Mathematics of microorganism swimming in micropolar fluids




Harinarasimha Prasad, Shankar Narasimha


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Typical representation of the swimming motion of microorganisms in fluid environments model the microorganisms as spherical squirmers in a viscous fluid (Newtonian Fluid) with prescribed surface velocities on the squirmer surface. There are numerous fluids in nature that deviate from the classical Newtonian fluid, such as human and animal blood. Modeling swimming mechanisms in these non-classical fluid settings can be very useful but are mathematically challenging. In this thesis, we develop and analyze a mathematical model for the swimming of microorganisms in micropolar fluids - the fluids that depart from the classical Newtonian fluid due to the microrotational effect. Specifically, micropolar fluid continuum equations involve both the velocity and internal spin vector fields resulting in antisymmetric and couple stresses. The mathematical problem of swimming in micropolar fluids is analyzed via a spherical squirmer model in the absence of inertial effects and assuming steady motion. The idealized configuration allows exact analytical solutions for the velocity and spin fields surrounding the squirmer via Stokes’s stream function formulation. Effects of normal and tangential modes induced on the surface of the squirmer are explained for the two-Mode squirmer. Closed-form expressions for the physical quantities involving the n-Mode general case are also reported. Our exact solutions to the boundary value problem (BVP) for the sixth-order partial differential equation (PDE) contain previously derived results for Stokes and Brinkman fluid squirmer models. It is observed that the propulsion speed, calculated using the force-free condition, depends on the first surface velocity mode only. Surprisingly, the swimming speed in a micropolar fluid is the same as that of the spherical microorganism swimming velocity in Newtonian (Stokes) fluids. The power dissipation and swimming efficiency results derived using non-zero spin boundary conditions on the squirmer surface, however, reveal the microrotational effects due to the inclusion of higher surface velocity modes. The two-mode analytical results are further utilized to inspect the structure of flow fields surrounding the spherical. Our exact mathematical results presented herein may be of interest in understanding microorganisms swimming mechanisms in fluids that exhibit angular momentum due to internal micro-rotation.



Analytical Solution, Brinkman medium, Fluid Mechanics, Micropolar Fluids, Partial Differential Equation, Spherical Squirmer



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