Sinking and swimming in viscoplastic fluids

A classical problem in fluid dynamics considers the drag on a translating particle. When the fluid is Newtonian and inertia is small, the method of matched asymptotics can be employed to determine the small-Reynolds number corrections to the Stokes drag, and to resolve the Stokes paradox that arises in two-dimensional flow (e.g. see Proudman & Pearson, 1957). Many real-world fluids, such as mud, cement and mucus, are non-Newtonian, exhibiting a yield stress, below which they act as solids. For a given material and typical strain-rate we can define the Bingham number, Bi, which measures the relative strength of the yield stress. Using matched asymptotic methods analogous to the inertial problem, I will present the weak yield stress (Bi<<1) correction to Stokes drag around a sphere and cylinder. This requires the numerical solution for the flow of a viscoplastic fluid driven by a point source (i.e. the "viscoplastic Stokeslet").

An extension to these ideas involves the modelling of microscopic swimmers, such as the single-celled organism, Paramecium, which swim by beating many small cilia covering the surface of their body. Lighthill (1952) and Blake (1970) modelled such microswimmers as particles on which a tangential surface velocity is imposed (capturing the effect of the beating cilia). The resulting swimming speed of the particle can then be determined from the condition that the particle is force (and torque) free. Using the ideas developed for the no-slip particle, we can now find the weak yield stress correction to the swimming speed in a viscoplastic fluid. Since the swimmer is force free, this requires a numerical solution of a higher order singularity, namely the "viscoplastic quadrapole". Finally, if time, I will discuss the regime of large yield stress (Bi>>1), for which a boundary layer analysis can employed to determine the swimming speed. For spheroidal swimmers, this provides an interesting result- below a critical aspect ratio of ~1.56, the swimming speed decays to zero as the Bingham number diverges, but at higher aspect ratios, the swimming speed remains O(1) at infinite Bi.