Average nutrient uptake by a self-propelled unsteady squirmer

We present results for the average mass transfer to a spherical squirmer, a model micro-organism whose surface oscillates tangentially to itself. The surface motion drives a low-Reynolds-number flow which enables the squirmer either to swim relative to the fluid at infinity, at an average speed proportional to a streaming parameter, W, or to stir the fluid around it while remaining, on average, at rest (if W = 0), as represented by a hovering parameter, b. We assume that the amplitude of the time-periodic surface distortions is scaled by a dimensionless small parameter epsilon, and consider only high Peclet numbers P - a measure of convection versus diffusion - by setting P-1 = epsilon(2)gamma, where gamma is a parameter of O(1). It is shown that the average mass concentration distribution satisfies a steady convection-diffusion equation with an effective velocity field that is different from the actual mean velocity field. The model is used to calculate the mass transfer across the surface of the squirmer, measured by the mean Sherwood number Sh. We find asymptotic solutions for small and large gamma and numerical results for the whole range of values. While the large-gamma expansions are reproduced well by the numerical results, there is a discrepancy between the two at small gamma. We believe this is due to very small recirculation regions, attached to the surface of the squirmer, which make boundary layer theory applicable only when 1/gamma is immense. For the parameters chosen in this study, results indicate that both hovering and streaming contribute to the mass transfer, although streaming has a greater effect. Also, energy dissipation considerations show that an optimum swimming mode exists, at least at small and large gamma, for any given uptake rate. However, other factors have still to be taken into account, and the model realism improved, if we want to make predictions for real aquatic micro-organisms.