ,  7.764544  )
(  8.128745  ,  7.752737  )
(  8.173244  ,  7.740776  )
(  8.217743  ,  7.728665  )
(  8.262241  ,  7.716409  )
(  8.30674  ,  7.704013  )
(  8.351239  ,  7.691482  )
(  8.395739  ,  7.67882  )
(  8.440236  ,  7.666035  )
(  8.484735  ,  7.653131  )
(  8.529234  ,  7.640114  )
(  8.573733  ,  7.626991  )
(  8.618232  ,  7.613769  )
(  8.66273  ,  7.600453  )
(  8.707229  ,  7.587052  )
(  8.751728  ,  7.573572  )
(  8.796228  ,  7.56002  )
(  8.840724  ,  7.546405  )
(  8.885226  ,  7.532734  )
(  8.929722  ,  7.519015  )
(  8.974223  ,  7.505256  )
(  9.018719  ,  7.491466  )
(  9.06322  ,  7.477653  )
(  9.107717  ,  7.463826  )
(  9.152218  ,  7.449994  )
(  9.196713  ,  7.436167  )
(  9.241215  ,  7.422352  )
(  9.285716  ,  7.408561  )
(  9.330213  ,  7.394802  )
(  9.374714  ,  7.381086  )
(  9.419209  ,  7.367421  )
(  9.463711  ,  7.353819  )
(  9.508206  ,  7.340289  )
(  9.552708  ,  7.326842  )
(  9.597204  ,  7.313488  )
(  9.641706  ,  7.300236  )
(  9.686201  ,  7.287099  )
(  9.730702  ,  7.274086  )
(  9.775198  ,  7.261207  )
(  9.8197  ,  7.248475  )
(  9.864202  ,  7.235898  )
(  9.908697  ,  7.22349  )
(  9.953198  ,  7.211259  )
(  9.997694  ,  7.199218  )
(  10.0422  ,  7.187376  )
(  10.08669  ,  7.175744  )
(  10.13119  ,  7.164333  )
(  10.17569  ,  7.153155  )
(  10.22019  ,  7.14222  )
(  10.26469  ,  7.131537  )
(  10.30919  ,  7.121119  )
(  10.35368  ,  7.110975  )
(  10.39818  ,  7.101116  )
(  10.44269  ,  7.091553  )
(  10.48718  ,  7.082294  )
(  10.53168  ,  7.073351  )
(  10.57618  ,  7.064734  )
(  10.62068  ,  7.056453  )
(  10.66518  ,  7.048516  )
(  10.70968  ,  7.040934  )
(  10.75417  ,  7.033715  )
(  10.79868  ,  7.02687  )
(  10.84317  ,  7.020407  )
(  10.88767  ,  7.014334  )
(  10.93217  ,  7.00866  )
(  10.97667  ,  7.003393  )
(  11.02117  ,  6.998542  )
(  11.06567  ,  6.994114  )
(  11.11016  ,  6.990117  )
(  11.15466  ,  6.986557  )
(  11.19916  ,  6.983442  )
(  11.24366  ,  6.980779  )
(  11.28816  ,  6.978573  )
(  11.33266  ,  6.97683  )
(  11.37716  ,  6.975557  )
(  11.42166  ,  6.974759  )
(  11.46616  ,  6.974441  )
(  11.51065  ,  6.974607  )
(  11.55515  ,  6.975261  )
(  11.59965  ,  6.976408  )
(  11.64415  ,  6.978051  )
(  11.68865  ,  6.980195  )
\end{pspicture}

\caption{Results for the case of a jet with the noozle inside the magnetic field.}
\label{fig20}
\end{figure}
 We see that over the extent of the new jet (from - 0.3 to 0.3~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 0.25 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 $4\%$ without surface tension, and less than 0.2\% with surface tension.
\ei

Beyond the target region (z=0.3 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.

We summarize in Tb.~\ref{tb301} these results
\begin{table}[thb!p]
\centering
\caption{Summary of maximum and minimum values of the quantities shown in Fig.~\ref{fig20} }
\label{tb301}
\begin{tabular}{|ccccccc|}
 \hline
 $B_z$ & $\Delta p$ & $< \Delta v>$ & $\hat{ \Delta v}$ & $\Delta y$ & $\Delta x$/r&$\Delta x$/r\\
&&&&&No ST&\\
 T     &  Atm.   &      m/s    &          m/s           &   mm        &    \%  &\%\\
\hline
18.8 &-0.240 &0.0 &-0.41 &0.0 &0.0 &0.0\\
20.0 &0.245 &0.058 &-0.019 &0.005 &0.37 &0.18\\
 \hline
 \end{tabular}
\end{table}

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

The coil dimensions are given in Tb.~\ref{tb4}.
\begin{table}
\caption{Coils used in the Study~II target magnet~\cite{ref7} }
\label{tb4}
\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
\multicolumn{8}{|c|}{\textbf{Fe}}\\
\hline \hline
 0.980 & 0.980 &  0.108 &  0 &  0.313 &   0 &  0 &  0  \\
 1.088 & 0 &  0.312 &  0 &  0.168 &   0 &  0 &  0  \\
\hline
\multicolumn{8}{|c|}{\textbf{Hollow}}\\
\hline \hline
 1.288 & -0.112 &  0.749 &  0.178 &  0.054 & -24.37 &  0.98 &  1.26  \\
 1.288 & -0.749 &  0.877 &  0.231 &  0.122 & -19.07 &  2.04 &  3.74  \\
 1.288 & -0.877 &  1.073 &  0.353 &  0.137 & -14.87 &  2.18 &  5.78  \\
\hline
\multicolumn{8}{|c|}{\textbf{SC}}\\
\hline \hline
 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 & -0.065 &  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}
\end{table}

%\section{\rr Using Calculated Fields}
The axial fields are shown in Figs.~\ref{fig26}. The left figure shows the components from: the use of iron (green), superconductors (blue), and the total (red). The Gaussian distribution used in the above calculations is also given in Fig.~\ref{fig26}-right (black), and is seen to be a good match to the total field over the target region (-0.6 m to 0).
\begin{figure}[htb!]
%\input{wegf.fig}

%\newpage
\includegraphics{wegf.ps}
\caption{Axial field $B_z$ vs. z. Left: It is shown the contribution of each component. Right: Comparison of the Gaussian field vs. a realistic one builded with coils (see Tb.~\ref{tb4})  }
\label{fig26}
\end{figure}

%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}

%%%


\section{\rr Conclusion}

\bi
\item It is not acceptable to have the jet nozzle outside the magnetic field.
\item With the Study~II 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}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} K.~McDonald, \textsl{Damping and Radial Pinching Effects},\\
%begin{verbatim}
http://www.hep.princeton.edu/{\~{}}mcdonald/mumu/target/radialpinch.ps
%end{verbatim}
\bibitem{ref6}P. Thieberger, \textsl{Estimated perturbations of the axial motion of a liquid-metal jet entering a strong magnetic field }, MUC-NOTE-TARGET-0182, Nov. 2000.
\bibitem{ref7}R.~Weggel, \textsl{ 20~T Hybrid magnet for Study II}
\end{thebibliography}

\end{document}
%\includegraphics*[bb=160 500 560 776,width=0.5\textwidth]{jet_fig20a.ps}
%\quad
%\includegraphics*[bb=70 333 486 670,width=0.5\textwidth]{jet_fig20b.ps}
%\includegraphics*[bb=70 333 486 670 ,width=0.5\textwidth]{jet_fig20c.ps}
%\quad 
%\includegraphics*[bb=70 333 486 670 ,width=0.5\textwidth]{jet_fig20d.ps}
%\includegraphics[bb=70 333 486 670,width=0.5\textwidth]{jet_fig20e.ps}
%\includegraphics[bb=70 333 486 670,width=0.5\textwidth]{jet_fig20f.ps}
%\input{jet30.fig}


