Slender body theory for a flagellated bacterium in a two-fluid model of a polymeric solution

We analyse the motion of a flagellated bacterium in a polymer solution modelled as a two-fluid Newtonian medium using slender body theory. The radius of the flagellar bundle is comparable or smaller than the microstructural length scale of the polymer so that the polymer and solvent satisfy different boundary conditions on the flagella surface. This is true for bacterial motion in biological fluids such as 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, the scenario can be effectively modelled as a medium composed of two interpenetrating Newtonian fluids, the solvent and polymer, with different viscosities. The interphase drag yields a screening length L_B, within which the relative tangential velocity between the two phases is screened. From our calculations, we observe either a monotonic or non-monotonic variation of swimming speed with the viscosity ratio, depending on L_B. The results are sensitive to the way the polymer interacts with the flagella and we discuss scenarios corresponding to different types of interaction and compare our predictions with experimental observations.