Manuel G. Velarde

Momentum transport at a fluid-porous interface.

Abstract The momentum balance at the interface between a liquid and a porous substrate is investigated for a configuration with forced flow parallel to the interface. An heterogeneous continuously varying transition layer between the two outer bulk regions is introduced. The stress jump coefficient earlier introduced in the jump interface condition is here derived as an explicit function of the variations of the velocity and effective properties of the transition layer. Agreement is found between our numerical results based on the single-domain approach and the existing ad hoc estimates in the literature. Further advantages of this non-homogeneous analysis are also provided.

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.

On weakly nonlinear modulation of waves on deep water

We propose a new approach for modeling weakly nonlinear waves, based on enhancing truncated amplitude equations with exact linear dispersion. Our example is based on the nonlinear Schrodinger (NLS) equation for deep-water waves. The enhanced NLS equation reproduces exactly the conditions for nonlinear four-wave resonance (the “figure 8” of Phillips) even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean; therefore, sideband instability cannot produce energy leakage to high-wave-number modes for the enhanced equation, as reported previously for the NLS equation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equation. Being solvable numerically at no additional cost in comparison with the NLS equation, the new model is physically and numerically attractive for investigation of wave evolution.

Surface forces and wetting phenomena

Conditions for thermodynamic equilibrium of liquid drops on solid substrates are presented. It is shown that if surface force (disjoining/conjoining Derjaguin pressure) action in a vicinity of the three-phase contact line is taken into account the condition of thermodynamic equilibrium is duly satisfied. Then the thermodynamic expressions for equilibrium contact angles of drops on solid substrates and menisci in thin capillaries are expressed in terms of the corresponding Derjaguin isotherm. It is shown that equilibrium contact angles of drops vary significantly depending on the vapour pressure in the ambient atmosphere, while there is a single, unique equilibrium contact angle in thin capillaries. It is also shown that the static advancing contact angle of a drop depends on its volume, in agreement with experimental data. In the case of menisci in capillaries, the expression for the receding contact angle is deduced, with results that are also in agreement with known experimental data.

Spreading of liquid drops over porous substrates.

The spreading of small liquid drops over thin and thick porous layers (dry or saturated with the same liquid) has been investigated in the case of both complete wetting (silicone oils of different viscosities) and partial wetting (aqueous SDS solutions of different concentrations). Nitrocellulose membranes of different porosity and different average pore size have been used as a model of thin porous layers, glass and metal filters have been used as a model of thick porous substrates. The first problem under investigation has been the spreading of small liquid drops over thin porous layers saturated with the same liquid. An evolution equation describing the drop spreading has been deduced, which showed that both an effective lubrication and the liquid exchange between the drop and the porous substrates are equally important. Spreading of silicone oils over different nitrocellulose microfiltration membranes was carried out. The experimental laws of the radius of spreading on time confirmed the theory predictions. The spreading of small liquid drops over thin dry porous layers has also been investigated from both theoretical and experimental points of view. The drop motion over a dry porous layer appears caused by the interplay of two processes: (a). the spreading of the drop over already saturated parts of the porous layer, which results in a growth of the drop base, and (b). the imbibition of the liquid from the drop into the porous substrate, which results in a shrinkage of the drop base and a growth of the wetted region inside the porous layer. As a result of these two competing processes the radius of the drop base goes through a maximum as time proceeds. A system of two differential equations has been derived to describe the time evolution of the radii of both the drop base and the wetted region inside the porous layer. This system includes two parameters, one accounts for the effective lubrication coefficient of the liquid over the wetted porous substrate, and the other is a combination of permeability and effective capillary pressure inside the porous layer. Two additional experiments were used for an independent determination of these two parameters. The system of differential equations does not include any fitting parameter after these two parameters were determined. Experiments were carried out on the spreading of silicone oil drops over various dry nitrocellulose microfiltration membranes (permeable in both normal and tangential directions). The time evolution of the radii of both the drop base and the wetted region inside the porous layer was monitored. In agreement with our theory all experimental data fell on two universal curves if appropriate scales were used with a plot of the dimensionless radii of the drop base and of the wetted region inside the porous layer using a dimensionless time scale. Theory predicts that (a). the dynamic contact angle dependence on the dimensionless time should be a universal function, (b). the dynamic contact angle should change rapidly over an initial short stage of spreading and should remain a constant value over the duration of the rest of the spreading process. The constancy of the contact angle on this stage has nothing to do with hysteresis of the contact angle: there is no hysteresis in our system. These predictions are in the good agreement with our experimental observations. In the case of spreading of liquid drops over thick porous substrates (complete wetting) the spreading process goes in two similar stages as in the case of thin porous substrates. In this case also both the drop base and the radii of the wetted area on the surface of the porous substrates were monitored. Spreading of oil drops (with a wide range of viscosities) on dry porous substrates having similar porosity and average pore size shows universal behavior as in the case of thin porous substrates. However, the spreading behavior on porous substrates having different average pore sizes deviates from the universal behavior. Yet, even in this case the dynamic contact angle remains constant over the duration of the second stage of spreading as in the case of spreading on thin porous substrates. Finally, experimental observations of the spreading of aqueous SDS solution over nitrocellulose membranes were carried out (case of partial wetting). The time evolution of the radii of both the drop base and the wetted area inside the porous substrate was monitored. The total duration of the spreading process was subdivided into three stages: in the first stage the drop base growths until a maximum value is reached. The contact angle rapidly decreases during this stage; in the second stage the radius of the drop base remains constant and the contact angle decreases linearly with time; finally in the third stage the drop base shrinks while the contact angle remains constant. The wetted area inside the porous substrate expands during the whole spreading process. Appropriate scales were used to have a plot of the dimensionless radii of the drop base, of the wetted area inside the porous substrate, and the dynamic contact angle vs. the dimensionless time. Our experimental data show: the overall time of the spreading of drops of SDS solutions over dry thin porous substrates decreases with the increase of surfactant concentration; the difference between advancing and hydrodynamic receding contact angles decreases with the surfactant concentration increase; the constancy of the contact angle during the third stage of spreading has nothing to do with the hysteresis of contact angle, but determined by the hydrodynamics. Using independent spreading experiments of the same drops on a non-porous nitrocellulose substrate we have shown that the static receding contact angle is equal to zero, which supports our conclusion on the hydrodynamic nature of the hydrodynamic receding contact angle on porous substrates.

MUTUAL SYNCHRONIZATION OF CHAOTIC SELF-OSCILLATORS WITH DISSIPATIVE COUPLING

In a case of two dissipative coupled electronic circuits with possible chaotic dynamics, results of experimental, analytical and computer studies are provided concerning bifurcations on the boundaries of the synchronization regime of chaotic self-oscillations.

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

Dissipative Korteweg–de Vries description of Marangoni–Bénard oscillatory convection

O(1)

A three‐dimensional description of solitary waves and their interaction in Marangoni–Bénard layers

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.

Exact periodic solutions of the complex Ginzburg–Landau equation

For a horizontal liquid layer open to air and heated either from the air side or the liquid side, threshold values for the onset of oscillatory interfacial instability are provided, as well as the nonlinear evolution equation describing the deformable open surface. The latter equation is the dissipation‐modified Korteweg–de Vries evolution equation for solitary excitations in the liquid layer.