ALBERT

Description: A contact line on a heated oscillating plate is investigated. The interface is a non-deformable plane and the contact angle is π/2. The amplitude of the oscillation and the temperature deviation of the plate from the ambient temperature of the fluid are assumed to be much smaller than the viscous velocity scale. This flow is then governed by the unsteady Stokes equations coupled to the heat equation in a frame of reference moving with the contact line. Evaporation is assumed to be neglible, but the effects of heat transfer across the interface and unsteadiness are assumed to be significant. For a stationary heated plate, there are two distinct regions of flow that is induced by Marangoni stresses. An outer stagnation-point-type flow is seen, which separates from the plate for non-zero Biot numbers. For an oscillatory, isothermal plate, vortices are generated at the plate during plate reversal and are propagated along the interface. Dissipation of these vortices occurs on the Stokes layer scale. The order-Péclet-number correction in the thermal field is also found, and the presence of the flow field leads to a heated region in the steady case along the separating streamline. For the unsteady case, a localized cooled region propagates into the bulk with a trajectory determined by the relative scale of the thermal diffusive scale and the rate of heat transfer across the interface.

Description: An efficient algorithm for hydrodynamical interaction of many deformable drops subject to shear flow at small Reynolds numbers with triply periodic boundaries is developed. The algorithm, at each time step, is a hybrid of boundary-integral and economical multipole techniques, and scales practically linearly with the number of drops N in the range N 〈 1000, for N Δ ~ 103 boundary elements per drop. A new near-singularity subtraction in the double layer overcomes the divergence of velocity iterations at high drop volume fractions c and substantial viscosity ratio λ. Extensive long-time simulations for N = 100-200 and NΔ = 1000-2000 are performed up to c = 0.55 and drop-to-medium viscosity ratios up to λ = 5, to calculate the non-dimensional emulsion viscosity μ* = Σ12⊤ (μe·γ), and the first N1 = (Σ11 - Σ22) ⊤ (μe·γ) and second N2 = (Σ22 - Σ33) ⊤ (μe·γ) normal stress differences, where ·γ is the shear rate, μe is the matrix viscosity, and Σij is the average stress tensor. For c = 0.45 and 0.5, μ* is a strong function of the capillary number Ca = μe·γaσ (where α is the non-deformed drop radius, and σ is the interfacial tension) for Ca ≪ 1, so that most of the shear thinning occurs for nearly non-deformed drops. For c = 0.55 and λ =1, however, the results suggest phase transition to a partially ordered state at Ca ≤ 0.05, and μ* becomes a weaker function of c and Ca; using λ = 3 delays phase transition to smaller Ca. A positive first normal stress difference, N1, is a strong function of Ca; the second normal stress difference, N2, is always negative and is a relatively weak function of Ca. It is found at c = 0.5 that small systems (N ~ 10) fail to predict the correct behaviour of the viscosity and can give particularly large errors for N1, while larger systems N ≥. O(102) show very good convergence. For N ~ 102 and N Δ ~ 103, the present algorithm is two orders of magnitude faster than a standard boundary-integral code, which has made the calculations feasible.

Description: Experiments were performed to measure the rebound velocities of small plastic and metal spheres dropped from various heights onto a smooth quartz surface coated with a thin layer of viscous fluid. The spheres stick without rebounding for low impact velocities, due to viscous dissipation in the thin fluid layer. Above a critical impact velocity, however, the lubrication forces in the thin layer cause elastic deformation and rebound of the spheres. The apparent coefficient of restitution increases with the ratio of the Stokes number to its critical value for rebound, where the Stokes number is a dimensionless ratio of the inertia of the sphere to viscous forces in the fluid. The critical Stokes number required for rebound decreases weakly with increasing values of a dimensionless elasticity parameter which is a ratio of the viscous forces which cause deformation to the elastic forces which resist deformation. The experimental results show good agreement with an approximate model based on lubrication theory for undeformed spheres and scaling relations for elastic deformation.

Description: We consider viscous shear flow of a monolayer of solid spheres and discuss the effect that microscopic particle surface roughness has on the stress in the suspension. We consider effects both within and outside the dilute régime. Away from jamming concentrations, the viscosity is lowered by surface roughness, and for dilute suspensions it is insensitive to friction between the particles. Outside the dilute region, the viscosity increases with increasing friction coefficient. For a dilute system, roughness causes a negative first normal stress difference (N1) at order c2 in particle area concentration. The magnitude of N1 increases with increasing roughness height in the dilute limit but the trend reverses for more concentrated systems. N1 is largely insensitive to interparticle friction. The dilute results are in accord with the three-dimensional results of our earlier work (Wilson and Davis 2000), but with a correction to the sign of the tangential friction force.

Description: Capillarity is an important feature in controlling the spreading of liquid drops and in the coating of substrates by liquid films. For thin films and small contact angles, lubrication theory enables the analysis of the motion to be reduced to single evolution equations for the heights of the drops or films, provided the inertia of the liquid can be neglected. In general, the presence of inertia destroys the major simplification provided by lubrication theory, but two special cases that can be treated are identified here. In the first example, the approach of a drop to its equilibrium position is studied. For sufficiently low Reynolds numbers, the rate of approach to the terminal state and the contact angle are slightly reduced by inertia, but, above a critical Reynolds number, the approach becomes oscillatory. In the latter case there is no simple relation connecting the dynamic contact angle and contact-line speed. In the second example, the spreading drop is supported by a plate that is forced to oscillate in its own plane. For the parameter range considered, the mean spreading is unaffected by inertia, but the oscillatory motion of the contact line is reduced in magnitude as inertia increases, and the drop lags behind the plate motion. The oscillatory contact angle increases with inertia, but is not in phase with the plate oscillation.

Description: Greenstockings was the name that the members of Sir John Franklin's first Arctic Land Expedition gave to a young Dene woman during their winter residence at Fort Enterprise in 1820–21. All the officers' journals remark on her physical beauty, a reputed beauty that subsequently put her at the centre of numerous rumours and accounts. Historians know little about her other than her physical attractiveness and her age, although Greenstockings might have borne a child to one of Franklin's officers, and male jealousy over her might have put the expedition at serious risk. In spite of the paucity of factual information, Greenstockings has been cast as a central character in a poem by Franklin's first wife and in an award-winning book by Canadian novelist Rudy Wiebe. These and other pieces of information show the way that different perspectives can allow for different representations of historical figures.

Description: Steady, finite-amplitude internal-wave disturbances, induced by nearly hydrostatic stratified flow over locally confined topography that is more elongated in the spanwise than the streamwise direction, are discussed. The nonlinear three-dimensional equations of motion are handled via a matched-asymptotics procedure: in an 'inner' region close to the topography, the flow is nonlinear but weakly three-dimensional, while far upstream and downstream the 'outer' flow is governed, to leading order, by the fully three-dimensional linear hydrostatic equations, subject to matching conditions from the inner flow. Based on this approach, non-resonant flow of general (stable) stratification over finite-amplitude topography in a channel of finite depth is analysed first. Three-dimensional effects are found to inhibit wave breaking in the nonlinear flow over the topography, and the downstream disturbance comprises multiple small-amplitude oblique wavetrains, forming supercritical wakes, akin to the supercritical free-surface wake induced by linear hydrostatic flow of a homogeneous fluid. Downstream wakes of a similar nature are also present when the flow is uniformly stratified and resonant (i.e. the flow speed is close to the long-wave speed of one of the modes in the channel), but, in this instance, they are induced by nonlinear interactions precipitated by three-dimensional effects in the inner flow and are significantly stronger than their linear counterparts. Finally, owing to this nonlinear-interaction mechanism, vertically unbounded uniformly stratified hydrostatic flow over finite-amplitude topography also features downstream wakes, in contrast to the corresponding linear disturbance that is entirely locally confined.

Description: Planar flow in the interfacial region of an open porous medium is investigated by finding solutions for Stokes flow in a channel partially filled with an array of circular cylinders beside one wall. The cylinders are in a square array oriented across the flow and are widely spaced, so that the solid volume fraction Φ is 0.1 or less. For this spacing, singularity methods are appropriate and so they are used to find solutions for both planar Couette flow and Poiseuille flow in the open portion of the channel. The solutions, accurate to O(Φ), are used to calculate the apparent slip velocity at the interface, Us, and results obtained for Us are presented in terms of a dimensionless slip velocity. For shear-driven flow, this dimensionless quantity is found to depend only weakly on Φ and to be independent of the height of the array relative to the height of the channel and independent of the cylinder size relative to the height of the channel. For pressure-driven flow, Us is found to be less than that under comparable shear-flow conditions, and dependent on cylinder size and filling fraction in this case. Calculations also show that the external flow penetrates the porous medium very little, even for sparse arrays, and that Us is about one quarter of the velocity predicted by the Brinkman model.