Stability of electromagnetically-driven flows in induction channels

X. Ai¹ - B. Q. Li^{1, 2} - O. Zikanov²

¹ School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164, USA
² Department of Mechanical Engineering, University of Michigan -- Dearborn, Dearborn, MI 48128, USA

Abstract
Results of linear and nonlinear analysis of the instability of an electromagnetically driven flow in an induction channel are presented. In the case of small magnetic Reynolds numbers considered here, the linearized stability equation reduces to the Orr-Sommerfeld equation with the base flow profile determined by the imposed traveling magnetic field. The equation is solved using a high-order finite difference technique, and the QR method is applied to obtain the eigenvalue spectra. The critical Reynolds numbers for the linear instability are found to be much higher than the critical numbers for the pressure driven Poiseuille flows. They are also found to depend upon the frequency of the applied magnetic field and the electrical properties of the fluid. The higher critical Reynolds number is correlated with a smaller skin depth, which is attributed to stronger suppression of Tollmien-Schlichting (T-S) waves generated near the solid walls. The spectrum of eigenvalues is populated in the v-gap, which is different from the spectrum of the plane Poiseuille flow. The nonlinear analysis of the resulting bifurcation is based on the direct numerical calculation of two-dimensional solutions of the magnetohydrodynamic equations using a high-order finite difference method. A Hopf bifurcation is observed near the critical Reynolds number. Based on the nonlinear analysis, the bifurcation diagram is constructed and the sub-critical range is determined. Figs 13, Refs 36.