%27 Sep 03

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\title{On Thin or Slender Bodies}
\author{E.~O. Tuck}
%\address{Emeritus and Adjunct Professor, School of Mathematical Sciences, The University of Adelaide, South Australia~5005, \textsc{Australia}, from his retirement in June 2002 until his death on 11 March 2009.}
\author{Y.~M. Stokes}
\address{School of Mathematical Sciences, The University of Adelaide, South Australia~ 5005, \textsc{Australia}}
\mailto{yvonne.stokes@adelaide.edu.au}
%\date{2004}
\begin{document}

%\\
%\\
%(to be presented as Lighthill Lecture, \\
%BAMC, Norwich, April 2004)
%\\
%\\
%DRAFT
\maketitle

\begin{abstract}
This is a review of thin-body and slender-body theories, with indications of some new
applications. Topics discussed include bodies with near-constant surface pressure,
subsonic and supersonic aerodynamics, ship hydrodynamics, slender bodies in Stokes flow,
slender footings in elastic media, and slender moonpools. Mathematical features of
the thin and slender-body approximations are also discussed, especially non-local
convolution terms modelling three-dimensionality in the otherwise two-dimensional near
field, end effects, and the role of the logarithm of the slenderness ratio. 

This review was presented by the author as the \textsc{IMA} Lighthill Memorial Lecture at the British Applied Mathematics Colloquium (\textsc{BAMC}) of 2004. Its posthumous publication now is with the consent of the \textsc{IMA}/\textsc{BAMC}.
      
\end{abstract}

\tableofcontents

\section{Introduction}

This paper mainly concerns fluids streaming past impermeable elongated bodies. This
topic has always been one of the most important in fluid mechanics, and one to which
Sir James Lighthill contributed substantially, from his earliest research days in the 1940s.
However, there are also potential or actual applications in other areas including solid
mechanics. 

Within fluid mechanics, most of this
paper concerns irrotational potential flow of an inviscid incompressible fluid, but again
there are many other fluid-mechanical applications, e.g. to subsonic and supersonic
compressible flows, and to low Reynolds number viscous flows. In all cases, the reference frame adopted is moving with the body and, in this reference frame, the flow is steady and streaming from left to right with unit velocity. The potential flow demands solution of
an exterior boundary-value problem for Laplace's equation, with a Neumann boundary condition
on the body surface.

Even when there is just one such boundary surface, solution of this type of boundary-value
problem for geometrically complex bodies presented major computational difficulties until
relatively late in the 20th century, and hence there was often a resort to approximations
based on simplifying geometrical features of the body. Fortunately, in many real
applications there are indeed such features present, namely the body is elongated in the flow
direction, or ``streamlined''.  The now-common English word ``streamlined''
has a technical origin involving the body geometry conforming to flow
streamlines, and this tells us that there is no element of luck about the presence of
geometrical simplifying features!  Streamlined bodies are usually either thin (approximated
by a plane) or slender (approximated by a line). 

Nowadays, the tools of computational fluid dynamics are such that this type of
boundary-value problem is almost trivially solvable, and one might question the
need to use such approximations. However, there are often other reasons for their
retention. One such reason is that there may be other boundaries -- other bodies,
or more importantly other boundaries such as free surfaces. The problem of flow past a
ship or submarine is one such example, and the nonlinear boundary conditions on
a free-surface that is not known in advance present serious difficulties to any
computational procedure, even with 21st century computers.

\suppressfloats
Just to set that scene, Figure \ref{ship_fig} shows some ship waves computed in a few minutes of
2GHz PC time using a thin-ship approximation \cite{TSL02}. % (Tuck, Scullen and Lazauskas 2002).  
These
computations are for a vessel which approximates in size, shape and
speed the ship in the photograph of Figure
\ref{photo_fig}. Capturing the full complexity of a ship-wave pattern is a
challenge even after the thin-ship approximation has been made --- without that approximation,
compromises have to be made that may be unacceptable in some applications, e.g. to detection.

%
\begin{figure*}
\begin{center}
\includegraphics[bb=40 50 330 220, width=0.8\textwidth,clip=true]{ship.eps}
\caption{Computed ship waves.}
\label{ship_fig}
\end{center}
\end{figure*}
%
%
\begin{figure*}
\begin{center}
\includegraphics[bb=40 50 330 220, width=0.8\textwidth,clip=true]{photo.eps}
\caption{Actual ship waves.}
\label{photo_fig}
\end{center}
\end{figure*}
%

We shall return to problems involving free surfaces and other generalisations later, but
first let us discuss the most straightforward flow situation, which requires solution of a
Neumann problem for Laplace's equation exterior to a single body. For definiteness we shall
call this the ``prototype'' problem.
        
\section{Interior sources}

Among the many ways to solve Neumann boundary-value problems, one of the oldest is to
place singularities inside the body, whose strength is to be determined. We shall only be
concerned with non-lifting bodies here, so these singularities can be assumed to
be sources. This concept generalises the Rankine body, which is generated by one source and
one equal sink. Good numerical methods can then be devised both in two and three dimensions
for arbitrary bodies, with a careful placement of many sources. The
source distribution may be discrete or continuous; let us here assume the latter.
In the most general case,
it is best that the sources lie not too far inside the body, and, in particular, if they
lie on the (inside of the) body surface itself, such methods reduce to ``boundary integral equation''
or ``panel'' methods. 

However, there is a significant class of smooth and not-too-fat bodies for which this
principle of closeness to the body surface can be violated without cost. For most laterally
symmetric bodies the sources can be placed on the plane of symmetry, and for most bodies of
revolution they can be placed on the axis. 

Thus for a body alone in two dimensions we may suggest 
% 
\be
\phi(x,y)=\frac{1}{2\pi}\int m(\xi) \log\sqrt{(x-\xi)^2+y^2}\;d\xi ,
\label{phi2D} 
\ee
%
and for a body of revolution (with $r$ as a polar coordinate normal to the axis)
% 
\be
\phi(x,r)=-\frac{1}{4\pi}\int \frac{m(\xi)}{\sqrt{(x-\xi)^2+r^2}}\;d\xi .
\label{phi3D} 
\ee
%
The integrals in each case take place over a suitable range of $\xi$ values inside the body.
For finite bodies, we shall often take this range to be $(-1,1)$, but in principle
$(-\infty,\infty)$ can be assumed, with truncation to a finite range if $m(\xi)$ vanishes
outside that range. Here
$\phi$ is the potential for the disturbance to a unit-magnitude
$x$-directed stream due to the body, such that the total fluid velocity is
${\bf q}=\nabla (x+\phi)$. The only remaining requirement is that
the Neumann boundary condition ${\bf q}.{\bf n}=0$ is satisfied on the body surface, leading
to an integral equation for the source-strength function 
$m(\xi)$. 

For example, in the 3D case on $(-1,1)$, the linearly-varying function 
% 
\be
m(\xi)=-2\pi \epsilon^2\;\xi
\ee
%
generates the exact solution for flow over a prolate spheroid which extends somewhat beyond the
sources, to 
$|x|=\zeta$ with $\zeta>1$, and has a maximum radius $r=\sqrt{\zeta^2-1}$, where
$\epsilon^2=-\left[Q_1'(\zeta)\right]^{-1}$, with 
%   
\be
Q_1(\zeta)=\frac{1}{2}\zeta\log\frac{\zeta+1}{\zeta-1}-1
\label{qleg}
\ee
%
a Legendre function of the second kind. We shall further
explore this exact solution later. For small
$\epsilon$ we have $\zeta \approx 1+\epsilon^2/2$, and the spheroid is slender, with
maximum radius/half-length approximately equal to $\epsilon$. 

One primitive but often quite successful version of slender-body theory simply
says the following. When the body is expanding its cross-section area, each unit-length
section occupies an extra volume equal to the rate of increase in section area, and that is
volume lost to the stream. To conserve mass, there must therefore be an outward flux 
or
source strength equal to the stream magnitude $U$ times that volume. 
Hence $m(\xi)$ is approximately equal to $U$ times the $x$-wise rate of change of body 
section area at station $x=\xi$. Thus, for a body of revolution $r=f(x)$ 
with section area $\pi f^2$ in a unit stream $U=1$, we have
% 
\be
m(\xi)=2\pi f(\xi)f'(\xi) .
\label{m3D}
\ee
%
In 2D the section area is replaced by body thickness, so for a symmetric 2D 
body
$y=\pm f(x)$ with thickness
$2f$ in a unit stream we have
% 
\be
m(\xi)=2f'(\xi) .
\label{m2D}
\ee
%    

Once $m(\xi)$ is so specified, in principle there is no need for further approximation,
and the flow is fully determined. However, let us explore systematic theories in which not
only are the source strengths correctly determined by the above formula to a leading-order
approximation, but further consistent approximations are
made in determining flow quantities of interest such as the pressure distribution on the
body.
  
\section{Thin symmetric bodies in 2D}

A consistent thin-body theory for the symmetric body $y=\pm f(x)$ demands that we consider
the limit as
$f(x)\to0$, in which the body shrinks to the plane $y=0$. The limiting form of the Neumann
boundary condition is then
% 
\be
\phi_y(x,0_\pm)= \pm f'(x)
\ee
%
on the top and bottom sides of the limiting plane boundary $y=0$. This boundary
condition is satisfied exactly by the expression (\ref{phi2D}), providing
$m(\xi)$ is given by (\ref{m2D}).

The main output is then the pressure distribution (excess over atmospheric) on the body,
which is given (neglecting second-order terms) by Bernoulli's equation as 
% 
\be
p(x)=-\phi_x(x,0_\pm) .
\ee
%
Both the fluid density and the stream velocity have been normalised to unity; the
non-dimensional quantity
$p(x)$ used here is then half of the usual pressure coefficient. Note that not only have we
neglected the square of the lateral velocity
$\phi_y$, but also we have consistently evaluated the pressure on the limiting plane boundary
$y=0_\pm$ rather than the actual boundary $y=\pm f$.  Then using (\ref{phi2D}) and
(\ref{m2D}) we find that the pressure is the Hilbert transform of the body slope, namely
% 
\be
p(x)=-\frac{1}{\pi}\int \frac{f'(\xi)}{x-\xi}d\xi .
\label{hilb1}
\ee
%
This is a ``small'' pressure disturbance, of the same (first-order) magnitude as the body
thickness $f$. The relationship (\ref{hilb1}) between shape $f$ and pressure $p$ is
linear, and if $f=O(\epsilon)$ for some small parameter $\epsilon$, then also
$p=O(\epsilon)$, with error $O(\epsilon^2)$.

For example, on $(-1,1)$ the elliptic cylinder 
% 
\be
f(x)=\epsilon\sqrt{1-x^2}
\label{ellcyl}
\ee
%
gives, according to (\ref{hilb1}), the constant negative pressure $p=-\epsilon$. This is
clearly not a good approximation near the ends of the body (where we should find
stagnation pressure $p=+1/2$), a matter to which we shall return later.

There are many properties of Hilbert transforms that can be
used to generate interesting results for pressures on thin bodies. For example, since an
analytic inverse Hilbert transform is known, we can immediately write down the body slope
generating a given pressure distribution. On the full infinite range, we have
% 
\be
f'(x)=\frac{1}{\pi}\int_{-\infty}^\infty \frac{p(\xi)}{x-\xi}d\xi
\label{hilb4}
\ee
%
and 
on the finite interval
$(-1,1)$, we have
% 
\be
f'(x)=\frac{1}{\pi\sqrt{1-x^2}}\int_{-1}^1 \frac{\sqrt{1-\xi^2}\;p(\xi)}{x-\xi}d\xi
\label{hilb5}
\ee
%
(assuming $f=0$ for all $|x|\ge 1$).


\section{Slender bodies of revolution}

Not everything follows as easily for slender bodies as it does for thin bodies!
Essentially this is because the limiting line is a one-dimensional object, and therefore
cannot serve as a boundary for the limiting 3D boundary-value problem. Instead of
letting the body shrink fully down to the limiting line
$r=0$, we must instead match with an ``inner'' flow for small but nonzero $r$.  

We therefore need the behaviour of the potential 
(\ref{phi3D}) for small $r$, namely
% 
\be
\phi(x,r)=\frac{m(x)}{2\pi}\log r + b(x) +O(m r^2\log r) ,
\label{sbt} 
\ee
%
where
% 
\be
b(x)=-\frac{1}{4\pi}\int m'(\xi)\;{\rm sgn}(x-\xi)\log2|x-\xi|\;d\xi .
\label{sbtb1}
\ee
%
The logarithmic part of (\ref{sbt}) is easy to understand -- a line of 3D
sources behaves near that line as if it was a 2D line source with the local source strength.
However, if the source strength $m(x)$ varies with $x$, then there will be non-zero and
non-trivial axial ($x$-wise) fluid motion which is determined by the non-local term $b(x)$.
Equation (\ref{sbtb1}) applies both to finite and infinitely-long bodies. If the body is
of finite length, e.g. occupies $|x|<1$, a careful integration by parts yields the
alternative formula
% 
\be
b(x)=-\frac{m(x)}{2\pi}\log2\sqrt{1-x^2}
+\frac{1}{4\pi}\int_{-1}^1 \frac{m(x)-m(\xi)}{|x-\xi|}\;d\xi .
\label{sbtb2}
\ee
%
Again, substitution of (\ref{sbt}) in the approximate Neumann boundary
condition
$\phi_r=f'(x)$ on $r=f(x)$ verifies the slender-body approximation (\ref{m3D})
for the source strength $m(x)$.

Note that $m=O(\epsilon^2)$ if $f=O(\epsilon)$,
i.e. if $\epsilon$ measures slenderness, and hence the full disturbance potential
given by (\ref{phi3D}) is also of second order, with $\phi=O(\epsilon^2)$ . The inner
approximation (\ref{sbt}) to $\phi$ with
$r=O(\epsilon)$ is strictly then of order $\epsilon^2\log\epsilon$. This type of extra
``$\log\epsilon$'' factor is common in slender-body theory;  for some
 order-of-magnitude estimation purposes it can be ignored, and we then view the disturbance due
to the body as simply ``second order'', even near the body. Similarly, it can be shown that
the error in the slender-body approximation (\ref{m3D}) to the source distribution is of
order
$\epsilon^4\log\epsilon$ or (loosely) ``fourth order'', and the error in the formula
(\ref{sbt}) for the disturbance potential near the body is then of order
$\epsilon^4\log^2\epsilon$, again loosely fourth order. We shall return to matters
involving these logarithms later.

%\end{document}
Now the Bernoulli equation determines the
pressure as
% 
\be
p=-\phi_x-\frac{1}{2} \phi_r^2 ,
\ee
%
where we have neglected the (fourth-order) term $\phi_x^2$, but now (in contrast to
the 2D case) cannot neglect the (second-order) term $\phi_r^2$. The whole pressure is then of
second order. Substituting the inner approximation (\ref{sbt}) gives, on the body $r=f(x)$,
% 
\be
p(x)=-[f(x)f'(x)]'\log f(x)-b'(x)-\frac{1}{2}f'(x)^2 .
\ee
%  
For finite-length bodies on $(-1,1)$ it is convenient to use (\ref{sbtb2}), with the
definitions
$A(x)=f(x)f'(x)$ and 
% 
\be
B(x)=\frac{1}{2}\int_{-1}^1 \frac{A(x)-A(\xi)}{|x-\xi|}\;d\xi,
\label{sbtb3}
\ee
% 
to write the pressure as
%   
\be
p(x)=-A'(x)\log\frac{f(x)}{2\sqrt{1-x^2}}-B'(x)
-\frac{xA(x)}{1-x^2}-\frac{1}{2}f'(x)^2 .
\label{p16}
\ee
%
This expression is in some ways equivalent to the Hilbert transform 
(\ref{hilb1}) for the corresponding thin-body pressure, and indeed the (linear) integral
transform (\ref{sbtb3}) connecting $A(x)$ and $B(x)$ plays a similar non-local role.
However, now there are significant local terms as well, and the full relationship (\ref{p16})
between body geometry and pressure distribution is not linear.

The approximation (\ref{p16}) clearly breaks down at the ends $x=\pm 1$, where it predicts a
(positive) infinite pressure. For most finite bodies, the ends will be stagnation points
where we expect $p=1/2$. It should of course be noted that this is not as dramatic a
failure as appears at first sight. After all, what (\ref{p16}) is saying is that the
pressure on a slender body is small, of second order in $\epsilon$. On such a scale, the
finite stagnation pressure appears effectively infinite. 

However, both in 2D and 3D it is also possible to construct uniformly valid approximations,
and Lighthill (1949) was one of the first to do this in 2D. For slender bodies of
revolution
$r=f(x)$ this simply involves dividing the pressure given by (\ref{p16}) by    
%
\be
1+\left(\frac{xf(x)}{1-x^2}\right)^2 ,
\label{endcorr} 
\ee
%
which is such as to guarantee stagnation pressure $p=1/2$ at $x=\pm 1$, but makes a
negligible $O(\epsilon^4\log\epsilon)$ change if $|x|$ is not near $1$.

%
\begin{figure*}[t]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(125,90)
\put(1,2){\includegraphics[bb=15 10 350 250,width=0.9\textwidth,clip=true]{press_exact.eps}}
\put(0,46){\makebox(0,0){$p$}}
\put(66,0){\makebox(0,0){$x$}}
\end{picture}
\caption{Exact non-dimensional pressure $p(x)$ on ellipse and spheroid, each with thickness/length
$\epsilon=0.1$.}
\label{exact_fig}
\end{center}
\end{figure*}
%%%%  
%%%%  THE EPS FILE ABOVE WILL NOT COMPILE WITH JNN PCTEX CODE.
%%%%  THE EPS FILE USED BELOW WAS CONVERTED FROM ERNIE'S JPG FILE
%%%%
%\begin{figure*}[t]
%\begin{center}
%\includegraphics[width=0.9\textwidth]{press_exact_copy2.eps}
%\caption{Exact non-dimensional pressure on ellipse and spheroid, each with thickness/length
%$\epsilon=0.1$.}
%\label{exact_fig}
%\end{center}
%\end{figure*}
%
 
\section{Exact pressure on ellipses and spheroids}

If either the elliptic cylinder 
% 
\be
y=\pm \epsilon\sqrt{1-x^2}
\ee
%
or the spheroid
% 
\be
r=\epsilon\sqrt{1-x^2}
\ee
%
is placed in a unit-magnitude
stream, the velocity magnitude on its surface is given (exactly) by
%   
\be
q=\frac{1+\lambda}{\sqrt{1+\epsilon^2x^2/(1-x^2)}} \, ,
\ee
%
where $1+\lambda$ is the (maximum) velocity, namely that at $x=0$. This result follows from
the exact solutions \cite{Lamb32}, %(Lamb 1932), 
or simply by careful tangential differentiation of 
the velocity potential $\phi$, which is (exactly) equal to $(1+\lambda)x$ 
on these bodies \cite{Tulin98}. The corresponding (non-dimensional) pressure disturbance $p$ is  
(exactly)   
%   
\bea
p(x)&=&\frac{1}{2}-\frac{1}{2}q^2
\nonumber\\
&=&\frac{-\lambda-\frac{1}{2}\lambda^2
+\frac{1}{2}\epsilon^2x^2/(1-x^2)}{1+\epsilon^2x^2/(1-x^2)} \, .
\label{4}
\eea
%
 
%
\begin{figure*}[t]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(125,90)
\put(1,2){\includegraphics[bb=15 10 350 250,width=0.9\textwidth,clip=true]{press2D.eps}}
\put(0,46){\makebox(0,0){$p$}}
\put(67,0){\makebox(0,0){$x$}}
\end{picture}
\caption{Pressure $p(x)$ on ellipse with thickness/length $\epsilon=0.1$.}
\label{ellipse_fig}
\end{center}
\end{figure*}
%

Although the same formula (\ref{4}) yields the surface pressure on ellipses in 2D and
spheroids in 3D, the maximum velocity $1+\lambda$ takes
different values in each case. For ellipses of width/length ratio $\epsilon$ in 2D flow
we have simply
%   
\be
\lambda=\epsilon\;.
\ee
%
The exact result in 3D for spheroids of diameter/length ratio $\epsilon$ is a little more
complicated, namely
%   
\be
\lambda=-\frac{Q_1(\zeta)}{\zeta Q_1'(\zeta)} \, ,
\label{7}
\ee
%
where $\zeta=(1-\epsilon^2)^{-1/2}$ and $Q_1(\zeta)$ is 
the Legendre function defined in (\ref{qleg}).

Figure \ref{exact_fig} shows example computations of
$p(x)$ in $x\ge 0$, for
$\epsilon=0.1$. Although these exact pressures eventually return to the (maximum
positive) stagnation value
$p=1/2$ at the extreme end $x=1$, they take small negative values
over more than 90\% of the length. 

%
\begin{figure*}[t]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(125,90)
\put(1,2){\includegraphics[bb=15 10 350 250,width=0.9\textwidth,clip=true]{press3D.eps}}
\put(0,46){\makebox(0,0){$p$}}
\put(66,0){\makebox(0,0){$x$}}
\end{picture}
\caption{Pressure $p(x)$ on spheroid with slenderness $\epsilon=0.1$.}
\label{spheroid_fig}
\end{center}
\end{figure*}
%

\section{Thin ellipses and slender spheroids}

We are mainly concerned with thin 2D or slender 3D bodies, for which we need a
small-$\epsilon$ approximation. 
Since such bodies increase the stream velocity by only a
small amount,   
$\lambda$ is small when $\epsilon$ is small; in fact it is {\em at least\/} as small as
$\epsilon$ itself. Hence even without further information about the dependence of
$\lambda$ on $\epsilon$, we can approximate the denominator of
(\ref{4}) by unity, to give  (in both 2D and 3D)
%   
\be
p(x) = -\lambda-\frac{1}{2}\;\lambda^2 +\frac{1}{2}\;\frac{\epsilon^2x^2}{1-x^2}\;,
\label{9}
\ee
%
with error of order at most $\epsilon^3$.  But
then, since
$\lambda$ is small, we can also neglect the term in
$\lambda^2$ relative to that in $\lambda$, giving
%   
\be
p(x) = -\lambda +\frac{1}{2}\;\frac{\epsilon^2x^2}{1-x^2}\;,
\label{10}
\ee
%
now with  errors of order at most the maximum of $\lambda^2$ and $\epsilon^3$.

This is most obvious in 2D for ellipses, since
$\lambda=\epsilon$ exactly and the error in (\ref{10}) is of order $\epsilon^2$.
But in that case the second term of (\ref{10}) is also small within the above error
range, and can and should be neglected relative to the first term (so long as $|x|$ is not too
close to unity), leading to the constant pressure 
%   
\be
p(x) = -\epsilon
\ee
%
on thin ellipses, a result with order $\epsilon^2$ error, as already obtained by the thin-body
theory.
If we revert to (\ref{9}), we obtain a more accurate but non-constant result for
ellipses, of the form
%   
\be
p(x)= -\epsilon -\frac{1}{2}\;\epsilon^2
+\frac{1}{2}\;\frac{\epsilon^2x^2}{1-x^2}\;,
\ee
%
now with formal error of order $\epsilon^3$, but which happens to be exact at $x=0$.
Both approximations are displayed (as the horizontal line ``FIRST APPROX'' and the curve
``SECOND APPROX'') relative to the ``EXACT'' solution, in Figure \ref{ellipse_fig} for
$\epsilon=0.1$. 

Again the 3D result for spheroids is somewhat more complicated. Now 
by small-$\epsilon$ expansion of the exact
result (\ref{7}) we find a much smaller relative increase in velocity, namely
%   
\be
\lambda=-\epsilon^2\left[\log\frac{\epsilon}{2}+1\right]
+O(\epsilon^4\log^2\epsilon) ,
\ee
%
so (\ref{10}) becomes
%   
\be
p(x)=\epsilon^2\left[\log\frac{\epsilon}{2}+1\right]
+\frac{1}{2}\;\frac{\epsilon^2x^2}{1-x^2} \, ,
\label{14}
\ee
%
which has an error of order $\epsilon^4\log^2\epsilon$. Equation (\ref{14}) is thus a
very accurate approximation so long as $|x|$ is not too close to unity, but does 
{\em not\/} predict constant pressure on slender spheroids. It is displayed for
$\epsilon=0.1$ as the ``SECOND APPROX'' curve of Figure \ref{spheroid_fig}, relative to the
``EXACT'' curve. 

The approximation (\ref{14}) also follows from the general slender-body formula (\ref{p16})
noting that for the spheroid $f(x)=\epsilon\sqrt{1-x^2}$, we have $A(x)=B(x)=-\epsilon^2x$,
and the argument of the logarithm in (\ref{p16}) is $\epsilon/2$, so the first two terms of
(\ref{p16}) yield the constant part of (\ref{14}), while the remaining
two terms yield the non-constant part. The denominator of (\ref{4}) is also restored if the
end correction factor (\ref{endcorr}) is applied. 

Is the pressure on a slender spheroid in 3D then not constant to a leading order of
approximation, as it was on a thin ellipse in 2D? Well, yes in a sense the pressure can be said
to be constant, but only as a very crude first approximation. If we drop the second
term of (\ref{14}), we leave a constant pressure 
%   
\be
p(x)= \epsilon^2\left[\log\frac{\epsilon}{2}+1\right]
\label{15}
\ee
%
on a slender spheroid, but we have now made an absolute error of order $\epsilon^2$. The
constant term (\ref{15}) that then remains is of order $\epsilon^2\log\epsilon$, so we have
made a relative error of order
$1/\log\epsilon$, and
$\epsilon$ must be exceedingly small for this to be justified. The approximation
(\ref{15}) at $\epsilon=0.1$ is shown as the horizontal line labelled ``FIRST APPROX'' in
Figure \ref{spheroid_fig}.

The danger of neglecting order $\epsilon^2$ terms relative to order
$\epsilon^2\log\epsilon$ terms is illustrated by the fact that if we do wish to drop
the second term of (\ref{14}), consistently we should also drop the ``$+1$'' inside the
square bracket of the first term and even the factor ``$\;1/2\;$'' inside the logarithm,
since these terms also contribute order
$\epsilon^2$ to the final answer. In fact, with the same formal order $\epsilon^2$
error, (\ref{15}) can be replaced by 
%   
\be
p(x)= \epsilon^2\big[\log\epsilon+k\big]
\label{152}
\ee
%
for any finite constant $k$. It is hardly comfortable to have
to deal with an approximation which has such a large element of non-uniqueness.

However, one redeeming feature of the approximation (\ref{15}), or of
(\ref{152}) with $k=1-\log2\approx0.31$, is that it is a good approximation
(with much smaller formal error, of order
$\epsilon^4\log^2\epsilon$) to the pressure at the centre station $x=0$, where the
second term of (\ref{14}) vanishes. Hence the FIRST APPROX and SECOND APPROX agree at
$x=0$ in Figure \ref{spheroid_fig}. This enables a quantitative measure of constancy of
pressure. In general as $x$ varies, the pressure coefficient rises from its (small negative)
value at $x=0$, and we may ask how small must
$\epsilon$ be in order that this rise is small relative to the pressure magnitude at
$x=0$.  

For example, suppose we fix attention on a point where $x^2=1/2$, or
$x\approx 0.7$. Then the absolute rise in $p$ between the centre station and
this station is
$\epsilon^2/2$, both for ellipses and spheroids. However, this is relatively much more
significant in the latter case than the former. Thus in order for the pressure
magnitude at this point to be at most
$10\%$ different from that at
$x=0$, it is only necessary for $\epsilon$ to be smaller than $0.2$ for ellipses, but
$\epsilon$ must be smaller than $2e^{-6}\approx0.005$ for spheroids. This comparison is
even more severe closer to the ends, e.g. if we require such pressure constancy 
as far from the centre as $x^2=2/3$ or
$x\approx 0.8$, where the absolute rise is $\epsilon^2$, then we require
$\epsilon<0.1$ for ellipses, but a quite unrealistic $\epsilon<2e^{-11}\approx0.000033$
for spheroids. 


\section{Slender bodies with near-constant pressure}

If the pressure is not constant on a spheroid, is there any body of revolution on which the
pressure is constant?  The exact answer is certainly
no, but we only require a slender-body approximation. Specifically we seek shapes $f(x)$ such
that the pressure $p(x)$ given by (\ref{p16}) takes a constant (small negative) value $p_0$,
for all
$x$ not close to the ends. This can be done in a straightforward numerical manner by forcing
$p(x_n)=p_0$, at a set of $N$ collocation points 
$x=x_n,\; n=1,\dots,N$, which can lie in $0\le x<1$ using symmetry. If we specify $f(x)$ by a
set of
$N$ parameters, this yields a set of
$N$ nonlinear algebraic equations in the $N$ unknown parameters, which
can be solved by Newton's method.

A convenient parametrisation of the body shape is obtained by the Legendre series
%
\be
A(x)=\sum_{n=1}^N a_n P_{2n-1}(x) ,
\label{19} 
\ee
%  
where $P_N(t)$ is the Legendre polynomial and $a_n$ are the $N$ coefficients to
be determined. Then 
%
\be
B(x)=\sum_{n=1}^N a_n \sigma_{2n-1} P_{2n-1}(x),
\label{20} 
\ee
% 
where $\sigma_{n}=1+1/2+1/3+\dots+1/n$, and 
%
\be
\frac{1}{2}f(x)^2=\int_1^xA(x)dx
=\sum_{n=1}^N \frac{a_n}{4n-1} \left[P_{2n}(x)-P_{2n-2}(x)\right] ,
\label{21} 
\ee
%  
so all quantities required for (\ref{p16}) can be computed readily, given the
coefficients $a_n$. The spheroid is the particular case $N=1$, with $a_1=-\epsilon^2$.

%
\begin{figure*}[t]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(125,90)
\put(1,2){\includegraphics[bb=15 10 350 250,width=0.9\textwidth,clip=true]{p_optimal22.eps}}
\put(0,46){\makebox(0,0){$p$}}
\put(66,0){\makebox(0,0){$x$}}
\end{picture}
\caption{Pressure $p(x)$ on ``constant'' pressure body with slenderness
$\epsilon=0.1$, compared to exact pressure on spheroid with the same slenderness.}
\label{p_body_fig}
\end{center}
\end{figure*}
%

The numerical results for large $N$ depend to a certain extent on the choice of the
collocation points
$x_n$, and ultimate refinement of this choice and of the present numerical method was not
sought. A satisfactory grid was $x_n=(n-1)/(N+1)$, for which convergence (e.g. of
$f(0)=\epsilon$ as $N$ increases) to about 4 figures was achieved by $N=22$. The coefficients
$a_n$ decrease in size quite rapidly, with, for example, $a_{20}$ generally of the order of
$10^{-8}$ compared to $a_1$ of the order of $10^{-2}$. The final pressure is constant
to more than four figures not only at the grid points, but also for all $x$ less than 
$x_{N-1}$. It is then constant to three figures (falling slightly) between $x_{N-1}$
and $x_N$, before increasing rapidly for $x>x_N$, i.e. for $x>0.91$ at
$N=22$. Even so, at $N=22$ the pressure remains within $10\%$ of
$p_0$ up to $x=0.93$. 

It is clearly impossible to maintain constancy of pressure right to the ends. This is
true whether or not the uniformising factor (\ref{endcorr}) is used; with it, the 
extreme end pressure is finite, without it infinite, but in neither case is the end pressure small
and negative. In practice, it made very little difference to the numerical
results whether (\ref{endcorr}) was or was not used, and loss of pressure constancy in about
the last 7\% of the length seems inevitable. Further work is needed to assess whether
this breakdown range can be reduced. In any case, the present extent of constancy of
pressure far exceeds that for a spheroid, as indicated by the comparison in Figure
\ref{p_body_fig} for $\epsilon=0.1$.


%
\begin{figure*}[t]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(125,90)
\put(1,2){\includegraphics[bb=15 10 350 250,width=0.9\textwidth,clip=true]{r_optimal.eps}}
\put(0,46){\makebox(0,0){$r$}}
\put(66,0){\makebox(0,0){$x$}}
\end{picture}
\caption{Shape $r(x)$ of ``constant'' pressure body compared to spheroid.}
\label{r_body_fig}
\end{center}
\end{figure*}
%

Figure \ref{r_body_fig} shows the final body shape $r=f(x)$ at $p_0=-0.018$, which generates a
body with width/length ratio $\epsilon=0.10$, compared to a spheroid of the same slenderness.
As we increase $|p_0|$, the body becomes less slender, and eventually the procedure
fails. This is signalled first by a loss of convexity at about $p_0=-0.030$ or 
$\epsilon=0.14$, with
$f'(0)>0$. All ``constant'' pressure bodies less slender than this are dumbell-shaped. 

Within the whole range of convex bodies, the computed relation between slenderness
$\epsilon$ and pressure
$p=p_0$ is quite accurately represented by (\ref{152}) with $k=0.50$,
which is to be compared to $k=0.31$ for the
spheroid result (\ref{15}).  
 
\section{Relationship to long cavities}

Although in principle the present work is independent of the particular application to
supercavitation, this area is one where there has been a significant amount of
experimental, empirical and computational effort; see, for example, Tulin \cite{Tulin98} and
several review articles in Michel \cite{Michel01}. In this application
the closed ``body'' of interest is a combination of a relatively short rigid ``cavitator''
or nose, which has a prescribed shape and on which the pressure is not necessarily
constant, a much longer attached cavity, on which the pressure is required to be
constant, and another short region of cavity collapse at
the trailing end. In supercavitation the constant negative non-dimensional pressure is written as
$p_0=-\sigma/2$, where
$\sigma$ is the ``cavitation number''.

Hence the present theory is relevant to the slender-body limit in which the
cavity length far exceeds the cavitator length, and the cavitation number is small. 
In a sense this is no different from the numerical consequences in the previous
section of seeking approximate solutions for ``constant'' pressure bodies, where we
must allow the pressure to be non-constant near the ends of the body. Those end
regions can then act as surrogates for the cavitator and collapse region, and indeed Reichardt's early
study of supercavitation \cite{Reichardt46} has  already used this approach.

Significant interest in supercavitation lies in relationships between cavitator geometry
and cavity dimensions. There are empirical formulae indicating that the length and
width of the cavity (relative to the length of the cavitator) are each proportional to
the square root of a drag coefficient which characterises the geometry of the cavitator.
Hence the slenderness $\epsilon$ or width/length ratio of the cavitator-cavity combination is (at
least according to these formulae) independent of the drag coefficient, and thus independent of
the shape of the cavitator, depending only on the cavitation number $\sigma$.

%
\begin{figure*}[t]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(125,90)
\put(1,2){\includegraphics[bb=15 10 350 250,width=0.9\textwidth,clip=true]{s_e.eps}}
\put(0,46){\makebox(0,0){$\sigma$}}
\put(66,0){\makebox(0,0){$\epsilon$}}
\end{picture}
\caption{Cavitation number $\sigma$ versus slenderness $\epsilon$ from (\ref{sigeps}) with
$k=0.3,0.5,0.7$,  compared to supercavitation results of
May \cite{May75}, Savchenko \cite{Savchenko01}  and Garabedian \cite{Garabedian56}.
 }
\label{s_e_fig1}
\end{center}
\end{figure*}
%

For two-dimensional supercavitation, the thin-body theory of Tulin \cite{Tulin53} indicates that
$\sigma=2\epsilon$, and that the cavity is asymptotically elliptical. This agrees with the
conclusion that, to leading order in
$\epsilon$, the pressure on a thin ellipse is constant and of magnitude $p_0=-\epsilon$, with a
consistently small $O(\epsilon^2)$ error.   

The equivalent result for axisymmetric three-dimensional supercavitation is a little more
problematic. If we are prepared to neglect terms with relative error $O(1/\log\epsilon)$, then
the cavity is asymptotically spheroidal and (\ref{152}) implies that
%
\be
\sigma=-2\epsilon^2\big(\log\epsilon+k\big) ,
\label{sigeps}
\ee
%
for some constant $k$. This constant $k$ is formally arbitrary, although if the cavity
were exactly spheroidal, $k=0.31$ might be preferred. However, the cavity is only spheroidal to
a very crude approximation, and if we abandon the spheroidal assumption, then there is no formal
asymptotic justification for (\ref{sigeps}) at any $k$. On the other hand, if
the cavity asymptotes to one of the ``constant''-pressure bodies constructed in the previous
section, then since we have found that these bodies have a pressure close to 
(\ref{152}) with
$k=0.50$, a case can be made for use of (\ref{sigeps}) with $k\approx0.5$.

Figure \ref{s_e_fig1} shows graphs of $\sigma$ versus $\epsilon$ from (\ref{sigeps}) for
$k=0.3,0.5,0.7$, compared to supercavitation results of May \cite{May75}, Savchenko \cite{Savchenko01} 
and Garabedian \cite{Garabedian56}.  The May and Savchenko curves are empirical fits to experimental
measurements. The May curve is very close to the present $k=0.5$ curve. The Savchenko curve
is  close to
$k=0.7$ at the more slender end of the experimental range, and to $k=0.3$ at the
less slender end, with $k=0.5$ being a reasonable compromise value. The Garabedian curve is a
theoretical formula 
%
%\be
%\epsilon^2=-\frac{\sigma}{\log\sigma}
%\label{garab}
%\ee
%
based on neglect of terms with 
relative error
$O(1/\log\sigma)$, which is almost (but not quite!) the same thing as neglect of terms with
relative error
$O(1/\log\epsilon)$, and seems to give results between 
$k=0.6$ and $k=0.7$, which are also very close to an empirical formula
of Reichardt \cite{Reichardt46}. Other experimental, empirical or computational results are quoted in
articles in Michel \cite{Michel01}, which also fall in the approximate range
$0.3<k<0.7$ of Figure
\ref{s_e_fig1}.  
  
%
%\begin{figure*}[t]
%\begin{center}
%\includegraphics[width=0.9\textwidth]{s_e.eps}
%\caption{$\sigma$ versus $\epsilon$ from Haese computations.(wrong figure as space filler)}
%\label{kepshaese_fig}
%\end{center}
%\end{figure*}
%

%Figure \ref{kepshaese_fig} is based on recent direct computations by Haese 2002, and also
%suggests $k\approx0.5$.        

\section{Other applications of slender-body theory}

\subsection{General inner limit and drag}
Let us turn to more general slender-body theories. There are a number of such
theories which use a distribution of 3D or point sources along the $x$-axis, with a varying
strength $m(x)$ which often can be shown to be equal to $US'(x)$ where $U$ is the free
stream velocity (normalised here to unity)
and $S(x)$ is the body section area. If the potential of a unit 3D
source at the origin is written
$G_3(x,y,z)$, we need the limit of such a line distribution as
$r=\sqrt{y^2+z^2}\to 0$, and expect this to be of the form
% 
\be
\int m(\xi) G_3(x-\xi,y,z) d\xi \quad\to\quad m(x)\; G_2(y,z) + b(x),
%+O(m r^2\log r) ,
\label{limit} 
\ee
%
where
% 
\be
b(x)=\int m'(\xi) F(x-\xi) d\xi  .
\label{convol} 
\ee
%
Here $G_2(y,z)$ is the corresponding unit 2D or line source potential in the $(y,z)$
cross-section plane, and
$F(x)$ is a suitable kernel function. At each fixed station $x$, the first term on the right of
(\ref{limit}) is ``local'', depending only on the current value of $m(x)$. If the source
strength $m(x)$ is not constant, there is an additional ``non-local'' term $b(x)$, whose value
at station
$x$ in principle depends on values of the axial rate of change of source strength 
$m'(\xi)$ at all other stations $\xi$, expressed as a convolution integral
(\ref{convol}) with  weighting function
$F(x-\xi)$. 

For the simple prototype problem
considered so far, namely irrotational flow of an incompressible inviscid fluid past a
single fixed impermeable body, we have
% 
\be
G_3(x,y,z)=-\frac{1}{4\pi}\frac{1}{\sqrt{x^2+r^2}} ,
\label{g3} 
\ee
%
% 
\be
G_2(y,z)=\frac{1}{2\pi}\log(r) 
\label{g2} 
\ee
%
and
% 
\be
F(x)=-\frac{1}{4\pi}{\rm sgn}(x)\log2|x| .
\label{fx}
\ee
%
Here the
non-local term captures the essential elliptic character of the prototype boundary-value
problem; what we do at one point of space affects what happens everywhere else.


Generalisations of the prototype problem sometimes involve asymptotic results
similar to (\ref{limit}) with different 3D source potentials
$G_3$ on the left, but the same or similar functions $G_2$ and $F$ on the right.
In particular there are a number of slender-body problems in which the 2D source $G_2$ is
still given by (\ref{g2}) even though $G_3$ and $F$ are different. Physically, this can be
interpreted as saying that although the 3D problem differs from the prototype, that
difference disappears in the 2D near field in the cross-section plane. However, axial
changes in the parameters of that (incompressible) 2D flow induce axial velocities via
$b(x)$ which are non-local and capture the unique three-dimensionality of the problem.

Various important output quantities follow from the above formalism. For example, local
terms (both that already in (\ref{limit}) and others arising from its use to generate the
pressure)   do not contribute to the drag on the body, which is then due to a contribution
$-b'(x)$ to the pressure 
depending on $x$ alone. Hence the drag is $D=\int S'(x)(-b'(x) dx$ where $S(x)$ is the
cross-section area, so after integration by parts,
% 
\be
D=\int dx\; S''(x) \int d\xi\; S''(\xi)\; F(x-\xi) .
\label{drag}
\ee
%
Hence  d'Alembert's paradox
$D=0$ is verified for the prototype problem, and indeed for any generalisation where $F(x)$
remains an odd function of $x$ as in (\ref{fx}). Otherwise, in evaluating the drag by
(\ref{drag}) we can replace $F(x)$ by its even part.

\subsection{Compressibility}
   
Inclusion of compressibility of the fluid provides a direct
generalisation of the prototype problem. Then at free-stream Mach number $M$, the 3D source
function is simply obtained by replacing $r^2$ with $(1-M^2)r^2$ in (\ref{g3}) for the
prototype (incompressible) problem. For subsonic flow with $M<1$ this can always be done, and the
net effect is essentially as if $F(x)$ was still given by (\ref{fx}). The boundary-value
problem remains elliptic, and d'Alembert's paradox $D=0$ still holds.

For supersonic flow with $M>1$, the 3D source potential must vanish outside of the Mach
cone $r=x/\beta$ where $\beta=\sqrt{M^2-1}$, so the range of integration is truncated to
$\xi<x-\beta r$. Then we find as we let $r\to 0$ that the local term is again given by the
incompressible 2D line source (\ref{g2}), while the non-local term has kernel
% 
\be
F(x)=-\frac{1}{2\pi}\log\frac{2}{\beta}x
\label{superfx}
\ee
%
if $x>0$, and zero otherwise. Now although $b(x)$ still describes non-local effects, 
it only carries a downstream
influence, the contribution at station $x$ only depending on $m(\xi)$ for
$\xi\le x$, thus capturing the essential hyperbolic nature of the original 3D boundary-value
problem. Now
$F(x)$ is no longer an odd function, and the drag is
% 
\be
D=-\frac{1}{4\pi}\int dx\; S''(x) \int d\xi\; S''(\xi)\;\log|x-\xi| .
\label{superdrag}
\ee
%
This supersonic result was obtained by von Karman and Moore \cite{KM32}, and improved upon by
Lighthill \cite{Lighthill45}; in fact the supersonic slender-body theory was obtained before 
corresponding incompressible or subsonic slender body theories were derived \cite{Ward55}.

\subsection{Slender ships} 

Another qualitatively straightforward generalisation is to slender ships moving
over a free surface, where the
linearised disturbance potential must satisfy, on the plane $z=0$, the Kelvin boundary condition
% 
\be
g \phi_z+ \phi_{xx}=0 ,
\label{kelvin}
\ee
%  
where $g$ is gravity.  Then the unit point source potential $G_3$ satisfying  (\ref{kelvin})
is given by a very complicated formula \cite[p.~484]{WL60}, involving a
double integral with respect to the wave number and direction of the water waves. However,
again, the local term in (\ref{limit}) is just given by the simple line source (\ref{g2}).
This now reflects the fact that the near-field limit of the Kelvin condition is $\phi_z=0$, so
the plane
$z=0$ is a plane of symmetry not supporting waves. 
This has an immediate local consequence, in that the
apparent source strength is related to the section area of the double body, the ship
section being reflected in
$z=0$. Thus if $S(x)$ now denotes the true submerged ship section area, $m(x)=2S'(x)$. 

However, the wave-like nature of
the original 3D flow is retained in the near field by the non-local term $b(x)$, which now
has the kernel function
% 
\be
F(x)=-\frac{1}{4\pi}{\rm sgn}(x)\log2|x|-\frac{1}{8}
\left[{\bf H_0}(gx)+(2+{\rm sgn}(x))Y_0(g|x|)\right]  ,
\label{fxwave}
\ee
%
where ${\bf H_0}$ is a Struve function and $Y_0$ a Bessel function. The first term of
(\ref{fxwave}) is given by (\ref{fx}) as in the prototype problem, and the remaining terms
are wave-like. If we formally let $g\to\infty$, these wave-like terms disappear, and in
that limit the Kelvin condition becomes (everywhere) the symmetry condition $\phi_z=0$, so
the problem has reduced to the prototype problem for the double body.

Since ${\bf H_0}$ is odd, the even part of the kernel $F(x)$ is $-\frac{1}{4}Y_0(g|x|)$,
and the drag (\ref{drag}) is
% 
\be
D=-\frac{1}{2}\int dx\; S''(x) \int d\xi\; S''(\xi)\; Y_0(g|x-\xi|) ,
\label{ssdrag}
\ee
%  
which is the slender-ship wave-resistance formula obtained independently by Vossers \cite{Vossers62}, Maruo \cite{Maruo62} and Tuck \cite{Tuck63}.

In fact Maruo obtained (\ref{ssdrag}) simply by approximating Michell's thin-ship
wave-resistance integral for small draft \cite{Michell1898}. That is, in ship hydrodynamics, to a leading order
of approximation, slender-body theory is subsumed in thin-body theory.
It is interesting that Michell's integral demands information about the complete offsets
$y=\pm Y(x,z)$ of the ship, whereas the slender-ship formula (\ref{ssdrag}) demands only a
knowledge of the ship's section area $S(x)=2\int Y(x,z)dz$.  If a computer program for
Michell's integral is available, there is no need for a separate
computer program for the slender-ship integral (\ref{ssdrag}), since one can simply run the
thin-ship code for artificial rectangular sections with offsets
$Y(x,z)=S(x)/(2D)$ and an arbitrary small draft $D$.  

However, this subsumption
does not apply at higher orders of approximation. Thus, as with the prototype
problem, the leading-order slender-ship potential is second order in the slenderness or
beam/length ratio $\epsilon$. In the prototype problem, the next correction to the inner-region
potential is of fourth order in $\epsilon$. However, in the presence of a free surface this is no
longer the case, and the next term is larger, of third order in $\epsilon$. Not only that, but it
depends on nonlinear free-surface terms neglected in the Kelvin boundary condition
(\ref{kelvin}). Hence this correction cannot be subsumed in (linear) thin-ship theory. The
effect on the drag is that the slender-ship formula (\ref{ssdrag}) gives a 4th-order result,
which is subject to a nonlinear 5th-order correction. It would be of considerable interest to
compute this correction to the wave resistance, but this does not seem to have been done.    

\subsection{Stokes flows past slender bodies}

We now turn to applications of slender-body theory where the
source strength $m(x)$ is no longer given by the axial rate of change of body-section area. 
One such application is to viscous flow, described by the Stokes
%(4th-order partial differential) 
equations \cite{Tuck64}. Then internal (irrotational) sources alone are not
sufficient to represent the flow about a fixed body, and we also need point
singularities generating rotational flow, and contributing to drag forces. For slender bodies of
revolution $r=f(x)$, the appropriate generators are then axial distributions of both point
sources and point ``Stokeslets''.

It turns out that, in this problem, the source distribution plays a minor role, not contributing
to the leading-order slender-body approximation. If the Stokeslet strength is now written
$m(x)$, satisfaction  of the no-slip boundary condition on $r=f(x)$ demands that  \cite{Tuck64}
%
\be
\frac{m(x)}{2\pi}\left[\log f(x)+\frac{1}{2}\right]+b(x)=-\frac{1}{2} ,
\ee
%
or on $(-1,1)$ that $m(x)$ satisfy the integral equation
%
\be
\frac{m(x)}{2\pi}\left[\log \frac{f(x)}{2\sqrt{1-x^2}}+\frac{1}{2}\right]+
\frac{1}{4\pi}\int_{-1}^1 \frac{m(x)-m(\xi)}{|x-\xi|}\;d\xi
=-\frac{1}{2} .
\label{stokes2}
\ee
%   
Although the Legendre series (\ref{9}) provides a convenient way to discretise (\ref{stokes2})
for numerical purposes, there are a number of serious difficulties, especially at the ends, of a
character similar to that encountered earlier when attempting to force the pressure in the
prototype problem to be everywhere constant. A simple alternative \cite{Tuck68} is to use inverse
methods, in which
$m(x)$ is given and the shape $f(x)$ is determined by (\ref{stokes2}).

\subsection{Solid mechanics for elongated loads}
 
A somewhat similar application in solid mechanics \cite{Kalker72,TM83} is to a ``slender punch'', i.e. a  load applied to a finite elongated region of the plane
free boundary of an elastic half-space. This has applications to elongated footings for
buildings. The similarity to Stokes flow arises from the 4th-order nature of the partial
differential equations describing the problem.

Suppose the punch lies in $|y|< f(x)$, and let $m(x)$ be the (downward)
force per unit length that it exerts. Then the downward displacement $w(x)$ under the punch can
be shown to be proportional to the potential $\phi$ of a line of sources $m(x)$ as in the
prototype problem, i.e. as given in the near field by (\ref{sbt}), evaluated for frictionless
punches at
$r=f(x)/2$.  Specifically
% 
\be
\frac{1}{2}\frac{\mu}{1-\nu}w(x)=\frac{m(x)}{2\pi}\log\left[\frac{1}{2}f(x)\right]
+b(x) ,
\label{punch1}
\ee
%
where
$\mu$ and $\nu$
are the elastic (Lame) constants. If there is adhesion, the point of evaluation differs from
$r=f(x)/2$, but one can then define an ``effective frictionless width'' \cite{TM83}
replacing and increasing the actual punch width $2f(x)$. 

If the load $m(x)$ is given, on a given punch width $2f(x)$, equation (\ref{punch1}) 
with $b(x)$ determined from $m(x)$ by (\ref{sbtb1}) or (\ref{sbtb2}) gives the resulting
displacement
$w(x)$ directly. However, again it is a much more difficult task to solve for the load
$m(x)$ given the displacement
$w(x)$, since 
we then have to solve the integral equation
%
\be
\frac{1}{2}\frac{\mu}{1-\nu}w(x)=\frac{m(x)}{2\pi}\log\left[\frac{f(x)}{4\sqrt{1-x^2}}\right]
+\frac{1}{4\pi}\int_{-1}^1 \frac{m(x)-m(\xi)}{|x-\xi|}\;d\xi
\label{punch2}
\ee
%
for $m(x)$. Equation (\ref{punch2}) is quite similar to  (\ref{stokes2}) for
slender-body Stokes flow, and leads to similar difficulties at the ends \cite{PK77, TM83}.   
  



\subsection{Slender moonpools}

Let me mention a final application of slender-body ideas, even more remote from the
original prototype problem, in that it does not involve a streaming flow, or indeed even a
``body''. The application is to sloshing of water waves in an elongated basin.
If this basin is fully enclosed, as in a swimming pool or lake, there are 
interesting asymptotic issues for slender basins, concerned with the seiching modes of motion
along the basin, but these issues have little in common with the source-distribution models
discussed in the present paper. In particular, if the water level at one station $x$ falls, it
must rise instantly at other stations to conserve mass, and the flow
in the bounded cross-section at station $x$ cannot have a source-like character. 

However, suppose the ``basin'' is not
closed, having access at each station $x$ to an ocean of infinite depth. Then when the water
level at that station falls, 
the outward volume flux so generated can pass to infinity in a source-like
manner. The water level then need not immediately rise at other stations.  

For example, the basin may be a ``moonpool'' \cite{Molin01}, a hole giving access to water
beneath an otherwise rigid boundary. This boundary could be the bottom of the hull of a drilling
ship, or perhaps an ice sheet. Here we idealise the boundary to be a rigid sheet of infinite
extent and zero thickness, and the water to be of infinite extent and depth, and let the moonpool
be elongated in the
$x$-direction.

Now if sloshing is taking place in this moonpool, with a water level varying with
$x$, there will be an apparent source at each station $x$, of some strength $m(x)$ to be
determined, and hence if the moonpool is slender, the flow will appear in the far field to be
generated by a line distribution of point sources of strength $m(x)$ as in (\ref{phi3D}). Note
that there are no waves in the far field, since
the ``free'' surface there is rigid!

On the other hand, if we move into the near field, with $r$ reduced to the
size of the width of the moonpool, we shall see water waves, with a rise or fall of the water
level in the moonpool. Hence we need again the inner limit (\ref{limit}). 
For details of the 2D wave-like inner flow see Tuck and Newman \cite{TN02}.
 
%
\begin{figure*}[t]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(125,90)
\put(1,2){\includegraphics[bb=15 10 350 250,width=0.9\textwidth,clip=true]{moon.eps}}
\put(0,46){\makebox(0,0){$\sigma$}}
\put(66,0){\makebox(0,0){$\epsilon$}}
\end{picture}
\caption{The first three longitudinal moonpool modes.}
\label{moonpool_fig}
\end{center}
\end{figure*}
%

However, what is clear without such detail is that, for moonpools with a
rectangular planform, there is a necessity for the {\em whole\/} disturbance potential $\phi$ in
the near field as given by (\ref{limit}) to have the identical $x$-variation along the pool. That
is, the non-local term $b(x)$ needs to be proportional to the local source strength $m(x)$. If
the proportionality constant is $\mu$, we therefore have to solve an
eigenvalue problem, for the integro-differential equation 
%
\be
\int m'(\xi) F(x-\xi) d\xi = \mu m(x) ,
\ee
%
with $F(x)$ given by (\ref{fx}). For a pool in $(-1,1)$, the truncated Legendre series (\ref{19})
is a convenient tool for numerical solution of this problem, and the
first three eigenvalues in increasing order are given by 
%
\be
2\pi\mu+\log 2=0.2332,1.4437,1.9409,\dots \ .
\ee
%

The first three eigenmodes $m(x)$ are shown in Figure \ref{moonpool_fig}. The lowest-order mode
is a symmetric ``pumping'' mode which does not have an equivalent for a closed pool, as there is a
non-zero net volume of water-surface displacement which would not conserve mass if the pool was
closed. The second (antisymmetric) mode has a nodal point at the centre station $x=0$ and is the
fundamental longitudinal ``sloshing'' mode. The third mode is symmetric with two nodal points,
and is also of a pumping nature. All of these modes seem numerically to have infinite slope at
the ends
$|x|=1$, though this matter has not yet been investigated analytically.

\section{Editorial Note}
This review was prepared for and presented as the IMA Lighthill Memorial Lecture at the British Applied Mathematics Colloquium (\textsc{BAMC}) of 2004. Subsequent to Professor E.O. Tuck's death (11 March 2009),  the editors of the \textsc{ANZIAM} Journal decided to invite the winners of the \textsc{ANZIAM} Medal (\url{www.anziam.org.au/The+ANZIAM+medal}) each to submit a paper for publication in the journal. Dr Y.M. Stokes was asked to look for unpublished work of the late Professor Tuck, who was awarded this medal in 1999, that might be suitable for posthumous publication, perhaps with some editing. This almost complete review article was found in a search of the excellent webpage he maintained and, with the help of Professor Tuck's wife, Mrs Helen Tuck, the source files were located on his computer. The \textsc{IMA}/\textsc{BAMC} was approached and consented to its publication. After a search for an appropriate editor an editorial committee of three was agreed, comprising Professor C.C. Mei, Professor J.N. Newman, and Dr Y.M. Stokes, all former colleagues of Professor Tuck.

Only minor editing was required and the paper is almost exactly as written by Professor Tuck. The conclusions and acknowledgements sections, which in the original draft contained only the words ``to be written'', have been removed. We did not think it possible to write these sections as Professor Tuck would have done. Rather we decided to add this editorial note explaining the origin of this paper and the reason for the lack of the usual conclusions and acknowledgments. Dr Stokes would here like to express her sincere thanks for the editorial assistance of Professors Mei and Newman, without which this paper would not have been published.

We trust that readers will, like us, find the paper interesting and enjoyable, written, as it is, in the much-loved, exuberant presentation style of the late Professor Tuck.


%\section{Conclusions}
%(to be written)

%\paragraph{Acknowledgements}
%(to be written)                  

\iffalse \bibliography{} \else
\begin{thebibliography}{99}
%\section{References}
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\fi

 
\end{document}

