Swimming dynamics of a micro-organism in a couple stress fluid : a rheological model of embryological hydrodynamic propulsion

Mathematical simulations of embryological fluid dynamics are fundamental to improving clinical understanding of the intricate mechanisms underlying sperm locomotion. The strongly rheological nature of reproductive fluids has been established for a number of decades. Complimentary to clinical studies, mathematical models of reproductive hydrodynamics provide a deeper understanding of the intricate mechanisms involved in spermatozoa locomotion which can be of immense benefit in clarifying fertilization processes. Although numerous non-Newtonian studies of spermatozoa swimming dynamics in non-Newtonian media have been communicated, very few have addressed the micro-structural characteristics of embryological media. This family of micro-continuum models include Eringen’s micro-stretch theory, Eringen’s microfluid and micropolar constructs and V.K. Stokes’ couple-stress fluid model, all developed in the 1960s. In the present paper we implement the last of these models to examine the problem of micro-organism (spermatozoa) swimming at low Reynolds number in a homogenous embryological fluid medium with couple stress effects. The micro-organism is modeled as with Taylor’s classical approach, as an infinite flexible sheet on whose surface waves of lateral displacement are propagated. The swimming speed of the sheet and rate of work done by it are determined as function of the parameters of orbit and the couple stress fluid parameter (α). The perturbation solutions are validated with a Nakamura finite difference algorithm. The perturbation solutions reveal that the normal beat pattern is effective for both couple stress and Newtonian fluids only when the amplitude of stretching wave is small. The swimming speed is observed to decrease with couple stress fluid parameter tending to its Newtonian limit as alpha tends to infinity. However the rate of work done by the sheet decreases with α and approaches asymptotically to its Newtonian value. The present solutions also provide a good benchmark for more advanced numerical simulations of micro-organism swimming in couple-stress rheological biofluids