Thin Viscous Film Research Articles

Instability of a thin viscous film flowing under an inclined substrate: the emergence and stability of rivulets – CORRIGENDUM

Instability of a thin viscous film flowing under an inclined substrate: the emergence and stability of rivulets – CORRIGENDUM

Deformation and relaxation of viscous thin films under bouncing drops

Abstract

Hydrodynamic instability and wave formation of a viscous film flowing down a slippery inclined substrate: Effect of odd-viscosity

Hydrodynamic instability and formation of waves for thin viscous film flowing down a slippery inclined substrate with broken time-reversal-symmetry have been discussed in this present study. The effect of slip on the substrate is modeled using the Navier slip boundary condition. We have derived a nonlinear evolution model in the framework of the long-wave expansion technique. The one equation model can track the free surface evolution and involve the viscosity, gravity, surface tension, and the effect of the slip of the wall. The odd part of the Cauchy stress tensor with an odd-viscosity coefficient significantly modifies the characteristic of the film flow. We performed the linear stability analysis with the Orr Sommerfeld technique and classical temporal analysis from the model. Comparing the results shows a satisfactory agreement between Orr Sommerfeld and the model when the velocity scale is chosen as twice the free surface velocity. Analyzing the traveling wave solutions and studying different bifurcation analyzes and phase diagrams, we focused on and discussed two significant wave families, namely γ1 (low-wave-speed) and γ2 (high-wave-speed). The spatio-temporal study of the model has been performed numerically for different odd-viscosity and slip parameters. Results show that the odd-viscosity plays a stabilizing role while the slip on the substrate gives a destabilizing effect within the parameter range of our consideration.

Levitation of a cylinder by a thin viscous film

Abstract

Instability of ultrathin viscoelastic freestanding films

The linear stability of freestanding thin films under the influence of attractive van der Waals forces is investigated for three rheological models, viz., Newtonian viscous films, viscoelastic solid films, and Jeffreys viscoelastic liquid films, with the aim of studying the role of rheology on the instability. Thin freestanding viscous films are unconditionally unstable, whereas the shear modulus in thin freestanding solid viscoelastic films governs the onset of instability. Interestingly, elasticity plays a dual role with regard to the stability of freestanding solid and liquid films: while it has a stabilizing influence on the former, it is destabilizing in the latter. Linear stability results of Jeffreys viscoelastic freestanding films are compared with those from supported films in the inertialess limit. The instability of Jeffreys viscoelastic freestanding film is unaffected by the relaxation time, but is enhanced with decrease in the viscosity ratio (μr, the ratio of solvent viscosity to total viscosity). The dominant length scale of instability in Jeffreys viscoelastic freestanding film shifts toward shorter wavelengths with decrease in μr. For μr→0, the maximum growth rate remains bounded in a freestanding viscoelastic film in the presence of inertia, but diverges in its absence, similar to supported viscoelastic films. In general, our results show that freestanding thin films exhibit faster dynamics than supported thin films. The mode of deformation of the freestanding film (viz., bending or squeezing) is not imposed a priori in our analysis and is found to be a squeezing (symmetric) mode with equal amplitudes at the interfaces.

Modeling of Poly(methylmethacrylate) Viscous Thin Films by Spin-Coating

A predictive film thickness model based on an accepted equation of state is applied to the spin-coating of sub-micron poly(methylmethacrylate) viscous thin films from toluene. Concentration effects on density and dynamic viscosity of the spin-coating solution are closely examined. The film thickness model is calibrated with a system-specific film drying rate and was observed to scale with the square root of spin speed. Process mapping is used to generate a three-dimensional design space for the control of film thickness.

Inertial drag-out problem: sheets and films on a rotating disc

Abstract

Instability of a thin viscous film flowing under an inclined substrate: the emergence and stability of rivulets

Abstract

Hydrodynamics and instabilities of falling liquid film over a non-uniformly heated inclined wavy bottom

Hydrodynamics and instabilities of a thin viscous liquid film flowing down an undulated inclined plate with linear temperature variation have been investigated. Using the long-wave expansion method, a non-linear evolution equation for the development of the free surface is derived under the assumption that the bottom undulations are of moderate steepness. A normal mode approach has been considered to take into account the linear stability of the film to investigate both the spatial and temporal instabilities, while the method of multiple scales is used to obtain the Ginzburg–Landau-type worldly equation for studying the weakly non-linear stability solutions. The numerical study has been carried out in python with a newly developed library Scikit–FDif. The entire investigation is done for a general bottom profile followed by a case study with a sinusoidal topography. The case study reveals that the Marangoni effect destabilizes the film flow throughout the domain, whereas the bottom steepness ζ gives a dual effect for the linear stability. In the “uphill” portion, an increase in ζ stabilizes the flow, and in the “downhill” portion, an increase in ζ gives a destabilizing effect. Furthermore, a weakly non-linear study shows that both supercritical and subcritical solutions are possible for the system. It is noted that the unconditional stable region decreases and all the other region increases in the “downhill” portion in comparison with the “uphill” portion for a fixed set of parameters. The stability analysis of a truncated bimodal system is investigated. The spatial uniform solution of the complex Ginzburg–Landau equation for sideband disturbances has also been discussed. Numerical simulation indicates that a different kind of finite-amplitude permanent wave exists. The amplitudes and the phase speeds of the wave are dependent on thermocapillary as well as the bottom steepness.

Instability of a thin viscous film flowing under an inclined substrate: steady patterns

The flow of a thin film coating the underside of an inclined substrate is studied. We measure experimentally spatial growth rates and compare them to the linear stability analysis of a flat film modelled by the lubrication equation. When forced by a stationary localized perturbation, a front develops that we predict with the group velocity of the unstable wave packet. We compare our experimental measurements with numerical solutions of the nonlinear lubrication equation with complete curvature. Streamwise structures dominate and saturate after some distance. We recover their profile with a one-dimensional lubrication equation suitably modified to ensure an invariant profile along the streamwise direction and compare them with the solution of a purely two-dimensional pendent drop, showing overall a very good agreement. Finally, those different profiles agree also with a two-dimensional simulation of the Stokes equations.

Symmetrization of Thin Freestanding Liquid Films via a Capillary-Driven Flow.

We present experiments to study the relaxation of a nanoscale cylindrical perturbation at one of the two interfaces of a thin viscous freestanding polymeric film. Driven by capillarity, the film flows and evolves toward equilibrium by first symmetrizing the perturbation between the two interfaces and eventually broadening the perturbation. A full-Stokes hydrodynamic model is presented, which accounts for both the vertical and lateral flows and which highlights the symmetry in the system. The symmetrization time is found to depend on the membrane thickness, surface tension, and viscosity.

Evolution of a thin film down an incline: A new perspective

A new model which accounts for energy balance while describing the evolution of a thin viscous, Newtonian film down an incline at high Reynolds numbers and moderate Weber numbers has been derived. With a goal to improve the predictions by the model in inertia dominated regimes, the study employs the Energy Integral Method with ellipse profile EIM(E) as a weight function and is motivated by the success of EIM in effectively and accurately predicting the squeeze film force in squeeze flow problems and in predicting the inertial effects on the performance of squeeze film dampers [Y. Han and R. J. Rogers, “Squeeze film force modeling for large amplitude motion using an elliptical velocity profile,” J. Tribol. 118(3), 687–697 (1996)]. The focus in the present study is to assess the performance of the model in predicting the instability threshold, the model successfully predicts the linear instability threshold accurately, and it agrees well with the classical result [T. Benjamin, “Wave formation in laminar flow down an inclined plane,” J. Fluid Mech. 2, 554–573 (1957)] and the experiments by Liu et al. [“Measurements of the primary instabilities of film flows,” J. Fluid Mech. 250, 69–101 (1993)]. The choice of the ellipse profile allows us to have a free parameter that is related to the eccentricity of the ellipse, which helps in refining the velocity profile, and the results indicate that as this parameter is increased, there is a significant improvement in the inertia dominated regimes. Furthermore, the full numerical solutions to the coupled nonlinear evolution equations are computed through approximations using the finite element method. Based on a measure {used by Tiwari and Davis [“Nonmodal and nonlinear dynamics of a volatile liquid film flowing over a locally heated surface,” Phys. Fluids 21, 102101 (2009)]} of the temporal growth rate of perturbations, a comparison of the slope of the nonlinear growth rate with the linear growth rate is obtained and the results show an excellent agreement. This confirms that the present model’s performance is as good as the other existing models, weighted residual integral boundary layer (WRIBL) by Ruyer-Quil and Manneville [“Improved modeling of flows down inclined planes,” Eur. Phys. J: B 15, 357–369 (2000)] and energy integral method with parabolic profile [EIM(P)] by Usha and Uma [“Modeling of stationary waves on a thin viscous film down an inclined plane at high Reynolds numbers and moderate Weber numbers using energy integral method,” Phys. Fluids 16, 2679–2696 (2004)]. Furthermore, for any fixed inclination θ of the substrate, 0 < θ < π/2, there is no significant difference between the EIM(E) and EIM(P) results for weaker inertial effects, but when the inertial effects become stronger, the EIM(E) results for energy contribution from inertial terms to the perturbation at any streamwise location is enhanced. More detailed investigation on the model’s performance due to this enhancement in energy contribution and the assessment of the model as compared to the other existing theoretical models, experimental observations, and numerical simulations, in the inertia dominated regimes, will be reported in a future study.

Asymptotic regimes in elastohydrodynamic and stochastic leveling on a viscous film

An elastic sheet that deforms near a solid substrate in a viscous fluid is a situation relevant to various dynamical processes in biology, geophysics and engineering. Here, we study the relaxation dynamics of an elastic plate resting on a thin viscous film that is supported by a solid substrate. By combining scaling analysis, numerical simulations and experiments, we identify asymptotic regimes for the elastohydrodynamic leveling of a surface perturbation of the form of a bump, when the flow is driven by either the elastic bending of the plate or thermal fluctuations. In both cases, two distinct regimes are identified when the bump height is either much larger or much smaller than the thickness of the pre-wetted viscous film. Our analysis reveals a distinct crossover between the similarity exponents with the ratio of the perturbation height to the film height.

Investigation the combined effects of wear and turbulent on the performance of hydrodynamic journal bearing operating with couple stress fluids

Purpose The purpose of this paper is to investigate the combined effect of wear and turbulence on the performance of a hydrodynamic journal bearing operating under Newtonian and couple stress fluids (CSF). Design/methodology/approach The analysis consists of a modified Reynolds equation of incompressible thin viscous films, and the film thickness model taking into account the wear effect. The governing equation was solved numerically using the finite difference approach. Findings The effect of both the wear parameter and the local Reynolds number on the performance characteristics of bearing has been presented and discussed. The obtained results observed that the characteristics of the intact and worn bearing in turbulent and laminar have been enhanced due to the non-Newtonian fluid (CSF) effect. Also, the results display that bearing worn and the turbulent regime cannot be neglected in calculating the performance characteristics of the bearing lubricated with Newtonian and non-Newtonian fluids. The results achieved from this study, specify that the bearing characteristics are significantly affected by these effects. Originality/value The paper investigates the behavior of hydrodynamic bearings considering different aspects simultaneously is interesting, and the application meets the current needs of improvement in modeling hydrodynamic bearings under different conditions.

Capillary ripples in thin viscous films

Capillary ripples in thin viscous films are important features of coating and lubrication flows. Here, we present experiments based on digital holographic microscopy, measuring with nanoscale resolution the morphology of capillary ripples ahead of a viscous drop spreading on a prewetted surface. Our experiments reveal that upon increasing the spreading velocity, the amplitude of the ripples first increases and subsequently decreases. Above a critical spreading velocity, the ripples even disappear completely and this transition is accompanied by a divergence of the ripple wavelength. These observations are explained quantitatively using linear wave analysis, beyond the usual lubrication approximation, illustrating that new phenomena arise when the capillary number becomes of the order of unity.

Texture shape effects on hydrodynamic journal bearing performances using mass-conserving numerical approach

ABSTRACTIt is a known fact that incorporating textures in the contact surfaces can significantly enhance bearing performances. The purpose of this paper is to outline the effects of texture bottom profiles and contour geometries on the performances of hydrodynamic textured journal bearings. The analysis was conducted using computational approach to test eight texture shapes: rectangular, cylindrical, spherical, triangular (TR, T1, T2, T3) and chevron. The steady-state Reynolds equation for modelling the hydrodynamic behaviour of thin viscous film was solved using finite difference technique and mass conservation algorithm (JFO boundary conditions), taking into account the presence of textures on both full film and cavitation regions. The comparison with the benchmark data shows good consistency and an enhancement in bearing performances (load carrying capacity and friction). The results clearly show that the mechanisms of wedge effect and micro-step bearing for the full/partial texturing feature are the main crucial parameters, where the convergent wedge effect present in T2 triangular texture shape can significantly enhance the load-carrying capacity, while the divergent wedge action causes a net load loss. Considering the right arrangement of textures on the contact surface, their surface contours can have a significant impact on the performance of hydrodynamic journal bearings at high eccentricity ratios.

Odd-viscosity-induced stabilization of viscous thin liquid films

Thin viscous liquid films sitting on a solid substrate support nonlinear capillary waves, driven by surface shear stresses at a liquid–gas interface. When surface tension is spatially dependent other mechanisms, such as the thermocapillary effect, influence the dynamics of thin films. In this article we show that in liquids with broken time-reversal symmetry the character of the aforementioned waves and of the thermocapillary effect are significantly modified due to the presence of odd or Hall viscosity in the liquid. This is because odd viscosity gives rise to new terms in the pressure gradient of the flow thus modifying the evolution equation of the liquid–gas interface accordingly. These terms in turn break the reflection symmetry of the evolution equation leading the system to evolve from a pitchfork to a Hopf bifurcation. The odd-viscosity incipient waves can stabilize unstable thin liquid films. For instance, we show that they can suppress the thermocapillary instability. We establish the parameter ranges that odd viscosity has to satisfy in order to initiate those waves that will lead to stability.

Deformation and removal of viscous thin film by submerged jet impingement

AbstractIn this work, for the first time, we characterize the deformation and removal dynamics of viscous thin films under submerged jet impingement using direct numerical simulation. The impinging jet deforms the thin film near the impingement region at the beginning until local removed region is formed. Subsequently, the removed region expands due to the shear stress imposed by the jet. At later times, we find that the film is removed in a quasi‐adhesive removal mechanism, and its removal radius as a function of time, follows a scaling law with a universal removal rate for films over a wide range of viscosities. However, the kinetic constant of the removal scaling is highly dependent on the viscosity of the film. We also find that larger initial film thickness may change the removal mechanism, although the scaling law describing the removal rate still applies.

Travelling wave solutions in dilatant non-Newtonian thin films with second-order viscosity

Given any positive speed, we prove the existence of a travelling wave solution for a viscous incompressible thin dilatant liquid film coated with an insoluble surfactant. Relying on first principles of fluid mechanics, lubrication theory leads to a coupled system of second-order partial differential equations for the film's height and the surfactant's concentration where a second-order approximation of the viscosity function is used. The travelling wave solution is -regular and generates a heteroclinic orbit which connects two distinct equilibrium points of the associated plane dynamical system.

Stability of a thin film flow on a weakly wavy wall

We consider a thin viscous liquid film flow down an inclined plane with sinusoidal topography. We use the integral boundary layer approach to derive governing equations. By use of Floquet theory, we study linear stability of steady-state flow. For small values of the topography amplitude, perturbation analysis gives an analytical expression for corrections to the growth rate of a normal mode disturbance. This correction is proportional to the second power of the topography amplitude. The topography can both stabilize and destabilize the flow, depending on the corrugation period and the plane inclination angle. We study the topology of neutral curves and the critical Reynolds number dependence on topography parameters.