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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. 

Abstract A key challenge underlying the design of miniature machines is encoding materials with time‐ and space‐specific functional behaviors that require little human intervention. Dissipative processes that drive materials beyond equilibrium and evolve continuously with time and location represent one promising strategy to achieve such complex functions. This work reports how internal nonequilibrium states of liquid crystal (LC) emulsion droplets undergoing chemotaxis can be used to time the delivery of a chemical agent to a targeted location. During ballistic motion, hydrodynamic shear forces dominate LC elastic interactions, dispersing microdroplet inclusions (microcargo) within double emulsion droplets. Scale‐dependent colloidal forces then hinder the escape of dispersed microcargo from the propelling droplet. Upon arrival at the targeted location, a circulatory flow of diminished strength allows the microcargo to cluster within the LC elastic environment such that hydrodynamic forces grow to exceed colloidal forces and thus trigger the escape of the microcargo. This work illustrates the utility of the approach by using microcargo that initiate polymerization upon release through the outer interface of the carrier droplet. These findings provide a platform that utilizes nonequilibrium strategies to design autonomous spatial and temporal functions into active materials. 

We perform three-dimensional numerical simulations to investigate the sedimentation of a single sphere in the absence and presence of a simple cross-shear flow in a yield stress fluid with weak inertia. In our simulations, the settling flow is considered to be the primary flow, whereas the linear cross-shear flow is a secondary flow with amplitude 10 % of the primary flow. To study the effects of elasticity and plasticity of the carrying fluid on the sphere drag as well as the flow dynamics, the fluid is modelled using the elastoviscoplastic constitutive laws proposed by Saramito ( J. Non-Newtonian Fluid Mech. , vol. 158 (1–3), 2009, pp. 154–161). The extra non-Newtonian stress tensor is fully coupled with the flow equation and the solid particle is represented by an immersed boundary method. Our results show that the fore–aft asymmetry in the velocity is less pronounced and the negative wake disappears when a linear cross-shear flow is applied. We find that the drag on a sphere settling in a sheared yield stress fluid is reduced significantly compared to an otherwise quiescent fluid. More importantly, the sphere drag in the presence of a secondary cross-shear flow cannot be derived from the pure sedimentation drag law owing to the nonlinear coupling between the simple shear flow and the uniform flow. Finally, we show that the drag on the sphere settling in a sheared yield stress fluid is reduced at higher material elasticity mainly due to the form and viscous drag reduction. 

We experimentally investigate particle migration in a non-Brownian suspension sheared in a Taylor–Couette configuration and in the limit of vanishing Reynolds number. Highly resolved index-matching techniques are used to measure the local particulate volume fraction. In this wide-gap Taylor–Couette configuration, we find that for a large range of bulk volume fraction, $$\unicode[STIX]{x1D719}_{b}\in [20\,\%{-}50\,\%]$$ , the fully developed concentration profiles are well predicted by the suspension balance model of Nott & Brady ( J. Fluid Mech. , vol. 275, 1994, pp. 157–199). Moreover, we provide systematic measurements of the migration strain scale and of the migration amplitude which highlight the limits of the suspension balance model predictions. 

We present a numerical study of non-colloidal spherical and rigid particles suspended in Newtonian, shear thinning and shear thickening fluids employing an immersed boundary method. We consider a linear Couette configuration to explore a wide range of solid volume fractions ( $$0.1\leqslant \unicode[STIX]{x1D6F7}\leqslant 0.4$$ ) and particle Reynolds numbers ( $$0.1\leqslant Re_{p}\leqslant 10$$ ). We report the distribution of solid and fluid phase velocity and solid volume fraction and show that close to the boundaries inertial effects result in a significant slip velocity between the solid and fluid phase. The local solid volume fraction profiles indicate particle layering close to the walls, which increases with the nominal $$\unicode[STIX]{x1D6F7}$$ . This feature is associated with the confinement effects. We calculate the probability density function of local strain rates and compare the latter’s mean value with the values estimated from the homogenisation theory of Chateau et al. ( J. Rheol. , vol. 52, 2008, pp. 489–506), indicating a reasonable agreement in the Stokesian regime. Both the mean value and standard deviation of the local strain rates increase primarily with the solid volume fraction and secondarily with the $$Re_{p}$$ . The wide spectrum of the local shear rate and its dependency on $$\unicode[STIX]{x1D6F7}$$ and $$Re_{p}$$ point to the deficiencies of the mean value of the local shear rates in estimating the rheology of these non-colloidal complex suspensions. Finally, we show that in the presence of inertia, the effective viscosity of these non-colloidal suspensions deviates from that of Stokesian suspensions. We discuss how inertia affects the microstructure and provide a scaling argument to give a closure for the suspension shear stress for both Newtonian and power-law suspending fluids. The stress closure is valid for moderate particle Reynolds numbers, $$O(Re_{p})\sim 10$$ . 

We present numerical simulations of laminar and turbulent channel flow of an elastoviscoplastic fluid. The non-Newtonian flow is simulated by solving the full incompressible Navier–Stokes equations coupled with the evolution equation for the elastoviscoplastic stress tensor. The laminar simulations are carried out for a wide range of Reynolds numbers, Bingham numbers and ratios of the fluid and total viscosity, while the turbulent flow simulations are performed at a fixed bulk Reynolds number equal to 2800 and weak elasticity. We show that in the laminar flow regime the friction factor increases monotonically with the Bingham number (yield stress) and decreases with the viscosity ratio, while in the turbulent regime the friction factor is almost independent of the viscosity ratio and decreases with the Bingham number, until the flow eventually returns to a fully laminar condition for large enough yield stresses. Three main regimes are found in the turbulent case, depending on the Bingham number: for low values, the friction Reynolds number and the turbulent flow statistics only slightly differ from those of a Newtonian fluid; for intermediate values of the Bingham number, the fluctuations increase and the inertial equilibrium range is lost. Finally, for higher values the flow completely laminarizes. These different behaviours are associated with a progressive increases of the volume where the fluid is not yielded, growing from the centreline towards the walls as the Bingham number increases. The unyielded region interacts with the near-wall structures, forming preferentially above the high-speed streaks. In particular, the near-wall streaks and the associated quasi-streamwise vortices are strongly enhanced in an highly elastoviscoplastic fluid and the flow becomes more correlated in the streamwise direction.