ALBERT

Description: An oscillatory instability mechanism is identified for a horizontal liquid layer with undeformable open surface heated from the air side. Although buoyancy and surface tension gradients are expected to play a stabilizing role in this situation, we show that, acting together, they may lead to the instability of the motionless state of the system. The instability is a consequence of the coupling between internal and surface waves, whose resonant interaction and resulting mode mixing are discussed. Predictions amenable to experimental test are given together with a thorough analytical and numerical study of the problem.

Description: We consider a thin layer of a viscous fluid flowing down a uniformly heated planar wall. The heating generates a temperature distribution on the free surface which in turn induces surface tension gradients. We model this thermocapillary flow by using the Shkadov integral-boundary-layer (IBL) approximation of the Navier Stokes/energy equations and associated free-surface boundary conditions. Our linear stability analysis of the flat-film solution is in good agreement with the Goussis and Kelly (1991) stability results from the Orr Sommerfeld eigenvalue problem of the full Navier Stokes/energy equations. We numerically construct nonlinear solutions of the solitary wave type for the IBL approximation and the Benney-type equation developed by Joo et al. (1991) using the usual long-wave approximation. The two approaches give similar solitary wave solutions up to an O(1) Reynolds number above which the solitary wave solution branch obtained by the Joo et al. equation is unrealistic, with branch multiplicity and limit points. The IBL approximation on the other hand has no limit points and predicts the existence of solitary waves for all Reynolds numbers. Finally, in the region of small film thicknesses where the Marangoni forces dominate inertia forces, our IBL system reduces to a single equation for the film thickness that contains only one parameter. When this parameter tends to zero, both the solitary wave speed and the maximum amplitude tend to infinity.

Description: A derivation is given of the amplitude equations governing pattern formation in surface tension gradient-driven Bénard-Marangoni convection. The amplitude equations are obtained from the continuity, the Navier-Stokes and the Fourier equations in the Boussinesq approximation neglecting surface deformation and buoyancy. The system is a shallow liquid layer heated from below, confined below by a rigid plane and above with a free surface whose surface tension linearly depends on temperature. The amplitude equations of the convective modes are equations of the Ginzburg-Landau type with resonant advective non-variational terms. Generally, and in agreement with experiment, above threshold solutions of the equations correspond to an hexagonal convective structure in which the fluid rises in the centre of the cells. We also analytically study the dynamics of pattern formation leading not only to hexagons but also to squares or rolls depending on the various dimensionless parameters like Prandtl number, and the Marangoni and Biot numbers at the boundaries. We show that a transition from an hexagonal structure to a square pattern is possible. We also determine conditions for alternating, oscillatory transition between hexagons and rolls. Moreover, we also show that as the system of these amplitude equations is non-variational the asymptotic behaviour (t → ∞) may not correspond to a steady convective pattern. Finally, we have determined the Eckhaus band for hexagonal patterns and we show that the non-variational terms in the amplitude equations enlarge this band of allowable modes. The analytical results have been checked by numerical integration of the amplitude equations in a square container. Like in experiments, numerics shows the emergence of different hexagons, squares and rolls according to values given to the parameters of the system.

Description: We analyse the regularized reduced model derived in Part 1 (Ruyer-Quil et al. 2005). Our investigation is two-fold: (i) we demonstrate that the linear stability properties of the model are in good agreement with the Orr-Sommerfeld analysis of the linearized Navier-Stokes/energy equations; (ii) we show the existence of nonlinear solutions, namely single-hump solitary pulses, for the widest possible range of parameters. We also scrutinize the influence of Reynolds, Prandtl and Marangoni numbers on the shape, speed, flow patterns and temperature distributions for the solitary waves obtained from the regularized model. The hydrodynamic and Marangoni instabilities are seen to reinforce each other in a non-trivial manner. The transport of heat by the flow has a stabilizing effect for small-amplitude waves but promotes the instability for large-amplitude waves when a recirculating zone is present. Nevertheless, in this last case, by increasing the shear in the bulk and thus the viscous dissipation, increasing the Prandtl number decreases the amplitude and speed of the waves. © 2005 Cambridge University Press.

Description: We consider the dynamics of a thin liquid film falling down a uniformly heated wall. The heating sets up surface tension gradients that induce thermocapillary stresses on the free surface, thus affecting the evolution of the film. We model this thermocapillary flow by using a gradient expansion combined with a Galerkin projection with polynomial test functions for both velocity and temperature fields. We obtain equations for the evolution of the velocity and temperature amplitudes at first- and second-order in the expansion parameter. These equations are fully compatible close to criticality with the Benney long-wave expansion. Models of reduced dimensionality for the evolution of the local film thickness, flow rate and interfacial temperature only, are proposed. © 2005 Cambridge University Press.

Description: Energy stability theory has been used to study Bénard convection in one- and two-component horizontal fluid layers heated from below or above when there is a deformable upper surface. To a first approximation in the crispation number, we provide sufficient conditions for stability of the motionless state of the layer, and delineate regions of possible subcritical instability. © 1982, Cambridge University Press. All rights reserved.