Mechanical effects of complex rheological liquid on a microorganism propelling through a rigid cervical canal: swimming at low Reynolds number

Abstract

The dynamics of microorganism is described by the frictional interactions between the microbodies and the mechanical properties of surrounding fluid. Here, we mathematically study such rheological interactions by considering an infinite wavy sheet (as an organism surface) swimming through a non-Newtonian cervical mucus, i.e., Johnson–Segalman fluid. Since, the flow is generated due to the organism movement in narrowly spaced boundaries, therefore flow equations are setup under long wavelength approximation. Moreover, because of the small mass of many such microorganisms like spermatozoa, the inertial forces could be neglected in the description. A perturbation technique is used to attain the analytic expressions for cervical fluid velocity and pressure gradient in both regions above and below the swimmer. These expressions are further utilized to obtain the swimming speed, flow rate of cervical liquid and energy expanded, which are valid for small rheological parameters. To estimate swimming speed for large rheological parameters, a hybrid numerical procedure based on implicit finite difference method along with modified Newton–Raphson method is employed. Our analysis reveals that microorganism could attain maximum speed by suitably adjusting the rheology of the surrounding liquid. By keeping the organism fixed, a special case (pumping problem) is also discussed at the end of the article. The analysis presented here further finds applications in more realistic bio-rheological simulations for non-invasive hemodynamics for intraocular surgery, fluid dynamic characterization of stress-engineered microrobots for pharmacological drug-targeting and microsensing/mapping for lymphatics and intestinal diagnosis.