Biological interactions between Carreau fluid and microswimmers in a complex wavy canal with MHD effects

The efficient magnetic swimming of actual or mechanically designed microswimmers within bounded regions is reliant on several factors: the actuation of these swimmers via magnetic field, rheology of surrounding liquid (with dominant viscous forces), nature of medium (either porous or non-porous), position (either straight, inclined or declined) and state (either active or passive) of the narrow passage. To witness these interactions, we utilize Carreau fluid with Taylor swimming sheet model under magnetic and porous effects. Moreover, the cervical canal is approximated as a two-dimensional complex wavy channel inclined at certain angle with the horizontal. The momentum equations are reduced by means of lubrication assumption, which finally leads to a fourth-order differential equation. MATLAB's built-in bvp4c function is employed to solve the resulting boundary value problem. The solution obtained via bvp4c is further verified by finite difference method. In both these methods, the refined values of flow rate and cell speed are computed by utilizing modified Newton-Raphson method. These realistic pairs are further utilized to calculate the energy delivered by the microswimmer. The numerical results are plotted and discussed at the end of the article. Our study explains that the optimum speed of the microorganism can be achieved by means of exploiting the fluid rheology and with the suitable application of the magnetic field. The peristaltic nature of the channel walls and porous medium may also serve as alternative factors to control the speed of the propeller.