The deformation of fluid elements governs many processes in porous media, including the transport, dispersion and mixing of solutes, chemical reactions and colloidal particles. Recently, it has been shown that even in strongly heterogeneous porous media, steady three-dimensional (3-D) Darcy flow with isotropic hydraulic conductivity admits only constrained kinematics. This is due to its inherently helicity-free nature, as the velocity is everywhere orthogonal to the vorticity. This property precludes braided streamlines and instead admits a pair of coherent 3-D streamfunctions, and the streamlines cannot wander freely throughout the flow domain. In this study, we consider the impact of these kinematic constraints upon fluid deformation at the Darcy scale. We show that the helicity-free condition corresponds to an orthogonal 3-D streamline coordinate system, which we use to derive an ab initio continuous time random walk framework for fluid deformation. We find that the helicity-free condition combined with the intermittent nature of shear events leads to fluid deformation that is limited to algebraic growth, with stretching ranging from sublinear to superlinear behaviour. Fluid deformation in 3-D isotropic Darcy flow is remarkably similar to that of two-dimensional (2-D) Darcy flow, and the structure of 3-D Darcy flow is fundamentally the same as two superposed 2-D Darcy flows. These results have implications for understanding flow and transport in heterogeneous porous media, and provide a basis for quantification of mixing and dispersion in isotropic 3-D Darcy flow.

Darcy's law is used widely to model flow in heterogeneous porous media via a spatially varying conductivity field. The isotropic Darcy equation imposes significant constraints on the allowable Lagrangian kinematics of the flow field and thus upon scalar transport. These constraints stem from the fact that the helicity density in these flows is identically zero and so the flow does not admit closed or knotted flow paths. This implies that steady Darcy flow possesses a particularly simple flow topology which involves streamlines that do not possess closed orbits, knots or linked vortex lines. This flow structure is termed ‘complex lamellar’ and consists of fully integrable (in the dynamical systems sense) streamlines which admit two analytic constants of motion and so preclude chaotic advection. In this study we show that these constants of motion correspond to a pair of streamfunctions which are single valued and topologically planar, and the intersections of the level sets of these invariants correspond to streamlines of the flow. We show that the streamfunctions and iso-potential surfaces of the flow form a semi-orthogonal coordinate system, that naturally recovers the topological constraints imposed on the Lagrangian kinematics of these flows. We use this coordinate system to investigate the impact of these constraints upon the kinematics of Darcy flow, including the deformation of fluid elements and transverse macrodispersion of solutes in the absence of local dispersion. These results shed new light on the relevance and limitations of isotropic Darcy flow as a model of transport, mixing and reaction in porous media.

Selective heating of a microparticle surface had been observed to cause its autonomous movement in a fluid medium due to self-generated temperature gradients. Here, we theoretically investigate the response of such an auto-thermophoretic particle near an isothermal planar wall. We derive an exact solution of the energy equation and employ the Lorentz reciprocal theorem to obtain the translational and rotational swimming velocities in the creeping-flow limit. We report fixed points for vertical movement of the micromotor for its specific orientations relative to the wall. The critical wall gap for fixed points shows unique non-monotonic dependence on the metallic coating coverage on the particle. Also, the micromotor trajectories can be switched either from wall-bound sliding or stationary state to escape from the near-wall zone by tuning the particle and the surrounding fluid pair thermal conductivity contrast. The scenario holds several exclusive distinguishing features from the otherwise extensively studied self-diffusiophoresis phenomenon near an inert wall, despite obvious analogies in the respective constitutive laws relating the fluxes with the gradients of the concerned forcing parameters. The most contrasting locomotion is the ability of a self-thermophoretic micromotor with a large heated cap to migrate towards the wall even if it is initially directed away from the wall. During the stationary states of swimming, the cold portion on the micromotor surface faces away from the wall under all conditions. Such unique aspects hold the potential of being harnessed in practice towards achieving intricate control over the autonomous motion of microparticles in thermally regulated fluidic environments.

The interaction of motile micro-organisms with a nearby solid substrate is a well-studied phenomenon. However, the effects of hydrodynamic slippage on the substrate have received little attention. In the present study, within the framework of the squirmer model, we impose a tangential velocity at the swimmer surface as a representation of the ciliary propulsion, and subsequently obtain an exact solution of the Stokes equation based on a combined analytical–numerical approach. We illustrate how the near-wall swimming velocities are non-trivially altered by the interaction of wall slip and hydrodynamic forces. We report a characteristic transition of swimming trajectories for both puller- and pusher-type microswimmers by hydrodynamic slippage if the wall slip length crosses a critical value. In the case of puller microswimmers that are propelled by a breaststroke-like action of their swimming apparatus ahead of their cell body, the wall slip can cause wall-bound trapping swimming states, as either periodic or damped periodic oscillations, which would otherwise escape from a no-slip wall. The associated critical slip length has a non-monotonic dependence on the initial orientation of the swimmer, which is represented by novel phase diagrams. Pushers, which get their propulsive thrust from posterior flagellar action, also show similar swimming state transitions, but in this case the wall-slip-mediated reorientation dynamics and the swimming modes compete in a different fashion from that of the pullers. Although neutral swimmers lack a sufficient reorientation torque to exhibit any wall-bound trajectory, their detention time near the substrate can be significantly increased by tailoring the extent of hydrodynamic slippage at the nearby wall. The present results pave the way for understanding the motion characteristic of biological microswimmers near confinements with hydrophobic walls or strategize the design of microfluidic devices used for sorting and motion rectification of artificial swimmers by tailoring their surface wettability.

We investigate the effects of surfactant coating on a deformable viscous drop under the combined action of shear flow and a uniform electric field. Employing a comprehensive three-dimensional approach, we analyse the non-Newtonian shearing response of the bulk emulsion in the dilute suspension regime. Our results reveal that the location of the peak surfactant accumulation on the drop surface may get shifted from the plane of shear to a plane orthogonal to it, depending on the tilt angle of the applied electric field and strength of the electrical stresses relative to their hydrodynamic counterparts. The surfactant non-uniformity creates significant alterations in the flow perturbation around the drop, triggering modulations in the bulk shear viscosity. Overall, the shear-thinning or shear-thickening behaviour of the emulsion appears to be greatly influenced by the interplay of surface charge convection and Marangoni stresses. We show that the balance between electrical and hydrodynamic stresses renders a vanishing surface tension gradient on the drop surface for some specific shear rates, rendering negligible alterations in the bulk viscosity. This critical condition largely depends on the electrical permittivity and conductivity ratios of the two fluids and orientation of the applied electric field. Also, the physical mechanisms of charge convection and surface deformation play their roles in determining this critical shear rate. As a consequence, we obtain new discriminating factors, involving electrical property ratios and the electric field configuration, which govern the same. Consequently, the surfactant-induced enhancement or attenuation of the bulk emulsion viscosity depends on the electrical conductivity and permittivity ratios. The concerned description of the drop-level flow physics and its connection to the bulk rheology of a dilute emulsion may provide a fundamental understanding of a more complex emulsion system encountered in industrial practice.

Electrical effects can impart a cross-stream component to drop motion in a pressure-driven flow, due to either an asymmetric charge distribution or shape deformation. However, surfactant-mediated alterations in such migration characteristics remain unexplored. By accounting for three-dimensionality in the drop motion, we analytically demonstrate here a non-trivial switching of drop migration with the aid of a surfactant coating on its surface. We establish this phenomenon as controllable by exploiting an interconnected interplay between the hydrodynamic stress, electrical stress and Marangoni stress, manifested so as to achieve a net interfacial force balance. Our results reveal that under different combinations of electrical conductivity and permittivity ratios, the relative strength of the electric stress with respect to the hydrodynamic stress, the applied electric field direction and the surfactants alter the longitudinal and cross-stream velocity components of the droplets differently. The effect of drop deformation on its speed is found to be altered with the increased sensitivity of the surface tension to the surfactant concentration, depending on the competing effects of the electrohydrodynamic flow modification and the tip stretching phenomenon. Further, with a suitable choice of electrical property ratios, the Marangoni effects can be exploited to direct the drop in reaching a final transverse position towards or away from the channel centreline. These results may turn out to be of immense consequence in providing an insight to the underlying complex physical mechanisms dictating an intricate control on the drop motion in different directions.

The sedimentation of a surfactant-laden deformable viscous drop acted upon by an electric field is considered theoretically. The convection of surfactants in conjunction with the combined effect of electrohydrodynamic flow and sedimentation leads to a locally varying surface tension, which subsequently alters the drop dynamics via the interplay of Marangoni, Maxwell and hydrodynamic stresses. Assuming small capillary number and small electric Reynolds number, we employ a regular perturbation technique to solve the coupled system of governing equations. It is shown that when a leaky dielectric drop is sedimenting in another leaky dielectric fluid, the Marangoni stress can oppose the electrohydrodynamic motion severely, thereby causing corresponding changes in the internal flow pattern. Such effects further result in retardation of the drop settling velocity, which would have otherwise increased due to the influence of charge convection. For non-spherical drop shapes, the effect of Marangoni stress is overcome by the ‘tip-stretching’ effect on the flow field. As a result, the drop deformation gets intensified with an increase in sensitivity of the surface tension to the local surfactant concentration. Consequently, for an oblate type of deformation the elevated drag force causes a further reduction in velocity. For similar reasons, prolate drops experience less drag and settle faster than the surfactant-free case. In addition to this, with increased sensitivity of the interfacial tension to the surfactant concentration, the asymmetric deformation about the equator gets suppressed. These findings may turn out to be of fundamental significance towards designing electrohydrodynamically actuated droplet-based microfluidic systems that are intrinsically tunable by varying the surfactant concentration.

Electrohydrodynamics of a leaky dielectric suspended drop subjected to the combined influence of a uniform electric field and linear velocity field is analysed analytically and numerically. In the limit of small charge convection and small shape deformation, an analytical solution is obtained for the deformed drop shape when the imposed linear flow is of uniaxial extensional type with the extensional component aligned in the direction of the electric field. This perturbation approach is then applied towards obtaining the effect of a uniform electric field on the effective extensional rheology of a dilute emulsion. Key results indicate that the magnitude and sense of drop deformation not only depends on the material properties of the drop and medium but is also governed by the strength of the applied electric field relative to the applied flow field. The interfacial charge convection is found to increase or decrease the drop deformation depending on the direction of electrohydrodynamic flow and relative strength of electric field. The electrohydrodynamic flow and drop deformation modulates the effective extensional viscosity of the emulsion. Importantly, the presence of the electric field leads to strain-rate-dependent effective extensional viscosity of the emulsion. The emulsion is found to exhibit strain-rate thinning/thickening behaviour depending on the drop to medium charge relaxation time scale. The analytically obtained drop shape and deformation are in excellent agreement with numerical simulations for the small deformation ranges.

The effect of a uniform electric field on the motion of a drop in an unbounded plane Poiseuille flow is studied analytically. The drop and suspending media are considered to be Newtonian and leaky dielectric. We solve for the two-way coupled electric and flow fields analytically by using a double asymptotic expansion for small charge convection and small shape deformation. We obtain two important mechanisms of cross-stream migration of the drop: (i) shape deformation and (ii) charge convection. The second one is a new source of cross-stream migration of the drop in plane Poiseuille flow which is due to an asymmetric charge distribution on the drop surface. Our study reveals that charge convection can cause a spherical non-deformable drop to migrate in the cross-stream direction. The combined effect of charge convection and shape deformation significantly alters the drop velocity, drop trajectory and steady state transverse position of the drop. We predict that, depending on the orientation of the applied uniform electric field and the relevant drop/medium electrohydrodynamic parameters, the drop may migrate either towards the centreline of the flow or away from it. We obtain that the final steady state transverse position of the drop is independent of its initial transverse position in the flow field. Most interestingly, we show that the drop can settle in an off-centreline steady state transverse position. Two-dimensional numerical simulations are also performed to study the drop motion in the combined presence of plane Poiseuille flow and a tilted electric field. The drop trajectory and steady state transverse position of the drop obtained from numerical simulations are in qualitative agreement with the analytical results.

We investigate the motion of a sedimenting drop in the presence of an electric field in an arbitrary direction, otherwise uniform, in the limit of small interface deformation and low-surface-charge convection. We analytically solve the electric potential in and around the leaky dielectric drop, and solve for the Stokesian velocity and pressure fields. We obtain the correction in drop velocity due to shape deformation and surface-charge convection considering small capillary number and small electric Reynolds number which signifies the importance of charge convection at the drop surface. We show that tilt angle, which quantifies the angle of inclination of the applied electric field with respect to the direction of gravity, has a significant effect on the magnitude and direction of the drop velocity. When the electric field is tilted with respect to the direction of gravity, we obtain a non-intuitive lateral motion of the drop in addition to the buoyancy-driven sedimentation. Both the charge convection and shape deformation yield this lateral migration of the drop. Our analysis indicates that depending on the magnitude of the tilt angle, conductivity and permittivity ratios, the direction of the sedimenting drop can be controlled effectively. Our experimental investigation further confirms the presence of lateral migration of the drop in the presence of a tilted electric field, which is in support of the essential findings from the analytical formalism.

In the present study, we attempt to analyse the electro-osmotic flow of two superimposed fluids through narrow confinements in the presence of axially modulated surface charges. We attempt to solve for the flow structure as well as the interface deformation by both analytical and numerical techniques. Approximate analytical solutions are obtained through asymptotic analysis for low deformations, whereas numerical solutions are obtained by applying the phase field formalism; the numerical solutions are obtained for small as well as large interfacial deformations. The analytical solutions are derived only for the transient deformation of the interface, neglecting the transience in the flow, i.e. the flow is assumed to be quasisteady. The numerical solutions, however, are derived including the effects of inertia and transients in the flow. We attempt to compare our analytical and numerical results and explore the effects of several physico-chemical parameters on the deformation of the interface as well as the nature of the flow. Our analysis reveals that parameters such as the modulation wavelength, surface tension (described through the capillary number), viscosity ratio, permittivity ratio and extent of asymmetry in the potential on the two walls are the major contributors to the deformation and the resulting flow features.