An existence result for a class of nonlinear magnetorheological composites

Overview & Synopsis

We prove existence of a weak solution for a nonlinear, multi-physics, multi-scale problem of magnetorheological suspensions introduced in Nika & Vernescu (Z. Angew. Math. Phys., 71(1):1-19, '20) in the three dimensional setting. The hybrid model couples the Stokes' equation with the quasi-static Maxwell's equations through the Lorentz force and the Maxwell stress tensor. The proof of existence is based on: i) the augmented variational formulation of Maxwell's equations, ii) the definition of a new function space for the magnetic induction and the proof of a Poincaré type inequality, iii) the Altman-Shinbrot fixed point theorem when the magnetic Reynold's number, R m , is small.