Swimming speed and power calculations for squirming cylinders in porous media
Nguyen, Khai Quang
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Several micro-organisms swim in heterogeneous fluid environments such as the porous media (with pores, fibers or polymers). The past decade has seen an increasing interest in understanding propulsion mechanisms of minute organisms on small scales, both theoretically and experimentally. Interest to theoretical models on locomotion strategies with homogeneous fluids in three-dimensions has a long record in applied mathematics (Lighthill, Commun. Pure and Appl. Math., (1952); Blake, J. Austr. Math. Soc., (1971)). In this thesis, we have studied mathematical models for the squirming motion of circular cylinders suspended in porous media governed by the two-dimensional Brinkman partial differential equations (PDE) subject to the varying surface velocity conditions on the surface of the squirmer. The vector boundary-value problem (BVP) is reduced to a scalar BVP for the fourth order PDE via Stokes stream-function formulation in polar coordinates. Exact analytical solutions for the mathematical model are found in terms of the modified Bessel functions of the integer order for various radial and tangential modes of surface oscillations. We performed the calculation of the swimming speed and power for the two-dimensional squirming motion of cylinders in heterogeneous fluids for various modes of the surface velocity changes. It is found that the non-dimensional permeability parameter, arising due to the heterogeneity of the fluid environment, has a significant impact on the velocity and pressure fields as well as the speed and power of the squirming cylinder. Our results show that the speed, which depends on the first mode only, is always less than that of the cylinder squirming in Stokes flow. The power can increase by the use of certain normalization factors. The stream line plots reveal the existence of quadrupolar flow patterns and saddle points in the vicinity of the cylindrical squirmer. Another illustration of our method for the problem of a cylinder swimming in a homogeneous fluid (Stokes fluid) surrounded by a heterogeneous fluid (Brinkman fluid) will also be shown in this talk. We believe that the results of this thesis will be crucial in the understanding of various biological functions including bacterial movement in mucous, motility in reproduction, and escaping from predators.