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\newcommand{\GaPQ}  {G_a(P,Q)}
\newcommand{\Gat}   {\widetilde{G_a}}
\newcommand{\GsPQ}  {G_s(P,Q)}
\newcommand{\kgd}   {k\gamma d}
\newcommand{\ndq}   {\frac{\mbox{$\partial$}}{\mbox{$\partial$} n_q}}
\newcommand{\p}     {\mbox{$\partial$}}
\newcommand{\sh}    {^{\shalf}}
\newcommand{\smh}   {^{-\shalf}}
\newcommand{\sumjm} {\sum_{j=1}^{M}}
\newcommand{\tti}   {\rightarrow\infty}
\newcommand{\ttz}   {\rightarrow 0}
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%-----title and author----------------------

\title[Various phenomena in fluid mechanics]{An interesting and seminal work
on various phenomena in fluid mechanics}

\author[A. N. Other, H.-C. Smith and J. Q. Public]%
{A\ls L\ls A\ls N\ns N.\ns O\ls T\ls H\ls E\ls R$^1$%
  \thanks{Present address: Fluid Mech Inc.,
24 The Street, Lagos, Nigeria},\ns
H.\ls-\ls C.\ns S\ls M\ls I\ls T\ls H$^1$\ns 
\and J.\ns Q.\ns P\ls U\ls B\ls L\ls I\ls C$^2$}

\affiliation{$^1$Department of Chemical Engineering, University of America,
Somewhere, IN 12345, USA\\[\affilskip]
$^2$Department of Aerospace and Mechanical Engineering, University of
Camford, Academic Street, Camford, CF3 5QL, UK}

\pubyear{1993}
\volume{538}
\pagerange{119--126}
\date{1 July 1992 and in revised form 28 February 1993}
\setcounter{page}{119}

% -----------------------------

\begin{document}

\maketitle

\begin{abstract}
Using Stokes flow between eccentric counter-rotating cylinders as a prototype
for bounded nearly parallel lubrication flow, we investigate the effect of a
slender recirculation region within the flow field on cross-stream heat or mass
transport in the important limit of high P\'{e}clet number \Pen\ where the
enhancement over pure conduction heat transfer without recirculation is most
pronounced.  The steady enhancement is estimated with a matched asymptotic
expansion to resolve the diffusive boundary layers at the separatrices which
bound the recirculation region.  The enhancement over pure conduction is
shown to vary as $\epsilon\sh$ at infinite \Pen, where $\epsilon\sh$
is the characteristic width of the recirculation region. The enhancement decays from this asymptote as $\Pen\smh$.
\end{abstract}

\section{Introduction}

The use of integral equations to solve exterior problems in linear acoustics,
i.e.\ to solve the Helmholtz equation $(\nabla^2+k^2)\phi=0$ outside
a surface $S$ given that $\phi$ satisfies certain boundary conditions
on $S$, is very common. A good description is provided by \cite{Martin80}. Integral equations have also been used to solve
the two-dimensional Helmholtz equation that arises in water-wave problems
where there is a constant depth variation.  The problem of wave
oscillations in arbitrarily shaped harbours using such techniques has
been examined (see for example Hwang \& Tuck 1970; Lee 1971).

In a recent paper \cite{Linton92} have shown how radiation
and scattering problems for vertical circular cylinders placed on
the centreline of a channel of finite water depth can be solved
efficiently using the multipole method devised originally by Ursell
(1950). This method was also used by \cite{Callan91}
to prove the existence of trapped modes in the vicinity of such a
cylinder at a discrete wavenumber $k<\upi/2d$ where $2d$ is the channel
width.

% NOTE use of \upi in above paragraph and subsequently throughout paper.
% The Greek constant character pi should be upright. See Input
% Guide section 4.2.3. \upi is redefined in the preamble above to use% Computer Modern sloping pi.

Many water-wave/body interaction problems in which the body is a
vertical cylinder with constant cross-section can be simplified by
factoring out the depth dependence. Thus if the boundary conditions
are homogeneous we can write the velocity potential
$\phi(x,y,z,t) =
 \hbox{Re}\{\phi(x,y)\cosh k(z+h)\rmn{e}^{-\rmn{i}\omega t}\}$,
where the $(x,y)$-plane corresponds to the undisturbed free surface and
$z$ is measured vertically upwards with $z=-h$ the bottom of the channel.


Subsequently Callan \etal\/ (1991) proved the existence of, and
computed the wavenumbers for, the circular cross-section case.
It should be noted however that experimental evidence for acoustic
resonances in the case of the circular cylinder is given by Bearman
\& Graham (1980, pp.~231--232).

\mbox{}\cite{Koch83} provided a theory for determining the trapped-mode
frequencies for the thin plate, based on a modification of the Wiener--Hopf
technique. Further interesting results can be found in Williams (1964) 
and Dennis (1985).

The use of channel Green's functions, developed in~\S\,2, allows
the far-field behaviour to be computed in an extremely simple manner,
whilst the integral equation constructed in~\S\,3 enables the trapped
modes to be computed in~\S\,4 and the scattering of an incident plane
wave to be solved in~\S\,5. Section~6 contains comparisons with 
experiments.


\section{Green's functions}

\subsection{Construction of equations}

We are concerned with problems for which the solution, $\phi$, is
either symmetric or antisymmetric about the centreline of the
waveguide, $y=0$. The first step is the construction of a
symmetric and an antisymmetric Green's function, $\GsPQ$ and $\GaPQ$.
Thus we require
\begin{equation}
  (\nabla^2_P+k^2)G_s=(\nabla^2_P+k^2)G_a=0
  \label{Helm}
\end{equation}
in the fluid, where $\bnabla_P$ is a gradient operator, except when $P\equiv Q$,
\begin{equation}
  G_s,G_a\sim 1 / (2\upi)\ln r
  \qquad \mbox{as\ }\qquad r\equiv|P-Q|\rightarrow 0,
  \label{singular}
\end{equation}
\begin{equation}
  \frac{\p G_s}{\p y}=\frac{\p G_a}{\p y}=0
  \qquad \mbox{on\ }\qquad y=d,
  \label{wallbc}
\end{equation}
\begin{equation}
  \frac{\p G_s}{\p y}=0
  \qquad \mbox{on\ }\qquad y=0,
  \label{symbc}
\end{equation}
\begin{equation}
  G_a=0
  \qquad \mbox{on\ }\qquad y=0,
  \label{asymbc}
\end{equation}
and we require $G_s$ and $G_a$ to behave like outgoing waves as
$|x|\tti$.

One way of constructing $G_s$ or $G_a$ is to replace (\ref{Helm})
and (\ref{singular}) by
\begin{equation}
  (\nabla^2_P+k^2)G_s=(\nabla^2_P+k^2)G_a
  = -\delta(x-\xi)\delta(y-\eta)
\end{equation}
and to assume initially that $k$ has a positive imaginary part.


\subsection{Further developments}

Using results from \cite{Linton92} we see that this has the integral
representation 
\begin{equation}
  -\frac{1}{2\upi} \int_0^{\infty} \gamma^{-1}[\rmn{e}^{-k\gamma|y-\eta|}
   + \rmn{e}^{-k\gamma(2d-y-\eta)}] \cos k(x-\xi)t\:\rmn{d} t,
   \qquad 0<y,\quad \eta<d,
\end{equation}
where 
\[
  \gamma(t) = \left\{
    \begin{array}{ll} 
      -\rmn{i}(1-t^2)\sh, & t\le 1 \\[2pt]
      (t^2-1)\sh,         & t>1.
    \end{array} \right. 
\]
In order to satisfy (\ref{symbc}) we add to this the function 
\[
  -\frac{1}{2\upi}
   \pvi B(t)\frac{\cosh k\gamma(d-y)}{\gamma\sinh k\gamma d}
   \cos k(x-\xi)t\:\rmn{d} t 
\]
which satisfies (\ref{Helm}), (\ref{wallbc}) and obtain 
\begin{equation} 
  B(t) = 2\rmn{e}^{-\kgd}\cosh k\gamma(d-\eta). 
\end{equation} 
Thus the function 
\begin{equation} 
  G = -\squart\rmn{i} (H_0(kr)+H_0(kr_1))
    - \frac{1}{\upi} \pvi\frac{\rmn{e}^{-\kgd}}%
    {\gamma\sinh\kgd} \cosh k\gamma(d-y) \cosh k\gamma(d-\eta)
\end{equation} 
satisfies (\ref{Helm})--(\ref{symbc}). By writing this function 
as a single integral which is even in $\gamma$, it follows that 
$G$ is real. Similar ideas have been developed in a variety of ways 
(Keller 1977; Rogallo 1981; van Wijngaarden 1968).


\section{The trapped-mode problem}

The unit normal from $D$ to $\p D$ is {\bit n}$_q=(-y^{\prime}(\theta),
x^{\prime}(\theta))/w(\theta)$.
Now $G_a=\squart Y_0(kr)+\Gat$ where
$r=\{[x(\theta)-x(\psi)]^2 + [y(\theta)-y(\psi)]^2\}\sh$ and $\Gat$ is
regular as $kr\ttz$. In order to evaluate
$\p G_a(\theta,\theta)/\p n_q$ we note that 
\[ 
  \ndq\left(\squart Y_0(kr)\right)\sim\ndq\left(\frac{1}{2\upi} 
  \ln r\right)=\frac{1}{2\upi r^2 w(\theta)}[x^{\prime}(\theta) 
  (y(\theta)-y(\psi))-y^{\prime}(\theta)(x(\theta)-x(\psi))] 
\] 
as $kr\ttz$.  Expanding $x(\psi)$ and $y(\psi)$ about the point 
$\psi=\theta$ then shows that 
\begin{eqnarray}
  \ndq\left(\squart Y_0(kr)\right) & \sim &
    \frac{1}{4\upi w^3(\theta)} 
    [x^{\prime\prime}(\theta)y^{\prime}(\theta)- 
    y^{\prime\prime}(\theta)x^{\prime}(\theta)] \nonumber\\
  & = & \frac{1}{4\upi w^3(\theta)} 
    [\rho^{\prime}(\theta)\rho^{\prime\prime}(\theta) 
    - \rho^2(\theta)-2\rho^{\prime 2}(\theta)] 
    \quad \mbox{as\ }\quad kr\ttz . \label{inteqpt}
\end{eqnarray}


\subsection{Computation}
 
For computational purposes we discretize (\ref{inteqpt}) by dividing 
the interval ($0,\upi$) into $M$ segments. Thus we write 
\begin{equation} 
  \shalf\phi(\psi) = \frac{\upi}{M} \sumjm\phi(\theta_j)\ndq 
  G_a(\psi,\theta_j)\,w(\theta_j),
  \qquad 0<\psi<\upi,
  \label{phipsi}
\end{equation} 
where $\theta_j=(j-\shalf)\upi/M$. Collocating at $\psi=\theta_i$ and 
writing $\phi_i=\phi(\theta_i)$ \etc gives
\begin{equation} 
  \shalf\phi_i = \frac{\upi}{M} \sumjm\phi_j K_{ij}^a w_j,
  \qquad i=1,\,\ldots,\,M,
\end{equation}
where 
\begin{equation} 
  K_{ij}^a = \left\{
    \begin{array}{ll} 
      \p G_a(\theta_i,\theta_j)/\p n_q, & i\neq j \\[2pt]
      \p\Gat(\theta_i,\theta_i)/\p n_q
      + [\rho_i^{\prime}\rho_i^{\prime\prime}-\rho_i^2-2\rho_i^{\prime 2}] 
      / 4\upi w_i^3, & i=j. 
  \end{array} \right. 
\end{equation} 
For a trapped mode, therefore, we require the determinant of the 
$M\times M$ matrix whose elements are 
\[ 
  \delta_{ij}-\frac{2\upi}{M}K_{ij}^a w_j, 
\] 
to be zero.
%
\begin{table} 
  \begin{center} 
  \begin{tabular}{cccc} 
      $a/d$ & $M=4$  & $M=8$   & Callan {\it et al.} \\[3pt] 
       0.1 & 1.56905 & 1.56905 & 1.56904\\ 
       0.3 & 1.50484 & 1.50484 & 1.50484\\ 
       0.5 & 1.39128 & 1.39131 & 1.39131\\ 
       0.7 & 1.32281 & 1.32287 & 1.32288\\ 
       0.9 & 1.34479 & 1.35160 & 1.35185\\ 
  \end{tabular}
  \caption{Values of $kd$ at which trapped modes occur when $\rho(\theta)=a$}
  \end{center} 
\end{table} 
%
Table 1 shows a comparison of results 
obtained from this method using two different truncation parameters 
with accurate values obtained using the method of Callan {\it et al.}\ (1991).

\begin{figure} 
  \vspace{16.5pc}
  \caption{Trapped-mode wavenumbers, $kd$, plotted against $a/d$ for 
    three ellipses:\protect\\
    \hbox{{--}\,{--}\,{--}\,{--}\,{--}}, $b/a=0.5$; ---$\!$---,
    $b/a=1$; ---\,$\cdot$\,---, $b/a=1.5$.} 
\end{figure} 
%
An example of the results that are obtained from our method is given 
in figure~1.
Figure~2\,({\it a},{\it b}) shows shaded contour plots of $\phi$ for these
modes, normalized so that the maximum value of $\phi$ on the body is 1. 
Symmetric (figure 2{\it a\/}) modes are shown, while the antisymmetric ones appear in figure 2({\it b\/}).
%
\begin{figure} 
  \vspace{15pc}
  \caption{Shaded contour plots of the potential $\phi$ for the two trapped 
    modes that exist for an ellipse with $a/d=1.5$, $b/d=0.75$.
    ({\it a}) Symmetric about $x=0$, $kd=0.96$;
    ({\it b}) antisymmetric about $x=0$, $kd=1.398$.} 
\end{figure}


\subsection{Basic properties}

Let
\refstepcounter{equation}
$$
  \rho_l = \lim_{\zeta \rightarrow Z^-_l(x)} \rho(x,\zeta), \quad
  \rho_{\itl{u}} = \lim_{\zeta \rightarrow Z^{+}_u(x)} \rho(x,\zeta)
  \eqno{(\theequation{\itl{a},\itl{b}})}
$$
be the fluid densities immediately below and above the cat's-eyes.  Finally
let $\rho_0$ and $N_0$ be the constant values of the density and the vorticity inside the cat's-eyes, so that
\begin{equation}
  (\rho(x,\zeta),\phi_{\zeta\zeta}(x,\zeta))=(\rho_0,N_0)
  \quad \mbox{for\ } Z_l(x) < \zeta < Z_u(x).
\end{equation}

The Reynolds number \Rey{\it \/} is defined by $u_\tau H/\nu$ ($\nu$ is the kinematic
viscosity), the length given in wall units is denoted by $(\;)_+$, and the
Prandtl number \Pran{\it \/} is set equal to 0.7. In (\ref{Helm}) and (\ref{singular}), $\tau_{ij}$ and $\tau^\theta_j$ are
\ifoldfss    
  \begin{subeqnarray}
    \tau_{ij} & = &
      (\overline{\overline{u}_i \overline{u}_j}
      - \overline{u}_i\overline{u}_j)
      + (\overline{\overline{u}_iu^{SGS}_j
      + u^{SGS}_i\overline{u}_j})
      + \overline{u^{SGS}_iu^{SGS}_j},\\[3pt]
    \tau^\theta_j & = &
      (\overline{\overline{u}_j\overline{\theta}}
      - \overline{u}_j \overline{\theta})
      + (\overline{\overline{u}_j\theta^{SGS}
      + u^{SGS}_j \overline{\theta}})
      + \overline{u^{SGS}_j\theta^{SGS}}.
  \end{subeqnarray}
\else
  % line up the equations by measuring the width of the two sides of both 
  % equations, then adding the difference to the second eqn. 
  % Not as complex as it looks!
  \newlength{\topleft}
  \newlength{\topright}
  \newlength{\bottomleft}
  \newlength{\bottomright}
  \newlength{\rightside} % difference in width on right
  \newlength{\leftside}  % difference in width on left
  \settowidth{\topleft}{$\tau_{ij}$}
  \settowidth{\topright}{$(\overline{\overline{u}_i \overline{u}_j}
      - \overline{u}_i\overline{u}_j)
      + (\overline{\overline{u}_iu^{SGS}_j
      + u^{SGS}_i\overline{u}_j})
      + \overline{u^{SGS}_iu^{SGS}_j},$}
  \settowidth{\bottomleft}{$\tau^\theta_j$}
  \settowidth{\bottomright}{$(\overline{\overline{u}_j\overline{\theta}}
      - \overline{u}_j \overline{\theta})
      + (\overline{\overline{u}_j\theta^{SGS}
      + u^{SGS}_j \overline{\theta}})
      + \overline{u^{SGS}_j\theta^{SGS}}.$}
  \def\leftside{\topleft} \advance\leftside -\bottomleft
  \def\rightside{\topright} \advance\rightside -\bottomright
\stepcounter{equation}
  $$
    \tau_{ij} = 
      (\overline{\overline{u}_i \overline{u}_j}
      - \overline{u}_i\overline{u}_j)
      + (\overline{\overline{u}_iu^{SGS}_j
      + u^{SGS}_i\overline{u}_j})
      + \overline{u^{SGS}_iu^{SGS}_j},
        \eqno{(\theequation{\itl{a}})}
  $$
  \vspace{-0.7\baselineskip}
  $$
    \hskip\leftside 
    \tau^\theta_j = 
      (\overline{\overline{u}_j\overline{\theta}}
      - \overline{u}_j \overline{\theta})
      + (\overline{\overline{u}_j\theta^{SGS}
      + u^{SGS}_j \overline{\theta}})
      + \overline{u^{SGS}_j\theta^{SGS}}.\hskip\rightside
        \eqno{(\theequation{\itl{b}})}
  $$
\fi


\subsubsection{Calculation of the terms}

The first terms in the right-hand side of (3.5{\it a}) and
(3.5{\it b}) are the Leonard terms explicitly calculated by
applying the Gaussian filter in the $x$- and $z$-directions in the
Fourier space.

The interface boundary conditions given by (\ref{Helm}) and (\ref{singular}),
which relate the displacement and stress state of the wall at the mean
interface to the disturbance quantities of the flow, can also be reformulated
in terms of the transformed quantities.  The transformed boundary conditions
are summarized below in a matrix form that is convenient for the subsequent
development of the theory:
%
% NOTE use of \slsQ below for the matrix Q. It has been defined in the
% preamble above to typeset the slanting sans serif Q that is available in
% Computer Modern. It should be the bold slanting sans serif and would be
% substituted by the printer. (See Input Guide section 4.2.4.)
%
\begin{equation}
\setlength{\arraycolsep}{0pt}
\renewcommand{\arraystretch}{1.3}
\slsQ_C = \left[
\begin{array}{ccccc}
  -\omega^{-2}V'_w  &  -(\alpha^t\omega)^{-1}  &  0  &  0  &  0  \\%&  0 \\
  \displaystyle
  \frac{\beta}{\alpha\omega^2}V'_w  &  0  &  0  &  0  &  \rmn{i}\omega^{-1} \\
  \rmn{i}\omega^{-1}  &  0  &  0  &  0  &  0  \\
  \displaystyle
  \rmn{i} R^{-1}_{\delta}(\alpha^t+\omega^{-1}V''_w)  &  0
    & -(\rmn{i}\alpha^tR_\delta)^{-1}  &  0  &  0  \\
  \displaystyle
  \frac{\rmn{i}\beta}{\alpha\omega}R^{-1}_\delta V''_w  &  0  &  0
    &  0  & 0 \\%&  R^{-1}_\delta \\
  (\rmn{i}\alpha^t)^{-1}V'_w  &  (3R^{-1}_{\delta}+c^t(\rmn{i}\alpha^t)^{-1})
    &  0  &  -(\alpha^t)^{-2}R^{-1}_{\delta}  &  0  \\
\end{array}  \right] .
\label{defQc}
\end{equation}
$\biS^t$ is termed the displacement-stress vector and $\slsQ_C$
the flow-wall coupling matrix.  Subscript $w$ in (\ref{defQc}) denotes
evaluation of the terms at the mean interface.  It is noted that $V''_w = 0$
for the Blasius mean flow.


\subsubsection{Wave propagation in anisotropic compliant layers}

From (\ref{Helm}), the fundamental wave solutions to (\ref{inteqpt}) and
(\ref{phipsi}) for a uniformly thick homogeneous layer in the transformed
variables has the form of
\begin{equation}
\etb^t = \skew5\hat{\etb}^t \exp [\rmn{i} (\alpha^tx^t_1-\omega t)],
\end{equation}
where $\skew5\hat{\etb}^t=\mbox{\bit b}\exp (\rmn{i}\gamma x^t_3)$.  For a
non-trivial wave, the substitution of (1.7) into (1.3) and (1.4) yields
the following determinantal equation:
\begin{equation}
\mbox{Det}[\rho\omega^2\delta_{ps}-C^t_{pqrs}k^t_qk^t_r]=0,
\end{equation}
where the wavenumbers $\langle k^t_1,k^t_2,k^t_3\rangle = \langle \alpha^t,0,\gamma\rangle $ and
$\delta_{ps}$ is the Kronecker delta.


\section{Torus translating along an axis of symmetry}

Consider a torus with axes $a,b$ (see figure 2), moving along the $z$-axis.
Symmetry considerations imply the following form for the stress function, given
in body coordinates:
\begin{equation}
\mbox{\bit f}(\theta,\psi) = (g(\psi) \cos \theta,g(\psi) \sin \theta,f(\psi)).
\label{eq41}
\end{equation}

Because of symmetry, one can integrate analytically in the $\theta$-direction
obtaining a pair of equations for the coefficients $f$, $g$ in (4.1),
\begin{eqnarray}
f(\psi_1) = \frac{3b}{\upi(2(a+b \cos \psi_1))^{\frac{3}{2}}}
  \int^{2\upi}_0 \frac{(\sin \psi_1 - \sin \psi)(a+b \cos \psi)\sh}%
  {(1 - \cos (\psi_1 - \psi))(2+\alpha)\sh},
\label{eq42}
\end{eqnarray}
\begin{eqnarray}
g(\psi_1) & = & \frac{3}{\upi(2(a+b \cos \psi_1))^{\frac{3}{2}}}
  \int^{2\upi}_0 \left(\frac{a+b \cos \psi}{2+\alpha}\right)\sh
  \left\{ \astrut f(\psi)[(\cos \psi_1 - \cos \psi - b \beta_1)S + \beta_1P]
  \right. \nonumber\\
&& \mbox{}\times \frac{\sin \psi_1 - \sin \psi}{1-\cos(\psi_1 - \psi)}
  + g(\psi) \left[\left(2+\alpha - \frac{(\sin \psi_1 - \sin \psi)^2}
  {1- \cos (\psi - \psi_1)} - b^2 \gamma \right) S \right.\nonumber\\
&& \left.\left.\mbox{} + \left( b^2 \cos \psi_1\gamma -
  \frac{a}{b}\alpha \right) F(\thalf\upi, \delta) - (2+\alpha)
  \cos\psi_1 E(\thalf\upi, \delta)\right] \astrut\right\} \rmn{d} \psi,
\label{eq43}
\end{eqnarray}
\begin{equation}
\alpha = \alpha(\psi,\psi_1) = \frac{b^2(1-\cos(\psi-\psi_1))}%
  {(a+b\cos\psi) (a+b\cos\psi_1)};
  \quad
  \beta - \beta(\psi,\psi_1) = \frac{1-\cos(\psi-\psi_1)}{a+b\cos\psi}.
\end{equation}

\section{Conclusions} 
 
We have shown how integral equations can be used to solve a particular class
of problems concerning obstacles in waveguides, namely the Neumann problem
for bodies symmetric about the centreline of a channel, and two such problems
were considered in detail.


\begin{acknowledgments}
We would like to acknowledge the useful comments of a referee concerning
the solution procedure used in \S\,5.  A.\,N.\,O. is supported by SERC
under grant number GR/F/12345.
\end{acknowledgments}

\appendix
\section{Boundary conditions}

It is convenient for numerical purposes to change the independent variable in
(\ref{eq41}) to $z=y/ \tilde{v}^{\frac{1}{2}}_T$ and to introduce the
dependent variable $H(z) = (f- \tilde{y})/ \tilde{v}^{\frac{1}{2}}_T$.
Equation (\ref{eq41}) then becomes
\begin{equation}
  (1 - \beta)(H+z)H'' - (2+ H') H' = H'''.
\end{equation}

Boundary conditions to (\ref{eq43}) follow from (\ref{eq42}) and the
definition of {\it H\/}:
\begin{equation}
\left. \begin{array}{l}
\displaystyle
H(0) = \frac{\epsilon \overline{C}_v}{\tilde{v}^{\frac{1}{2}}_T
(1- \beta)};\quad H'(0) = -1+\epsilon^{\frac{2}{3}} \overline{C}_u
+ \epsilon \skew5\hat{C}_u'; \\[16pt]
\displaystyle
H''(0) = \frac{\epsilon u^2_{\ast}}{\tilde{v}^{\frac{1}{2}}
_T u^2_P};\quad H' (\infty) = 0.
\end{array} \right\}
\end{equation}

\section{}

A simple sufficient condition for the method of separation of variables to hold
for the convection problem is derived.  This criterion is then shown to be
satisfied for the ansatz described by~(3.27), thus justifying the approach used
in~\S\,3.  The basic ingredient of our argument is contained in the following
estimate for a Rayleigh Ritz ratio:
\begin{lemma}
Let $f(z)$ be a trial function defined on $[0,1]$.  Let $\varLambda_1$ denote the
ground-state eigenvalue for $-\rmn{d}^2g/\rmn{d} z^2=\varLambda g$, where $g$ must
satisfy $\pm\rmn{d} g/\rmn{d} z+\alpha g=0$ at $z=0,1$ for some non-negative
constant~$\alpha$.  Then for any $f$ that is not identically zero we have
\begin{equation}
\frac{\displaystyle
  \alpha(f^2(0)+f^2(1)) + \int_0^1 \left(
  \frac{\rmn{d} f}{\rmn{d} z} \right)^2 \rmn{d} z}%
  {\displaystyle \int_0^1 f^2\rmn{d} z}
\ge \varLambda_1 \ge
\left( \frac{-\alpha+(\alpha^2+8\upi^2\alpha)\sh}{4\upi} \right)^2.
\end{equation}
\end{lemma}

Before proving it, we note that the first inequality is the standard
variational characterization for the eigenvalue~$\varLambda_1$.

\begin{corollary}
Any non-zero trial function $f$ which satisfies the boundary condition
$f(0)=f(1)=0$ always satisfies
\begin{equation}
  \int_0^1 \left( \frac{\rmn{d} f}{\rmn{d} z} \right)^2 \rmn{d} z.
\end{equation}
\end{corollary}

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\end{document}
