Viscous Film Research Articles

Evaporation of a thin viscous liquid film sheared by gas in a microchannel

In the present paper a 3D non-stationary two-sided mathematical model of joint motion of evaporating liquid film and cocurrent gas flow in a microchannel with local heating has been developed. This model takes into account a deformable gas–liquid interface, convective heat transfer in the liquid and the gas phases as well as temperature dependence of surface tension and liquid viscosity. Assuming the lubrication theory to be valid, the problem has been reduced to five governing equations for the film thickness, temperature fields in the gas and liquid, vapor concentration in the gas phase and gas pressure. Numerically it is shown that for films sheared by gas in microchannels vapor is transported by forced convection and diffusion, and diffusion plays the most considerable role in vapor transport at low gas velocities. Also, it is shown that concentration and thermal boundary layers are formed. The boundary layers have a specific S-shaped form. The width of the vapor track increases along the gas flow direction in front of the heater and decreasesdownstream the heater. The distance over which the width decreases is an order of magnitude higher than the heater length. This fact can be explained by the condensation of the vapor.

Relaxation and intermediate asymptotics of a rectangular trench in a viscous film

The surface of a thin liquid film with nonconstant curvature flattens as a result of capillary forces. While this leveling is driven by local curvature gradients, the global boundary conditions greatly influence the dynamics. Here, we study the evolution of rectangular trenches in a polystyrene nanofilm. Initially, when the two sides of a trench are well separated, the asymmetric boundary condition given by the step height controls the dynamics. In this case, the evolution results from the leveling of two noninteracting steps. As the steps broaden further and start to interact, the global symmetric boundary condition alters the leveling dynamics. We report on full agreement between theory and experiments for the capillary-driven flow and resulting time dependent height profiles, a crossover in the power-law dependence of the viscous energy dissipation as a function of time as the trench evolution transitions from two noninteracting to interacting steps, and the convergence of the profiles to a universal self-similar attractor that is given by the Green's function of the linear operator describing the dimensionless linearized thin film equation.

Modeling of wave modes on a vertical film of a viscous ferromagnetic fluid flowing down a cylindrical electric conductor

The present study considers axially symmetrical waves on the surface of a ferromagnetic viscous fluid film flowing down a cylindrical conductor with a non-constant current. In addition to the gravitational force, the film is affected by a uniform magnetic field. Using the long-wave approximation, a model equation is obtained for the evolution of the film thickness from its undisturbed value. By showing that solutions with and without a constant magnetic field are self-similar, scaling parameters are identified and presented. Numerical solutions of the time-dependent non-linear partial differential equation are presented when current in the conductor is described by a ramp function (i.e., magnetic field onset).

Wavy flow of a liquid film in the presence of a cocurrent turbulent gas flow

Wavy downflow of viscous liquid films in the presence of a cocurrent turbulent gas flow is analyzed theoretically. The parameters of two-dimensional steady-state traveling waves are calculated for wide ranges of liquid Reynolds number and gas flow velocity. The hydrodynamic characteristics of the liquid flow are computed using the full Navier-Stokes equations. The wavy interface is regarded as a small perturbation, and the equations for the gas are linearized in the vicinity of the main turbulent flow. Various optimal film flow regimes are obtained for the calculated nonlinear waves branching from the plane-parallel flow. It is shown that for high velocities of the cocurrent gas flow, the calculated wave characteristics correspond to those of ripple waves observed in experiments.

Nonlinear Wave Simulation on a Surface of Liquid Film Entrained by Turbulent Gas Flow at Weightlessness

The article considers the joint flow of a liquid film entrained by turbulent gas. Since liquid velocity is small in comparison with gas velocity, the problem is reduced to the calculation of pressure and shear stresses produced by the gas flowing over a wavy wall with small amplitude. Further, these data are used at the boundary conditions, when the flow of a liquid film is considered separately. As a result, we obtain a new system of equations for modeling the dynamics of long-wave perturbations on the surface of a viscous liquid film entrained by turbulent gas at microgravity. At small Reynolds numbers typical to the condition of microgravity, it was proved that this system is reduced to one evolution equation for the film thickness. Some numerical solutions of this equation have been received in this work.

Stability of film flow over inclined topography based on a long-wave nonlinear model

AbstractThe stability of a viscous liquid film flowing under gravity down an inclined wall with periodic corrugations is investigated. A long-wave model equation valid at near-critical Reynolds numbers is used to study the film dynamics, and calculations are performed for either sinusoidal or rectangular wall corrugations assuming either a fixed flow rate in the film or a fixed volume of fluid within each wall period. Under the two different flow assumptions, steady solution branches are delineated including subharmonic branches, for which the period of the free surface is an integer multiple of the wall period, and the existence of quasi-periodic branches is demonstrated. Floquet–Bloch theory is used to determine the linear stability of steady, periodic solutions and the nature of any instability is analysed using the method of exponentially weighted spaces. Under certain conditions, and depending on the wall period, the flow may be convectively unstable for small wall amplitudes but undergo transition to absolute instability as the wall amplitude increases, a novel theoretical finding for this class of flows; in other cases, the flow may be convectively unstable for small wall amplitudes but stable for larger wall amplitudes. Solutions with the same spatial period as the wall become unstable at a critical Reynolds number, which is strongly dependent on the period size. For sufficiently small wall periods, the corrugations have a destabilizing effect by lowering the critical Reynolds number above which instability occurs. For slightly larger wall periods, small-amplitude corrugations are destabilizing but sufficiently large-amplitude corrugations are stabilizing. For even larger wall periods, the opposite behaviour is found. For sufficiently large wall periods, the corrugations are destabilizing irrespective of their amplitude. The predictions of the linear theory are corroborated by time-dependent simulations of the model equation, and the presence of absolute instability under certain conditions is confirmed. Boundary element simulations on an inverted substrate reveal that wall corrugations can have a stabilizing effect at zero Reynolds number.

A two-parameter model of wave regimes for viscous liquid film flows

Wave regimes of viscous liquid film flows are considered when the viscosity coefficients vary in a wide range. An approximate model system of differential equations with two external governing parameters for the film layer thickness and the local flow rate is derived. The viscous dissipation of a film layer is taken into account in this system more accurately than in the well-known one-parameter Shkadov model. New properties of linear and nonlinear waves caused by the hydrodynamic instability of high-viscous liquid flows under gravity and surface tension are found.

Numerical investigations on self-similar solutions of the nonlinear diffusion equation

In this paper, we present the numerical investigations of self-similar solutions for the nonlinear diffusion equation ht=−(h3hxxx)x, which arises in the context of surface-tension-driven flow of a thin viscous liquid film. Here, h=h(x,t) is the liquid film height. A self-similar solution is h(x,t)=h(α(t)(x−x0)+x0,t0)=f(α(t)(x−x0)) and α(t)=[1−4A(t−t0)]−1/4, where A and x0 are constants and t0 is a reference time. To discretize the governing equation, we use the Crank–Nicolson finite difference method, which is second-order accurate in time and space. The resulting discrete system of equations is solved by a nonlinear multigrid method. We also present efficient and accurate numerical algorithms for calculating the constants, A, x0, and t0. To find a self-similar solution for the equation, we numerically solve the partial differential equation with a simple step-function-like initial condition until the solution reaches the reference time t0. Then, we take h(x,t0) as the self-similar solution f(x). Various numerical experiments are performed to show that f(x) is indeed a self-similar solution.

Viscous liquid films on a porous vertical cylinder: Dynamics and stability

In this paper, liquid films flowing down a porous vertical cylinder were investigated by an integral boundary layer model. Linear stability and nonlinear evolution were studied. Linear stability results of the integral boundary layer model were in good agreement with the linearized Navier-Stokes equations which indicated that the permeability of the porous medium enhanced the instability of the flow system. The growth rate and cut-off wave number increased with increasing the permeability and the Reynolds number. Linear stability analysis showed that the system was more unstable for a larger Reynolds number Re. Nonlinear studies showed that, for a very small Re, the film evolved with time while a saturated state was not observed. In addition, it was observed that the film ruptured when the permeability parameter β > 0, and the rupture time decreased with increasing β. However, for a moderate Reynolds number, a small finite harmonic disturbance evolved to a saturated traveling wave. Further investigation was conducted on the droplet-like wave solution. Results showed that the wave speed increased as the permeability parameter increased.

Heat Transfer in a Liquid Film Due to an Unsteady Stretching Surface With Variable Heat Flux

The heat transfer characteristics of a viscous liquid film flow over an unsteady stretching sheet subject to variable heat flux are investigated numerically. The effect of thermal radiation applying to an optically thick medium is also considered. The governing boundary layer equations are transformed into a set of nonlinear ordinary differential equations using an efficient fifth-order step-adapted Runge–Kutta integration scheme together with Newton–Raphson method. The dimensionless temperature is plotted for various governing parameters; say, unsteadiness parameter, effective Prandtl number, distance index, as well as time index. It is found that the heat transfer aspects are strongly influenced by the relevant parameters.

Quartz Crystal Microbalance Technique for Analysis of Cooling Crystallization

A quartz crystal microbalance (QCM) technique is developed for the in situ analysis of the cooling crystallization processes of crystal nucleation and growth. In contrast to conventional techniques based on property changes in the solid or solution phase, the proposed QCM technique simultaneously exploits property changes in both the solid and solution phases, such as the solid mass and liquid viscosity, to analyze the crystallization processes. When initially cooling the solution, an increase in the solution viscosity is reflected in the QCM responses for the resonant frequency and resonant resistance. With further cooling, the resonant frequency and resonant resistance sharply change at the induction point of crystal nucleation, as the viscous liquid film on the sensor suddenly shifts to an elastic solid phase. Thereafter, the QCM responses are mainly controlled by the suspension viscosity due to simultaneous crystal nucleation and growth with further cooling. As a result, the QCM responses allow accurate measurement of the induction point and metastable zone width during the cooling crystallization. Additional mechanistic information on the crystallization, including molecular cluster formation, crystal nucleation, and crystal growth, is also extracted from a resonant frequency-resistance plot (F-R plot) of the QCM responses when varying the cooling conditions.

Buckling of a thin, viscous film in an axisymmetric geometry

By adapting the Föppl-von Kàrmàn equation, which describes the deformation of a thin elastic membrane, we present an analysis of the buckling pattern of a thin, very viscous fluid layer subject to shear in an axisymmetric geometry. A linear stability analysis yields a differential eigenvalue problem, whose solution, obtained using spectral techniques, yields the most unstable azimuthal wave-number, m⋆. Contrary to the discussion of Slim et al. [J. Fluid Mech. 694, 5–28 (2012)]10.1017/jfm.2011.437, it is argued that the axisymmetric problem shares the same degeneracy as its rectilinear counterpart, i.e., at the onset of instability, m⋆ is indefinitely large. Away from this point, however, a comparison with analogue experimental results is both possible and generally favorable. In this vein, we describe the laboratory apparatus used to make new measurements of m⋆, the phase speed and the wave amplitude; note that no prediction concerning the latter two quantities can be made using the present theory. Experiments reveal a limited range of angular velocities wherein waves of either small or large amplitude may be excited. Transition from one to the other regime does not appear to be associated with a notable change in m⋆.

Second Law Analysis of a Gas-Liquid Absorption Film

This paper reports an analytical study of the second law in the case of gas absorption into a laminar falling viscous incompressible liquid film. Velocity, temperature, and concentration profiles are determined and used for the entropy generation calculation. Irreversibilities due to heat transfer, fluid friction, and coupling effects between heat and mass transfer are derived. The obtained results show that entropy generation is mainly due to coupling effects between heat and mass transfer near the gas-liquid interface. Total irreversibility is minimum at the diffusion film thickness. On approaching the liquid film thickness, entropy generation is mainly due to viscous irreversibility.

NUMERICAL GRADIENT FLOW DISCRETIZATION OF VISCOUS THIN FILMS ON CURVED GEOMETRIES

The evolution of a viscous thin film on a curved geometry is numerically approximated based on the natural time discretization of the underlying gradient flow. This discretization leads to a variational problem to be solved at each time step, which reflects the balance between the decay of the free (gravitational and surface) energy and the viscous dissipation. Both dissipation and energy are derived from a lubrication approximation for a small ratio between the characteristic film height and the characteristic length scale of the surface. The dissipation is formulated in terms of a corresponding flux field, whereas the energy primarily depends on the fluid volume per unit surface, which is a conserved quantity. These two degrees of freedom are coupled by the underlying transport equation. Hence, one is naturally led to a PDE-constrained optimization problem, where the variational time stepping problem has to be solved under the constraint described by the transport equation. For the space discretization a discrete exterior calculus approach is investigated. Various applications demonstrate the qualitative and quantitative behavior of one- and two-dimensional thin films on curved geometries.

Dynamics of particle settling and resuspension in viscous liquid films

AbstractWe develop a dynamic model for suspensions of negatively buoyant particles on an incline. Our model includes settling due to gravity and resuspension of particles by shear-induced migration. We consider the case where the particles settle onto the solid substrate and two distinct fronts form: a faster liquid and a slower particle front. The resulting transport equations for the liquid and the particles are of hyperbolic type and we study the dilute limit for which we compute exact solutions. We also carry out systematic laboratory experiments, focusing on the motion of the two fronts. We show that the dynamic model predictions for small to moderate values of the particle volume fraction and the inclination angle of the solid substrate agree well with the experimental data.

Crucial flow stabilization and multiple instability branches of gravity-driven films over topography

In this paper, we present experimental results on the linear instability of gravity-driven viscous films flowing down a strongly undulated incline. To systematically investigate the relation between the eddies which form in the troughs of sufficiently steep undulations and the corresponding stability maps, we vary the liquid's viscosity experimentally. We report on a rich variety of phenomena, which is provoked by the corrugation of the underlying substrate, including: (a) transitions from long-wave to short-wave type instability, (b) a disjoining of the instability branches leading to formation of isles in the stability map, (c) flow destabilization, but also, (d) very strong stabilization of the flow up to a factor of two for arbitrary linear disturbances and even up to a factor of four for linear short-wave disturbances. To our knowledge, this is the first experimental work, which reports on any of these phenomena for gravity-driven liquids flowing down an undulated incline. Since structured substrates turn out to hold an enormous potential for crucial film flow stabilization, we present a first approach for a topography shape, based on the shape of the eddy, which is optimized in terms of flow stability.

A variational approach to a differential equation modeling thin-film flows and pertinent to Tanner's Law

This study is intended to provide a modified variational algorithm for the numerical solution of the third-order ordinary differential equation y″′ = y−2 which arises in the modeling of thin viscous films with surface tension. The resulting solution is used to solve a problem relevant to Tanner's Law for the speed of a moving three-phase contact line. Numerical results, computational comparisons, suitable error measures and illustrations are provided to testify and demonstrate the convergence and efficiency of the method.

Ring waves as a mass transport mechanism in air-driven core-annular flows

Air-driven core-annular fluid flows occur in many situations, from lung airways to engineering applications. Here we study, experimentally and theoretically, flows where a viscous liquid film lining the inside of a tube is forced upwards against gravity by turbulent airflow up the center of the tube. We present results on the thickness and mean speed of the film and properties of the interfacial waves that develop from an instability of the air-liquid interface. We derive a long-wave asymptotic model and compare properties of its solutions with those of the experiments. Traveling wave solutions of this long-wave model exhibit evidence of different mass transport regimes: Past a certain threshold, sufficiently large-amplitude waves begin to trap cores of fluid which propagate upward at wave speeds. This theoretical result is then confirmed by a second set of experiments that show evidence of ring waves of annular fluid propagating over the underlying creeping flow. By tuning the parameters of the experiments, the strength of this phenomenon can be adjusted in a way that is predicted qualitatively by the model.

Numerical solutions of thin-film equations for polymer flows

We report on the numerical implementation of thin-film equations that describe the capillary-driven evolution of viscous films, in two-dimensional configurations. After recalling the general forms and features of these equations, we focus on two particular cases inspired by experiments: the leveling of a step at the free surface of a polymer film, and the leveling of a polymer droplet over an identical film. In each case, we first discuss the long-term self-similar regime reached by the numerical solution before comparing it to the experimental profile. The agreement between theory and experiment is excellent, thus providing a versatile probe for nanorheology of viscous liquids in thin-film geometries.

Electric-field-induced interfacial instabilities of a soft elastic membrane confined between viscous layers

We explore the electric-field-induced interfacial instabilities of a trilayer composed of a thin elastic film confined between two viscous layers. A linear stability analysis (LSA) is performed to uncover the growth rate and length scale of the different unstable modes. Application of a normal external electric field on such a configuration can deform the two coupled elastic-viscous interfaces either by an in-phase bending or an antiphase squeezing mode. The bending mode has a long-wave nature, and is present even at a vanishingly small destabilizing field. In contrast, the squeezing mode has finite wave-number characteristics and originates only beyond a threshold strength of the electric field. This is in contrast to the instabilities of the viscous films with multiple interfaces where both modes are found to possess long-wave characteristics. The elastic film is unstable by bending mode when the stabilizing forces due to the in-plane curvature and the elastic stiffness are strong and the destabilizing electric field is relatively weak. In comparison, as the electric field increases, a subdominant squeezing mode can also appear beyond a threshold destabilizing field. A dominant squeezing mode is observed when the destabilizing field is significantly strong and the elastic films are relatively softer with lower elastic modulus. In the absence of liquid layers, a free elastic film is also found to be unstable by long-wave bending and finite wave-number squeezing modes. The LSA asymptotically recovers the results obtained by the previous formulations where the membrane bending elasticity is approximately incorporated as a correction term in the normal stress boundary condition. Interestingly, the presence of a very weak stabilizing influence due to a smaller interfacial tension at the elastic-viscous interfaces opens up the possibility of fabricating submicron patterns exploiting the instabilities of a trilayer.