We study the swimming motion of a flagellated bacterium in a concentrated polymer solution with a finite yield stress numerically using a method that successfully combines slender body theory (for the flagellar bundle) and a finite different solver for the spheroidal head of the bacterium. The effect of microstructure present in the concentrated polymer solution is captured using a two-fluid model that allows for relative motion between the solvent and polymer. 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. From our simulations, we find that microstructure is a more relevant aspect that affects the motion of the bacterium for small polymer concentration, relaxation times (quantified by the non-dimensional Deborah number De) and yield stress (quantified by the non-dimensional Bingham number Bi). Particularly, we note that the speed enhancements observed in experiments are easily explained by the microstructure alone in this limit. For a polymer solution without yield stress, at large De, the non-Newtonian effects lead to slight enhancements in swimming velocity and we elucidate the role of shear-dependent viscosity and viscoelasticity in this observation. For a polymer solution with yield stress (finite Bi), yield stresses hinder bacterial motility more at small De than at large De, suggesting that higher fluid elasticity helps bacteria overcome the resistance due to yield stress. Our simulations also capture other features observed in experiments and motivate further experimental investigations.
We study the swimming motion of a flagellated bacterium in a concentrated polymer solution with a finite yield stress numerically using a method that successfully combines slender body theory (for the flagellar bundle) and a finite different solver for the spheroidal head of the bacterium. The effect of microstructure present in the concentrated polymer solution is captured using a two-fluid model that allows for relative motion between the solvent and polymer. 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. From our simulations, we find that microstructure is a more relevant aspect that affects the motion of the bacterium for small polymer concentration, relaxation times (quantified by the non-dimensional Deborah number De) and yield stress (quantified by the non-dimensional Bingham number Bi). Particularly, we note that the speed enhancements observed in experiments are easily explained by the microstructure alone in this limit. For a polymer solution without yield stress, at large De, the non-Newtonian effects lead to slight enhancements in swimming velocity and we elucidate the role of shear-dependent viscosity and viscoelasticity in this observation. For a polymer solution with yield stress (finite Bi), yield stresses hinder bacterial motility more at small De than at large De, suggesting that higher fluid elasticity helps bacteria overcome the resistance due to yield stress. Our simulations also capture other features observed in experiments and motivate further experimental investigations.