(311i) A Swimming Bacterium in a Two-Fluid Model of a Polymer Solution

We analyse the swimming of a bacterium in a polymer solution modelled as a two-fluid Newtonian medium. The motivation for the study stems from the fact that, in the polymer solution, the flagella and the cell of the bacterium effectively interact with media of different viscosity, owing to the difference in their characteristic length scales. This is true of several biological fluids like mucus where the polymer forms a mesh with typical pore sizes larger than the flagellar radius but smaller than the size of the cell. If the polymer relaxes rapidly, this scenario can be effectively modelled as a bacteria swimming in a medium composed of two interpenetrating Newtonian fluids, the solvent and polymer, where the polymer is also modelled as a Newtonian fluid, albeit with a different viscosity (the viscosity ratio, being given by $\lambda$) and satisfying different boundary conditions (slip or partial slip at the bacterial surface) than the solvent. As a result, the governing equations for the two fluids are Stokes equations with an additional drag term describing the momentum transfer between the fluids due to their relative velocity. We derive a slender body theory for the flagella bundle motion and solve equations for the motion of the spherical cell in this two-fluid medium. The drag gives rise to a screening length $L_B$, within which, the relative tangential velocity between the polymer and solvent is screened. From our calculations, we observe either a monotonic or non-monotonic variation of swimming speed with the viscosity ratio $\lambda$, depending on $L_B$. The predictions will be compared to previous experimental observations of a bacterium swimming in polymer solution. Our results, indicate that the observed variations in the experiment could be due to relative motion between the polymer and solvent, which motivates further experimental investigation. We also discuss the implications of these results for a bacterium swimming in biological fluids like mucus, in which the polymer exhibits elastoviscoplastic behavior.