Stability of plane-parallel flow of magnetic fluids under external magnetic fields

In this work, we present a theoretical study on the stability of a two-dimensional plane Poiseuille flow of magnetic fluids in the presence of externally applied magnetic fields. The fluids are assumed to be incompressible, and their magnetization is coupled to the flow through a simple phenomenological equation. Dimensionless parameters are defined, and the equations are perturbed around the base state. The eigenvalues of the linearized system are computed using a finite difference scheme and studied with respect to the dimensionless parameters of the problem. We examine the cases of both the horizontal and vertical magnetic fields. The obtained results indicate that the flow is destabilized in the horizontally applied magnetic field, but stabilized in the vertically applied field. We characterize the stability of the flow by computing the stability diagrams in terms of the dimensionless parameters and determine the variation in the critical Reynolds number in terms of the magnetic parameters. Furthermore, we show that the superparamagnetic limit, in which the magnetization of the fluids decouples from hydrodynamics, recovers the same purely hydrodynamic critical Reynolds number, regardless of the applied field direction and of the values of the other dimensionless magnetic parameters.