In this note, we study the effect of viscosity gradients on the energy dissipated by the motion of microswimmers and the associated efficiency of that motion. Using spheroidal squirmer model swimmers in weak linearly varying viscosity fields, we find that efficiency depends on whether they generate propulsion from the back (pushers) or the front (pullers). Pushers are faster and more efficient when moving down gradients but slower and less efficient moving up viscosity gradients, and the opposite is true for pullers. However, both pushers and pullers display negative viscotaxis, therefore pushers dynamically tend to the most efficient orientation while pullers the least. We also evaluate the effect of shape on power expenditure and efficiency when swimming in viscosity gradients and find that in general the change in both due to gradients monotonically decreases with increasing slenderness. This work shows how shape and gait play an important role in determining dynamics and efficiency in inhomogeneous environments, and demonstrating that both efficiency minimizing and maximizing stable dynamical states are possible.
We apply the geometric theory of swimming at low Reynolds number to the study of nearly circular swimmers in two-dimensional fluids with non-vanishing Hall, or "odd", viscosity. The Hall viscosity gives an off-diagonal contribution to the fluid stress-tensor, which results in a number of striking effects. In particular, we find that a swimmer whose area is changing will experience a torque proportional to the rate of change of the area, with the constant of proportionality given by the coefficient $η^o$ of odd viscosity. After working out the general theory of swimming in fluids with Hall visc…
Microswimmers in nature often experience spatial gradients of viscosity. In this work we develop theoretical results for the dynamics of active particles, biological or otherwise, swimming through viscosity gradients. We model the active particles using the squirmer model, and show how viscosity gradients lead to viscotaxis for squirmers, and how the effects of viscosity gradients depend on the swimming gait of the microswimmers. We also show how such gradients in viscosity can be used to control active particles and suggest a mechanism to sort them based on their swimming style.
When a microorganism begins swimming from rest in a Newtonian fluid such as water, it rapidly attains its steady-state swimming speed since changes in the velocity field spread quickly when the Reynolds number is small. However, swimming microorganisms are commonly found or studied in complex fluids. Because these fluids have long relaxation times, the time to attain the steady- state swimming speed can also be long. In this article we study the swimming startup problem in the simplest liquid crystalline fluid: a two-dimensional hexatic liquid crystal film. We study the dependence of startup t…
The behavior of flagellated bacteria swimming in non-Newtonian media remains an area with contradictory and conflicting results. We report on the behavior of wild-type and smooth-swimming E. coli in Newtonian, shear thinning and viscoelastic media, measuring their trajectories and swimming speed using a three dimensional real-time tracking microscope. We conclude that the speed enhancement in Methocel solution at higher concentration is due to shear-thinning and an analytical model is used to support our experimental result. We argue that shear-induced normal stresses reduce the wobbling behav…
The swimming of a deformable planar slab in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations. A continuum of plane wave displacements, symmetric on both sides of the slab and characterized by a polarization angle, allows optimization of the swimming efficiency with respect to polarization. The mean swimming velocity and mean rate of dissipation are calculated to second order in the amplitude of the stroke. The optimum efficiency depends on the ratio of viscosity and mass density of the fluid. For high viscosity a stroke is found with significantly higher ef…
The effects of fluid viscosity on the kinematics of a small swimmer at low Reynolds number are investigated in both experiments and in a simple model. The swimmer is the nematode Caenorhabditis elegans, which is an undulating roundworm approximately 1 mm long. Experiments show that the nematode maintains a highly periodic swimming behavior as the fluid viscosity is varied from 1.0 mPa-s to 12 mPa-s. Surprisingly, the nematode's swimming speed (~0.35 mm/s) is nearly insensitive to the range of fluid viscosities investigated here. However, the nematode's beating frequency decreases to an asympto…
The swimming behaviour of microorganisms can be strongly influenced by the rheology of their fluid environment. In this manuscript, we experimentally investigate the effects of shear-thinning viscosity on the swimming behaviour of an undulatory swimmer, the nematode Caenorhabditis elegans. Tracking methods are used to measure the swimmer's kinematic data (including propulsion speed) and velocity fields. We find that shear-thinning viscosity modifies the velocity fields produced by the swimming nematode but does not modify the nematode's speed and beating kinematics. Velocimetry data show signi…
Numerous natural processes are contingent on microorganisms' ability to swim through fluids with non-Newtonian rheology. Here, we use the model organism Caenorhabditis elegans and tracking methods to experimentally investigate the dynamics of undulatory swimming in shear-thinning fluids. Theory and simulation have proposed that the cost of swimming, or mechanical power, should be lower in a shear-thinning fluid compared to a Newtonian fluid of the same zero-shear viscosity. We aim to provide an experimental investigation into the cost of swimming in a shear-thinning fluid from (i) an estimate …
Swimming of a sphere in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations for wave-type distortions of the spherical shape. At sizable values of the dimensionless scale number the mean swimming velocity is the result of a delicate balance between the net time-averaged flow generated directly by the surface distortions and the flow generated by the mean Reynolds force density. Depending on the stroke, this can lead to a surprising dependence of the mean swimming velocity on the kinematic viscosity of the fluid. The net flow pattern is calculated as a function…
The effects of temperature and viscosity on the metachronal swimming of crustaceans
Center for Fluid Mechanics, School of Engineering, Brown University（布朗大学工程学院流体力学中心）；Department of Mechanical and Aerospace Engineering, Case Western Reserve University（凯斯西储大学机械与航空航天工程系）
The swimming of a two-sphere system oscillating in a viscous fluidis studied on the basis of simplified equations of motion which take account of both friction and inertial effects. In the model the friction follows from an Oseen approximation to the mobility matrix, and the inertial effects follow from a dipole approximation to the added mass matrix. The resulting mean swimming velocity is evaluated analytically in a first harmonics approximation. For specific choices of the parameters this is compared with the exact result following from a numerical calculation including higher harmonics. Th…
The ciliary locomotion and feeding of an axisymmetric micro-swimmer in a complex fluid whose viscosity depends on nutrient concentration are investigated numerically. The micro-swimmer is modeled as having spheroidal geometry, and ciliary beating is modeled by a slip velocity; i.e. a squirmer is adapted. Looking at the coupling between swimming and feeding of spheroidal squirmers, it is found that swimming speed and feeding are most affected by a non-uniform viscosity environment when the ratio of advection-forces to diffusion-transport, characterized by the nondimensional Peclét number, is mo…
We experimentally and theoretically study the dynamics of a low-Reynolds number helical swimmer moving across viscosity gradients. Experimentally, a double-layer viscosity is generated by superposing two miscible fluids with similar densities but different dynamic viscosities. A synthetic helical magnetically-driven swimmer is then made to move across the viscosity gradients along four different configurations: either head-first (pusher swimmer) or tail-first (puller), and through either positive (i.e. going from low to high viscosity) or negative viscosity gradients. We observe qualitative di…
Active particles (living or synthetic) often move through inhomogeneous environments, such as gradients in light, heat or nutrient concentration, that can lead to directed motion (or taxis). Recent research has explored inhomogeneity in the rheological properties of a suspending fluid, in particular viscosity, as a mechanical (rather than biological) mechanism for taxis. Theoretical and experimental studies have shown that gradients in viscosity can lead to reorientation due to asymmetric viscous forces. In particular, recent experiments with Chlamydomonas reinhardtii algae swimming across sha…
A mechanical model of swimming and flying in an incompressible viscous fluid in the absence of gravity is studied on the basis of assumed equations of motion. The system is modeled as an assembly of rigid spheres subject to elastic direct interactions and to periodic actuating forces which sum to zero. Hydrodynamic interactions are taken into account in the virtual mass matrix and in the friction matrix of the assembly. An equation of motion is derived for the velocity of the geometric center of the assembly. The mean power is calculated as the mean rate of dissipation. The full range of visco…
In this work, we analyze the motion of an active particle, modeled as a spherical squirmer, in linearly varying viscosity fields. In general, the presence of a particle will disturb a background viscosity field and the disturbance generated depends on the boundary conditions imposed by the particle on the viscosity field. We find that, irrespective of the details of the disturbance, active squirmer-type particle tend to align down viscosity gradients (negative viscotaxis). However, the rate of rotation and the swimming speed along the gradient do depend on the details of the interaction of the…
There is increasing evidence that fish gain energetic benefits when they swim in a school. The most recent indications of such benefits are a lower tail (or fin) beat at the back of a school and reduced oxygen consumption in schooling fish versus solitary ones. How such advantages may arise is poorly understood. Current hydrodynamic theories concern either fish swimming side by side or in a diamond configuration and they largely ignore effects of viscosity and interactions among wakes and individuals. In reality, however, hydrodynamic effects are complex and fish swim in many configurations. S…
Recent research has shown that motile cells can adapt their mode of propulsion to the mechanical properties of the environment in which they find themselves--crawling in some environments while swimming in others. The latter can involve movement by blebbing or other cyclic shape changes, and both highlysimplified and more realistic models of these modes have been studied previously. Herein we study swimming that is driven by membrane tension gradients that arise from flows in the actin cortex underlying the membrane, and does not involve imposed cyclic shape changes. Such gradients can lead to…
An axisymmetric squirmer in a Bingham viscoplastic fluid is studied numerically to determine the effect of a yield stress environment on locomotion. The nonlinearity of the governing equations necessitates numerical methods, which is accomplished by solving a variable-viscosity Stokes equation with a Finite Element approach. The effects of stroke modes, both pure and combined, are investigated and it is found that for the treadmill or "neutral" mode, the swimmer in a yield stress fluid has a lower swimming velocity and uses more power. However, the efficiency of swimming reaches its maximum at…
It has been recently shown that a viscosity gradient could drive electrical current through a negatively charged nanochannel (Wiener and Stein, arXiv: 1807.09106). To understand the physics underlying this phenomenon, we employed the Maxwell-Stefan equation to obtain a relation between the flux of solvent species and the driving forces. Our 1D model, which was derived for both ideal and non-ideal solvents, shows that the ionic current depends on the ideality of the solvent, though both scenarios demonstrated good agreement with experimental data. We employed the model to understand the impact …
Janus particles self-propel by generating local tangential concentration gradients along their surface. These gradients are present in a thin layer whose thickness is small compared to the particle size. Chemical asymmetry along the surface is a prerequisite to generate tangential chemical gradient, which gives rise to diffusioosmotic flows in a thin region around the particle. This results in an effective slip on the particle surface. This slip results in the observed "swimming" motion of a freely suspended particle even in the absence of externally imposed concentration gradients.Motivated b…
Gradients of voltage, pressure, temperature, and salinity can transport objects in micro- and nanofluidic systems by well known mechanisms. Here we report the discovery of a transport effect driven by viscosity gradients, which cause an ionic current to flow inside a glass nanofluidic channel. Measurements of the current are well described by a simple model wherein counterions in the electric double layers near the surfaces drift in the direction of decreasing viscosity with a drift speed equal to the gradient of the ions' local diffusivity. Drift in a viscosity gradient is a consequence of mu…
We present a versatile perturbative calculation scheme to determine the mobility matrix for two and more particles in a low Reynolds number fluid with spatially variant viscosity. Assuming an asymptotic non-constant viscosity perturbation superimposed on a constant viscosity background, we exploit the Lorentz reciprocal theorem and a reflection method to obtain the leading order correction to the mobility matrix. We apply our approach firstly to an interface-like linear viscosity gradient with an extension much larger than the length scale of the particle separation. We find that the viscosity…
We examine the vertical mixing induced by the swimming of microorganisms at low Reynolds and Péclet numbers in a stably stratified ocean, and show that the global contribution of oceanic microswimmers to vertical mixing is negligible. We propose two approaches to estimating the mixing efficiency, $η$, or the ratio of the rate of potential energy creation to the total rate-of-working on the ocean by microswimmers. The first is based on scaling arguments and estimates $η$ in terms of the ratio between the typical organism size, $a$, and an intrinsic length scale for the stratified flow, $\ell = …
Motivated by the relevance of edge state solutions as mediators of transition, we use direct numerical simulations to study the effect of spatially non-uniform viscosity on their energy and stability in minimal channel flows. What we seek is a theoretical support rooted in a fully non-linear framework that explains the modified threshold for transition to turbulence in flows with temperature-dependent viscosity. Consistently over a range of subcritical Reynolds numbers, we find that decreasing viscosity away from the walls weakens the streamwise streaks and the vortical structures responsible …
We investigate the effect of a temperature gradient on oil recovery in a two-dimensional pore-network model. The oil viscosity depends on temperature as, $μ_o=exp(B/T)$, where $B$ is a physico-chemical parameter depending on the type of oil, and $T$ is the temperature. A temperature gradient is applied across the medium in the flow direction. Initially, the porous medium is saturated with oil and, then, another fluid is injected. We have considered two cases representing different injection strategies. In the first case, the invading fluid viscosity is constant (finite viscosity ratio) while i…
The influence of near-wall density and viscosity gradients on near-wall turbulence in a channel are studied by means of Direct Numerical Simulation (DNS) of the low-Mach number approximation of the Navier--Stokes equations. Different constitutive relations for density and viscosity as a function of temperature are used in order to mimic a wide range of fluid behaviours and to develop a generalised framework for studying turbulence modulations in variable property flows. Instead of scaling the velocity solely based on local density, as done for the van Driest transformation, we derive an extens…
Active diffusiophoresis - swimming through interaction with a self-generated, neutral, solute gradient - is a paradigm for autonomous motion at the micrometer scale. We study this propulsion mechanism within a linear response theory. Firstly, we consider several aspects relating to the dynamics of the swimming particle. We extend established analytical formulae to describe small swimmers, which interact with their environment on a finite lengthscale. Solute convection is also taken into account. Modeling of the chemical reaction reveals a coupling between the angular distribution of reactivity…
We perform more than 6000 steady-state simulations with a dynamic pore network model, corresponding to a large span in viscosity ratios and capillary numbers. From these simulations, dimensionless quantities such as relative permeabilities, residual saturations, mobility ratios and fractional flows are computed. Relative permeabilities and residual saturations show many of the same qualitative features observed in other experimental and modeling studies. However, while other studies find that relative permeabilities converge to straight lines at high capillary numbers we find that this is not …
In this work, the motion of a 2-D drop on a surface with given wettability gradient is studied numerically by a hybrid lattice-Boltzmann finite-difference method using the multiple-relaxation-time collision model. We incorporate the geometric wetting boundary condition that allows accurate implementation of a contact angle hysteresis model. The method is first validated through three benchmark tests, including the layered Poiseuille flow with a viscosity contrast, the motion of a liquid column in a channel with specified wettability gradient and the force balance for a static drop attached to …
A homogeneous thin polymer film with a stepped height profile levels due to the presence of Laplace pressure gradients. Here we report on studies of polymeric samples with precisely controlled, spatially inhomogeneous molecular weight distributions. The viscosity of a polymer melt strongly depends on the chain length distribution; thus, we learn about thin-film hydrodynamics with viscosity gradients. These gradients are achieved by stacking two films with different molecular weights atop one another. After a sufficient time these samples can be well described as having one dimensional viscosit…
We perform a linearized local stability analysis for short-wavelength perturbations of a circular Couette flow with the radial temperature gradient. Axisymmetric and nonaxisymmetric perturbations are considered and both the thermal diffusivity and the kinematic viscosity of the fluid are taken into account. The effect of the asymmetry of the heating both on the centrifugally unstable flows and on the onset of the instabilities of the centrifugally stable flows, including the flow with the Keplerian shear profile, is thoroughly investigated. It is found that the inward temperature gradient dest…
