Linear and Nonlinear Stability of Melt Flows in Induction Channels

Abstract

A linear and nonlinear stability analysis is presented of melt flows in induction channels, which are induced by traveling electromagnetic waves. The problem formulation within the framework of magnetodynamics for flow stability studies is discussed. The linear analysis entails the numerical solution of the Orr-Summerfield equation, which is solved by using the high order finite difference technique and combined with the QR method for eigenvalue solutions. The eigenvalue spectrum, the linear stability characteristics and time average Reynolds stresses are obtained from the linear stability analysis. Based on the linear analysis, the nonlinear stability is studied by direct numerical solution of the magnetohydrodynamic equations using the high order finite difference method. Starting from the linear critical Reynolds number, the weakly nonlinear analysis is performed to determine the nonlinear bifurcation phenomena associated with the induction channel flows. Results show that the flow exhibits the typical Hopf bifurcation behavior and the subcritical instability occurs with high perturbation amplitudes.