Viscous Liquid Film Research Articles

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.

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.

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.

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.

Nonlinearwaves in a liquid film-gas flow system

The instability and regular nonlinear waves in the film of a heavy viscous liquid flowing along the wall of a round tube and interacting with a gas flow are investigated. The solutions for the wave film flows are numerically obtained in the regimes from free flow-down in a counter-current gas stream to cocurrent upward flow of the film and the gas at fairly large gas velocities. Continuous transition from the counter-current to the cocurrent flow via the state with a maximum amplitude of nonlinear waves and zero values of the liquid flow rate and the phase velocity is investigated. The Kapitsa-Shkadov method is used to reduce a boundary value problem to a system of evolutionary equations for the local values of the layer thickness and the liquid flow rate.

Capillary-driven flow induced by a stepped perturbation atop a viscous film

Thin viscous liquid films driven by capillarity are well described in the lubrication theory through the thin film equation. In this article, we present an analytical solution of this equation for a particular initial profile: a stepped perturbation. This initial condition allows a linearization of the problem making it amenable to Fourier analysis. The solution is obtained and characterized. As for a temperature step in the heat equation, self-similarity of the first kind of the full evolution is demonstrated and a long-term expression for the excess free energy is derived. In addition, hydrodynamical fields are described. The solution is then compared to experimental profiles from a model system: a polystyrene nanostep above the glass transition temperature which flows due to capillarity. The excellent agreement enables a precise measurement of the capillary velocity for this polymeric liquid, without involving any numerical simulation. More generally, as these results hold for any viscous system driven by capillarity, the present solution may provide a useful tool in hydrodynamics of thin viscous films.

Nonstationary periodic spatial waves on the surface of a viscous liquid film falling down a vertical cylinder

The flows of viscous liquid film over the outer surface of a vertical cylinder are examined. Investigation of wave regimes in the case of low flow rates and large cylinder radii is reduced to the analysis of solutions to a nonlinear evolution equation for the film thickness. There are countable numbers of steady-state traveling solution families in the considered model. In turn, most of them are unstable to 2D and 3D perturbations. Thus, evolution of initial perturbations in different ranges of parameter values differs significantly.

Experiment on Vibration Reducing Capacities of Pore Viscous Damping and Liquid Film Damping

Recently, we develop a new method for reducing vibration named liquid film damping (LFD) which combines the mechanism of vibrational energy dissipation of liquid viscous damping (LVD) and air film damping (AFD). This paper use experiment to compare the capacities of vibration decreasing of LFD and the pore viscous damping (PVD) which is an extensive applied type of LVD. The results of experiment present LFD has better ability than traditional LVD in reducing vibration of structures in not only higher frequency but lower frequency.

Asymptotic behavior of a retracting two-dimensional fluid sheet

Two-dimensional (2D) capillary retraction of a viscous liquid film is studied using numerical and analytical approaches for both diphasic and free surface flows. Full 2D Navier-Stokes equations are integrated numerically for the diphasic case, while one-dimensional (1D) free surface model equations are used for free surface flows. No pinch-off is observed in the film in any of these cases. By means of an asymptotic matching method on the 1D model, we derive an analytical expansion of the film profile for large times. Our analysis shows that three regions with different timescales can be identified during retraction: the rim, the film, and an intermediate domain connecting these two regions. The numerical simulations performed on both models show good agreement with the analytical results. Finally, we report the appearance of an instability in the diphasic retracting film for small Ohnesorge number. We understand this as a Kelvin-Helmholtz instability arising due to the formation of a shear layer in the neck region during the retraction.

Frothability and surface behavior of a rhamnolipid biosurfactant produced by Pseudomonas aeruginosa MA01

In this work, surface activity and frothability of rhamnolipid biosurfactant produced by a Pseudomonas aeruginosa MA01 were studied and compared with conventional flotation frothers, i.e. methyl isobutyl carbinol (MIBC), pine oil, Dowfroth-250 (DF-250), and Aerofroth-65 (A-65). FTIR and ES-MS analysis indicated that the product contained two types of commonly found rhamnolipids: l-rhamnosyl-β-hydroxydecanoyl-β-hydroxydecanoate (RL-1) and l-rhamnosyl l-rhamnosyl-β-hydroxydecanoyl-β-hydroxydecanoate (RL-2). Surface tension measurements showed that the rhamnolipid product reduces surface tension more effectively than frothers due to higher molecular weight and the presence of multiple oxygenated groups in its structure, producing more viscous liquid film, which was confirmed with film elasticity calculations. Both surface tension and elasticity values followed the order: rhamnolipid > A-65 > DF-250 > pine oil > MIBC. Frothability of the tested surfactants gave the order: rhamnolipid > A-65 > DF-250 > MIBC > pine oil.

Entropy generation in gas absorption into a falling liquid film

The study of mass transfer into falling films constitutes a significant aspect for numerous applications in the chemical technology and is considered the subject of many theoretical and experimental researches. Evaluating the second law of thermodynamics is one of the contemporarily used methods to determine the performances of an industrial process and to study various sources of irreversibility. Expressions of the liquid velocity, the gas concentration, the entropy generation rate as well as the main sources of irreversibility in the case of gas absorption (carbon dioxide) into a laminar falling viscous incompressible liquid film (water) without chemical reaction, are analytically derived and graphically presented and discussed.

HPM for the slip velocity effect on a liquid film over an unsteady stretching surface with variable heat flux

This article presents a numerical solution for the unsteady slip velocity flow of a thin viscous liquid film over a heated horizontal stretching surface in the presence of variable heat flux. Similarity transformations are used to transform the governing equations to a set of coupled non-linear ordinary differential equations. The obtained differential equations are solved approximately by the homotopy perturbation method (HPM). The effects of various parameters governing the flow and heat transfer in this study are discussed and presented graphically. Comparison of numerical results is made with the earlier published results under limiting cases.

Investigation of the Conservative System of Equations for a Vertically Flowing Liquid Film

A new system of equations for modeling the dynamics of long-wave perturbations on the surface of a viscous liquid film flowing down a vertical plane was investigated. It is shown that the system obtained agrees with a known system derived by the variable changing method but unlikely has a conservative form that is perfect for designing of numerical efficient conservative difference schemes. The system is reduced to a single conservative equation for the function analogous to the hydrodynamic stream function with the appropriate boundary conditions. For the case of moderate Reynolds number it is noted that the equation with boundary conditions coincide with the well known Shkadov’s model under the assumption of self-similar profile of longitudinal velocity. At small Reynolds numbers typical to the condition of microgravity it was proved that the system is reduces to one evolution equation for the film thickness.

Shear-Driven Flow of Locally Heated Viscous Liquid Film in a Minichannel

This paper considers a flow of a liquid sheared by gas in a flat mini-channel with two identical heaters arranged in a row one after another in a streamwise direction at the bottom wall. The present study is focused on the investigation of influence of local heaters arrangement and size on thermocapillary deformations in a viscous film, gravity effect is also investigated. 3D one-sided model is considered, viscosity of the liquid is supposed to be temperature dependent. Numerical analysis reveals that interaction and mutual influence of 3D structures takes place. Film pattern changes qualitatively depending on the heaters arrangement and form. For rectangular heaters a middle stream exists. Minimum film thickness value increases and its location moves to heater edges for rectangular heaters. A critical backlash between two heaters, at which film thinning is the largest, exists. Gravity significantly affects on the film deformations. Decreasing of gravity level leads to a flow destabilization and film deformations, especially film thinning, essentially increases.

Thin film flow over a heated nonlinear stretching sheet in presence of uniform transverse magnetic field

The unsteady flow of a thin viscous liquid film over a heated horizontal stretching surface permitted by uniform transverse magnetic field is studied by considering the stretching velocity and the temperature distribution in their general functional forms. An evolution equation for the film thickness is derived using long-wave approximation theory of thin liquid film and this nonlinear PDE is solved numerically for some representative values of non-dimensional parameters. It is observed that the magnetic field resists the film thinning process for all types of velocity and temperature distribution. But thermocapillarity enhances the film thinning rate even in presence of magnetic field. Further, effect of Marangoni and Prandtl numbers are explored in presence of magnetic field. Physical explanations are provided to understand the different effects on film thinning.

Asymptotic analysis of the dewetting rim

Consider a film of viscous liquid covering a solid surface, which it does not wet. If there is an initial hole in the film, the film will retract further, forming a rim of fluid at the receding front. We calculate the shape of the rim as well as the speed of the front using lubrication theory. We employ asymptotic matching between the contact line region, the rim, and the film. Our results are consistent with simple ideas involving dynamic contact angles and permit us to calculate all free parameters of this description, previously unknown.

Effects of variable fluid properties on unsteady thin-film flow over a non-linear stretching sheet

The effects of the variation of viscosity and surface tension with temperature on unsteady flow of a thin viscous liquid film over a heated horizontal stretching surface are analyzed considering general form of the stretching velocity and the temperature distribution. Using perturbation technique, an evolution equation is derived from the governing equations and this evolution equation (non-linear PDE) is solved numerically by using a combination of modified Euler method, the Newton–Kantorovich method, and the finite difference method. This leads to a five-parameter problem for some representative value of the parameters. It is shown that the film thickness decreases with the decrease of viscosity of the fluid as temperature increases and more or less heat flows out of the liquid through the stretching surface. Film thins faster both for thermo-capillary force and decrease of viscous force provided temperature decreases along the stretching direction. Physical explanations are furnished where it is necessary to justify the results.