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Many biological fluids display shear-thinning rheology, where the viscosity decreases with an increasing shear rate. To better understand how this non-Newtonian rheology affects the motion of biological and artificial micro-swimmers, recent efforts have begun to seek answers to fundamental questions about active bodies in shear-thinning fluids. Previous analyses based on a squirmer model have revealed non-trivial variations of propulsion characteristics in a shear-thinning fluid via the reciprocal theorem. However, the reciprocal theorem approach does not provide knowledge about the flow surrounding the squirmer. In this work, we fill in this missing information by calculating the non-Newtonian correction to the flow analytically in the asymptotic limit of small Carreau number. In particular, we investigate the local effect due to viscosity reduction and the non-local effect due to induced changes in the flow; we then quantify their relative importance to locomotion in a shear-thinning fluid. Our results demonstrate cases where the non-local effect can be more significant than the local effect. These findings suggest that caution should be exercised when developing physical intuition from the local viscosity distribution alone around a swimmer in a shear-thinning fluid.

Some micro-organisms and artificial micro-swimmers propel at low Reynolds numbers (Re) via the interaction of their flexible appendages with the surrounding fluid. While their locomotion has been extensively studied with a Newtonian fluid assumption, in realistic biological environments these micro-swimmers invariably encounter rheologically complex fluids. In particular, many biological fluids such as blood and different types of mucus have shear-thinning viscosities. The influence of this ubiquitous non-Newtonian rheology on the performance of flexible swimmers remains largely unknown. Here, we present a first study to examine how shear-thinning rheology alters the fluid-structure interaction and hence the propulsion performance of elastic swimmers at low Re. Via a simple elastic swimmer actuated magnetically, we demonstrate that shear-thinning rheology can either enhance or hinder elastohydrodynamic propulsion, depending on the intricate interplay between elastic and viscous forces as well as the magnetic actuation. We also use a reduced-order model to elucidate the mechanisms underlying the enhanced and hindered propulsion observed in different physical regimes. These results and improved understanding could guide the design of flexible micro-swimmers in non-Newtonian fluids.

Cell motility plays important roles in a range of biological processes, such as reproduction and infections. Studies have hypothesized that the ulcer-causing bacterium Helicobacter pylori invades the gastric mucus layer lining the stomach by locally turning nearby gel into sol, thereby enhancing its locomotion through the biological barrier. In this work, we present a minimal theoretical model to investigate how heterogeneity created by a swimmer affects its own locomotion. As a generic locomotion model, we consider the swimming of a spherical squirmer in a purely viscous fluid pocket (representing the liquified or degelled region) surrounded by a Brinkman porous medium (representing the mucus gel). The use of the squirmer model enables an exact, analytical solution to this hydrodynamic problem. We obtain analytical expressions for the swimming speed, flow field, and power dissipation of the swimmer. Depending on the details of surface velocities and fluid properties, our results reveal the existence of a minimum threshold size of mucus gel that a swimmer needs to liquify in order to gain any enhancement in swimming speed. The threshold size can be as much as approximately 30% of the swimmer size. We contrast these predictions with results from previous models and highlight the significant role played by the details of surface actuations. In addition to their biological implications, these results could also inform the design of artificial microswimmers that can penetrate into biological gels for more effective drug delivery.

Micro-organisms encounter heterogeneous viscous environments consisting of networks of obstacles embedded in a viscous fluid medium. In this paper we analyse the characteristics of swimming in a porous medium modelled by the Brinkman equation via a spherical squirmer model. The idealized geometry allows an analytical and exact solution of the flow surrounding a squirmer. The propulsion speed obtained agrees with previous results using the Lorentz reciprocal theorem. Our analysis extends these results to calculate the power dissipation and hence the swimming efficiency of the squirmer in a Brinkman medium. The analytical solution enables a systematic analysis of the structure of the flow surrounding the squirmer, which can be represented in terms of singularities in Brinkman flows. We also discuss the spatial decay of flows due to squirming motion in a Brinkman medium in comparison with the decay in a purely viscous fluid. The results lay the foundation for subsequent studies on hydrodynamic interactions, nutrient transport and uptake by micro-organisms in heterogeneous viscous environments.

In this work we quantify the effects of surfactant transport on the deformation of a viscous drop under a DC electric field. We study how convective and diffusive transport of surfactants at drop surfaces influence the equilibrium and dynamic deformation of a leaky dielectric drop and a conducting drop. Focusing on the prolate drop shape (elongates along the electric field), we show the differences in equilibrium deformation and flow circulation between a leaky dielectric drop and a conducting drop. We quantify the drop electrodeformation via its dependence on the interior flow circulation and the dominant surfactant transport regime (characterized by the surface Péclet number Pe_{s}). For a leaky dielectric drop with dominant surfactant diffusion (Pe_{s}≪1), equator-to-pole (pole-to-equator) circulation yields smaller (larger) equilibrium deformation with increasing surfactant coverage, compared to a clean drop. However, when convection dominates (Pe_{s}≫1), the equilibrium drop deformation increases (decreases) with larger surfactant coverage for equator-to-pole (pole-to-equator) circulation. Larger equilibrium drop deformation is found for a leaky dielectric drop than a conducting drop when the interior flow is from equator to pole. For an interior flow from pole to equator, we identify cases where larger deformation is found for a conducting interior fluid. Finally, we study the effect of the surfactant transport on the dynamic evolution of drop shape. We found the drop undergoes an overshoot in the early deformation phase, before settling to its equilibrium shape-similar to the overshoot observed for unsteady Stokes flow.

Micro-organisms and artificial microswimmers often move in biological fluids displaying complex rheological behaviors, including viscoelasticity and shear-thinning viscosity. A comprehensive understanding of the effectiveness of different swimming gaits in various types of complex fluids remains elusive. The squirmer model has been commonly used to represent different types of swimmers and probe the effects of different types of complex rheology on locomotion. While many studies focused only on squirmers with surface velocities in the polar direction, a recent study has revealed that a squirmer with swirling motion can swim faster in a viscoelastic fluid than in Newtonian fluids [Binagia et al., J. Fluid Mech. 900, A4, (2020)]. Here, we consider a similar setup but focus on the sole effect due to shear-thinning viscosity. We use asymptotic analysis and numerical simulations to examine how the swirling flow affects the swimming performance of a squirmer in a shear-thinning but inelastic fluid described by the Carreau constitutive equation. Our results show that the swirling flow can either increase or decrease the speed of the squirmer depending on the Carreau number. In contrast to swimming in a viscoelastic fluid, the speed of a swirling squirmer in a shear-thinning fluid does not go beyond the Newtonian value in a wide range of parameters considered. We also elucidate how the coupling of the azimuthal flow with shear-thinning viscosity can produce the rotational motion of a swirling pusher or puller.

A numerical scheme basedon the immersed interfacemethod (IIM) is devel- oped to simulate the dynamics of an axisymmetric viscous drop under an electric field. In this work, the IIM is used to solve both the fluid velocity field and the electric po- tential field. Detailed numerical studies on the numerical scheme show a second-order convergence. Moreover, our numerical scheme is validated by the good agreement with previous analytical models (1,31,39), and numerical results from the boundary integral simulations (17). Our method can be extended to Navier-Stokes fluid flow with nonlinear inertia effects. AMS subject classifications: 76W05, 65M06, 65M12

Micro-organisms expend energy moving through complex media. While propulsion speed is an important property of locomotion, efficiency is another factor that may determine the swimming gait adopted by a micro-organism in order to locomote in an energetically favorable manner. The efficiency of swimming in a Newtonian fluid is well characterized for different biological and artificial swimmers. However, these swimmers often encounter biological fluids displaying shear-thinning viscosities. Little is known about how this nonlinear rheology influences the efficiency of locomotion. Does the shear-thinning rheology render swimming more efficient or less? How does the swimming efficiency depend on the propulsion mechanism of a swimmer and rheological properties of the surrounding shear-thinning fluid? In this work, we address these fundamental questions on the efficiency of locomotion in a shear-thinning fluid by considering the squirmer model as a general locomotion model to represent different types of swimmers. Our analysis reveals how the choice of surface velocity distribution on a squirmer may reduce or enhance the swimming efficiency. We determine optimal shear rates at which the swimming efficiency can be substantially enhanced com- pared with the Newtonian case. The non-trivial variations of swimming efficiency prompt questions on how micro-organisms may tune their swimming gaits to exploit the shear-thinning rheology. The findings also provide insights into how artificial swimmers should be designed to move through complex media efficiently.

Most of the existing numerical and theoretical investigations on the electrohydrodynamics of a viscous drop have focused on the creeping Stokes flow regime, where nonlinear inertia effects are neglected. In this work we study the inertia effects on the electrodeformation of a viscous drop under a DC electric field using a novel second-order immersed interface method. The inertia effects are quantified by the Ohnesorge number $\text{Oh}$, and the electric field is characterized by an electric capillary number ${\text{Ca}}_{E}$. Below the critical ${\text{Ca}}_{E}$, small to moderate electric field strength gives rise to steady equilibrium drop shapes. We found that, at a fixed ${\text{Ca}}_{E}$, inertia effects induce larger deformation for an oblate drop than a prolate drop, consistent with previous results in the literature. Moreover, our simulations results indicate that inertia effects on the equilibrium drop deformation are dictated by the direction of normal electric stress on the drop interface: Larger drop deformation is found when the normal electric stress points outward, and smaller drop deformation is found otherwise. To our knowledge, such inertia effects on the equilibrium drop deformation has not been reported in the literature. Above the critical ${\text{Ca}}_{E}$, no steady equilibrium drop deformation can be found, and often the drop breaks up into a number of daughter droplets. In particular, our Navier-Stokes simulations show that, for the parameters we use, (1) daughter droplets are larger in the presence of inertia, (2) the drop deformation evolves more rapidly compared to creeping flow, and (3) complex distribution of electric stresses for drops with inertia effects. Our results suggest that normal electric pressure may be a useful tool in predicting drop pinch-off in oblate deformations.

In this work, we develop a theoretical model to explain the equilibrium spheroidal deformation of a giant unilamellar vesicle (GUV) under an alternating (ac) electric field. Suspended in a leaky dielectric fluid, the vesicle membrane is modeled as a thin capacitive spheroidal shell. The equilibrium vesicle shape results from the balance between mechanical forces from the viscous fluid, the restoring elastic membrane forces, and the externally imposed electric forces. Our spheroidal model predicts a deformation-dependent transmembrane potential, and is able to capture large deformation of a vesicle under an electric field. A detailed comparison against both experiments and small-deformation (quasispherical) theory showed that the spheroidal model gives better agreement with experiments in terms of the dependence on fluid conductivity ratio, permittivity ratio, vesicle size, electric field strength, and frequency. The spheroidal model also allows for an asymptotic analysis on the crossover frequency where the equilibrium vesicle shape crosses over between prolate and oblate shapes. Comparisons show that the spheroidal model gives better agreement with experimental observations.

Cetacean strandings display a marked geographical clustering. We propose a simple, two-dimensional ray-dynamics model of cetacean echolocation to examine the role played by coastline topography in influencing the location and clustering of stranding sites. We find that a number of coastlines known to attract cetacean strandings produce acoustical "Dead Zones" where echolocation signals are severely distorted by purely geometric effects. Using available cetacean stranding data bases from four disparate areas, we show that the geographical clusters in the observations correlate strongly with the regions of distorted echolocation signals as predicted by the model.

We theoretically investigate the deformation of a viscous drop covered with non-diffusing insoluble surfactant under a uniform DC electric field. At equilibrium, surfactant immobilizes the spheroidal drop surface and completely suppresses the fluid flow. In this work we focus on the equilibrium electro-deformation of a surfactant-laden drop in the leaky dielectric framework by developing (1) a second-order small-deformation analysis and (2) a spheroidal model for a highly deformed (prolate or oblate) drop. Both models are compared against experimental data and numerical simulation results in the literature. Our analysis shows how the existence of equilibrium spheroidal drop depends on the permittivity ratio, conductivity ratio, surfactant coverage, and the elasticity number. Furthermore, the spheroidal model highlights that differences between surfactant effects, such as tip stretching and surface dilution effects, are greatly amplified at large surfactant coverage and high electric capillary number. These surfactant effects are well captured in the spheroidal model, but cannot be described in the second-order small-deformation theory.