Deformation of a Sheared Magnetic Droplet in a Viscous Fluid

A fully three-dimensional numerical study of the dynamics and field-induced deformation of a sheared, superparamagnetic ferrofluid droplet immersed in a New-tonian viscous fluid is presented. The system is a three-dimensional, periodic channel with top and bottom walls displaced to produce a constant shear rate and with an external, uniform magnetic field perpendicular to the walls. The model consists of the incompressible Navier-Stokes equations with the extra magnetic stress coupled to the static Maxwell's equations. The coupled system is solved with unprecedented resolution and accuracy using a fully adaptive, Immersed Boundary Method. For small droplet distortions, the numerical results are compared and validated with an asymptotic theory. For moderate and strong applied fields, relative to surface tension, and weak flows a large field-induced droplet deformation is observed. Moreover, it is found that the droplet distortion in the vorticity direction can be of the same order as that occurring in the shear plane. This study highlights the importance of the three-dimensional character of a problem of significant relevance to applications, where a dispersed magnetic phase is employed to control the rheology of the system.