Viscous Liquid Film Research Articles

Stability analysis of the rimming flow inside a uniformly heated rotating horizontal cylinder

The stability analysis is presented for a thin viscous liquid film flowing inside a uniformly heated horizontal cylinder that is rotating about its axis. The free surface evolution equation for the liquid-gas interface is obtained by simplifying the Navier-Stokes and energy equations within the lubrication approximation. Various dimensionless numbers are obtained that quantify the effect of gravity, viscous drag, inertia, surface tension, and thermocapillary stress. The film thickness evolution equation is solved numerically to obtain two-dimensional, steady state solutions neglecting axial variations. A liquid pool forms at the bottom of the cylinder when gravity dominates other forces. This liquid pool is shifted in the direction of rotation when inertia or viscous drag is increased. Small axial perturbations are then imposed to the steady solutions to study their stability behavior. It is found that the inertia and capillary pressure destabilize whereas the gravity and thermocapillary stress stabilize the rimming flow. The influence of Marangoni number is reported by computing the stable and unstable parametric regions. Thicker films are shown to be more susceptible to become unstable.

Research of non-contact measurement for high viscous fluid falling film thickness on spherical series surface

Abstract In the heat exchange system, the thickness of viscous liquid film on the surface of corrugated tube formed by fluid is an important parameter to control the performance of heat exchange. Because the surface of corrugated tube is a spherical series surface, it makes it difficult for measuring the thickness of liquid film. Based on the measuring object of liquid film formed by fluid on the surface of the corrugated tube, we specially designed a capacitive sensor for this kind of abnormity surface in the paper. We detailedly introduced the structure and operational principle of capacitive sensor, the circuit design and software system analysis, finally we displayed the waveform chart formed from the field data. Experimental results showed that the measuring system realized high precise and non-contact measurement for the liquid film thickness on the surface of the corrugated tube.

Production of nanostructured molecular liquids by supercritical CO2 processing

Stable molecular clusters of ibuprofen and naproxen were prepared in dry ice, by supersonic jet expansion of their supercritical CO2 drug formulations into a liquid nitrogen cooled collection vessel, with up to 80% yield. Mixing the “dry ice” in water, resulted in the solubilization of the clusters and in the case of ibuprofen, we were able to create solutions, with concentrations of up to 6mg/ml, a 300-fold increase over previously reported values. Drop casting and ambient drying of these solutions on silicon substrate resulted in a stable, viscous liquid film, referred to as nanostructured molecular liquids. These liquids exhibited a highly aligned, fine (self-assembled) super lattice features. In vitro cancer cell viability studies of these formulations exhibited similar cytotoxicity to that of the original raw materials, thus retaining their original potency. Besides its scientific importance, this invention is expected to open up new drug delivery platforms.

A three-equation model for thin films down an inclined plane

We derive a new model for thin viscous liquid films down an inclined plane. With an asymptotic expansion in the long-wave limit, the Navier–Stokes equations and the work–energy theorem are averaged over the fluid depth. This gives three equations for the mass, momentum and energy balance which have the mathematical structure of the Euler equations of compressible fluids with relaxation source terms, diffusive and capillary terms. The three variables of the model are the fluid depth, the average velocity and a third variable called enstrophy, related to the variance of the velocity. The equations are numerically solved by classical schemes which are known to be reliable and robust. The model gives satisfactory results both for the neutral stability curves and for the depth profiles of wavy films produced by a periodical forcing or by a random noise perturbation. The numerical calculations agree fairly well with experimental measurements of Liu & Gollub (Phys. Fluids, vol. 6, 1994, pp. 1702–1712). The calculation of the wall shear stress below the waves indicates a flow reversal at the first depth minimum downstream of the main hump, in agreement with experiments of Tihon et al. (Exp. Fluids, vol. 41, 2006, pp. 79–89).

Experimental Investigation of Ligament Formation Dynamics of Thin Viscous Liquid Film at Spinning Disk Edge

The ligament formation and break-up process of thin viscous liquid film at the spinning disk edge were performed experimentally. The regression models for ligament number, tail end diameter of ligament, and mean diameter of droplets were proposed. Despite the velocity slippage of rotary cup atomizers being smaller than that of the disk atomizer, it will not improve the atomizing characteristics. The tail end diameter of ligaments and the size of droplets decrease with increasing rotational speed, while liquid flow rate shows smaller influence on the tail end diameter and droplet sizes. For the break-up of a ligament in the centrifugal field, the connected region between joints presents as dumbbell structures, and the dumbbell structures may contract from both ends to the middle for further integration. Despite the capillary wavelength for the break-up of a ligament being about 3.15 times of the ligament diameters due to ligament stretching, the actual dominant mode is the short-wave mode.

Viscous liquid film flow down an inclined corrugated surface. Calculation of the flow stability to arbitrary perturbations using an integral method

Viscous liquid film flow along an inclined corrugated (sinusoidal) surface has been studied. Calculations were performed using an integral model. The stability of nonlinear steady-state flows to arbitrary perturbations was examined using the Floquet theory. It has been shown that for each type of corrugation there is a critical Reynolds number for which unstable perturbations occur. It has been found that this value greatly depends on the physical properties of the liquid and geometric parameters of the flow. In particular, in the case of film flow down a smooth wall, the critical waveformation parameter depends only on the angle of inclination of the flow surface. The values of the corrugation parameters (amplitude and period) were obtained for which the film flow down a wavy wall is stable to arbitrary perturbations up to moderate Reynolds numbers. Such parameter values exist for all investigated angles of inclination of the flow surface.

Dynamics and equilibria of thin viscous coating films on a rotating sphere

We examine the dynamics of a thin viscous liquid film on the outer surface of a solid sphere rotating around its vertical axis in the presence of gravity. An asymptotic model describing the evolution of the film thickness is derived in the rotating frame based on the lubrication approximation. The model includes the centrifugal and gravity forces and the stabilizing effect of surface tension. Depending on the values of the parameters, the problem admits different types of steady states: one with a uniformly positive film thickness, or those with one or two dry zones on the sphere. We prove that all steady states are energy minimizers and hence global attractors for axisymmetric states.

Стекание вязкой пленки жидкости по наклонной гофрированной поверхности. Расчет устойчивости течений к произвольным возмущениям с использованием интегрального метода

Стекание вязкой пленки жидкости по наклонной гофрированной поверхности. Расчет устойчивости течений к произвольным возмущениям с использованием интегрального метода

Effect of tangential and normal forces on the wave formation in the flow of a liquid film and a gas

The effect of surface forces on nonlinear waves induced by the hydrodynamic instability in the flow of a viscous liquid film along the inner surface of a tube blown with a gas.

Generalized linear stability of non-inertial rimming flow in a rotating horizontal cylinder.

The stability of a thin film of viscous liquid inside a horizontally rotating cylinder is studied using modal and non-modal analysis. The equation governing the film thickness is derived within lubrication approximation and up to first order in aspect ratio (average film thickness to radius of the cylinder). Effect of gravity, viscous stress and capillary pressure are considered in the model. Steady base profiles are computed in the parameter space of interest that are uniform in the axial direction. A linear stability analysis is performed on these base profiles to study their stability to axial perturbations. The destabilizing behavior of aspect ratio and surface tension is demonstrated which is attributed to capillary instability. The transient growth that gives maximum amplification of any initial disturbance and the pseudospectra of the stability operator are computed. These computations reveal weak effect of non-normality of the operator and the results of eigenvalue analysis are recovered after a brief transient period. Results from nonlinear simulations are also presented which also confirm the validity of the modal analysis for the flow considered in this study.

Stability of a film flowing down an inclined corrugated plate: The direct Navier-Stokes computations and Floquet theory

The paper is devoted to a theoretical analysis of the linear stability of the viscous liquid film flowing down an inclined wavy surface. The study is based on the Navier-Stokes equations in their full statement. The developed numerical algorithm allows us to compute both the steady state solution of the nonlinear equations and the rates of growing or damping in time of the arbitrary two-dimensional disturbances of the solution which are bounded in space. The wall corrugations have a great influence on the disturbances behaviour. There is a critical Reynolds number Recr when the steady-state viscous flow over an undulated surface becomes unstable. It is found that the value of Recr depends essentially both on the topography parameters and the liquid's physical properties. In the case of the flat plate, the critical Reynolds number depends only on the value of the inclination angle. For different values of the Kapitza number, the inclination angle, and the Reynolds number we obtained the regions of the corrugation's parameters (amplitude and period) where all two-dimensional disturbances decay in time.

Wavy liquid film in the presence of co- or counter-current turbulent gas flow

The wavy downflow of a viscous liquid film in the presence of the turbulent gas flow was analyzed theoretically. Two-dimensional stationary running waves are calculated in a wide range of Reynolds numbers of liquid and gas. Hydrodynamics of liquid is calculated based on complete Navier-Stokes equations. The wave interface surface is considered as a small perturbation and equations in gas are linearized near the main turbulent flow. Different optimal downflow regimes are determined, and the main wave characteristics are compared in detail with and without the co- and counter-current gas flows. It is shown that at high velocities of the co-current gas flow, the calculated waves correspond to ripples observed in experiments.

Viscous film flow coating the interior of a vertical tube. Part 1. Gravity-driven flow

AbstractThe gravity-driven flow of a viscous liquid film coating the inside of a tube is studied both theoretically and experimentally. As the film moves downward, small perturbations to the free surface grow due to surface tension effects and can form liquid plugs. A first-principles strongly nonlinear model based on long-wave asymptotics is developed to provide simplified governing equations for the motion of the film flow. Linear stability analysis on the basic solution of the model predicts the speed and wavelength of the most unstable mode, and whether the film is convectively or absolutely unstable. These results are found to be in remarkable agreement with the experiments. The model is also solved numerically to follow the time evolution of instabilities. For relatively thin films, these instabilities saturate as a series of small-amplitude travelling waves, while thicker films lead to solutions whose amplitude becomes large enough for the liquid surface to approach the centre of the tube in finite time, suggesting liquid plug formation. Next, the model’s periodic travelling wave solutions are determined by a continuation algorithm using the results from the time evolution code as initial seed. It is found that bifurcation branches for these solutions exist, and the critical turning points where branches merge determine film mean thicknesses beyond which no travelling wave solutions exist. These critical thickness values are in good agreement with those for liquid plug formations determined experimentally and numerically by the time-evolution code.

Stability of heated liquid film on an uneven substrate

This paper studies mainly the evolution and linear stability of the nonlinear surface waves of a two-dimensional viscous liquid film along an uneven inclined non-uniformly heated wall. A long wave perturbation method is used to derive zero- and first-order evolutions equations of the nonlinear surface wave flowing on an uneven substrate. Based on the obtained evolution equations, the diagram of evolution progress for film surface wave on a sinusoidal corrugated substrate is drawn, the linear stability analysis is also studied, and the effect of various parameters on the flow stability of liquid membrane is analysed. Theoretical results demonstrate that the free surface of the film shows sine wave and has the same frequency as the substrate, and the film thickness will decrease gradually along the flow direction. Marangoni number gives a stabilizing effect, the stable zone increases with the increase of Marangoni number. While, Peclet number and the angle theta are unstable factors, the stable region decreases with the increase of them. Besides, the trends that Marangoni number, Peclet number and angle theta may impact on the stability of the film whice are consistent with one another, so a liquid film is easy to destabilize at the trough of the substrate.

Capillary-gravity waves on a liquid film of arbitrary depth: Analysis of the wave resistance

We discuss the wave resistance in the case of an externally perturbed viscous liquid film of arbitrary thickness. Emphasis is placed on the dependence of the wave resistance on the film thickness H, the length scale b characterizing the external perturbation, and its velocity V. In particular, the effectiveness of the mechanisms of capillary-gravity waves and the viscous dissipation localized in the vicinity of the perturbation are compared and discussed as functions of H and V. We show that, in general, the wave resistance is a nonmonotonous function of H with a maximum whose amplitude and position depend on b and V. In the case of small H the wave resistance depends on a parameter S proportional V/H(3). We find three different regimes of this parameter in which the wave resistance behaves like S(r) with the exponent r equal to 1, 1/3, and -1. These results are also obtained independently within the thin liquid film approximation. This allows us to assess the range of validity of the thin liquid film approximation in various cases, in particular its dependence on the perturbation length scale b.

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.

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.