ALBERT

Description: A steady, two-dimensional cellular convection modifies the morphological instability of a binary alloy that undergoes directional solidification. When the convection wavelength is far longer than that of the morphological cells, the behaviour of the moving front is described by a slow, spatial-temporal dynamics obtained through a multiple-scale analysis. The resulting system has a parametric-excitation structure in space, with complex parameters characterizing the interactions between flow, solute diffusion, and rejection. The convection in general stabilizes two-dimensional disturbances, but destabilizes three-dimensional disturbances. When the flow is weak, the morphological instability is incommensurate with the flow wavelength, but as the flow gets stronger, the instability becomes quantized and forced to fit into the flow box. At large flow strength the instability is localized, confined in narrow envelopes. In this case the solutions are discrete eigenstates in an unbounded space. Their stability boundaries and asymptotics are obtained by a WKB analysis. The weakly nonlinear interaction is delivered through the Lyapunov-Schmidt method.

Description: Journal of Fluid Mechanics, vol. 268 (1994), pp. 231–265It has recently come to our attention that our paper, which describes Marangoni-driven flow near a contact line, overlooks solutions involving a general thermal boundary condition on the free surface (private communication, S. J. Tavener 1997). These new solutions are applicable for non-isothermal flows in a corner region where one boundary is a rigid plane (and either perfectly insulating or perfectly conducting) and the other is a free surface upon which a general thermal boundary condition is applied. We describe these additional solutions below.

Description: In an attempt to model the growth and collapse of a vapour bubble in nucleate boiling this paper investigates the unsteady expansion and contraction of a long two-dimensional vapour bubble confined between superheated or subcooled parallel plates whose motion is driven by mass-transfer effects due to evaporation from the liquid to the vapour and condensation from the vapour to the liquid. It is shown that in the asymptotic limit of strong surface tension (small capillary number) the solution consists of two capillary-statics regions (in which the bubble interface is semicircular at leading order) and two thin films attached to the plates, connected by appropriate transition regions. This generalization of the steady and isothermal problem addressed by Bretherton (1961) has a number of interesting physical and mathematical features. Unlike in Bretherton's problem, the bubble does not translate but can change in size. Furthermore, the thin films are neither spatially nor temporally uniform and may dry out locally, possibly breaking up into disconnected patches of liquid. Furthermore, there is a complicated nonlinear coupling with a delay character between the profiles of the thin films and the overall expansion or contraction of the bubble which means that the velocity with which the bubble expands or contracts is typically not monotonic. This coupling is investigated for three different combinations of thermal boundary conditions and two simple initial thin-film profiles. It is found that when both plates are superheated equally the bubble always expands, and depending on the details of the initial thin-film profiles, this expansion may either continue indefinitely or stop in a finite time. When both plates are subcooled equally the bubble always contracts, and the length of the thin-film region always approaches zero asymptotically. When one plate is superheated and the other subcooled with equal magnitude the bubble may either expand or contract initially, but eventually the bubble always contracts just as in the pure-condensation case.

Description: We consider a long-wave evolution equation that governs a draining film on a heated plate and hence is capable of describing both surface-wave and thermocapillary instabilities. When the flow and heat transfer rates are moderate, we show, via weakly nonlinear analysis of a truncated system and numerical simulation of the full nonlinear evolution equation, that coupled temporal instabilities can create surface deformations that lead to an array of rivulets aligned with the flow. This work thus demonstrates a mechanism of rivulet formation based solely on instability phenomena.

Description: Long axisymmetric liquid zones are subject to axial temperature gradients which induce steady viscous flows driven by thermocapillarity. The approximately parallel flow in a cylindrical zone is examined for linearized instabilities. Capillary, surface-wave and thermalmodes are found. Capillary breakup can be retarded or even suppressed for small Prandtl number and large Biot number B, whichmeasures heat transfer from the liquid to the surrounding atmosphere. In the limiting case B→∞ the zone becomes an isothermal jet subject to axial 'wind stress' on its interface. It is then possible to suppress capillary breakup entirely so that one canmaintain long coherent jets. © 1985, Cambridge University Press. All rights reserved.

Description: We consider the oscillatory motion of a solid plate into and out of a bath of liquid. Assuming that the displacement amplitude of the plate motion is small and that the capillary number is small, the problem reduces to solving an interfacial boundary-value problem for the response of the contact line. The characteristic contact angle versus contact-line speed relationship includes contact-angle hysteresis which is assumed small and comparable to the amplitude of the plate motion. Sinusoidal and square-wave plate motions are considered. We find that the contact line moves with the plate if the contact line is fixed, but has relative motion otherwise. It would then advance part of the time, recede part of the time, and remain stationary in the transition periods. Further, we find that both contact-angle hysteresis and steepening of the contact angle with increasing contact-line speed are dissipative effects. © 1987, Cambridge University Press. All rights reserved.

Description: An energy stability theory is formulated for systems having moving contact lines. The method derives from criteria obtained from the integral mechanical-energy balance manipulated to reflect general material and dynamical properties of moving-contact-line regions. The method yields conditions for both stability and instability and is applied to the two-dimensional Rayleigh-Taylor problem in a vertical slot. © 1986, Cambridge University Press. All rights reserved.