Viscous Liquid Film Research Articles

Stability of a viscous liquid film flowing down an inclined plane with respect to three-dimensional disturbances

An analysis is presented for the stability of a viscous liquid film flowing down an inclined plane with respect to three-dimensional disturbances under the action of gravity and surface tension. Using momentum-integral method, the nonlinear free surface evolution equation is derived by introducing the self-similar semiparabolic velocity profiles along the flow (x- and y-axis) directions. A normal mode technique and the method of multiple scales are used to obtain the theoretical (linear and nonlinear stability) results of this flow problem, which conceive the physical parameters: Reynolds number Re, Weber number We, angle of inclination of the plane θ and the angle of propagation of the interfacial disturbances ϕ. The temporal growth rate ωi+ and second Landau constant J2, based on which various (explosive, supercritical, unconditional, subcritical) stability zones of this flow problem are categorized, contain the shape factors B and β owing to the non-zero steady basic flow along the y-axis direction. A novel result which emerges from the linear stability analysis is that for any given value of Re, We and θ, any stability that arises in two-dimensional disturbances (ϕ = 0) must also be present in three-dimensional disturbances. For ϕ = 0, there exists a second explosive unstable zone (instead of unconditional stable zone) after a certain value of Re (or θ) due to the involvement of B and β in the expression of J2. This explosive unstable zone vanishes after a certain value of ϕ depending upon the values of Re, We and θ, which confirms the stabilizing influence of ϕ on the thin film flow dynamics irrespective of the values of Re, We and θ.

Faraday instability of viscous liquid films on a heated substrate with Maxwell–Cattaneo heat flux

Faraday instability of viscous liquid films with Maxwell–Cattaneo (MC) heat flux on an infinite, heated horizontal substrate subject to vertical time-varying periodic vibration is investigated theoretically. The MC effect means that the response of the heat flux to a temperature gradient obeys a relaxation time law rather than a classical Fourier time law. Applying the classic Floquet theory to linear analysis, a recursive relation is obtained. When considering the MC effect, a new phenomenon appears at a large wave number k. The neutral stability curves branch new tongues that turn left rather than right as before, but the tongues still move up and right as the wave number increases. Furthermore, typical harmonic (H) and subharmonic (SH) alternation behavior continues to exist. Interestingly, the first tongue of a branch is H or SH, implying that there is a transition following the branches. However, near the critical wave number kc of a branch, the SH and H almost overlap. As Cattaneo number C increases, the tongue-like unstable zones of branches become wider, and the critical wave number kc of the appeared branch becomes small. As the driving frequency ω decreases, the branch tongues become elongated and the critical wave number kc of the appeared branch becomes small.

A Compressible Formulation of the One-Fluid Model for Two-Phase Flows

In this paper, we introduce a compressible formulation for dealing with 2D/3D compressible interfacial flows. It integrates a monolithic solver to achieve robust velocity–pressure coupling, ensuring precision and stability across diverse fluid flow conditions, including incompressible and compressible single-phase and two-phase flows. Validation of the model is conducted through various test scenarios, including Sod’s shock tube problem, isothermal viscous two-phase flows without capillary effects, and the impact of drops on viscous liquid films. The results highlight the ability of the scheme to handle compressible flow situations with capillary effects, which are important in computational fluid dynamics (CFD).

Steady Wind-Generated Gravity-Capillary Waves on Viscous Liquid Film Flows

Steady Wind-Generated Gravity-Capillary Waves on Viscous Liquid Film Flows

Instability of viscous annular liquid film jet under high air-to-fluid velocity rate

Based on the surface wave dispersion relation equation of viscous annular liquid film jet established by the research group in the early stage, a dispersion relation equation calculation program is written according to Mathematica. A linear stability analysis is carried out for the high air-fluid velocity ratio viscous annular liquid film jet entering a compressible gas medium environment. The relationship between the surface wave growth rate of an annular liquid film jet and the surface wave number under different characteristic parameters is obtained. The connection based on the dominant wave’s growth rate or the dominant wave’s number with dimensionless parameters such as air-fluid velocity rate, air-fluid density rate, Weber number of liquid, liquid Reynolds number, and gas Mach number was analyzed. The consequences reveal that under the high air-fluid velocity ratio, the increase of the above dimensionless parameters promotes the fragmentation course of the annular liquid film jet. Moreover, the augment of the air-fluid velocity rate has the greatest influence on the stability of the viscous annular liquid film jet. In contrast, the Reynolds number of the liquid has less influence than the reliability of the annular liquid film jet. Linear stability analysis under this condition lays the foundation for the prediction model of the most unstable parameters of annular liquid film jet in practical engineering applications.

Stability of a Viscous Liquid Film in a Rotating Cylindrical Cavity under Angular Vibrations

Stability of a Viscous Liquid Film in a Rotating Cylindrical Cavity under Angular Vibrations

Rimming flow in a rotating horizontal cylinder with phase change at the interface

The two-dimensional solutions and stability analysis are presented for an evaporating thin viscous liquid film flowing inside a uniformly heated rotating horizontal cylinder. A non-linear, fourth-order, partial differential evolution equation is obtained by simplifying mass, momentum, and energy conservation equations within the lubrication approximation. The effect of evaporation, gravity, viscous drag, surface tension, thermocapillary stress, and intermolecular forces has been taken into account. The numerical solutions of the model are validated against the existing experimental as well as the numerical results, along with the analytical result for the limiting cases of the present model. The film thickness model is solved to elucidate two-dimensional spatiotemporal solutions and their stability for a wide range of thermal and other parameters. The evaporative mass flux at the liquid–air interface results in unsteady solutions which are oscillatory in nature, and the amplitude of the oscillations increases with an increase in the evaporative flux. The film ruptures after some time and the rupture time is found to be inversely proportional to the evaporation number, a non-dimensional number quantifying the rate of evaporation. The linear stability analysis shows that the thermocapillary stresses as well as the long-range molecular forces destabilize the film. A negative disjoining pressure is shown to reduce the rupture time and vice versa. Evaporation (condensation) plays a destabilizing (stabilizing) role in the thin film flow. Non-linear computations are carried out for the steady profiles, validating the growth rates obtained from the linear stability analysis.

Carbon dioxide foam bubbles enhance skin penetration through the stratum corneum layer with mechanical mechanism

Topical skin formulations often include penetration enhancers that interact with the outer stratum corneum (SC) layer to chemically enhance diffusion. Alternatively, penetration can be mechanically enhanced with simple rubbing in the presence of solid particles sometimes included to exfoliate the top layers of the SC. Our goal was to evaluate micron-sized carbon dioxide bubbles included in a foamed moisturizing formulation as a mechanical penetration enhancement strategy. We show that moisturizing foam bubbles cause an increase in SC formulation penetration using both mechanical and spectroscopic characterization. Our results suggest viscous liquid film drainage between coalescing gaseous bubbles creates local regions of increased hydrodynamic pressure in the foam liquid layer adjacent to the SC surface that enhances treatment penetration. An SC molecular diffusion model is used to rationalize the observed behavior. The findings indicate marked increased levels of treatment concentration in the SC at 2 h and that persists to 18 h after exposure, far exceeding non-foamed treatments. The study suggests an alternate strategy for increasing formulation penetration with a non-chemical mechanism.

Nonlinear periodic and solitary rolling waves in falling two-layer viscous liquid films

Nonlinear periodic and solitary rolling waves in falling two-layer viscous liquid films

Influence of wind on a viscous liquid film flowing down a thread

A liquid film flowing down a fiber or thread is well known to destabilize into an axisymmetric bead-like pattern in stagnant air. We demonstrate experimentally how side wind affects such a film flow. Our results show that side wind increases the descent speed of the liquid beads, highlighting that the film flow regime can be changed by controlling the magnitude of the wind speed.

Temporal instability of a viscous liquid film on an oscillating plane in the presence of static electric field

This work examined the linear instability of a two-dimensional liquid film on an oscillating plane theoretically in the presence of a static electric field perpendicular to the plane. The liquid was considered to be viscous and perfectly conducting, while the gas was considered to be viscous and dielectric. The viscous potential theory and Floquet theory were adopted to obtain the dispersion equation. The influences of electric field intensity, forcing amplitude, forcing frequency, and viscosity were obtained. The results showed that with there is more than one instability region with a non-zero forcing amplitude. The first instability region is called inherent instability, which is due to the electric force overcoming the viscous dissipation, surface tension, and gravity, while the other regions are due to the parametric instability induced by the oscillating plane, forming famous Faraday waves. It was found that increasing the forcing amplitude stabilizes the inherent instability but has a destabilizing impact on Faraday waves. In addition, the increase in the electric field intensity and forcing frequency enhances the inherent instability but has a complex impact on the Faraday waves, depending on the intensity of the viscous dissipation, which is approximately proportional to the square of the wavenumber.

Drop impact on viscous liquid films

When a liquid drop falls on a solid substrate, the air layer between them delays the occurrence of liquid–solid contact. For impacts on smooth substrates, the air film can even prevent wetting, allowing the drop to bounce off with dynamics identical to that observed for impacts on superamphiphobic materials. In this paper, we investigate similar bouncing phenomena, occurring on viscous liquid films, that mimic atomically smooth substrates, with the goal to probe their effective repellency. We elucidate the mechanisms associated with the bouncing to non-bouncing (floating) transition using experiments, simulations, and a minimal model that predicts the main characteristics of drop impact, the contact time and the coefficient of restitution. In the case of highly viscous or very thin films, the impact dynamics is not affected by the presence of the viscous film. Within this substrate-independent limit, bouncing is suppressed once the drop viscosity exceeds a critical value, as on superamphiphobic substrates. For thicker or less viscous films, both the drop and film properties influence the rebound dynamics and conspire to inhibit bouncing above a critical film thickness. This substrate-dependent regime also admits a limit, for low-viscosity drops, in which the film properties alone determine the limits of repellency.

Rayleigh-Taylor instability of viscous liquid film under an inclined substrate with time modulated temperature

The Rayleigh-Taylor instability of viscous liquid film under a temperature-modulated inclined substrate is studied. Long wave approximation and weighted residual model are used respectively to derive the nonlinear evolution equations of liquid film, and it is shown that the two evolution equations by the above two approaches are equivalent. The recurrence relation and damped Mathieu equation are obtained by employing Floquet theory and linear analysis, respectively. When amplitude a and oscillation frequency ω of the temperature exceed critical values, the film surface loses its stability. The stability diagram is obtained by linear analysis based on Floquet theory. Moreover, based upon the line approach, the nonlinear evolution equation is numerically simulated to give the spatiotemporal evolution of the free surface. Heated capillary tension might accelerate the production of suspended droplets due to the appearing of Marangoni number Ma produced by heating inclined wall. The results reveal that the larger Reynolds number Re, Marangoni number Ma and Biot number Bi will result in the instability of free surface. Moreover, the liquid film becomes more unstable at the smaller Froude number Fr and Weber number We.

Derivation of Nonlinear Equations for Surface of Fluid Adhering to a Moving Plate Withdrawn From Liquid Pool

The processes of the magnetic tape producing, wire adhering, as well as many other important technological processes, include preparing some special materials’ adhering to a product surface. For a surface withdrawn from the molten metal or the other liquid material there is a problem to determine a profile of a film surface. In this paper, the mathematical model developed for simulation of the adhering process of viscous liquid film to a slowly moving plate, which is vertically withdrawn from the molten metal or the other fluid capacity. The Navier-Stokes equations for a film flow on a surface of the withdrawn plate are considered with the corresponding boundary conditions, and the polynomial approximation is used for the film flow profile. The equations, after integration across the layer of a film flow, result in the system of partial differential equations for the wavy surface ζ(t,x) of a film flow, of flow rate q(t,x) and of flow energy Q(t,x).The derived equations are used for analysis of the nonlinear film flow that determines the quality of a fluid adhering on a surface of the withdrawn plate.

Puncture of a Viscous Liquid Film Due to Droplet Falling

Droplet impact may rupture a liquid film on a non-wettable surface. The formation of a stable dry spot has only been studied in the inviscid case. Here, we examine the break-up of viscous films, and demonstrate the importance and role of the viscous dissipation in both film and droplet. A new model was therefore proposed to predict the necessary droplet energy to create a dry spot. It also showed that the dissipation contribution in film dominates when the ratio of the thicknesses to drop diameter is larger than 7/4.

Thermocapillary patterning of non-Newtonian thin films

Deformation of thin viscous liquid films exposed to a transverse thermal gradient results in Bénard–Marangoni instability, which would lead to the formation of micro- and nano-sized features. Linear and nonlinear analyses are performed to investigate the thermally induced pattern formation in shear thinning and shear thickening liquid films. The so-called thin film (TF) equation is re-derived to include viscosity variations using the power-law (PL) model. The characteristic wavelength for the growth of instabilities is found using a linear stability analysis of the PL-TF equation. A finite-difference-based discretization scheme and adaptive time step solver are used to solve the PL-TF equation for the nonlinear numerical model. The results show that the rheological property affects the timescale of the process and the size and final shape of the formed features. The fastest growth pillar reaching the top substrate in a shear thickening fluid is shorter than both the shear thinning and the Newtonian fluid cases. Moreover, morphological changes between patterns of shear thinning and shear thickening fluids are correlated with local viscosity variations. The number of formed pillars considerably increases with the increasing flow behavior index. The existing model also predicts the formation of pillars and bicontinuous structures at very low and high filling ratios.

Numerical simulation of flow pattern for non-Newtonian flow in agitated thin film evaporator

Agitated thin film evaporator (ATFE) is a new type of high-efficiency evaporator where a film is forced to form through a rotating scraper and the high-viscosity non-Newtonian flow materials can be evaporated smoothly. The flow distribution and transmission mechanism of the material in the evaporator directly determine the evaporation efficiency and power consumption of the evaporator. Unlike previous study that was based mainly on Newtonian flow, this paper establishes a three-dimensional computational fluid dynamics model of ATFE for non-Newtonian flow with different viscosities, and systematically probes the flow field distribution characteristics and film forming mechanism in the evaporator. The results show that the flow field distribution characteristics of low-viscosity non-Newtonian flow are similar to those of Newtonian flow, the material can form a uniform and continuous liquid film on the wall; as the viscosity increases, the uniformity and continuity of the liquid film gradually deteriorate. Through studying the flow field distribution and transmission form of the materials, and combining the liquid film distribution, velocity distribution, shear strain rate distribution, and viscosity distribution, it is found that the shear field and viscosity distribution formed by the internal structure and operating state of the evaporator are the key to the good film formation. In addition, it is proposed that the bending of the leading edge of the scraper can assist the spreading of high viscous fluid liquid film, and the best bending angle is explored. This research provides theoretical guidance and basis for the design and application of ATFE.

Electrohydrodynamic instability of viscous liquid films: Electrostatically modified second-order Benney and Kuramoto–Sivashinsky models

Long wave evolution on viscous films flowing under gravity down an inclined plate in the presence of an electrostatic field acting normal to the plate is described using new nonlinear models. First, an asymptotic expansion is applied to derive a strongly nonlinear evolution equation for the interfacial position in the long-wave limit, which retains terms up to second order in a small film parameter (ε) and thus brings in dispersive effects, inclination-angle corrections for hydrostatic pressure, second-order contributions to inertia and electric stresses as well as the interaction of the last two. Two weakly nonlinear models (WNMs) for the disturbances with O(ε2) amplitude are then derived, which account for distinct effects of hydrostatic pressure, inertia and dispersion for shallow and steep angles. We compare the linear stability results based on the second-order electrified Benney equation (BE) and the WNMs with realistic parameter values. It is shown that the growth rates coincide, while the phase speed from the WNMs is a good approximation of that from the BE for small wavenumbers only. The comparison of exact dispersion relations from the Orr–Sommerfeld (OS) problem with the linear results of the second-order BE takes a step towards understanding the validity range of the BE with non-local terms that approximates the full electrohydrodynamic coupling system describing the inclined electrified film flow. The pertinent OS results demonstrate that a decrease in electrode distance can diminish the critical Reynolds number (Re). Furthermore, at subcritical Re with a slightly supercritical electric field, the finite wavelength of the least stable mode has been estimated using an energy balance, whose amplitude will be governed by two coupled Ginzburg–Landau (GL) equations, derived using a multiple-scale analysis by taking higher-order problems into account. The numerical solutions of the leading-order GL equation demonstrate that close to the bifurcation, it can fully govern the temporal behaviour of the system.

Hydrodynamic instabilities of a viscous liquid film flowing down an inclined or vertical plane

In this paper, we have investigated theoretically linear as well as weakly nonlinear stability of a viscous liquid film flowing down an inclined or vertical plane under the action of gravity. The classical momentum-integral method, which is applicable for small as well as large values of Reynolds number Re, has been used to formulate the single nonlinear free surface equation in terms of the dimensionless perturbed film thickness η(x,t). Using sinusoidal perturbation in the linearized part of the surface evolution equation, we obtain the stability criterion and the critical value of the wave number kc which conceives the physical parameters Re, inclination angle θ and Weber number We. However, the linear stability analysis reveals the stabilizing influence of We as well as the destabilizing influence of Re and θ on this flow dynamics. The multiple-scale analysis has been used to derive the complex Ginzburg–Landau type nonlinear equation for investigating the weakly nonlinear stability analysis. We demarcate all the four states of the flow in the Re-k (or θ-k)-plane which are found after the critical value of Rec (or θc) depending upon the values of the other parameters. A novel result which emerges from the nonlinear stability analysis is a simple relationship among the parameters k, Re, We, and θ. This relationship essentially gives us the conditions needed for the existence of an explosive unstable zone when (3Re−3cot θ−4ReWek2)= 0; otherwise, the flow system will be free from this zone. Indeed, this zone decreases with the increase in We, whereas it increases with the increase in Re and θ confirming the stabilizing role of We and destabilizing role of Re and θ as found in linear stability analysis.

Liquid Films Falling Down a Vertical Fiber: Modeling, Simulations and Experiments

We provide a new framework for analyzing the flow of an axisymmetric liquid film flowing down a vertical fiber, applicable to fiber coating flows and those in similar geometries in heat exchangers, water treatment, and desalination processes. The problem considered is that of a viscous liquid film falling under the influence of gravity and surface tension on a solid cylindrical fiber. Our approach is different from existing ones in that we derive our mathematical model by using a control-volume approach to express the conservation of mass and axial momentum in simple and intuitively appealing forms, resulting in a pair of equations that are reminiscent of the Saint-Venant shallow-water equations. Two versions of the model are obtained, one assuming a plug-flow velocity profile with a linear drag force expression, and the other using the fully-developed laminar velocity profile for a locally uniform film to approximate the drag. These can, respectively, model high- and low-Reynolds number regimes of flow. Linear stability analyses and fully nonlinear numerical simulations are presented that show the emergence of traveling wave solutions representing chains of identical droplets falling down the fiber. Physical experiments with safflower oil on a fishing line are also undertaken and match the theoretical predictions from the laminar flow model well when machine learning methods are used to estimate the parameters.