J.D. Sherwood

Breakup of fluid droplets in electric and magnetic fields

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.

Vane rheometry of bentonite gels

Abstract We describe the use of a four-bladed vane to measure the yield stress τ0 of a series of aqueous bentonite clay suspensions. An extension of the vane technique is given, showing how the shear modulus G and the yield strain γy can also be obtained. These techniques are validated by comparing the results with those obtained from conventional concentric cylinder rheometry at low clay concentrations where wall-slip is absent. A study of the clay concentration dependence is made for volume fractions (φ) up to 0.1. The magnitudes of τ0, G and γy are found to scale closely as aφx over the volume fraction range 0.01–0.1, with x approximating to 3 for τ0, to 4 for G and to −1 for γy. Whereas x is largely time-independent, a increases with time for both τ0 and G as the dispersions spontaneously gel.

A similarity solution for steady-state crossflow microfiltration

A similarity solution is presented for the convective-diffusion equation governing the steady-state concentration—polarization boundary layer in crossflow microfiltration of the particles, under conditions where a thin stagnant layer of particles deposited on the microporous membrane surface provides the controlling resistance to filtration. The analysis employs concentration-dependent shear viscosities and shear-induced hydrodynamic diffusivities based on empirical correlations for suspensions of rigid spheres. The resulting permeate flux is vw(x) = τw(a4/3x)13νt-w/μ0, where x is filter entrance, τw is the wall shear stress exerted on the boundary layer by the tangential flow of bulk suspension through the filter channel, a is the particle radius, and μ0 is the characteristic viscosity. The dimensionless permeate flux, ν−w (φb), depends only on the particle volume fraction in the bulk suspension, φb, and is given by ν−w = 0.0581φb−13 when the suspension is dilute (φb < 0.10). The results for the permeate flux and for the concentration and velocity profiles show that the approximate solution of Davis and Leighton [Chem. Engng Sci.42, 275–281 (1987)] and Romero and Davis [J. Membrane Sci.39, 157–185 (1988)], which neglects axial convection in the differential particle mass balance but retains it when integrating across the entire boundary layer, is exact in the dilute limit and accurate to within a few percent for nondilute suspensions. The solution may easily be extended to other suspensions having different dependencies of viscosity and diffusivity on concentration.

Biot poroelasticity of a chemically active shale

The Biot theory of poroelasticity relates the strain ε of a porous material to changes of the applied stress σ and of the pore pressure p. Additional osmotic effects are present in some rocks, such as shales. This paper modifies the thermodynamic arguments used by Biot in order to include the chemical potentials μr of all the chemical species within the pore fluid. In the limit in which salt is unable to move into or out of the shale, the deformation depends only on the chemical potential μw of the water and the applied stress. In the limit of a chemically inert rock, the standard Biot analysis is obtained, and the pore pressure p is again the important variable. Real shales lie somewhere between these two limits.

Squeeze-flow of a Herschel–Bulkley fluid

Abstract Squeeze-flow experiments of a Herschel–Bulkley material between two rigid plates, investigated both experimentally and computationally by Adams et al. (J. Non-Newtonian Fluid Mech. 71 (1997) 41) are compared against an approximate analysis for generalised Newtonian fluids presented by Sherwood and Durban (J. Non-Newtonian Fluid Mech. 62 (1996) 35), for the case in which the interface between the material and the plates is lubricated. The analysis presented here assumes a rigid-viscoplastic material, rather than the elastic-viscoplastic material of Adams et al., but the viscoplastic model for flow is identical. However, the shear stress boundary condition at the plates differs from the Coulomb friction law of Adams et al.: the shear stress is here assumed to be a constant fraction of the effective stress (and consequently turns out to be independent of position). A simple expression for the total force required to push the plates together is obtained for the case when friction at the plates is small. Agreement between this expression and the experimentally-measured force is good at high strain, though the elastic deformation observed prior to yield is not captured by the rigid-viscoplastic analysis.

Squeeze flow of a power-law viscoplastic solid

Abstract A cylinder of height h is squeezed between two parallel circular plates of radius R >>h. The cylinder is assumed to behave as a generalised Newtonian material in which the stress and strain rate are coaxial: the particular cases of a rigid-plastic solid and power-law fluid are considered in detail. It is assumed that the frictional stress at the walls is a fixed fraction m of the yield stress in shear, k, in the case of the plastic material, and a fixed fraction of the effective Mises stress in the case of the power-law fluid. This boundary condition, often used in plasticity analysis, leads in both cases to a constant shear stress at the walls, rather than a no-slip boundary condition. Hoop stresses are included in an approximate analysis in which stresses and velocities are expanded as series in inverse powers of the radial coordinate r: these expansions break down near the axis r = 0 of the cylinder. The force required to compress the rigid-plastic cylinder is F= 2 3 mkϵR 3 h − + 1 2 3 kϵR 2 [(1−m 2 ) 1 2 + m − sin − m]+ O(kRh) independent of the speed of compression. The analysis can be extended to other solids and fluids characterised by a coaxial constitutive relation: by way of example, results are presented for the Bingham fluid.

Motion of a drop along the centreline of a capillary in a pressure-driven flow

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). 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.

Swelling of Shale around a Cylindrical Wellbore

A modified form of Biot’s linear theory of poroelasticity is applied to shale swelling in contact with an aqueous electrolyte. The shale is assumed to behave as an isotropic, perfect ion exclusion membrane, and in this limit swelling depends only upon the total stress and on the chemical potential of water within the pores of the rock. An axisymmetric, plane-strain analysis of swelling around a wellbore is first presented, and this is subsequently extended to include swelling of a cylindrical hole in a finite, cylindrical shale sample. It is predicted that swelling is prevented if the chemical potential of water within the shale equals that within the wellbore. The predictions of the analysis are compared with experimental results obtained when drained outcrop shale swelled in contact with aqueous solutions of KCl or NaCl. The experimental swelling did indeed vary with water chemical potential, and could be prevented if the salt concentration within the wellbore fluid was sufficiently high. However, post-mortem chemical analysis of the shale showed that ion-exchange had taken place, with consequent modification of the shale’s mechanical and chemical properties. Ion exclusion was therefore imperfect, and an analysis that incorporates the chemical potentials of components other than water is necessary.

The primary electroviscous effect in a suspension of spheres with thin double layers

We study the primary electroviscous effect in a suspension of spheres when the double layer thickness K-~ is small compared with the particle radius a. The case of a 1-1 symmetric electrolyte is examined using the methods of Dukhin 6 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.

Squeeze-film rheometry of non-uniform mudcakes

The standard lubrication analysis for a squeeze flow of a Bingham fluid is reviewed, and then extended to the case of a fluid in which the yield stress varies as a function of depth. These analyses are used to obtain the yield stress τ0 from squeeze-film measurements made on uniform mudcakes formed by filtration of a typical bentonite mud. Over the solids volume fraction range 0.09 < φ < 0.6 we find τ0 (bar) = 3.9φ1.9 for this mud. If the cake remains under an applied differential pressure p∞ for a sufficiently long time, the volume fraction φ eventually reaches a limiting equilibrium value, and the yield stress may then be expressed in terms of p∞ as ln (φ0 (Pa)) = 0.72 ln (p∞ (bar)) + 9.3. Non-uniform mudcakes are also investigated, both directly, by the squeeze-film technique, and also indirectly, by combining measured filtercake concentration profiles with results for τ0(φ) obtained from uniform cakes. There is some agreement between the two sets of results, though further work would have to be done to make the direct measurement technique useful. Once deformation ceases, the relaxation of the stress within the cake is monitored; the relaxation time increases as the bentonite concentration within the cake increases.