CONSERVATION LAWS OF A NONLINEAR INCOMPRESSIBLE TWO-FLUID MODEL

Abstract

We study the conservation laws of the Choi-Camassa two-fluid model (1999) which is developed by approximating the two-dimensional (2D) Euler equations for incompressible motion of two non-mixing fluids in a channel. As preliminary work of this thesis, we compute the basic local conservation laws and the point symmetries of the 2D Euler equations for the incompressible fluid, and those of the vorticity system of the 2D Euler equations. To serve the main purpose of this thesis, we derive local conservation laws of the Choi-Camassa equations with an explicit expression for each locally conserved density and corresponding spatial flux. Using the direct conservation law construction method, we have constructed seven conservation laws including the conservation of mass, total horizontal momentum, energy, and irrotationality. The conserved quantities of the Choi-Camassa equations are compared with those of the full 2D Euler equations of incompressible fluid.

We review periodic solutions, solitary wave solutions and kink solutions of the Choi-Camassa equations. As a result of the presence of Galilean symmetry for the Choi-Camassa model, the solitary wave solutions, the kink and the anti-kink solutions travel with arbitrary constant wave speed. We plot the local conserved densities of the Choi-Camassa model on the solitary wave and on the kink wave. For the solitary waves, all the densities are finite and decay exponentially, while for the kink wave, all the densities except one are finite and decay exponentially.