On the Field-Induced Transport of Magnetic Nanoparticles in Incompressible Flow: Existence of Global Solutions - Grün, Günther; Weiß, Patrick (Journal of Mathematical Fluid Mechanics, 2021)
We prove global-in-time existence of weak solutions to a pde-model for the motion of dilute superparamagnetic nanoparticles in fluids influenced by quasi-stationary magnetic fields. This model has recently been derived in Grün and Weiß(On the field-induced transport of magnetic nanoparticles in incompressible flow: modeling and numerics, Mathematical Models and Methods in the Applied Sciences, in press). It couples evolution equations for particle density and magnetization to the hydrodynamic and magnetostatic equations. Suggested by physical arguments, we consider no-flux-type boundary conditions for the magnetization equation which entails H(div , curl) -regularity for magnetization and magnetic field. By a subtle approximation procedure, we nevertheless succeed to give a meaning to the Kelvin force (m· ∇) h and to establish existence of solutions in the sense of distributions in two space dimensions. For the three-dimensional case, we suggest two regularizations of the system which each guarantee existence of solutions, too.
CRIS-Team Scopus Importer:2020-12-04
Keywords: Existence results in continuous and discrete settings; Ferrohydrodynamics; Magnetic nanoparticles; Magnetization; System of partial differential equations

On the field-induced transport of magnetic nanoparticles in incompressible flow: Modeling and numerics - Grün, Günther; Weiß, Patrick (Mathematical Models & Methods in Applied Sciences, 2019)
By methods from non-equilibrium thermodynamics, we derive a class of nonlinear pde-models to describe the motion of magnetizable nanoparticles suspended in incompressible carrier fluids under the influence of external magnetic fields. Our system of partial differential equations couples Navier-Stokes and magnetostatic equations to evolution equations for the magnetization field and the particle number density. In the second part of the paper, a fully discrete mixed finite-element scheme is introduced which is rigorously shown to be energy-stable. Finally, we present numerical simulations in the 2D-case which provide first information about the interaction of particle density, magnetization and magnetic field.
CRIS-Team Scopus Importer:2019-11-01
Keywords: energy stable finite element scheme; Ferrohydrodynamics; magnetic nanoparticles; magnetization; system of partial differential equations