Viscous Liquid Film Research Articles

Stability of a thin viscous fluid film flowing down a rotating non-uniformly heated inclined plane

A thin viscous liquid film flowing down a rotating non-uniformly heated inclined plane is investi- gated. It is assumed that the rotation is small and the region of investigation is very far from the axis of rotation, so that the centrifugal force has a dominant role in the instability of the investigated region. Therefore, we have derived a Benney-like free surface evolution equation on the basis of a long-wave (small wave number) approximation and not included the Coriolis effects in the expansion of the dependent variables. Further, a linear and a weakly nonlinear study have been carried out. The linear study reveals that the growth of linear perturbation is independent of the Rossby number Ro, that is invariant with the Coriolis effect, while the linear phase speed cr depends on Ro as well as on the Taylor number Ta. It also reveals that as Ta increases the stable zone decreases, and the influence of Ta is stronger for greater inclination of the rotating inclined plane. Again, it is found that the Marangoni number Mn has similar qualitative (destabilizing) behavior as Ta, but the destabilizing behavior of Ta is more at high Mn than at low Mn. The relation between the parameters Ta and Ro gives a unified parameter Tarot, which reflects the effect of rotation, and we found that the linear phase speed cr first decreases with Tarot up to a critical value and then increases with Tarot. This is due to the fact that the Coriolis force is dominant at very small rotation, while for relatively large rotation the centrifugal force dominates the flow field. The weakly nonlinear study reveals that the effect of rotation appears in the form of both Coriolis and centrifugal force into the growth of finite amplitude perturbations in contrast to the growth of the infinitesimal perturbations where the rotation arrives in the form of centrifugal force only. Also, it plays significant role in the different stability zones, amplitudes of sub/super critical disturbances and nonlinear phase speed.

Steady flow of Viscous Thin films on solid surface

In This paper we consider two dimensional flow to a viscous symmetric thin liquid film flow on solid surface with no inertia force . Navier – stokes equation has been used to find the equation that governs this flow we solve these equations analytically.

Thin film flow over a non-linear stretching sheet in presence of uniform transverse magnetic field

A thin viscous liquid film flow is developed over a stretching sheet under different non-linear stretching velocities in presence of uniform transverse magnetic field. Evolution equation for the film thickness is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. It is observed that all types of stretching produces film thinning, but non-monotonic stretching produces faster thinning at small distance from the origin. Effect of the transverse magnetic field is to slow down the film thinning process. Observed flow behavior is explained physically.

The nonlinear dynamics of pendent drops on a thin film coating the underside of a ceiling

This paper considers the dynamics of a thin film of viscous liquid of density ρ coating the underside of a horizontal rigid boundary under the action of surface tension σ and gravity g, and in the lubrication limit. Gravitational instability for inverse wavenumbers larger than the capillary length ℓ = (σ/ρg)1/2) leads to the formation of quasi-static pendent drops of radius ≈3.83ℓ. If the boundary conditions are such as to pin the positions of the drops then the drops slowly drain fluid from the regions between them through thin annular trenches around each drop. A similarity solution is derived and verified numerically in which the film thickness in the intervening regions scales like t−1/4 and that in the trenches like t−1/2. A single drop placed far from boundaries on an otherwise uniform film, and given an initial perturbation, undergoes self-induced quasi-steady translation during which it grows slowly in amplitude by leaving a wake where the film thickness is reduced by an average of 90. It is driven by release of gravitational potential energy as fluid is collected from the film into the lower lying drop. Analysis of Landau–Levich regions around the perimeter of the translating drop predicts its speed and the profile of the wake. Two translating drops may coalesce if they collide, in contrast with the non-coalescence of colliding collars in the analogous one-dimensional problem (Lister et al., J. Fluid Mech. vol. 552, 2006b, p. 311). Colliding drops may also bounce off each other, the outcome depending on the angle of incidence through complex interactions between their surrounding capillary wave fields.

Improved Spray Model for Viscous Annular Sheets in a Swirl Injector

A modified sheet breakup model was applied to a thin, viscous liquid film generated by a swirl injector similar to that installed in a liquid propellant rocket engine combustor. The sheet breakup model consists of three steps: determination of the swirl injector characteristics for the prediction of initial sheet conditions at the injector exit as input, linear stability analysis for primary sheet breakup, and the Taylor analogy breakup model for final drop formation. Under atmospheric pressure, the liquid sheet breakup occurs under a long-wave regime, sometimes according to simple theoretical analysis. But in high ambient pressure conditions, like a liquid propellant rocket engine combustor, the sheet breakup regime changes from a long wave to a short-wave regime due to a high gas Weber number (We 2 > 27/16), although the same injector was used. The sheet breakup model was, therefore, modified to be applicable to both long- and short-wave regimes and validated by the comparison of breakup length and Sauter mean diameter to experimental results. In both experimental and computational results, the spray cone angle and breakup length decreased as the ambient pressure increased, even though the pressure difference of the injector was constant. Local Sauter mean diameters, predicted by computation, were smaller at high ambient pressures. The comparative results show that the computational model is able to accurately predict sheet breakup length, spray cone angle, local Sauter mean diameter, and overall spray shape. Therefore, the model can be used as a design tool, ahead of analyzing spray characteristics of an injector in both atmospheric and high ambient pressure conditions.

Viscous liquid film flow on dune slopes of Mars

It is shown that viscous liquid film flow (VLF-flow) on the surfaces of slopes of martian dunes can be a low-temperature rheological phenomenon active today on high latitudes. A quantitative model indicates that the VLF-flows are consistent with the flow of liquid brines similar to that observed by imaging at the Phoenix landing site. VLF-flows depend on the viscosity, dynamics, and energetics of temporary darkened liquid brines. The darkening of the flowing brine is possibly, at least partially, attributed to non-volatile ingredients of the liquid brines. Evidence of previous VLF-flows can also be seen on the dunes, suggesting that it is an ongoing process that also occurred in the recent past.

The effect of evaporation on fingering instabilities

We investigate the flow of evaporating thin films of viscous liquid on inclined solid substrates under the influence of gravity. A lubrication-type approach is used to develop a three-dimensional model of the flow including physical effects such as capillarity, gravity, Marangoni stresses, disjoining pressure, and evaporation. Numerical simulations are then carried out based on the model. The effect of evaporation on the so-called fingering instability that develops along the contact line in the transverse direction of the flow is studied. It is found that evaporation acts to suppress the instability if the evaporation number, a nondimensional measure of the mass flow rate across the interface, is above a critical value. The critical value decreases as the inclination angle is decreased. For the values of evaporation number below the critical one, the fingers grow initially but then saturate at a length that depends on the evaporation conditions. It is also shown that thermocapillarity acts to enhance the instability.

Effect of electric field on the stability of an oscillatory contaminated film flow

The stability of the viscous liquid film on an oscillating plane is investigated in the presence of both insoluble surfactant on the film surface and uniform electric field, acting normal to the plane. In this problem main motivation is to study the combine effect of surfactant and electric field on the stability of the liquid film. Here liquid is treated as a perfect conductor and the air above the liquid film is also treated as a perfect dielectric. The linear stability analysis is performed using the long-wave perturbation method based on Floquet theory. It is observed that two Floquet modes exist due to the presence of surfactant and both modes can be unstable. The growth rate corresponding to the Floquet modes increase with the presence of an electric field and decrease with the presence of surfactant.

Inertial thin film flow on planar surfaces featuring topography

A range of problems is investigated, involving the gravity-driven inertial flow of a thin viscous liquid film over an inclined planar surface containing topographical features, modelled via a depth-averaged form of the governing unsteady Navier–Stokes equations. The discrete analogue of the resulting coupled equation set, employing a staggered mesh arrangement for the dependent variables, is solved accurately using an efficient full approximation storage (FAS) algorithm and a full multigrid (FMG) technique; together with error-controlled automatic adaptive time-stepping and proper treatment of the associated nonlinear convective terms. An extensive set of results is presented for flow over both one- and two-dimensional topographical features, and errors quantified via detailed comparisons drawn with complementary experimental data and predictions from finite element analyses where they exist. In the case of one-dimensional (spanwise) topography, moderate Reynolds numbers and shallow/short topographical features, the results obtained are in close agreement with corresponding finite element solutions of the full free-surface problem. For the case of flow over two-dimensional (localised) topography, it is shown that the free-surface disturbance is influenced significantly by the presence of inertia leading, as in the case of spanwise topography, to an increase in the magnitude and severity of the resulting capillary ridge and trough formations: the effect of inclination angle and topography aspect ratio are similarly explored.

Heat transfer in locally heated liquid films moving under the action of gravity and gas flow

Heat transfer in a viscous liquid film moving under the action of gravity and a gas flow on a substrate with a locally heated rectangular area is investigated. The heat exchange coefficient is given on the liquid-gas surface; the heat flux to the liquid is given on the heated area; the substrate surface outside the heated area is heat-insulated. An analytical solution in a form of a convergent series is obtained for the liquid temperature distribution in the film. The influence of the dimensionless criteria on the obtained solution is analyzed. The effect of heat flux inhomogeneity on the temperature distribution is considered.

Thin-film flow over a nonlinear stretching sheet

A thin viscous liquid film flow is developed over a stretching sheet under different nonlinear stretching velocities. An evolution equation for the film thickness, is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. A comparison is made with the analytic solution obtained in [B. S. Dandapat, A. Kitamura, B. Santra, “Transient film profile of thin liquid film flow on a stretching surface”, ZAMP, 57, 623-635 (2006)]. It is observed that all types of stretching produce film thinning but non-monotonic stretching produces faster thinning at small distance from the origin. The velocity u along the stretching direction strongly depends on the distance along the stretching direction and the Froude number.

Primary instabilities of liquid film flow sheared by turbulent gas stream

Evolution of excited waves on a viscous liquid film has been investigated experimentally for the annular gas–liquid flow in a vertical tube. For the first time the dispersion relations are obtained experimentally for linear waves on liquid film surface in the presence of turbulent gas flow. Both cocurrent and countercurrent flow regimes are investigated. As an example of comparison with theory, the experimental data are compared to the results of calculations based on the Benjamin quasi-laminar model for turbulent gas flow. The calculation results are found to be in good agreement with experiments for moderate values of film Reynolds number.

Unsteady thin-film flow over a heated stretching sheet

The unsteady flow in a thin viscous liquid film over a heated horizontal stretching surface are analyzed considering the stretching velocity and the temperature distribution in their general forms. An evolution equation for the film thickness, that retains the convective heat transport effects, is derived using long-wave theory of thin liquid film and is solved numerically for some representative values of non-dimensional parameters. It is observed that the thermocapillary effects are responsible in shaping the film thickness. Further the thermocapillary effects are more pronounced for lower values of Prandtl number and Biot number.

Spreading of a viscous liquid film over the corrugated surface and the heat and mass transfer calculations

There are a lot of industrial applications of structured packing. Distillation columns are one of the examples where the liquid flows over the corrugated surface as a thin film to provide a good mass-transfer surface between the liquid and vapor phase. The purpose of the present paper is to study the hydrodynamics and the heat-mass transfer of the liquid film spreading down the corrugated surfaces when the corrugation amplitude is comparable with Nusselt’s film thickness (the amplitude corresponds to a small texture of the structured packing). As a result, a nonlinear type diffusion equation is obtained to describe the evolution of the film thickness profile. The nonlinear diffusion coefficient is obtained for three cases: a smooth inclined plate, a corrugated plate with large ribs, and an inclined corrugated plate with small ribs. The equations are solved numerically. As a result, it has been obtained that the small texture significantly increases the rate of the film thickness evolution in comparison with a smooth plate. To obtain the nonlinear diffusion coefficient in the case of a small texture, the hydrodynamics of the film flow over an inclined corrugated surface are studied. The viscosity, inertia, and surface tension forces are taken into account. The calculations were carried out on the basis of the Navier-Stokes equations. The influence of the microcorrugations on both the heat transfer from the wall and the mass transfer through the free surface was investigated.

Solitary waves in a viscous liquid film flowing down a thin vertical cylinder

Several equations to describe the flow of a viscous liquid film on a thin cylinder are derived. The solitary-wave solutions to these equations are studied. The families of solutions are constructed for the first two eigenvalues that correspond to single-humped and double-humped waves. It is found that these families become similar as the similarity parameter increases. The dependencies of phase velocities and wave amplitudes on the free parameters of the problem are analyzed. The resulting solutions are compared with solitary waves in films on a flat surface.

Three-dimensional non-stationary processes in a vertically falling film of a viscous liquid

Three-dimensional wave processes in vertically falling films of viscous liquid are considered. The 3D localized perturbations, which are studied insufficiently, are of a particular interest. The numerical method for watching evolution of initial perturbations was developed. The final stage of this evolution is formation of the 3D localized structures: solitons. The boundaries of 3D soliton stability were determined.

Formation of a vortex flow at the laser cutting of sheet metal with low pressure of assisting gas

Specific features of subsonic jet gas flows in narrow channels geometrically similar to the laser cut are studied experimentally and theoretically. Such flows are visualized by a technique based on prior application of a viscous liquid film onto the side walls of the channel made of transparent glass. The gas flow inside the channel induces a liquid flow on the glass wall in the form of extremely small filaments, which coincide with the streamlines of the gas flow. Filming of these filaments by a CCD camera allows one to capture the specific features of these gas-dynamic flows. Mathematical modelling of the dynamics of a viscous compressible heat-conducting gas was performed by solving full three-dimensional Navier–Stokes equations. Numerical calculations and experiments reveal vortex structures in the flow at the entrance and exit of the channel, which may directly affect the surface quality in real gas-laser cutting of metals. The largest vortex, which arises at the channel exit, collects and accumulates the liquid flowing down the channel walls. Jet flows are generated by sonic nozzles with conical or cylindrical exit sections or by a double coaxial nozzle. The double nozzle includes the central conical nozzle and the side concentric nozzle, which allows additional side injection of the gas to be organized. The study with the double nozzle shows that the vortices disappear as the pressure in the external nozzle is increased, and a stable vortex-free attached gas flow is formed.

Thick Films of Viscous Fluid Coating a Plate Withdrawn from a Liquid Reservoir

We consider the deposition of a film of viscous liquid on a flat plate being withdrawn from a bath, experimentally and theoretically. For any plate speed U, there is a range of "thick" film solutions whose thickness scales like U{1/2} for small U. These solutions are realized for a partially wetting liquid, while for a perfectly wetting liquid the classical Landau-Levich-Derjaguin film is observed, whose thickness scales like U{2/3}. The thick film is distinguished from the Landau-Levich-Derjaguin film by a dip in its spatial profile at the transition to the bath. We calculate the phase diagram for the existence of stationary film solutions as well as the film profiles and find excellent agreement with experiment.

Instabilities and pattern miniaturization in confined and free elastic-viscous bilayers

We present an analysis of the instabilities engendered by van der Waals forces in bilayer systems composed of a soft elastic film (<10 microm) and a thin (<100 nm) viscous liquid film. We consider two configurations of such systems: (a) Confined bilayers, where the bilayer is sandwiched between two rigid substrates, and (b) free bilayers, where the viscous film is sandwiched between a rigid substrate and the elastic film. Linear stability analysis shows that the time and length scales of the instabilities can be tuned over a very wide range by changing the film thickness and the material properties such as shear modulus, surface tension, and viscosity. In particular, very short wavelengths comparable to the film thickness can be obtained in bilayers, which is in contrast to the instability wavelengths in single viscous and elastic films. It is also shown that the instabilities at the interfaces of the free bilayers are initiated via an in-phase "bending" mode rather than out-of-phase "squeezing" mode. The amplitudes of deformations at both the elastic-air and elastic-viscous interfaces become more similar as the elastic film thickness decreases and its modulus increases. These findings may have potential applications in the self-organized patterning of soft materials.

Existence and stability of regularized shock solutions, with applications to rimming flows

This paper is concerned with regularization of shock solutions of nonlinear hyperbolic equations, i.e., introduction of a smoothing term with a coefficient ɛ, then taking the limit ɛ → 0. In addition to the classical use of regularization for eliminating physically meaningless solutions which always occur in non-regularized equations (e.g. waves of depression in gas dynamics), we show that it is also helpful for stability analysis. The general approach is illustrated by applying it to rimming flows, i.e., flows of a thin film of viscous liquid on the inside of a horizontal rotating cylinder, with or without surface tension (which plays the role of the regularizing effect). In the latter case, the spectrum of available linear eigenmodes appears to be continuous, but in the former, it is discrete and, most importantly, remains discrete in the limit of infinitesimally weak surface tension. The regularized (discrete) spectrum is fully determined by the point where the velocity of small perturbations vanishes, with the rest of the domain, including the shock region, being unimportant.