S. B. G. O'Brien

On the shape of small sessile and pendant drops by singular perturbation techniques

The problem of obtaining asymptotic expressions describing the shape of small sessile and pendant drops is revisited. Both cases display boundary-layer behaviour and the method of matched asymptotic expansions is used to obtain solutions. These give good agreement when compared with numerical results. The sessile solutions are relatively straightforward, while the pendant drop displays a behaviour which is both rich and interesting.

A mechanism for episodic subduction on Venus

We propose a mechanism previously developed as a hypothetical cause of the initiation of subduction in the Earths mantle, to describe a situation where such subduction may occur transiently, at irregular intervals of time. It has been suggested that tectonics on Venus may be described by such a scenario. In our model, a subduction event is followed by resumption of high Rayleigh number mantle convection below a stagnant lithosphere which thickens due to conductive cooling. As it thickens, differential buoyancy causes large lithospheric stresses which eventually lead to (plastic) failure in the upper portions of the lithosphere. This plastic zone thickens faster than the lithosphere, so that at some critical time, it reaches the base of the lithosphere. At this point, the effective lithosphere viscosity decreases to that of the underlying mantle, and subduction can occur. We suggest that this is mechanistically consistent with the postulated Venusian tectonic style.

On Marangoni drying : nonlinear kinematic waves in a thin film

Alcohol vapour, soluble in water, is introduced above a drying film and as a result of diffusion through the air and water phases a favourable concentration gradient gives rise to the required shear flow. We consider here a simple process driven by this mechanism, and by means of asymptotic simplification and the concepts of singular perturbation theory a leading-order approximation is obtained in which the alcohol concentration in the water is a specified function of space and time. The evolution of the free surface thus reduces to a single nonlinear partial differential equation, higher-derivative terms corresponding to surface tension and gravity effects. Numerical solutions of this equation are obtained and are compared to the application of first order nonlinear kinematic wave theory with corresponding shock solutions

Lithospheric failure on Venus

We develop a predictive model which has the ability to explain a postulated style of episodic plate tectonics on Venus, through the periodic occurrence of lithospheric subduction events. Present‐day incipient subduction zones are associated with the existence of arcuate trenches on the Venusian lithosphere. These trenches resemble terrestrial subduction zones, and occur at the rim of coronae, uplift features thought to be due to deep‐mantle convective plumes. The model we adopt represents the lithosphere as the thermal boundary layer which lies above a convective plume. We assume a temperature‐dependent nonlinear viscoelastic rheology, and we assume a stress‐based criterion for plastic yield. In developing this latter criterion, we are led to a re‐interpretation of the strength envelope which is commonly used in analysing lithospheric stress, and we propose that the plastic yield strength has meaning (and is finite) below the lithosphere, using behaviour in the Earth as our ‘laboratory’ justification for this view. An inferred yield stress on the Earth is ca.300 bar (30 MPa). Our model then shows that a thickening lithosphere becomes progressively more fluid as the stresses induced by the buoyant convective plume become large. Failure occurs when the effective lithosphere viscosity becomes equal to that of the underlying mantle. We show that reasonable expected values of yield stress in the range 100–200 bar (10–20 MPa) for Venusian mantle rocks are consistent within the framework of the model with radii of coronal trenches in the range 100–1200 km, and with the approximate time (200–800 Myr) which they may take to develop.

AN IMPLICIT SURFACE TENSION ALGORITHM FOR PICARD SOLVERS OF SURFACE–TENSION–DOMINATED FREE AND MOVING BOUNDARY PROBLEMS

One of the methods for solving a free or moving boundary problem is the use of Picard solvers which solve the geometry and the velocity field successively. When, however, the kinematic condition is used for updating the geometry in this technique, numerical stability problems occur for surface-tension-dominated flow. These problems are shown here to originate from the unstable integration of the local smoothing of the surface by surface tension. By an extension of the surface tension contribution to the flow field an implicit treatment of surface tension is obtained which overcomes these stability problems. The algorithm is applicable to both free and moving boundary problems, as will be shown by examples in this paper.

A mathematical model for the cleansing of silicon substrates by fluid immersion

Abstract The importance of cleanliness for the efficient manufacture of integrated circuits is well known in the microprocessor industry. The efficiency of traditional cleansing methods is known to decrease dramatically for dirt particles smaller than about 1 μm in radius. Experimentally it has been shown that the passage of a liquid-gas phase boundary along a substrate can be exploited to effect particle removal. A model is derived here which explains the success of this process, illustrates the dependency of the method on the speed of immersion, and further shows that it is theoretically independent of particle size.

Asymptotic, numerical and approximate techniques for a free boundary problem arising in the diffusion of glassy polymers

This paper considers approximate solution methods for a one dimensional Stefan problem describing solvent diffusion in glassy polymers. Similar to the classic Stefan problem, the region initially has zero thickness and so must be analysed carefully before performing a numerical computation. A small-time analysis gives the correct starting solution which is then incorporated into the second order accurate Keller box finite difference scheme. We also consider a detailed analysis of small and large time expansions, as well as the large control parameter limit, and show that our generalised approach enables us to obtain higher order terms than given in previous studies. Finally, we apply the combined integral method (CIM) to this problem, which is a refinement of the popular heat balance integral method (HBIM), and compare both the CIM and asymptotic solutions to the numerical results.

Waves in Guinness

We describe a simple model of a bubbly two-phase flow which is able to explain why waves propagate downward when a pint of Guinness is poured, and also how the waves are generated. Our theory involves a physically based regularization of the basic equations of the two-phase flow, using interphasic pressure difference and virtual mass terms, together with bulk or eddy viscosity terms. We show that waves can occur through an instability analogous to that which forms roll waves in inclined fluid flows, and we provide a description of the form of these waves, and compare them to observations. Our theory provides a platform for the description of waves in more general bubbly two-phase flows, and the way in which the flow breaks down to form slug flow.

An investigation of the hammocking effect associated with interface pressure measurements using pneumatic tourniquet cuffs

A simple experimental arrangement is used to investigate the influence of sensor height on the pressure indicated by an inherently linear sensor sandwiched between a rigid curved surface and a pneumatic tourniquet cuff. The sensor-indicated pressure is monitored for sensor heights in the range 0-3 mm and for cuff inflation pressures of 0-40 kPa ( approximately 0-300 mmHg). The sensor response is found to be non-linear with a saturation tendency at high applied pressures. A model which treats the cuff as an elastic membrane draped over the sensor is shown to be successful in accounting for the general form of the sensor characteristic particularly at cuff pressures greater than about 5 kPa. The model is of use in estimating the errors that are likely to arise in intrusive sensors used to measure interface pressures under tourniquets.

Shape of a small sessile drop and the determination of contact angle

Free liquid surfaces in equilibrium are described by the Laplace capillary equation with suitable boundary conditions generally given in terms of the contact angle. By a fortuitous formulation in the axisymmetric case, the second order ordinary differential equation can be reduced to a pair of coupled first-order equations. For the case of a small liquid drop, the present formulation allows perturbation solutions to second order to be derived in closed form. Furthermore the solutions obtained can be used to calculate contact angles, if the height and maximum width of the drop is known, the method being equally simple whether the contact angle is less than or greater than 90°.