WIAS Preprint No. 2804, (2021)

An existence result for a class of nonlinear magnetorheological composites


  • Nika, Grigor
    ORCID: 0000-0002-4403-6908

2020 Mathematics Subject Classification

  • 35A15 35J60 74F10


  • Magnetorheological fluids, augmented variational formulation, fixed point methods, weak solutions




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). 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, Rm, is small.

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