\documentclass[10pt]{article} \usepackage[english]{babel} \usepackage{graphicx} \usepackage{makeidx} \setlength{\textwidth}{125mm} \setlength{\textheight}{200mm} \begin{document} \title{\bf{Numerical Modelling of bubble liquid in a teapot} \thanks{ The Russian foundation for Basic Research supported this investigation by grants No.99-99-99999} } \author{ V.V. Vasya \index{V.V. Vasya} \thanks{Institute of Computational Mathematics and Mathematical Geophysics SB RAS}, \, P.P. Petro \index{P.P. Petrov} \thanks{Novosibirsk State University } } \date{} \maketitle \begin{abstract} The numerical results of computer modelling of bubble liquid motion in a teapot. The stability of the system at small influence. \end{abstract} \bigskip \section{Introduction} A mathematical model of bubble liquid motion in a teapot has been created in paper \cite{1}. The model is based in three general principles. Validity of the conservation laws for mass,momentum, energy and entropy is postulated in the continual approximation. \bigskip \section{Basic equations of bubble liquid model} The equations of conservation laws: \begin{eqnarray} \frac{\partial \rho}{\partial t} + \nabla\cdot{\bf j} = 0, \label{eq1} \\ \frac{\partial j_i}{\partial t} + \partial_k\Pi_{ik} = 0, \label{eq2} \\ \frac{\partial S}{\partial t} + \nabla\cdot(\frac{S}{\rho}{\bf j}) = 0. \label{eq3} \end{eqnarray} In equations (\ref{eq1}) - (\ref{eq3}), $\rho$ is density of a liquid, ${\bf j}$ is impulse, $S$ is entropy. \bigskip \section{Numerical solution} Complete results of the numerical modelling of bubble liquid in a teapot in Fig. 1. \begin{figure}[!htbp] \includegraphics[width=8cm,height=5cm,draft=true]{Penkov1.bmp} \caption{The density of bubble liquid in an teapot} \end{figure} \begin{thebibliography}{99} \bibitem{1} Rockafellar~R.T. Convex Analysis. -- Princeton: University Press, 1970. \bibitem{2} Tocher K.D. The application of automatic computers to sampling experiments~// J. Royal Statist. Soc. Ser. B. -- 1954. -- Vol.~16, N~1. -- P.~39--61. \bibitem{3} Marsaglia G. Random numbers fall mainly in the planes~// Proc. Nat. Acad. Sci. USA. -- 1968. -- Vol.~61. -- P.~23--25. \end{thebibliography} \end{document}