Packing a flexible fiber into a cavity.

The insertion of an elastic rod or fiber into a confining cavity is studied. Such an insertion is a feature of a variety of problems, including packing and unpacking of DNA in viral capsids and the insertion of catheters during surgery. We consider a simplified geometry in which the container is a smooth (frictionless) circular cylinder of radius a. The fiber is pushed through a hole in the curved surface of the cylinder and is then assumed to stay in a cross-sectional plane perpendicular to the cylinder axis. A solution is found for the fiber shape in which most of the fiber lies against the curved interior surface of the cylinder, apart from the final end section of the fiber, of length 2.0888a, which crosses the interior of the cylinder before ending at the opposite side, which it meets at an angle 1.15 rad to the normal. The force required to push the fiber into the cylinder is EI/2a^{2}, where E is the fiber's Young's modulus and I its cross-sectional moment of inertia. The shape of the final end section of the fiber is confirmed by experiment.

Effect of Nonzero Solid Permittivity on the Electrical Repulsion between Charged Surfaces.

The electrical repulsion between two charged solid surfaces separated by an electrolyte is studied as a function of the permittivity ϵs of the solid in the limit in which potentials are small, and the gap between the plane solid surfaces is small compared to the Debye length κ-1 within the electrolyte. The solid surfaces are uniformly charged in a central region |x|< L outside which they are uncharged. When ϵs = 0, ions from the charge cloud between the charged surfaces spill out into regions of length O(κ-1) beyond x = ± L, thereby reducing the pressure between the surfaces from that predicted by Derjaguin-Landau-Verwey-Overbeek theory for infinite, uniformly charged surfaces. When ϵs>0, ions spill out over much larger O(L) regions, thereby reducing still further both the electrical potential between the solid surfaces and the repulsive force between them. However, this reduction becomes smaller as κL becomes large.

Electrically generated eddies at an eightfold stagnation point within a nanopore.

Electrically generated flows around a thin dielectric plate pierced by a cylindrical hole are computed numerically. The geometry represents that of a single nanopore in a membrane. When the membrane is uncharged, flow is due solely to induced charge electroosmosis, and eddies are generated by the high fields at the corners of the nanopore. These eddies meet at stagnation points. If the geometry is chosen correctly, the stagnation points merge to form a single stagnation point at which four streamlines cross at a point and eight eddies meet.

Electroosmosis in a finite cylindrical pore: simple models of end effects.

A theoretical model of electroosmosis through a circular pore of radius a that traverses a membrane of thickness h is investigated. Both the cylindrical surface of the pore and the outer surfaces of the membrane are charged. When h ≫ a, end effects are negligible, and the results of full numerical computations of electroosmosis in an infinite pore agree with theory. When h = 0, end effects dominate, and computations again agree with analysis. For intermediate values of h/a, an approximate analysis that combines these two limiting cases captures the main features of computational results when the Debye length κ(-1) is small compared with the pore radius a. However, the approximate analysis fails when κ(-1) ≫ a, when the charge cloud due to the charged cylindrical walls of the pore spills out of the ends of the pore, and the electroosmotic flow is reduced. When this spilling out is included in the analysis, agreement with computation is restored.

Non-Newtonian stress in an electrolyte.

Analogies are drawn between the dynamics of electrolyte solutions and those of dilute suspensions of charged colloidal particles. The viscosity of both electrolytes and suspensions is a function of the ionic concentration c and of the Peclet number Pe characterizing the ratio of applied shear rate that tends to deform the ionic charge clouds, to diffusion that allows them to relax back to equilibrium. In particular, previously published results on the rheology of colloidal suspensions ( Lever , D. A. J. Fluid Mech. 1979 , 92 , 421 - 433 ) imply not only that the Falkenhagen O(c(1/2)) electrical contribution to viscosity is shear thinning, as shown by H. Wada ( J. Stat. Mech.: Theory Exp. 2005 , P01001 ), but also that this contribution to the stress is elastic, with normal stress differences appearing at O(Pe(2)). In practice, the shear rates required for substantial departure from a Newtonian rheology are large, typically 10(9) s(-1).

Streaming potential generated by two-phase flow in a polygonal capillary.

Theoretical predictions of the streaming potential generated by two-phase flow in a polygonal capillary are presented. The capillary walls are wetted by the continuous water phase, in which is suspended a long non-conducting oil drop. It is assumed that when fluid flows along the capillary a thin film of fluid is created between the drop and the capillary walls, which allows ions in the charge cloud adjacent to the wall to be convected. Current returns via conduction along the menisci of wetting fluid in the corners of the polygonal capillary. These menisci have finite cross-sectional area even when fluid is at rest, so that streaming potentials are predicted to grow linearly with the applied pressure gradient.

Streaming potential generated by a long viscous drop in a capillary.

The streaming potential generated by motion of a long drop of viscosity mu(d) = lambdamu in a uniform circular capillary filled with fluid of viscosity mu is investigated by means of a model previously used to study electrophoresis of a charged mercury drop in water. The capillary wall is at potential zeta c relative to the bulk fluid within it, and the surface of the drop is at potential zeta(d). Potentials are assumed to be sufficiently small so that the charge cloud is described by the linearized Poisson-Boltzmann equation, and the Debye length characterizing the thickness of the charge cloud is assumed to be thin compared with the gap h(0) between the drop and the capillary wall. Ions in the external fluid are not allowed to discharge at the surface of the drop, and the wall of the capillary has a nonzero surface conductivity sigma c. The drop is assumed to be sufficiently long so that end effects can be neglected. Recirculation of fluid within the drop gives rise to an enhanced streaming current when zeta(d) is nonzero, leading to an anomalously high streaming potential. This vanishes as the drop viscosity becomes large. If V is the velocity of the drop and gamma is the coefficient of interfacial tension between the two fluids, then the capillary number is Ca = mu V/gamma, and the gap varies as h(0)planck'sCa(2/3). When Ca is small, the gap h(0) is small and electrical conduction along the narrow gap is dominated by the surface conductivity sigma(c) of the capillary wall, which is constant. The electrical current convected by flowing fluid is proportional to Ca, as is the change in streaming potential caused by the presence of the drop. If sigma(c) = 0, then the electrical conductance of the gap depends on its width h(0) and on the bulk fluid conductivity sigma and becomes small as h(0) approximately equal to Ca(2/3) --> 0. The streaming potential required to cancel the O(Ca) convection current therefore varies as Ca(1/3). If sigma(c) = 0 and the drop is rigid (lambda --> infinity), then the change in streaming potential over and above that expected due to the change in pressure gradient is proportional to the difference in potentials zeta(c)-zeta(d).

Cell models for the primary electroviscous effect.

The primary electroviscous effect in a nondilute suspension of charged spherical particles is studied by means of cell models. The governing equations are derived, and then analytic results are obtained by restricting attention to the limit of thin double layers, small Hartmann and Peclet numbers, and small potentials. Previous work has assumed that the velocity at the outer boundary of the cell is identical to the imposed flow, as proposed by Simha (J. Appl. Phys. 1952, 23, 1020). Results with this boundary condition are compared against those predicted when the tangential shear stress on the outer boundary is assumed to be unperturbed, as proposed by Happel (J. Appl. Phys. 1957, 28, 1288). Both the hydrodynamic and electroviscous contributions to the effective viscosity are smaller with the Happel boundary condition, showing that such cell models offer a range of predictions and should be used with caution.

Filtration of deformable emulsion droplets.

Oil-in-water (o/w) emulsions of different droplet size were filtered on membranes of various pore sizes to investigate the growth and behaviour of o/w filter cakes. The cake desorptivity S and the filter membrane resistance R were measured at various filtration pressures P. The variation of S with P shows that filter cake oil droplets of radius a are effectively rigid for P << gamma/a and fully deformable for P >> gamma/a, where gamma is the oil-water interfacial tension. For the largest P, when S became P-independent, the filter cake remained water-permeable as expected from theory.

Rates of transport through a capsule membrane to attain Donnan equilibrium.

The swelling of a capsule consisting of salt solution and polyelectrolyte, surrounded by a membrane, is studied. The membrane allows salt and water to pass, but is impermeable to polyelectrolyte molecules. Equilibrium swelling of the capsule is governed by Donnan equilibrium. Transport rates of a salt and water through the membrane are expressed in terms of a Darcy permeability and a salt diffusivity. The governing equations predict that the rate at which equilibrium is attained as the external salt concentration varies is controlled by the timescale for diffusion of salt, rather than by that for Darcy flow. Experiments were performed using capsules with membranes made of covalently linked HSA and alginate. The capsule volume varied with a single relaxation rate when the external salt concentration was changed, as predicted by theory. This constitutes the first step toward a simple method for determining the membrane properties of capsules by measuring rates of change of capsule volume.