Search

Kang et al. (J. Fluid Mech., vol. 874, 2019, pp. 339–358) studied viscous dissipation within a permeable body with a view to maximizing the damping of oscillations of the body. They found that dissipation is maximal when the length scale for diffusion of vorticity in the fluid outside the body is similar to the length scale for decay of fluid motion within the body. Their results are examined in the context of the simpler problem of a porous half-space oscillating parallel to the interface between porous solid and fluid. The analysis is then extended to consider the impulsive start-up from rest of a porous plane surface adjacent to unbounded fluid.

We investigate electrophoresis of a tightly fitting sphere of radius $R-h_{0}$ on the axis of a circular tube of radius $R$, using lubrication theory and ideas due to Schnitzer & Yariv (Phys. Fluids, vol. 26, 2014, 122002). The electrical charge clouds on both the cylindrical wall and the surface of the sphere are assumed thin compared to the gap between the sphere and cylinder, so that charge clouds do not overlap and ion exclusion effects are minimal. Nevertheless, non-uniform pumping of counter-ions within the charge clouds leads to a change in the ionic concentration outside the charge clouds in the narrow gap between sphere and cylinder. The electro-osmotic slip velocities at the two surfaces are modified, leading to a decrease in the electrophoretic velocity of the sphere at low Péclet numbers and an increase in the velocity at high Péclet numbers. When the field strength $E_{0}$ is low, it is known that the electrophoretic velocity $U_{0}$ is proportional to $E_{0}(\unicode[STIX]{x1D701}_{s}-\unicode[STIX]{x1D701}_{c})$ which is zero when the zeta potential $\unicode[STIX]{x1D701}_{s}$ on the sphere surface is equal to the zeta potential $\unicode[STIX]{x1D701}_{c}$ on the cylinder. The perturbation to the above low field strength electrophoretic velocity (at high Péclet number) is predicted to be proportional to $E_{0}^{3}(\unicode[STIX]{x1D701}_{s}+\unicode[STIX]{x1D701}_{c})^{2}(\unicode[STIX]{x1D70E}_{s}+\unicode[STIX]{x1D70E}_{c})$, where $\unicode[STIX]{x1D70E}_{s}$ and $\unicode[STIX]{x1D70E}_{c}$ are the surface charge densities on the sphere and cylinder. The choice of materials with similar or identical zeta potentials (and surface charge densities) for the cylinder and sphere should therefore facilitate the observation of velocities nonlinear in the field strength $E_{0}$, since the reference linear electrophoretic velocity will be small.

The electrical streaming potential generated by a two-phase pressure-driven Stokes flow in a cylindrical capillary is computed numerically. The potential difference ΔΦ between the two ends of the capillary, proportional to the pressure difference Δp for single-phase flow, is modified by the presence of a suspended drop on the centreline of the capillary. We determine the change in ΔΦ caused by the presence of an uncharged insulating neutrally buoyant drop at a small electric Hartmann number, i.e. when the perturbation to the flow field caused by electric stresses is negligible.The drop velocity and deformation, and the consequent changes in the pressure difference Δp and streaming potential ΔΦ, depend upon three independent parameters: the size a of the undeformed drop relative to the radius R of the capillary; the viscosity ratio λ between the drop phase and the continuous phase; and the capillary number Ca which measures the ratio of viscous to capillary forces. We investigate how the streaming potential depends on these parameters: purely hydrodynamic aspects of the problem are discussed by Lac & Sherwood (J. Fluid Mech., doi:10.1017/S0022112009991212).The potential on the capillary wall is assumed sufficiently small so that the electrical double layer is described by the linearized Poisson–Boltzmann equation. The Debye length characterizing the thickness of the charge cloud is taken to be small compared with all other length scales, including the width of the gap between the drop and the capillary wall. The electric potential satisfies Laplace's equation, which we solve by means of a boundary integral method. The presence of the drop increases |ΔΦ| when the drop is more viscous than the surrounding fluid (λ > 1), though the change in |ΔΦ| can take either sign for λ < 1. However, the difference between ΔΦ and Δp (suitably non-dimensionalized) is always positive. Asymptotic predictions for the streaming potential in the case of a vanishingly small spherical droplet, and for large drops at high capillary numbers, agree well with computations.

The deformation of a drop as it flows along the axis of a circular capillary in low Reynolds number pressure-driven flow is investigated numerically by means of boundary integral computations. If gravity effects are negligible, the drop motion is determined by three independent parameters: the size a of the undeformed drop relative to the radius R of the capillary, the viscosity ratio λ between the drop phase and the wetting phase and the capillary number Ca, which measures the relative importance of viscous and capillary forces. We investigate the drop behaviour in the parameter space (a/R, λ, Ca), at capillary numbers higher than those considered previously. If the fluid flow rate is maintained, the presence of the drop causes a change in the pressure difference between the ends of the capillary, and this too is investigated. Estimates for the drop deformation at high capillary number are based on a simple model for annular flow and, in most cases, agree well with full numerical results if λ ≥ 1/2, in which case the drop elongation increases without limit as Ca increases. If λ < 1/2, the drop elongates towards a limiting non-zero cylindrical radius. Low-viscosity drops (λ < 1) break up owing to a re-entrant jet at the rear, whereas a travelling capillary wave instability eventually develops on more viscous drops (λ > 1). A companion paper (Lac & Sherwood, J. Fluid Mech., doi:10.1017/S002211200999156X) uses these results in order to predict the change in electrical streaming potential caused by the presence of the drop when the capillary wall is charged.

A drop of fluid, initially held spherical by surface tension, will deform when an electric or magnetic field is applied. The deformation will depend on the electric/magnetic properties (permittivity/permeability and conductivity) of the drop and of the surrounding fluid. The full time-dependent low-Reynolds-number problem for the drop deformation is studied by means of a numerical boundary-integral technique. Fluids with arbitrary electrical properties are considered, but the viscosities of the drop and of the surrounding fluid are assumed to be equal.Two modes of breakup have been observed experimentally: (i) tip-streaming from drops with pointed ends, and (ii) division of the drop into two blobs connected by a thin thread. Pointed ends are predicted by the numerical scheme when the permittivity of the drop is high compared with that of the surrounding fluid. Division into blobs is predicted when the conductivity of the drop is higher than that of the surrounding fluid. Some experiments have been reported in which the drop deformation exhibits hysteresis. This behaviour has not in general been reproduced in the numerical simulations, suggesting that the viscosity ratio of the two fluids can play an important role.

The streamline patterns of some simple two-dimensional Stokes flows are studied and the results used both to understand and to predict the streamlines of flows in more complicated geometries, in particular the streamlines of flows that contain eddies or regions of closed streamlines. Initially, streamline patterns are studied locally, either around some special point, such as a stagnation point or a point where a streamline meets a wall, or in a special region, such as a corner. The use of these local analyses is illustrated by finding the streamlines for shear flow around a rotating cylinder; the illustration also shows how fluid in Stokes flow can be turned back on itself (‘blocked’). The local analyses of flow in a corner are used to understand the eddy patterns that have been discovered in a variety of flows. The eddies occur in corner-like regions and in these regions the flow can be regarded as the superposition of two components. One component, the eddy flow, is the result of flow outside the corner stirring the fluid in the corner, while the other is the direct result of local conditions in the corner. The competition between these two components determines whether eddies actually appear in a given flow. Finally, the approach developed here is applied to a new flow situation, namely a shear flow which is bounded by a moving wall and which contains a stationary cylinder touching the wall. The streamlines deduced for different ratios of the shear strength to the wall velocity show both new eddy patterns and unexpected regions of blocked flow.

We study the primary electroviscous effect in a suspension of spheres when the double layer thickness κ−1 is small compared with the particle radius a. The case of a 1–1 symmetric electrolyte is examined using the methods of Dukhin & coworkers (1974), whilst the asymmetric electrolyte is studied along lines similar to those of O'Brien (1983). Sherwood's (1980) asymptotic results for high surface potentials and high Hartmann numbers are extended and complemented.

The deformation of inviscid and slightly viscous drops is studied using slender-body theory. The imposed axisymmetric flow is a combination of a linear extensional flow, with velocity uz = G1 z along the axis of symmetry, together with a cubic flow uz = G3z3. When G3/G1 is sufficiently small the viscous drop breaks in a manner similar to that described by Acrivos & Lo (1978). For larger G3 > 0 the drop breaks by a rapid growth at its end. Steady-state experiments in a 4-roll mill show the ejection of a column of liquid from the tip of the drop, though this is probably caused by a change in the pressure gradient rather than the mechanism described above. The ejected column then breaks into droplets via the Rayleigh instability. It is hypothesized that one or other of these mechanisms corresponds to tip streaming as observed by Taylor (1934).

Previous studies of the distortion of the electric double layer around a charged sphere have assumed that the electric stresses are small compared with the viscous stresses. The flow around the particle is therefore changed only slightly by the presence of the charge cloud. This change is measured by the Hartmann number, and in § 6 we remove the restriction that it should be small. It is found that the previous linearized theory is sufficiently accurate for typical experimental values of the Hartmann number. Previous studies have also assumed that the potential at the surface of the particle is small. This assumption is removed in § 7 of this paper. For values of the non-dimensional surface potential smaller than 2 the predictions are altered by less than 10 %. For higher values the differences between linear and nonlinear theory are not negligible, especially when the charge cloud is thin compared with the radius of the charged sphere.

The electrophoretic velocity of a gas bubble is difficult to measure, since we must also contend with velocities due to buoyancy. One way to avoid this problem is to spin the electrophoretic cell about a horizontal axis. Centrifugal forces keep the bubble on the centreline of the cell. The price to be paid is the creation of Taylor columns which alter the hydrodynamic drag on the bubble. Here we modify two analyses by Moore & Saifman (1968, 1969) to include electrical effects. Motion of the body is assumed to be slow and steady, and the Ekman number small. The electrical double-layer thickness is small compared with the thickness of the Ekman layer. It is assumed that the presence of surfactants makes the gas-water interface rigid, and a no-slip boundary condition is applied. We predict that the electrophoretic velocity U should be proportional to e E ψ 0 ( a ν) −l (ν/Ω) ½ , where E is the applied electric field, e the permittivity of the suspending fluid, ψ 0 the ζ potential at the surface of the bubble, a the bubble radius, ν the fluid viscosity, ν the kinematic viscosity and Ω the rate of rotation. There is reasonable agreement with some, but not all, published experimental results.