Numerical study of a thin liquid film flowing down an inclined wavy plane

We investigate the stability of a thin liquid film flowing down an inclined wavy plane using a direct numerical solver based on a finite element/arbitrary Lagrangian Eulerian approximation of the free-surface Navier-Stokes equations. We study the dependence of the critical Reynolds number for the onset of surface wave instabilities on the inclination angle, the waviness parameter, and the wavelength parameter, focusing in particular on mild inclinations and relatively large waviness so that the bottom does not fall monotonously. In the present parameter range, shorter wavelengths and higher amplitude for the bottom undulation stabilize the flow. The dependence of the critical Reynolds number evaluated with the Nusselt flow rate on the inclination angle is more complex than the classical relation (5/6 times the cotangent of the inclination angle), but this dependence can be recovered if the actual flow rate at critical conditions is used instead.     

Stability of liquid film falling down a vertical non-uniformly heated wall

The main object in this paper is to study the stability of a viscous film flowing down a vertical non-uniformly heated wall under gravity. The wall temperature is assumed linearly distributed along the wall and the free surface is taken to be adiabatic. A long wave perturbation method is used to derive the nonlinear evolution equation for the falling film. Using the method of multiple scale, the nonlinear stability analysis is studied for travelling wave solution of the evolution equation. The complex Ginzburg-Landau equation is determined to discuss the bifurcation analysis of the evolution equation. The results indicate that the supercritical unstable region increases and the subcritical stable region decreases with the increase of Peclet number. It has been also shown that the spatial uniform solution corresponding to the sideband disturbance may be stable in the unstable region.     

Linear Stability of Taylor-Couette Flows with Axial Heat Buoyancy

The linear stability is studied of flows confined between two concentric cylinders, in which the radial temperature gradient and axial gravity are considered for an incompressible Newtonian fluid. Numerical method based on the Petrov-Galerkin scheme is developed to deal with the buoyancy term in momentum equations and an additional temperature perturbation equation. Computations of the neutral stability curves are performed for different rotation cases. It is found that the flow instability is influenced by both centrifugal and axial shear instabilities, and the two instability mechanisms interact with each other. The outer cylinder rotation plays dual roles of stabilizer and destabilizer under different rotating stages with the inner cylinder at rest. For the heat buoyancyinduced axial flow, spiral structures are found in the instability modes.     

The inertial dynamics of thin film flow of non-Newtonian fluids

Consider the flow of a thin layer of non-Newtonian fluid over a solid surface. I model the case where the viscosity depends nonlinearly on the shear-rate; power law fluids are an important example, but the analysis here is for general nonlinear dependence. The modelling allows for large changes in film thickness provided the changes occur over a relatively large enough lateral length scale. Modifying the surface boundary condition for tangential stress forms an accessible foundation for the analysis where flow with constant shear is a neutral critical mode, in addition to a mode representing conservation of fluid. Perturbatively removing the modification then constructs a model for the coupled dynamics of the fluid depth and the lateral momentum. For example, the results model the dynamics of gravity currents of non-Newtonian fluids when the flow is not creeping.     

Modeling film flows down inclined planes

A new model of film flow down an inclined plane is derived by a method combining results of the classical long wavelength expansion to a weighted-residuals technique. It can be expressed as a set of three coupled evolution equations for three slowly varying fields, the thickness h, the flow-rate q, and a new variable that measures the departure of the wall shear from the shear predicted by a parabolic velocity profile. Results of a preliminary study are in good agreement with theoretical asymptotic properties close to the instability threshold, laboratory experiments beyond threshold and numerical simulations of the full Navier-Stokes equations. Received: 16 April 1998 / Revised: 29 June 1998 / Accepted: 2 July 1998     

非平整基底上受热液膜流动稳定性研究

本文研究了二维黏性流体薄膜沿非平整不均匀加热基底流动时非线性表面波的演化及其流动稳定性.利用长波摄动法推导出非平整线性加热基底上非线性表面波的零阶和一阶演化方程,基于所得演化方程,绘制出正弦波纹基底上液膜的表面波形图,并研究液膜流动的线性稳定性,分析了各无量纲参数对液膜线性稳定性的影响.分析结果表明:在正弦波纹基底上,液膜自由表面随同壁面作相同频率的正弦型波动,且液膜厚度沿流动方向逐渐变小;Marangoni数为稳定影响因素,随Marangoni数的增大,液膜稳定区域增大;Peclet数和倾角θ均为不稳定影响因素,随Peclet数和倾角θ的增大,液膜稳定区域减小;在非平整基底的波峰和波谷处,Peclet数、Marangoni数和倾角θ对稳定性的影响趋势一致,但基底波谷处的液膜稳定区域小于波峰处区域,流动更易失稳.     

A simplified model for flows with eddies in symmetrically expanding channels

We derive a simplified model for two-dimensional (2D) channel flows with recirculated regions at moderate Reynolds numbers based on an extension of the boundary layer (BL) theory and averaging across the channel. The model reproduces symmetry-breaking bifurcations and resulting flow structures accurately. Analytical estimates for the decay rates toward the parabolic profile before and after a sudden change in the walls agree well with the full numerical simulations. A seemingly chaotic steady flow is also discovered in a channel with periodic expansions and contractions.     

Stability of fluid flow past a membrane

The stability of the flow of a fluid past a solid membrane of infinitesimal thickness is investigated using a linear stability analysis. The system consists of two fluids of thicknesses R and H R and bounded by rigid walls moving with velocities and , and separated by a membrane of infinitesimal thickness which is flat in the unperturbed state. The fluids are described by the Navier-Stokes equations, while the constitutive equation for the membrane incorporates the surface tension, and the effect of curvature elasticity is also examined for a membrane with no surface tension. The stability of the system depends on the dimensionless strain rates and in the two fluids, which are defined as and for a membrane with surface tension , and and for a membrane with zero surface tension and curvature elasticity K. In the absence of fluid inertia, the perturbations are always stable. In the limit , the decay rate of the perturbations is O(k ³ ) smaller than the frequency of the fluctuations. The effect of fluid inertia in this limit is incorporated using a small wave number asymptotic analysis, and it is found that there is a correction of smaller than the leading order frequency due to inertial effects. This correction causes long wave fluctuations to be unstable for certain values of the ratio of strain rates and ratio of thicknesses H. The stability of the system at finite Reynolds number was calculated using numerical techniques for the case where the strain rate in one of the fluids is zero. The stability depends on the Reynolds number for the fluid with the non-zero strain rate, and the parameter , where is the surface tension of the membrane. It is found that the Reynolds number for the transition from stable to unstable modes, , first increases with , undergoes a turning point and a further increase in the results in a decrease in . This indicates that there are unstable perturbations only in a finite domain in the plane, and perturbations are always stable outside this domain. Received: 29 May 1997 / Revised: 9 October 1997 / Accepted: 26 November 1997     

Thin liquid films on a slightly inclined heated plate

The behavior of a thin liquid film on a uniformly heated substrate is considered. When the substrate is horizontal and the Marangoni number sufficiently large the film breaks up into a periodic array of drops. When the substrate is slightly inclined this drop-like state slides down the substrate. The relation between these states is discussed and their stability properties with respect to longitudinal perturbations are determined. The results shed light on the multiplicity of states accessible to systems of this type and on the possible transitions among them.     

Essence of Inviscid Shear Instability: a Point View of Vortex Dynamics

The essence of shear instability is reviewed both mathematically and physically, which extends the instability theory of a sheet vortex from the viewpoint of vortex dynamics. For this, the Kelvin-Arnol＇d theorem is retrieved in linear context, i.e., the stable flow minimizes the kinetic energy associated with vorticity. Then the mechanism of shear instability is explored by combining the mechanisms of both Kelvin Helmholtz instability （K-H instability） and resonance of waves. The waves, which have the same phase speed with the concentrated vortex, have interactions with the vortex to trigger the instability. The physical explanation of shear instability is also sketched by extending Batchelor＇s theory. These results should lead to a more comprehensive understanding on shear instabilities.     

Instability and dynamics of thin viscoelastic liquid films

The instability, rupture, and subsequent growth of holes in a thin Jeffreys-type viscoelastic film under the influence of long-range van der Waals force are investigated using both linear stability analysis and nonlinear numerical solutions. The linear stability analysis of full governing equations valid for arbitrary wave numbers shows that although fluid rheology does not influence the dominant length scale of the instability, it significantly affects the growth rate. It is shown that neglect of inertia and solvent dynamics results in a nonphysical singularity in the growth rate beyond a critical value of relaxation time. We further carry out numerical simulations of a set of long-wave, nonlinear differential equations (also derived in Rauscher et al., Eur. Phys. J. E 17, 373 (2005)) governing the evolution of the free surface. The nonlinear simulations, in their domain of validity, confirm the results of the linear analysis. Interestingly, results from nonlinear simulations further show that both for Newtonian and viscoelastic liquids, the shape and the dewetting dynamics of a hole are identical when examined in terms of a rescaled time which depends on rheological parameters. Thus, viscoelasticity of Jeffreys type merely accelerates the growth rate, without however affecting the important morphological characteristics.     

Stability of a dry patch in a viscous flowing film

We investigate the dewetting of liquid films flowing down an incline. At low flow rate we observe the formation of stationary dry patches edged with a liquid rim. Their shape can be predicted by a simple model in which the rim weight is balanced by surface tension. Above a critical flow rate per unit length Γ_c of typical scale U_cl_c (U_c capillary velocity, l_c capillary length), these dry patches cannot remain stationary and are swept away. An improved model taking into account capillary effects linked to contact line curvature, hydrostatic pressure in the film and inertial effects predicts this loss of stability in good agreement with experiments for sufficiently high viscosity values.     

Experimental study of the instability of a film flowing down a vertical fiber

The wavy dynamics of a viscous film flowing axisymmetrically on a vertical fiber is characterized experimentally. The study of the initial response of the flow to natural noise shows a well-defined transition between a convective and an absolute instability [1]. In the convective case, disturbances of controlled frequencies have been applied at inlet. The flow responds to inlet excitation at frequencies lower than a well-defined cut-off frequency. A good agreement has been found with the linear stability analysis and with the solution to a two-equation model [2] in the nonlinear regime.     

Absolute and convective instabilities of a viscous film flowing down a vertical fiber

The stability of a viscous film flowing down a vertical fiber under the action of gravity is analyzed both experimentally and theoretically. At large or small film thicknesses, the instability is convective, whereas an absolute instability mode is observed in an intermediate range of film thicknesses for fibers of small enough radius. The onset of the experimental irregular wavy regime corresponds precisely to the theoretical prediction of the threshold of the convective instability.     

Stability of the heated liquid film in the presence of the thermocapillary effect

Stability of the flow of the heated liquid film is studied in the presence of the thermocapillary effect. To describe the waves in the film, the integral model is used. According to results of linear analysis of stability, the thermocapillary effect expands the area of instability only at low values of Peclet number Pe, and at high values of Pe, the instability area narrows. Wave evolution in the film on a substrate with the fixed temperature was simulated numerically. Results of numerical simulations agree with the linear theory of stability.     

Formation of thermocapillary structures in a heated liquid film

The conditions of formation of a three-dimensional thermocapillary structure on the surface of a liquid film flowing along a heater with the constant temperature were studied numerically based on the derived system of equations. Formation of the thermocapillary structure was modeled by periodic perturbations in the transverse direction superimposed on the two-dimensional flow. It is shown in calculations that transversal perturbations can develop into periodic rivulet structures on the film surface if the Marangoni number exceeds some threshold value. It is revealed that the rivulet structure develops when the period belongs to a certain range, which is determined by the value of Marangoni number. The results of calculations are in a good agreement with experimental data.     

Improved modeling of flows down inclined planes

New models of film flows down inclined planes have been derived by combining a gradient expansion at first or second order to weighted residual techniques with polynomials as test functions. The two-dimensional formulation has been extended to account for three-dimensional flows as well. The full second-order two-dimensional model can be expressed as a set of four coupled evolution equations for four slowly varying fields, the thickness h, the flow rate q and two other quantities measuring the departure from the flat-film semi-parabolic velocity profile. A simplified model has been obtained in terms of h and q only. Including viscous dispersion effects properly, it closely sticks to the asymptotic expansion in the appropriate limit. Our new models improve over previous ones in that they remain valid deep into the strongly nonlinear regime, as shown by the comparison of our results relative to travelling-wave and solitary-wave solutions with those of both direct numerical simulations and experiments. Received 14 September 1999 and Received in final form 6 January 2000