% iaus2esa.tex -- sample pages for Proceedings IAU Symposium document class % (based on v1.0 cca2esam.tex) % v1.04 released 17 May 2004 by TechBooks %% small changes and additions made by KAvdH/IAU 4 June 2004 % Copyright (2004) International Astronomical Union \NeedsTeXFormat{LaTeX2e} \documentclass{iaus} \usepackage{graphicx} \title[short title of paper] %% give here short title %% {An interesting and seminal work on various phenomena in fluid mechanics} \author[short author list] %% give here short author list %% {Alan N. Other$^1$% \thanks{Present address: Fluid Mech Inc., 24 The Street, Lagos, Nigeria.}, H.-C. Smith$^1$ \break \and J.Q. Public$^2$} \affiliation{$^1$Department of Chemical Engineering, University of America, Somewhere, IN 12345, USA \break email: ...........\\[\affilskip] $^2$Department of Aerospace and Mechanical Engineering, University of Camford, \break Academic Street, Camford, CF3 5QL, UK \break email: ...........} \pubyear{2004} \volume{xxx} %% insert here IAU Symposium No. \pagerange{119--126} \date{?? and in revised form ??} \setcounter{page}{119} \jname{Proceedings Title IAU Symposium} \editors{A.C. Editor, B.D. Editor \& C.E. Editor, eds.} \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 $\epsilon^{1/2}$ at infinite \Pen, where $\epsilon^{1/2}$ is the characteristic width of the recirculation region. The enhancement decays from this asymptote as $\Pen^{-1/2}$. \keywords{Keyword1, keyword2, keyword3, etc.} %% add here a maximum of 10 keywords, to be taken form the file \end{abstract} \firstsection % if your document starts with a section, % remove some space above using this command. \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 \cite[Hwang \& Tuck 1970]{Hwang70}; \cite[Lee 1971]{Lee71}; \cite{Figer02}). 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 \cite{Ursell50}. 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. 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 $\phi(x,y,z,t) = \hbox{Re}\{\phi(x,y)\cosh k(z+h)\mathrm{e}^{-\mathrm{i}\omega t}\}$, 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 \cite[Callan \etal\ (1991)]{Callan91} 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 \cite[Bearman \& Graham (1980, pp.~231--232)]{Bearman80}. \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 \cite{Williams64} and \cite{Dennis85}. The use of channel Green's functions allows the far-field behaviour to be computed in an extremely simple manner, whilst the integral equation constructed in~\S\,\ref{sec:trapmode} enables the trapped modes to be computed in~\S\,\ref{sec:torustrans} and the scattering of an incident plane wave to be solved in~\S\,\ref{sec:concl}. Appendix~\ref{ap:boundcon} contains comparisons with experiments. The Galactic Center region proves to be an area very rich in WR stars. The VIIth Catalogue lists within 50 pc from the Galactic Center 15 WNL and 11 WCL stars, at near-IR wavelengths discovered by \cite{Krabbe95} in the Galactic Center Cluster. \section{Green's functions}\label{sec:greenfun} \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+k^2)G_s=(\nabla^2+k^2)G_a=0 \label{Helm} \end{equation} in the fluid, where $\bnabla$ is a gradient operator, \[ \bnabla\bcdot\boldsymbol{v} = 0,\quad \nabla^{2}P= \bnabla\bcdot(\boldsymbol{v}\times \boldsymbol{w}). \] In (\ref{Helm}) \begin{equation} G_s,G_a\sim 1 / (2\upi)\ln r \quad \mbox{as\ }\quad 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 \quad \mbox{on\ }\quad y=d, \label{wallbc} \end{equation} \begin{equation} \frac{\p G_s}{\p y}=0 \quad \mbox{on\ }\quad y=0, \label{symbc} \end{equation} \begin{equation} G_a=0 \quad \mbox{on\ }\quad 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+k^2)G_s=(\nabla^2+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}[\mathrm{e}^{-k\gamma|y-\eta|} + \mathrm{e}^{-k\gamma(2d-y-\eta)}] \cos k(x-\xi)t\:\mathrm{d} t, \qquad 01. \end{array} \right. \] In order to satisfy (\ref{symbc}) we add to this the function \[ -\frac{1}{2\upi} \pvi B(t)\frac{\cosh k\gamma(d-y)}{\gamma\sinh k\gamma d} \cos k(x-\xi)t\:\mathrm{d} t \] which satisfies (\ref{Helm}), (\ref{wallbc}) and obtain \begin{equation} B(t) = 2\mathrm{e}^{-\kgd}\cosh k\gamma(d-\eta). \end{equation} Thus the function \begin{equation} G = -\squart\mathrm{i} (H_0(kr)+H_0(kr_1)) - \frac{1}{\upi} \pvi\frac{\mathrm{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 (\cite[Keller 1977]{Keller77}; \cite[Rogallo 1981]{Rogallo81}; \cite[van Wijngaarden 1968]{Wijngaarden68}). \section{The trapped-mode problem}\label{sec:trapmode} The unit normal from $D$ to $\p D$ is $\boldsymbol{n}_q=(-y^{\prime}(\theta), x^{\prime}(\theta))/w(\theta)$. Now $G_a=\squart Y_0(kr)+\Gat$ where $r=\{[x(\theta)-x(\psi)]^2 + [y(\theta)-y(\psi)]^2\}^{1/2}$ 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} \thalf\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 $\theta_j=(j-\thalf)\upi/M$. Collocating at $\psi=\theta_i$ and writing $\phi_i=\phi(\theta_i)$ \etc gives \begin{equation} \thalf\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}\def~{\hphantom{0}} \begin{center} \caption{Values of $kd$ at which trapped modes occur when $\rho(\theta)=a$} \label{tab:kd} \begin{tabular}{lccc}\hline $a/d$ & $M=4$ & $M=8$ & Callan \etal \\\hline%\\%[3pt] 0.1~~ & 1.56905 & ~~1.56~ & 1.56904\\ 0.3~~ & 1.50484 & ~~1.504 & 1.50484\\ 0.55~ & 1.39128 & ~~1.391 & 1.39131\\ 0.7~~ & 1.32281 & ~10.322 & 1.32288\\ 0.913 & 1.34479 & 100.351 & 1.35185\\\hline \end{tabular} \end{center} \end{table} % Table~\ref{tab:kd} shows a comparison of results obtained from this method using two different truncation parameters with accurate values obtained using the method of \cite[Callan \etal\ (1991)]{Callan91}. \begin{figure} \includegraphics{fig01.eps} \caption{Trapped-mode wavenumbers, $kd$, plotted against $a/d$ for three ellipses:\protect\\ \hbox{{--}\,{--}\,{--}\,{--}\,{--}}, $b/a=0.5$; ---$\!$---, $b/a=1$; ---\,$\cdot$\,---, $b/a=1.5$.}\label{fig:wave} \end{figure} % An example of the results that are obtained from our method is given in figure~\ref{fig:wave}. Figure~\ref{fig:contour}\,(\textit{a},\textit{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~\ref{fig:contour}\textit{a}) modes are shown, while the antisymmetric ones appear in figure~\ref{fig:contour}(\textit{b}). \subsection{Basic properties} Let \refstepcounter{equation} $$ \rho_l = \lim_{\zeta \rightarrow Z^-_l(x)} \rho(x,\zeta), \quad \rho_{u} = \lim_{\zeta \rightarrow Z^{+}_u(x)} \rho(x,\zeta) \eqno{(\theequation{\mathit{a},\mathit{b}})}\label{eq35} $$ 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}\quad Z_l(x) < \zeta < Z_u(x). \end{equation} The Reynolds number \Rey\ 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\ is set equal to 0.7. In (\ref{Helm}) and (\ref{singular}), $\tau_{ij}$ and $\tau^\theta_j$ are \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} \begin{figure} %\includegraphics[height=2in,width=3in,angle=10]{fig02.eps} \includegraphics{fig02.eps} \caption{Shaded contour plots of the potential $\phi$ for the two trapped modes that exist for an ellipse with $a/d=1.5$, $b/d=0.75$. (\textit{a}) Symmetric about $x=0$, $kd=0.96$; (\textit{b}) antisymmetric about $x=0$, $kd=1.398$.}\label{fig:contour} \end{figure} \subsubsection{Calculation of the terms} The first terms in the right-hand side of (\ref{eq35}\textit{a}) and (\ref{eq35}\textit{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. % \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 \\ \displaystyle \frac{\beta}{\alpha\omega^2}V'_w & 0 & 0 & 0 & \mathrm{i}\omega^{-1} \\ \mathrm{i}\omega^{-1} & 0 & 0 & 0 & 0 \\ \displaystyle \mathrm{i} R^{-1}_{\delta}(\alpha^t+\omega^{-1}V''_w) & 0 & -(\mathrm{i}\alpha^tR_\delta)^{-1} & 0 & 0 \\ \displaystyle \frac{\mathrm{i}\beta}{\alpha\omega}R^{-1}_\delta V''_w & 0 & 0 & 0 & 0 \\ (\mathrm{i}\alpha^t)^{-1}V'_w & (3R^{-1}_{\delta}+c^t(\mathrm{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 = \skew2\hat{\etb}^t \exp [\mathrm{i} (\alpha^tx^t_1-\omega t)], \end{equation} where $\skew2\hat{\etb}^t=\boldsymbol{b}\exp (\mathrm{i}\gamma x^t_3)$. 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}\label{sec:torustrans} Consider a torus with axes $a,b$ (see figure~\ref{fig:contour}), moving along the $z$-axis. Symmetry considerations imply the following form for the stress function, given in body coordinates: \begin{equation} \boldsymbol{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 (\ref{eq41}), \begin{eqnarray} f(\psi_1) = \frac{3b}{\upi[2(a+b \cos \psi_1)]^{{3}/{2}}} \int^{2\upi}_0 \frac{(\sin \psi_1 - \sin \psi)(a+b \cos \psi)^{1/2}}% {[1 - \cos (\psi_1 - \psi)](2+\alpha)^{1/2}}, \label{eq42} \end{eqnarray} \begin{eqnarray} g(\psi_1) & = & \frac{3}{\upi[2(a+b \cos \psi_1)]^{{3}/{2}}} \int^{2\upi}_0 \left(\frac{a+b \cos \psi}{2+\alpha}\right)^{1/2} \left\{ \astrut f(\psi)[(\cos \psi_1 - 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\} \mathrm{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}\label{sec:concl} 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. \appendix \section{Boundary conditions}\label{ap:boundcon} It is convenient for numerical purposes to change the independent variable in (\ref{eq41}) to $z=y/ \tilde{v}^{{1}/{2}}_T$ and to introduce the dependent variable $H(z) = (f- \tilde{y})/ \tilde{v}^{{1}/{2}}_T$. 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 \textit{H}: \begin{equation} \left. \begin{array}{l} \displaystyle H(0) = \frac{\epsilon \overline{C}_v}{\tilde{v}^{{1}/{2}}_T (1- \beta)};\quad H'(0) = -1+\epsilon^{{2}/{3}} \overline{C}_u + \epsilon \skew5\hat{C}_u'; \\[16pt] \displaystyle H''(0) = \frac{\epsilon u^2_{\ast}}{\tilde{v}^{{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\,\ref{sec:trapmode}. 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 $-\mathrm{d}^2g/\mathrm{d} z^2=\varLambda g$, where $g$ must satisfy $\pm\mathrm{d} g/\mathrm{d} z+\alpha g=0$ 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{\mathrm{d} f}{\mathrm{d} z} \right)^2 \mathrm{d} z}% {\displaystyle \int_0^1 f^2\mathrm{d} z} \ge \varLambda_1 \ge \left( \frac{-\alpha+(\alpha^2+8\upi^2\alpha)^{1/2}}{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{\mathrm{d} f}{\mathrm{d} z} \right)^2 \mathrm{d} z. \end{equation} \end{corollary} \begin{acknowledgments} We would like to acknowledge the useful comments of a referee concerning the solution procedure used in \S\,\ref{sec:concl}. A.\,N.\,O. is supported by SERC under grant number GR/F/12345. \end{acknowledgments} \begin{thebibliography}{} \bibitem[Bearman \& Graham (1980)]{Bearman80} {Bearman, P.W. \& Graham, J.M.R.} 1980, \textit{J.~Fluid Mech.} 99, 225 \bibitem[Callan, Linton \& Evans (1991)]{Callan91} {Callan, M., Linton, C.M. \& Evans D.V.} 1991, \textit{J.~Fluid Mech.} 229, 51 \bibitem[Dennis (1985)]{Dennis85} {Dennis, S.C.R.} 1985, in: X. Soubbaramayer \& J.P. Boujot (eds.), \textit{Ninth Intl. Conf. on Numerical Methods in Fluid Dynamics}, Lecture Notes in Physics (Heidelberg: Springer), vol.\ 218, p.\ 23 \bibitem[Figer, Najarro, Gilmore, \etal\ (2002)]{Figer02} {Figer, D.F., Najarro, F., Gilmore, D., Morris, M., Kim, S.S., Serabyn, E., McLean, I.S., Gilbert, A.M., Graham, J.R., Larkin, J.E., Levenson, N.A., Teplitz, H.I.} 2002, \textit{ApJ} 581, 258 \bibitem[Hwang \& Tuck (1970)]{Hwang70} {Hwang, L.-S. \& Tuck, E.O.} 1970, \textit{J.~Fluid Mech.} 42, 447 \bibitem[Keller (1977)]{Keller77} {Keller, H.B.} 1977 in: P.H. Rabinovich (ed.), \textit{Applications of Bifurcation Theory} (Academic), p.\ 359 \bibitem[Koch (1983)]{Koch83} {Koch, W.} 1983, \textit{J.~Sound Vib.} 88, 233 \bibitem[Krabbe, Genzel, Eckart, \etal\ (1995)]{Krabbe95} {Krabbe, A., Genzel, R., Eckart, A., Najarro, F., Lutz, D., Cameron, M., Kroker, H., Tacconi-Garman, L. E., Thatte, N., Weitzel, L., Drapatz, S., Geballe, T., Sternberg, A., Kudritzki, R.-P.} 1995, \textit{ApJ} (Letters) 447, L95 \bibitem[Lee (1971)]{Lee71} {Lee, J.-J.} 1971, \textit{J.~Fluid Mech.} 45, 375 \bibitem[Linton \& Evans (1992)]{Linton92} {Linton, C.M. \& Evans, D.V.} 1992, \textit{Phil.\ Trans.\ R. Soc.\ Lond.} A338, 325 \bibitem[Martin (1980)]{Martin80} {Martin, P.A.} 1980, \textit{Q.~J. Mech.\ Appl.\ Maths} 33, 385 \bibitem[Rogallo (1981)]{Rogallo81} {Rogallo, R.S.} 1981, \textit{NASA Tech.\ Mem.} 81835. \bibitem[Ursell (1950)]{Ursell50} {Ursell, F.} 1950, \textit{Proc.\ Camb.\ Phil.\ Soc.} 46, 141 \bibitem[van Wijngaarden (1968)]{Wijngaarden68} {van Wijngaarden, L.} 1968, \textit{J.~Engng Maths} 2, 225 \bibitem[Williams (1964)]{Williams64} {Williams, J.A.} 1964, PhD thesis, University of California, Berkeley. \end{thebibliography} \begin{discussion} \discuss{Massey}{I’m wondering if you have considered the expected intrinsic dispersion in absolute magnitude of WRs --— if you consider the (large) mass range that becomes an early WN or late WC according to the evolutionary models, wouldn’t you expect a large dispersion in M$_v$?} \discuss{van der Hucht}{Indeed, we will be always left with some intrinsic scatter in M$_v$ due to mass differences within the same spectral subtype. But in my opinion, the current large dispersion is for a large fraction due to incertainties of the adopted distances of open clusters and OB associations.} \discuss{Walborn}{I think that the scatter in WNL absolute magnitudes is dominated by intrinsic spread rather than errors. In the LMC, one finds a range of $–5$ to nearly $–8$. This in turn likely reflects different formation channels: mass-transfer binaries, post-RSG, and extremely massive stars in giant H {\scshape ii} regions.} \discuss{van der Hucht}{As said above, there is likely to be intrinsic scatter. But, I wonder whether a scatter of 3 magnitudes perhaps reflects undetected multiplicity.} \discuss{Ma\'{\i}z-Apell\'{a}niz}{I could not agree more with your comment on the need for an updated catalogue of O-type stars (as a follow up of that of Garmany {\itshape et al.} 1982). We are currently working on precisely that (see our poster, these Proceedings) and we will soon make it available.} \discuss{van der Hucht}{Wonderful.} \discuss{Koenigsberger}{Is the ratio WR/O-stars in clusters similar or different from this ratio for the field stars?} \discuss{van der Hucht}{I think it is different because incompleteness among field stars is even larger than that among cluster stars. But perhaps it should also be different because WR stars are older and could have drifted away from clusters, more than O-type stars.} \discuss{Gies}{How many of the WR stars in your catalogue might be low mass objects?} \discuss{Walborn}{Comment: PN central stars in the WR sample would be only [WC].} \discuss{van der Hucht}{Among the WR stars in our VIIth Catalogue we doubt only one: WR109 (V617 Sgr), which is a peculiar object (not even a [WR] central star of a PN). All other stars in our cataloge are true massive Population I WR stars, and properly classified as such. We have not listed known Population II [WC] objects, as we did separately in our VIth Catalogue (van der Hucht {\itshape et al.} 1981). [WN] objects are not known to exist, see the comment by Nolan.} \discuss{Zinnecker}{Are all Galactic WR stars in open clusters and OB association or are there many WR stars in the field?} \discuss{van der Hucht}{See the VIIth WR catalogue (van der Hucht 2001): of the listed 227 Galactic WR stars, only 53 are in open culsters and OB associations, or believed to be. The other 184 are supposedly field stars.} \end{discussion} \end{document}