Benoit Scheid

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.

Wave patterns in film flows: Modelling and three-dimensional waves

In a previous work, two-dimensional film flows were modelled using a weightedresidual approach that led to a four-equation model consistent at order e2. A twoequation model resulted from a subsequent simplification but at the cost of lowering the degree of the approximation to order e only. A Pade approximant technique is applied here to derive a refined two-equation model consistent at order e2. This model, formulated in terms of coupled evolution equations for the film thickness h and the flow rate q, accounts for inertia effects due to the deviations of the velocity profile from the parabolic shape, and closely follows the asymptotic long-wave expansion in the appropriate limit. Comparisons of two-dimensional wave properties with experiments and direct numerical simulations show good agreement for the range of parameters in which a two-dimensional wavy motion is reported in experiments. The stability of two-dimensional travelling waves to three-dimensional perturbations is investigated based on the extension of the models to include spanwise dependence. The secondary instability is found to be not very selective, which explains the widespread presence of the synchronous instability observed in the experiments by Liu et al. (1995) whereas Floquet analysis predicts a subharmonic scenario in most cases. Three-dimensional wave patterns are computed next assuming periodic boundary conditions. Transition from two- to three-dimensional flows is shown to be strongly dependent on initial conditions. The herringbone patterns, the synchronously deformed fronts and the three-dimensional solitary waves observed in experiments are recovered using our regularized model, which is found to be an excellent compromise between the complete model, which has seven equations, and the simplified model, which does not include the second-order inertia corrections. Those corrections are found to play a role in the selection of the type of secondary instability as well as of the spanwise wavelength of the emerging pattern.

On the instability of a falling film due to localized heating

We analyse the stability of a thin film falling under the influence of gravity down a locally heated plate. Marangoni flow, due to local temperature changes influencing the surface tension, opposes the gravitationally driven Poiseuille flow and forms a horizontal band at the upper edge of the heater. The thickness of the band increases with the surface tension gradient, until an instability forms a rivulet structure periodic in the transverse direction. We study the dependence of the critical Marangoni number, a non-dimensional measure of the surface tension gradient at the onset of instability, on the associated Bond and Biot numbers, non-dimensional measures of the curvature pressure and heat-conductive properties of the film respectively. We develop a model based on long-wave theory to calculate base-state solutions and their linear stability. We obtain dispersion relations, which give us the wavelength and growth rate of the fastest growing mode. The calculated film profile and wavelength of the most unstable mode at the instability threshold are in quantitative agreement with the experimental results. We show via an energy analysis of the most unstable linear eigenmode that the instability is driven by gravity and an interaction between base-state curvature and the perturbation thickness. In the case of non-zero Biot number transverse variations of the temperature profile also contribute to destabilization.

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

The Benney equation including thermocapillary effects is considered to study a liquid film flowing down a homogeneously heated inclined wall. The link between the finitetime blow-up of the Benney equation and the absence of the one-hump travelling-wave solution of the associated dynamical system is accurately demonstrated in the whole range of linearly unstable wavenumbers. Then the blow-up boundary is tracked in the whole space of parameters accounting for flow rate, surface tension, inclination and thermocapillarity. In particular, the latter two effects can strongly reduce the validity range of the Benney equation. It is also shown that the subcritical bifurcation found for falling films with the Benney equation is related to the blow-up of solutions and is unphysical in all cases, even with the thermocapillary effect though in contrast to horizontally heated films. The accuracy of bounded solutions of the Benney equation is determined by comparison with a reference weighted integral boundary layer model. A distinction is made between closed and open flow conditions, when calculating travelling-wave solutions; the former corresponds to the conservation of mass and the latter to the conservation of flow rate. The open flow condition matches experimental conditions more closely and is explored for the first time through the associated dynamical system. It yields bounded solutions for larger Reynolds numbers than the closed flow condition. Finally, solutions that are conditionally bounded are found to be unstable to disturbances of larger periodicity. In this case, coalescence is the pathway yielding finite-time blow-up.

Nonlinear evolution of nonuniformly heated falling liquid films

The present theoretical study focuses on the dynamics of a thin liquid film falling down a vertical plate with a nonuniform, sinusoidal temperature distribution. The results are compared to those obtained in the case of the uniform temperature distribution. The governing evolution equation for the film thickness profile based on long-wave theory accounts for two instability mechanisms related to thermocapillarity. The first mechanism is due to an inhomogeneity of the temperature at the liquid–gas interface induced by perturbations of the film thickness, when heat transfer to the gas phase is present, while the second one is due to the nonuniform heating imposed at the plate and leads to steady-state deformations of the liquid–gas interface. For a moderate nonuniform heating the traveling waves obtained in the case of a uniform heating are modulated by an envelope. When the temperature modulation along the plate increases the shape of the liquid–gas interface becomes “frozen” and the oscillatory traveling ...

Heat transfer and rivulet structures formation in a falling thin liquid film locally heated

An experimental investigation of the heat transfer from a local heat source to a liquid film falling down a vertical plate is performed. The thermocapillary counterflow, induced by non-uniform heating, causes a deformation of the film surface having a horizontal bump-shape. This shape becomes unstable above a critical value of the imposed heat flux and deforms into vertical downstream rivulets. This variation of patterns is expected to modify significantly the heat transfer through the film. Experiments are carried out at atmospheric pressure with three varying parameters: the streamwise heater length, the Reynolds number and the imposed heat flux density. Velocimetry, shadowgraphy and infrared thermography are used to study the behavior of the interface and the heat transfer. We put in evidence the presence of a thermocapillary counter flow producing a stagnation line at the upper edges of the horseshoe structures, beyond the instability threshold, and observe a decrease of the heat transfer with the Reynolds number.

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

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.

The role of surface rheology in liquid film formation

The role of surface rheology in fundamental fluid dynamical systems, such as liquid coating flows and soap film formation, is poorly understood. We investigate the role of surface viscosity in the classical film-coating problem. We propose a theoretical model that predicts film thickening based on a purely surface-viscous theory. The theory is supported by a set of new experimental data that demonstrates slight thickening even at very high surfactant concentrations for which Marangoni effects are irrelevant. The model and experiments represent a new regime that has not been identified before.

Heated Falling Films

We present new insights and results for the problem of a film falling down a heated wall: (i) treatment of a mixed heat flux boundary condition on the substrate; (ii) development of a long-wave theory for large Peclet numbers; (iii) refined treatment of the energy equation based on a high-order Galerkin projection in terms of polynomial test functions which satisfy all boundary conditions; (iv) time-dependent computations for the free-surface height and interfacial temperature; (v) numerical solution of the full energy equation; (vi) demonstration of the existence of a thermal boundary layer at the front stagnation point of a solitary pulse; (vii) development of models that prevent negative temperatures and are in good agreement with the numerical solution of the full energy equation.

Microgravity investigations of instability and mixing flux in frontal displacement of fluids

The goal of the present study is to investigate analytically, numerically and experimentally the instability of the displacement of viscous fluid by a less viscous one in a two-dimensional channel, and to determine characteristic size of entrapment zones. Experiments on miscible displacement of fluids in Hele-Shaw cells were conducted under microgravity conditions. Extensive direct numerical simulations allowed to investigate the sensitivity of the displacement process to variation of values of the main governing parameters. Validation of the code was performed by comparing the results of model problems simulations with experiments and with the existing solutions published in literature. Taking into account non-linear effects in fluids displacement allowed to explain new experimental results on the pear-shape of fingers and periodical separation of their tip elements from the main body of displacing fluid. Those separated blobs of less viscous fluid move much faster than the mean flow of the displaced viscous fluid. The results of numerical simulations processed based on the dimensions analysis allow to introduce criteria characterizing the quality of displacement. The functional dependence of the dimensionless criteria on the values of governing parameters needs further investigations.