Research Express@NCKU - Articles Digest
Research Express@NCKU Volume 24 Issue 10 - September 27, 2013 [ http://research.ncku.edu.tw/re/articles/e/20130927/2.html ]
Self-similar solutions of the Euler equations with
spherical symmetry
Wen-Ching Lien
*1, Chen-Chang Peng
21Department of Mathematics, National Cheng Kung University 2Department of Applied Mathematics, National Chiayi University [email protected]
Nonlinear Analysis, Vol 75, No 17, 6370-6378 (2012)
M
any phenomena in nature or engineering when seen at an appropriate-intermediate-distance, in space or time, exhibit the remarkable property of self-similarity: they reproduce themselves as the scale changes. One of the most famous results in fluid mechanics was performed by G. I. Taylor who solved the nuclear explosion problem in World War II. The derivation of the self-similar solution to G.I. Taylor's ideal problem was essentially based on dimensional analysis. This is the fundamental principle: all physical relationships can be represented in a form that is equally valid for all observers.Recently, there has been a great deal of activity in developing the techniques of scaling or dimensional analysis for nonlinear differential equations and its variants. The rigorous analysis of the self-similar solution for nonlinear partial differential equations is an issue of importance and full of challenges mathematically and it has been the subject of research in the last 30 years. Although there are very rich results concerning the self-similar solutions of the nonlinear heat or Schrödinger equations, there are still very few results concerning the coupled nonlinear partial equations, fluid dynamical equations for example.
In this paper, we reformulate G.I. Taylor's ideal problem in a simple form in order to present the analytic proof of the global existence of positive smooth solutions. We consider a simplified model of explosion: a spherical piston motion. The spherical piston moves outward at a constant speed and the gas flow is headed by a weak shock moving at a constant speed. To describe the fluid motion, we study the Euler system with spherical symmetry (derived from the conservation laws of mass and momentum). With the assumption of self-similarity, the Euler system can be transformed into a system of nonlinear ordinary differential equations. It is known that there are two families of elementary waves associated with the Euler system. Due to the Rankine–Hugoniot condition and the entropy condition, we can obtain the flow velocity and density immediately behind the two-shock wave, which is the initial condition for the ODE system. Furthermore, the kinematic condition at the piston requires that the flow velocity on the piston surface is the same as the piston velocity. This boundary condition gives the fixed point of the velocity function. Hence, the problem is treated as an initial value problem and the system is integrated backward until the fixed point is reached. We notice that the system is singular when the shock Mach number is equal to 1.
In this simple model, the motion is supposed to be so small that only weak shocks are produced; hence, the changes in entropy are ignored. Besides, to maintain constant speed of the piston in three-dimensional space, an energy supply at an increasing rate is required. However, in the actual situation of explosion, the total energy is given (usually by a given mass of explosive), and the strength of the shock and the change of entropy rapidly decrease; therefore, the flow is no longer isentropic. Even with such limitations, one can still see the present model as a proper formulation of the problem in intermediate time and space intervals.
Research Express@NCKU - Articles Digest
References
1. G.I. Barenblatt, Scaling, Cambridge University Press, New York, 2003.
2. M.H. Giga, Y. Giga, Nonlinear Partial Differential Equations, Asymptotic Behavior of Solutions and Self-Similar Solutions, Birkhauser, 2010.
3. G.I. Taylor, The air wave surrounding an expanding sphere, Proc. R. Soc. Lond. Ser. A 186 (1946) 273– 292.
4. G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.
