(268e) Nutrient Uptake By a Spherical Squirmer: Role of Closed Streamlines

Mobility of microorganisms is important for many of their functions such as reproduction, survival, and nutrition. Microorganisms propel using a variety of means: some use long flagella attached to their body which either rotate or generate waves to produce thrust while others use waves generated by tiny hair-like cilia protruding from their body. To understand how their swimming may affect their nutrient uptake or how it may lead to collective motion and reorganization of a community of microorganisms through hydrodynamic interactions, one may use simple models for flow induced by microorganisms. Lighthill [Comm. Pure and Appl. Math. 1952] and, subsequently, Blake [J. Fluid Mech. 1971] introduced the so-called spherical squirmer, a spherical particle with time-dependent motion along its surface, as a model for microorganisms such as protozoan whose body is covered with beating cilia. An even simpler model, known as the steady squirmer, in which the surface velocity of the sphere is steady, is sometimes used to model processes on a time scale larger than the inverse of cilia beating frequency. In a review article on spherical squirmers, Pedley [J. Appl. Math. 2016] describes a colony consisting of tens of thousands of green alga volvox corteri that together form a nearly spherical aggregate that may be modeled as a squirmer.

We consider the problem of determining the rate of nutrient uptake by a self-propelled spherical squirmer. The tangential velocity induced by the squirmer varies along the surface in such a manner that a region of fluid recirculation is formed on the rear part of the squirmer. This problem was formulated by Magar et al. [J. Mech. Appl. Math. 2003] who investigated it in detail numerically covering large range of Peclet numbers, Pe=aU/D, a being the radius of the squirmer, U its swimming speed, and D the diffusivity of the nutrient in the surrounding liquid. Their numerical results for large Pe suggested that Sherwood number Sh, the non-dimensional mass transfer rate, increases in proportion to but they were unable to determine the constant of proportionality. The present study reexamines this problem in the limit of large Pe and small Reynolds numbers. The concentration in the bulk of the recirculating region approaches a constant whose value is determined by requiring that the mass transfer of nutrient across the streamline dividing the open and closed streamline regions must equal the mass transfer to the squirmer from the closed streamlines. The analysis of the nutrient concentration in the vicinity of the dividing streamline is complicated by the fact that this streamline originates from the stagnation point at the squirmer surface. The boundary layers formed at the squirmer surface on the either side of this stagnation point continue along the dividing streamline resulting in a discontinuity in the nutrient flux. Consequently, new boundary layers form within these two larger boundary layers. When the squirmer motion induces the recirculation region ahead of itself, the distribution near the dividing streamline is simpler to determine but that near the squirmer surface involves formation of new boundary layers inside the boundary layers returning from the dividing streamline. Analytical expressions for large Pe are derived in both cases and shown to be in agreement with the numerical results of Magar et al.