Flow of a viscoplastic fluid around a particle

We study the force exerted by the uniform flow of a Bingham fluid around two- and three-dimensional particles in the regime of slow creeping flow and relatively weak yield stress. Matched asymptotic expansions are employed to couple a viscously-dominated Stokes flow close to the particle with a far field in which the yield stress and viscous stresses are comparable. The far-field region is therefore modelled as a Bingham fluid driven by a point force at the origin (i.e a viscoplastic Stokeslet). It features the full nonlinearity of the viscoplastic rheology, and its solution is computed through direct numerical simulation. Asymptotic matching then leads to a quasi-analytical expression for the drag force in terms of the dimensionless Bingham number, $Bi$, which measures the magnitude of the yield stress relatively to viscous effects at the particle scale. We deploy this methodology to determine the drag force on a sphere in three dimensions and circular and elliptic cylinders in two-dimensions, confirming our asymptotic predictions by comparison with full numerical simulations of the motion. We also generalise the three-dimensional result to arbitrary particles. The viscoplastic correction to the Newtonian drag in three-dimensions scales as Bi^1/2. In two-dimensions, however, the effects of viscoplasticity are non-negligible at leading order. The drag varies with [ln(1/Bi)]^-1, but this asymptotic result is only approached very slowly. Instead, an accurate representation of the drag is derived in terms of a single algebraic relation between the drag and the Bingham number.