M. G. Blyth

Electrified viscous thin film flow over topography

The gravity-driven flow of a liquid film down an inclined wall with periodic indentations in the presence of a normal electric field is investigated. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. In particular, the interaction between the electric field and the topography is examined by predicting the shape of the film surface under steady conditions. A nonlinear, non-local evolution equation for the thickness of the liquid film is derived using a long-wave asymptotic analysis. Steady solutions are computed for flow into a rectangular trench and over a rectangular mound, whose shapes are approximated with smooth functions. The limiting behaviour of the film profile as the steepness of the wall geometry is increased is discussed. Using substantial numerical evidence, it is established that as the topography steepness increases towards rectangular steps, trenches, or mounds, the interfacial slope remains bounded, and the film does not touch the wall. In the absence of an electric field, the film develops a capillary ridge above a downward step and a slight depression in front of an upward step. It is demonstrated how an electric field may be used to completely eliminate the capillary ridge at a downward step. In contrast, imposing an electric field leads to the creation of a free-surface ridge at an upward step. The effect of the electric field on film flow into relatively narrow trenches, over relatively narrow mounds, and down slightly inclined substrates is also considered.

Effect of surfactants on the stability of two-layer channel flow

The effect of an insoluble surfactant on the stability of two-layer viscous flow in an inclined channel confined by two parallel walls is considered. A lubrication-flow model applicable to long waves and low-Reynolds-number-flow is developed, and pertinent nonlinear evolution equations for the interface position and surfactant concentration are derived. Linear stability analysis based on the lubrication-flow model and the inclusive equations of Stokes flow confirm the recent findings of Frenkel & Halpern (2002) and Halpern & Frenkel (2003) that the presence of an insoluble surfactant induces a Marangoni instability in certain regions of parameter space defined by the layer thickness and viscosity ratios. Numerical simulations based on both approaches show that the interfacial waves may grow and saturate into steep profiles. The lubrication-flow model is adequate in capturing the essential features of the instability for small and moderate wavenumbers.

Effect of surfactant on the stability of film flow down an inclined plane

The effect of an insoluble surfactant on the stability of the gravity-driven flow of a liquid film down an inclined plane is investigated by a normal-mode analysis. Numerical solutions of the Orr–Sommerfeld equation reveal the occurrence of a stable Marangoni mode and a possibly unstable Yih mode, and demonstrate that the primary role of the surfactant is effectively to raise the critical Reynolds number at which instability is first encountered.

Film flow down an inclined plane over a three-dimensional obstacle

The low-Reynolds-number flow of a liquid film down an inclined plane wall over a particle attached to the wall is considered. The effect of the particle is described on the assumption that the free surface suffers only a minor disturbance from its basic flat state, and the disturbance velocity over the free surface is small, even though the particle size may not be small compared to the unperturbed film thickness. The problem is formulated using the boundary integral method for Stokes flow to describe the surface velocity, the deformation of the film surface, and the distribution of the traction over the particle surface. Results are presented for small particles following the earlier asymptotic analysis of Pozrikidis and Thoroddsen, and for moderate-sized particles with semispherical, spherical, and semispheroidal shapes. The simulations reveal that, in all cases, the free surface causes an upstream hump and a horseshoe type of deformation downstream whose intensity depends on the Bond number and is largely insensitive to the specific particle shape.

Effect of an electric field on film flow down a corrugated wall at zero Reynolds number

The effect of an electric field on a liquid layer flowing down an inclined, corrugated wall at zero Reynolds number is investigated. The layer is taken to be either a perfect conductor or a perfect dielectric. The region above the layer is assumed to be a perfect dielectric. Steady flow down a wall with small-amplitude sinusoidal corrugations is considered, and it is shown how the electric field can be used to control the amplitude of the free-surface deflection and the phase shift between the free surface and the wall profile. Steady flow over walls with large amplitude sinusoidal corrugations or other-shaped indentations is studied by using the boundary-element method. Results for flow into a wide rectangular trench are compared to previous model predictions based on the lubrication approximation. For a perfect-conductor film, the results confirm that the height of the capillary ridge, which appears above a downward step, monotonically decreases as the electric field strength increases. Solutions for a ...

Stability of axisymmetric core–annular flow in the presence of an insoluble surfactant

The effect of an insoluble surfactant on the stability of the core-annular flow of two immiscible fluids is investigated by a normal-mode linear analysis and by numerical simulations based on the immersed-interface method for axisymmetric perturbations. The results reveal that, although the Marangoni stress due to surfactant concentration variations is unable to initiate a new type of instability as in the case of two-dimensional two-layer channel flow, it does destabilize the interface by broadening the range of growing wavenumbers and by raising the growth rate of unstable perturbations. Numerical simulations for large-amplitude disturbances reveal that the surfactant plays an important role in determining the morphology of the interfacial structures developing in the nonlinear stages of the motion.

A note on oblique stagnation-point flow

Previous analyses of oblique stagnation-point flow at a plane wall are discussed and unified with reference to a free parameter. The oblique flow consists of orthogonal stagnation-point flow to which is added a shear flow whose vorticity is fixed at infinity. Physically the free parameter may be viewed as altering the structure of the shear flow component by varying the magnitude of the pressure gradient. For large adverse pressure gradients, the shear component has a region of reversed flow near the wall. Remarkably, combining an orthogonal flow with shear flows featuring different levels of reversed flow always produces the same oblique flow but with the stagnation-point of attachment shifted along the wall by a predictable amount.

Motion of a two-dimensional elastic capsule in a branching channel flow

The transit of a two-dimensional elastic fluid-filled capsule through a channel with a side branch is investigated numerically. The mathematical formulation allows for a capsule carried in a pressure-driven flow of fluid of generally different viscosity to that inside the capsule. Far upstream and downstream in the main channel, and downstream in the side branch, the fluid velocity profiles are assumed to adopt those of unidirectional Poiseuille flow with prescribed flow rates. The capsule boundary is treated as a two-dimensional elastic membrane developing elastic tensions and bending moments according to simple constitutive laws. A boundary-integral formulation allows for the explicit computation of the fluid pressures upstream and downstream of the branching. The novelty of the approach is the inclusion of a notional boundary at the entrance to the side branch, which avoids the need to collocate the channel ends. The deformation experienced by the capsule in the region of the junction is found to depend strongly on the branch angle. The deformation is ameliorated by increasing the membrane stiffness or lowering the viscosity of the suspending fluid relative to the encapsulated fluid. When a capsule exits the branch region, a distance of many decades of capsule diameters is required before the capsule relaxes to an equilibrium shape. Capsule residence times in the vicinity of the branch region can be considerable, depending on the line of approach into the junction and the capsule deformability. The path selection of a cell at a branch junction can depend crucially on capsule deformability: capsules with different elastic properties may follow different routes out of the junction in otherwise identical flow conditions.

Steady flow in a dividing pipe

The high Reynolds number flow through a circular pipe divided along a diameter by a semi-infinite splitter plate is considered. Matched asymptotic expansions are used to analyse the developing flow, which is decomposed into four regions: a boundary layer of Blasius type growing along the plate, an inviscid core, a viscous layer close to the curved wall and a nonlinear corner region. The core solution is found numerically, initially in the long-distance down-pipe limit and thereafter the full problem is solved using down-pipe Fourier transforms. The accuracy in the corners of the semicircular cross-section is improved by subtracting out the singularity in the velocity perturbation. The linear viscous wall layer is solved analytically in terms of a displacement function determined from the core. A plausible structure for the corner region and equations governing the motion therein are presented although no solution is attempted. The presence of the plate has little effect ahead of the bifurcation, but wall shear on the curved wall is found to increase from its undisturbed value downstream

Viscous electrified film flow over step topography

The steady, gravity-driven flow of a liquid film over a topographically structured substrate is investigated. The analysis is based on a model nonlinear equation for the film thickness derived on the basis of long-wave asymptotics. The free-surface shape is expanded in a regular asymptotic expansion in powers of the topography amplitude, and solutions are obtained up to second order. Solutions are constructed for downward steps, upward steps, and rectangular trenches, and the results are compared favorably with numerical solutions of the nonlinear model equation. The results indicate that all of the salient features previously found for film flows over steps and into trenches are captured by the small-step asymptotics, including the capillary ridge formed just above a downward step and oscillations upstream of an upward step. We derive analytical expressions for the period and amplitude of these oscillations. The effect of a normal electric on the film surface shape is also investigated on the assumption ...