Thermocapillary instability in self-rewetting liquid films flowing down a heated soft vertical fibre.

The stability of a free vertical liquid film under the combined action of gravity and thermocapillary forces has been studied. An exact solution of the Navier-Stokes and thermal conductivity equations is obtained for the case of plane steady flow with constant film thickness. It is shown that if the free surfaces of the film are perfectly heat insulated, the liquid flow rate through the cross section of the layer is zero. It is found that to close the model with consideration of the heat exchange with the environment, it is necessary to specify the liquid flow rate and the derivative of the temperature with respect to the longitudinal coordinate or the flow rate and the film thickness. The stability of the solution with constant film thickness at small wave numbers is studied. A solution of the spectral problem for perturbations in the form of damped oscillations is obtained.

The stability analysis of a gravity-driven thin liquid film with an insoluble surfactant flowing over a surface with embedded, regularly spaced heaters is investigated. At the leading edge of a heater, the presence of a temperature gradient induces an opposing Marangoni stress at the interface leading to the formation of a capillary ridge. This ridge has been shown to be susceptible to thermocapillary (oscillating in the flow direction) and rivulet (spanwise periodic pattern) instabilities. The presence of an insoluble surfactant is shown to have a stabilizing effect on this system. The governing equations for the evolution of the film thickness and surfactant concentration are obtained within the lubrication approximation. The coupled two-dimensional base solutions for the film thickness and surfactant concentration show that there is no significant change in the height of the capillary ridge at the subsequent heaters downstream. The height of the capillary ridge is reduced by the presence of the surfactant. For very small Peclet number, the presence of multiple heaters has almost no significant effect on the film stability as compared to a single heater and similar trends are observed between the two configurations in the presence of the surfactant as for the case of a clean interface. However, for large Peclet number, the effect was observed on both types of instabilities for certain heater configurations. The Biot number is shown to have a strong effect on the stability results wherein the dominant mode of instability is altered (from rivulet to thermocapillary instability) for a passive or no surfactant case with increase in the Biot number. For an active surfactant thermocapillary instability is found to remain the dominant mode of instability for all the values of the Biot number. It is shown that increasing the number of heaters beyond a couple does not further affect the stability results.

In the present study, the effects of variable fluid properties on the dynamics of a falling liquid film over a solid cylinder are analyzed. Fluid properties, such as density, viscosity, thermal conductivity, and surface tension, are treated as functions of temperature. The mass, momentum, and energy equations are simplified using the lubrication approximation. A film thickness evolution equation is derived, which governs the temporal and spatial variations of the film's shape. Steady state results are presented to show the variation of fluid properties with temperature. Linear stability analysis is performed for various limiting cases. The thin film flow is found to be stable (unstable) for positive (negative) thermoviscous numbers. The growth rate of perturbations increases (decreases) for positive (negative) thermoviscous numbers as the rate of change of thermal conductivity (Λ) rises. The Rayleigh–Plateau instability is dominated by thermocapillary instability for Marangoni number (Ma)>0. Convective heat transfer at the free surface stabilizes the film by suppressing thermocapillary instability through thermoviscous effects. Heat transfer at the free surface and variable density significantly influence the transition from convective to absolute instability. An increase in the value of α (or a decrease in density), which corresponds to a lower flow rate, promotes the transformation of convective instability into absolute instability, consistent with the experimental findings of Duprat et al. [“Absolute and convective instabilities of a viscous film flowing down a vertical fiber,” Phys. Rev. Lett. 98(24), 244502 (2007)]. Furthermore, an increase in heat transfer at the free surface leads to a reduction in the convective regime and an expansion of the absolute regime. Nonlinear computations are also performed, showing good agreement with the results of linear temporal and spatiotemporal stability analyses.

A liquid film flowing down an inclined heated plane subject to surface wave and thermocapillary instabilities is studied. Three mechanisms exist by which energy can be transferred to the disturbance. Two of these mechanisms are associated with the thermocapillary forces and one with the shear stress of the basic flow at the deformed free surface. Depending on which mechanism is dominant, the instability can assume the form of either long transverse waves or short longitudinal rolls.

Effect of the crucible/crystal rotation on thermocapillary instability in a shallow Czochralski configuration

Investigations on the effects of the wettability on the liquid metal free surface film flow states have been performed by numerical simulations and experiments to establish a film flow which can cover the whole bottom solid surface. The effects of the fluid density, the inlet film thickness, and the width of the bottom surface on the film flow states under poor wetting conditions have been investigated by numerical simulations. The results show that the rivulet flow is easily developed when the inlet film thickness is small. The covering bottom surface of the film flow increases with the increase of the bottom surface width. For the liquid lithium, it is difficult to get a full covering film flow by increasing the bottom surface width and the inlet film thickness. A new method by changing the bottom surface shape to obtain a full covering film flow has been proposed by us. First, an experiment of the Galinstan film flow through a chute with a wavy bottom surface has been done to validate the effective of the proposed method. The experimental results indicate that this method is effective for the Galinstan film flow and the experimental results are consistent with numerical simulation results. Second, numerical simulations have been carried out to get full covering lithium film flows. The results show that a full covering liquid lithium film flow can be obtained by optimizing the dimension of the wavy bottom surface.

We study three-dimensional wave patterns on the surface of a film flowing down a uniformly heated wall. Our starting point is a model of four evolution equations for the film thickness h , the interfacial temperature theta , and the streamwise and spanwise flow rates, q and p , respectively, obtained by combining a gradient expansion with a weighted residual projection. This model is shown to be robust and accurate in describing the competition between hydrodynamic waves and thermocapillary Marangoni effects for a wide range of parameters. For small Reynolds numbers, i.e., in the "drag-gravity regime," we observe regularly spaced rivulets aligned with the flow and preventing the development of hydrodynamic waves. The wavelength of the developed rivulet structures is found to closely match the one of the most amplified mode predicted by linear theory. For larger Reynolds numbers, i.e., in the "drag-inertia regime," the situation is similar to the isothermal case and no rivulets are observed. Between these two regimes we observe a complex behavior for the hydrodynamic and thermocapillary modes with the presence of rivulets channeling quasi-two-dimensional waves of larger amplitude and phase speed than those observed in isothermal conditions, leading possibly to solitarylike waves. Two subregions are identified depending on the topology of the rivulet structures that can be either "ridgelike" or "groovelike." A regime map is further proposed that highlights the influence of the Reynolds and the Marangoni numbers on the rivulet structures. Interestingly, this map is found to be related to the variations of amplitude and speed of the two-dimensional solitary-wave solutions of the model. Finally, the heat transfer enhancement due to the increase of interfacial area in the presence of rivulet structures is shown to be significant.

An engineering model of a turbulent liquid film flow

An eight-channel capacitive sensor is used for the first time, which enables one to investigate the dynamics of three-dimensional wave flows and the variation of the transverse profile of a nonisothermal film of liquid during formation of jets. Measurements are performed of the wave characteristics of the flow of a film of water on a vertical plate with a heater 150 × 150 mm in size. During the heating of falling liquid, the thermocapillary forces cause the formation of jets and of a thin film between them. The film thickness and wave amplitude in the interjet region decrease with increasing heat flux. Two ranges of the effect of the heat flux on the characteristics of wave flow are identified. Under conditions of low heat fluxes, the film flow hardly differs from isothermal. Under significant heat loads, an intensive formation of jets occurs. Three-dimensional waves propagate over the jet crests, where the film thickness and wave amplitude increase with increasing heat flux. In the interjet region of the film being heated, the average relative amplitude of waves increases with decreasing average thickness, and in the isothermal region this amplitude decreases. Comparison of the obtained results with experimental data for isothermal film reveals that the values of relative amplitude differ significantly in the interjet region at high densities of heat fluxes. Transverse temperature gradients cause a decrease in the liquid film thickness, and longitudinal gradients cause an increase in the relative amplitude of waves compared to isothermal flows. In the end, this leads to the emergence of dry spots and breakdown of film. The relative amplitude of waves on the jet surface decreases with increasing heat flux; this is true of isothermal film flows.

Numerical simulation of falling film flow hydrodynamics over round horizontal tubes

A theoretical analysis of the thermal effects on the free-surface film flow on a flat rotating disk is presented. Assuming a small aspect ratio of the initial film thickness to the disk radius and neglecting peripheral effects of the liquid film, the evolution equation describing the shape of thin liquid film interface is obtained as a function of space and time and is solved using perturbation analysis. The results reveal the effects of inertial, gravitational, surface-tension and thermocapillary forces and of variable viscosity on the film planarization and thinning. Among these, it was found that the variable viscosity has the most profound effect on the transient film thickness.

Reply to comment by N. Shokri and D. Or on “A simple model for describing hydraulic conductivity in unsaturated porous media accounting for film and capillary flow”

Thermocapillary instability is one of the primary causes of a spontaneous rupture of thin films on heated walls. The film rupture may lead to an appearance of uncontrolled dry patches that significantly deteriorate the heat and mass transfer. In the present paper the thermocapillarity-induced film flow on a microstructured wall is studied in the framework of the long-wave theory. When the wall is heated or cooled, the solution predicts a film deformation caused by thermocapillarity. The linear stability analysis shows that the films on heated microstructured walls are less stable to the long-wave disturbances compared to the films on flat walls. The time-dependent film evolution is simulated and the effect of the wall structure on the film thinning and rupture is analyzed. It is shown that the wall topography exerts a profound effect on the dynamics of the film deformation and rupture, as well as on the size and the location of the dry patches. The full-scale direct volume-of-fluid simulations are used to verity the predictions of the long-wave theory. Good agreement is found for the small ratios between the groove depth and period. The agreement is further improved by including the effect of the convection heat transfer into the long-wave model.

Gravitationally driven flow of a thin film down an arbitrarily curved wall is analyzed for moderate Reynolds number by generalizing equations previously developed for flow on a planar wall. In the analysis, the ratio of the characteristic film thickness to the characteristic dimension of the wall is presumed small, and terms estimated to be first order in this parameter are retained. Partial differential equations are reduced to ordinary differential equations by the method of von Ka´rma´n and Pohlhausen; namely, an expression for the velocity profile is assumed, and the equation for conservation of linear momentum is averaged across the film. The assumed velocity profile changes shape in the flow direction because a self-similar profile, one of fixed shape but variable magnitude, leads to an equation that typically fails under critical conditions. The resulting equations for film thickness routinely accommodate subcritical-to-supercritical transitions and supercritical-to-subcritical transitions as classified by the underlying wave propagation. The more severe supercritical-to-subcritical transition is manifested by a standing wave where the film noticeably thickens; this standing wave is a simple analogue of a hydraulic jump. Predictions of the film-thickness profile and variations in the velocity profile compare favorably with those from the Navier-Stokes equation obtained by the finite element method.

In this paper, we investigate the quadratic Marangoni instability along with inertia in a self-rewetting fluid film that has a nonmonotonic variation of surface tension with temperature. The dynamics of such a thin self-rewetting fluid film flowing along an inclined heated substrate is examined by deriving an evolution equation for the film thickness using long-wave theory and asymptotic expansions. By adopting the derived long-wave model that includes the inertial and thermocapillary effects, we perform a linear stability analysis of the flat film solution. Two cases of the nonlinear flow are explored in depth using Tm (temperature corresponding to the minimum of surface tension) as the cutoff point. One is the case of (Ti,s−Tm)<0, and the other is (Ti,s−Tm)>0, where Ti,s is the interface temperature corresponding to the flat film. The Marangoni effect switches to the anomalous Marangoni effect as (Ti,s−Tm) shifts from a negative value to a positive value. Our calculations reveal that the Marangoni effect augments the flat film instability when (Ti,s−Tm)<0, whereas the stability of the flat film is promoted for (Ti,s−Tm)>0. Our further analysis demonstrates that the destabilizing inertial forces can be entirely compensated by the stabilizing anomalous thermocapillary forces. We verify the linear stability predictions of the long-wave Benney-type model with the solution to the Orr–Sommerfeld problem in the long-wave limit. Our time-dependent computations of the long-wave model establish the modulation of interface deformation in the presence of inertia and temperature gradients in the conventional Marangoni regime, whereas such deformations are suppressed in the anomalous Marangoni regime. A comparison of the numerical computations with the linear theory shows good agreement.