Magnetic particle microrheometric dynamics in Newtonian fluids: numerical simulations on the early motion and beyond

摘要

Rotating magnetic particle microrheometry has been a promising technique in measuring material properties in limited-sample high-viscosity fluids. Experimental limitations in the early motion require further theoretical exploration. In this work, the rotation of a ferromagnetic particle is considered under the influence of an external uniform magnetic field in an infinite highly viscous Newtonian fluid. The motion is restricted at the very low Reynolds number limit. Early-time analytical approximations are utilised to initiate numerical calculations in an attempt to describe the azimuthal velocity dependency on scaled time and radius. The equation of motion is solved by implementing a Crank-Nicholson finite-difference scheme, while the driving time-dependent boundary condition is discretised according to a Lax-Wendroff scheme. Stability and convergence criteria for the PDE are also discussed. It is demonstrated that the step function form of the applied magnetic field does not cause finite displacement other than that expected from Newtonian fluid flow for the typical magnetic field magnitude ranges encountered in micro-rheometric studies. The numerical solution is compared against analytical values available for particle 'zero-total-torque' condition and it was found to be second-order accurate in time and radial dimension.