Continuum modeling of polarizable systems

Abstract

The basic equations embodying macroscopic conservation and balance statements applicable to polarizable media are derived phenomenologically. These general balance statements are then applied to the particular case of ferrofluids, thereby obtaining the governing equations for isothermal incompressible ferrofluid flows. The governing equations thus obtained are subsequently used to extend previous models of plane-Poiseuille and Couette flow of ferrofluids in alternating and rotating uniform magnetic fields, including the effects of couple stresses. We obtain analytical expressions for the translational and spin velocity profiles, vorticity profiles, volumetric flow rate, and the shear force on a moving duct surface, comparing the effects of various boundary conditions for the spin velocity at the duct walls. Simple representative shearing experiments are proposed to differentiate between wall boundary conditions and to determine key rheological parameters of the fluid. Extending the plane-flow analysis to the analogous cylindrical geometry, a regular perturbation scheme is used to solve the coupled ferrohydrodynamic problem of spin-up flow of ferrofluid in a uniform rotating magnetic field. Our approach properly accounts for the spin-magnetization coupling term in the magnetization equation, neglected in prior analyses. The features and contributions of terms up to third order in the expansion parameter are obtained and discussed. The predicted flow profiles possess the major characteristics of the observed flow phenomena, as compared to results in the literature. The torque required to restrain the cylindrical container wall is evaluated and the limits of applicability of the analysis are explored.

(cont.) Finally, the thermodynamics of polarizable systems in external force fields is examined by considering "exact" and "coarse-grained" descriptions of mass-, charge- and magnetically-polarized continua in external gravitational, electric, and magnetic fields. Starting from an exact micromechanical model of the energetics for a change of state of said system, we derive corresponding coarse-grained expressions where the internal distribution of the relevant scalar measure is represented by a series of polarization moments, arbitrarily truncated at a given order. As such, the analysis introduces a new canonical thermodynamic state variable, the polarization T3 (of mass, charge or "magnetic monopoles"). The irreversible thermodynamics of ferrofluids is subsequently considered. Simple experimental systems are introduced and discussed.