%0 Journal Article
%T Some Nonlinear Vortex Solutions
%A Michael C. Haslam
%A Christopher J. Smith
%A Ghada Alobaidi
%A Roland Mallier
%J International Journal of Differential Equations
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/929626
%X We consider the steady-state two-dimensional motion of an inviscid incompressible fluid which obeys a nonlinear Poisson equation. By seeking solutions of a specific form, we arrive at some interesting new nonlinear vortex solutions. 1. Introduction The study of wakes behind bodies has generated significant interest due to the problem¡¯s important applications. An excellent review of current theories on the evolution of such wakes is given by Chomaz [1]. Numerical models of these flows are typically based on the full three-dimensional Navier-Stokes equations. Still, two-dimensional exact solutions of the Euler equation add significantly to our understanding of such flows [2]. Recently, vortex solutions of the steady two-dimensional inviscid problem have been used as initial conditions for solvers applied to the full three-dimensional time-dependent problem. This approach was adopted by Faddy and Pullin [3] to model the three-dimensional wake behind an aircraft wing. Flows which bear strong resemblance to exact vortex solutions occur in many other applications; a recent experimental study [4] involving cavity flows provides some good examples of this. In the present contribution we examine a class of exact solutions of the two-dimensional inviscid problem which are related to vortex flows. Similar theoretical efforts are well documented; in particular, we note two recent contributions [5, 6]. In plane two-dimensional hydrodynamics, the equation governing the motion of an inviscid incompressible fluid can be written in terms of a stream function as where is a Jacobian and is the two-dimensional Laplacian. The stream function equation (1.1) admits steady-state solutions of the form for any function . Equation (1.2) is a nonlinear Poisson or an elliptic Klein-Gordon equation, and a number of solutions are known for two-dimensional hydrodynamics. In particular, the solution corresponding to a row of corotating Stuart vortices [7] is , while the solution corresponding to the counter-rotating Mallier-Maslowe (M&M) vortices [8] is Stuart vortices satisfy Liouville¡¯s equation [9], , and a number of additional solutions of Liouville¡¯s equation are known, with some of these given in [10¨C12]. M&M vortices satisfy the sinh-Poisson, or elliptic sinh-Gordon, equation, . Again, a number of additional solutions are known [5, 13, 14], some of which involve Jacobian elliptic functions [15] such as and , which are doubly periodic functions of . The goal of this study is to find solutions to (1.2), that are nonlinear vortex solutions of the form with , , constants and and .
%U http://www.hindawi.com/journals/ijde/2012/929626/