A novel numerical method for a swimming bacterium in a two-fluid model of a polymer solution

We develop a novel numerical method for studying the motion of a swimming bacterium in a concentrated, entangled polymer solution modeled as a two-fluid medium composed of a solvent and a polymer phase. The two-fluid model captures the non-continuum effects at the scale of the flagellar bundle, arising due to the underlying microstructure of an entangled polymer solution. The motivation for the problem is to gain a mechanistic understanding of the motion of bacteria in complex biological liquids (e.g. mucus) and is therefore useful in understanding the spread of bacterial diseases. The numerical scheme combines slender body theory, boundary element method, and a finite-difference solver for the flow of an inertialess, viscoelastic polymer medium. Additionally, the method exploits a novel decomposition of the problem into Newtonian and non-Newtonian parts, where the Newtonian part is linear with non-linearities arising in the non-Newtonian part through the time-dependent polymer constitutive equation. We show that this decomposition results in a linear system of equations for the unknown swimming parameters of the bacterium, which are easily solved. The method is validated by comparing the results of a bacterium swimming in a Newtonian liquid, with previous numerical studies. We then analyse the motion of our model bacterium, with a helical flagellar bundle and a spheroidal head, in a polymer solution using this method and comment on the effect of elasticity and microstructure on its motility.