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We analyse the motion of a flagellated bacterium in a two-fluid medium using slender body theory. The two-fluid model is useful for describing a body moving through a complex fluid with a microstructure whose length scale is comparable to the characteristic scale of the body. This is true for bacterial motion in biological fluids (entangled polymer solutions), where the entanglement results in a porous microstructure with typical pore diameters comparable to or larger than the flagellar bundle diameter, but smaller than the diameter of the bacterial head. Thus, the polymer and solvent satisfy different boundary conditions on the flagellar bundle and move with different velocities close to it. This gives rise to a screening length$$L_B$$within which the fluids exchange momentum and the relative velocity between the two fluids decays. In this work, both the solvent and polymer of the two-fluid medium are modelled as Newtonian fluids with different viscosities$$\mu _s$$and$$\mu _p$$(viscosity ratio$$\lambda = \mu _p/\mu _s$$), thereby capturing the effects solely introduced by the microstructure of the complex fluid. From our calculations, we observe an increased drag anisotropy for a rigid, slender flagellar bundle moving through this two-fluid medium, resulting in an enhanced swimming velocity of the organism. The results are sensitive to the interaction between the bundle and the polymer, and we discuss two physical scenarios corresponding to two types of interaction. Our model provides an explanation for the experimentally observed enhancement of swimming velocity of bacteria in entangled polymer solutions and motivates further experimental investigations. 

A recent experiment by Wanget al.(Soft Matt., vol. 17, 2021, pp. 2985–2993) shows that a self-propelled compound drop in a surfactant-laden solution can autonomously change its motion from a straight line to a spiraling trajectory, enhancing its capability for chemical detection, catalytic reaction and pollutant removal in a large fluid region. To understand the underlying physics of this peculiar motion, we develop a two-dimensional minimal model to study the swimming dynamics of a compound droplet driven by a self-generated Marangoni stress. We find that, depending on the Péclet number ($$Pe$$) and the viscosity and volume ratios of the two compound phases, the drop can swim in a variety of trajectories, including straight lines, circles, zigzag curves and chaotic trajectories. The drop moves in circles when its two components have comparable volumes. Otherwise, it shows other types of motions depending on$$Pe$$. Our simulation results for the circular motion at small$$Pe$$are qualitatively comparable to the experiment. The transition between zigzag and circular trajectories is mainly determined by the orientation of high-order modes with respect to the drop's swimming direction. For most compound drops, the speed decays as$$Pe^{-1/3}$$at high Péclet numbers as it does for a single-phase drop. A drop with two equal components undergoes a run-and-reorient motion due to the competition between the even and odd modes.