Serafim Kalliadasis

Falling liquid films

Falling Liquid Films gives a detailed review of state-of-the-art theoretical, analytical and numerical methodologies, for the analysis of dissipative wave dynamics and pattern formation on the surface of a film falling down a planar inclined substrate. This prototype is an open-flow hydrodynamic instability, that represents an excellent paradigm for the study of complexity in active nonlinear media with energy supply, dissipation and dispersion. It will also be of use for a more general understanding of specific events characterizing the transition to spatio-temporal chaos and weak/dissipative turbulence. Particular emphasis is given to low-dimensional approximations for such flows through a hierarchy of modeling approaches, including equations of the boundary-layer type, averaged formulations based on weighted residuals approaches and long-wave expansions. Whenever possible the link between theory and experiment is illustrated, and, as a further bridge between the two, the development of order-of-magnitude estimates and scaling arguments is used to facilitate the understanding of basic, underlying physics.

Steady free-surface thin film flows over topography

We consider the slow motion of a thin viscous film flowing over a topographical feature (trench or mound) under the action of an external body force. Using the lubrication approximation, the equations of motion simplify to a single nonlinear partial differential equation for the evolution of the free surface in time and space. It is shown that the problem is governed by three dimensionless parameters corresponding to the feature depth, feature width and feature steepness. Quasi-steady solutions for the free surface are reported for a wide range of these parameters. Our computations reveal that the free surface develops a ridge right before the entrance to the trench or exit from the mound and that this ridge can become large for steep substrate features of significant depth. Such capillary ridges have also been observed in the contact line motion over a planar substrate where the buildup of pressure near the contact line is responsible for the ridge. For flow over topography, the ridge formation is a mani...

Marangoni instability of a thin liquid film heated from below by a local heat source

We consider the motion of a liquid film falling down a heated planar substrate. Using the integral-boundary-layer approximation of the Navier-Stokes/energy equations and free-surface boundary conditions, it is shown that the problem is governed by two coupled nonlinear partial differential equations for the evolution of the local film height and temperature distribution in time and space. Two-dimensional steady-state solutions of these equations are reported for different values of the governing dimensionless groups. Our computations demonstrate that the free surface develops a bump in the region where the wall temperature gradient is positive. We analyse the linear stability of this bump with respect to disturbances in the spanwise direction

Thermocapillary instability and wave formation on a film falling down a uniformly heated plane

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 & 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

Rayleigh–Taylor instability of reaction-diffusion acidity fronts

O(1)

Thin films of soft matter

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.

Optimal leveling of flow over one-dimensional topography by Marangoni stresses

We consider the buoyancy driven Rayleigh–Taylor instability of reaction-diffusion acidity fronts in a vertical Hele–Shaw cell using the chlorite–tetrathionate (CT) reaction as a model system. The acid autocatalysis of the CT reaction coupled to molecular diffusion yields isothermal planar reaction-diffusion fronts separating the two miscible reactants and products solutions. The reaction is triggered at the top of the Hele–Shaw cell and the resulting front propagates downwards, invading the fresh reactants, leaving the product of the reaction behind it. The density of the product solution is higher than that of the reactant solution, and hence a hydrodynamic instability develops due to unfavorable density stratification. We examine the linear stability of the isothermal traveling wavefront with respect to disturbances in the spanwise direction and demonstrate the existence of a preferred wavelength for the developed fingering instability. Our linear stability analysis is in excellent agreement with two-di...

Thermocapillary long waves in a liquid film flow: Part 2. Linear stability and nonlinear waves

Structure Formation in Thin Liquid Films: Interface Forces Unleashed.- Structure Formation in Thin Liquid Films.- Singularities and Similarities.- Three-Phase Capillarity.- Falling Films Under Complicated Conditions.- Miscible Fingering in Electrokinetic Flow: Symmetries and Zero Modes.

Fingering instabilities of exothermic reaction-diffusion fronts in porous media

A thin viscous film flowing over a step down in topography exhibits a capillary ridge preceding the step. In applications, a planar liquid surface is often desired and hence there is a need to level the ridge. This paper investigates optimal leveling of the ridge by means of a Marangoni stress such as might be produced by a localized heater creating temperature variations at the film surface. The differential equation for the free surface based on lubrication theory and incorporating the effects of topography and temperature gradients is solved numerically for steps down in topography with different temperature profiles. Both rectangular “top-hat” and parabolic profiles, chosen to model physically realizable heaters, were found to be effective in reducing the height of the capillary ridge. Leveling the ridge is formulated as an optimization problem to minimize the maximum free-surface height by varying the heater strength, position, and width. With the optimized heaters, the variation in surface height is...

General Dynamical Density Functional Theory for Classical Fluids

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.