Transient film profile of thin liquid film flow on a stretching surface

In this paper we have studied a non-planar thin liquid film flow on a planar stretching surface. The stretching surface is assumed to stretch impulsively from rest and the effect of inertia of the liquid is considered. Equations describing the laminar flow on the stretching surface are solved analytically. It is observed that faster stretching causes quicker thinning of the film on the stretching surface. Velocity distribution in the liquid film and the transient film profile as functions of time are obtained. (Received: May 4, 2004; revised: February 2/August 24, 2005)     

A model for gravity driven flow of a thin liquid-solid solution with evaporation effects

A vertical substrate is coated with a thin film of a solution consisting of a volatile viscous liquid and a solid solute. The liquid film thins under gravity while the volatile component simultaneously evaporates. We develop a model to predict the evolving film thickness and in so doing we develop an approximation for the later stages of the well-known dip-coating process.     

Coherent Reflection of Electromagnetic Plane Waves from Infinite Rough Slabs

We adopt the prospect of an observer interested to optimise the signal-to-noise ratio in the reception of the backward radiation coming from a surface illuminated by an electromagnetic wave with a wavelength chosen to minimize the diffuse scattering so that he has just to point his receiver in the direction of the coherent reflection. Then, to analyse the coherent reflection for harmonic plane waves impinging on a dielectric infinite film deposited on a metallic substrate we develop a formalism generalizing the customary angular spectrum representation used to tackle this kind of problem. This new approach whose efficiency is proved in the easier situation of a dielectric film endowed with an impedance, is used to get the coherent reflection from a structured 1D-dielectric film illuminated by TE and TM electromagnetic plane waves when the rough amplitude h is small enough to justify 0(h ²) approximations. The Idemen technique is used to get the boundary conditions needed to tackle these scattering problems.     

Global smooth solutions of the compressible Navier-Stokes equations with density-dependent viscosity

In this paper we study a free boundary problem for the viscous, compressible, heat conducting, one-dimensional real fluids. More precisely, the viscosity is assumed to be a power function of density, i.e., μ(ρ)=ρα, where ρ denotes the density of fluids and α is a positive constant. In addition, the equations of state include and are more general than perfect flows which only depend linearly on temperature. The global existence (uniqueness) of smooth solutions is established with for general, large initial data, which improves the previous results. Moreover, it is also shown that the solutions will not develop vacuum, mass concentration or heat concentration in a finite time provided the initial data are bounded and smooth, and do not contain vacuum.     

Free boundary value problem for a viscous two-phase model with mass-dependent viscosity

In this paper, we study a free boundary value problem for two-phase liquid-gas model with mass-dependent viscosity coefficient when both the initial liquid and gas masses connect to vacuum with a discontinuity. This is an extension of the paper [S. Evje, K.H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, http://www.irisresearch.no/docsent/emp.nsf/wvAnsatte/SEV]. Just as in [S. Evje, K.H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, http://www.irisresearch.no/docsent/emp.nsf/wvAnsatte/SEV], the gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid. We give the proof of the global existence and uniqueness of weak solutions when β∈(0,1], which have improved the previous result of Evje and Karlsen, and get the asymptotic behavior result, also we obtain the regularity of the solutions by energy method.     

Stability and linear oscillations of liquid bridges with varied boundary conditions under zero gravity

This article deals with stability and small linear oscillations of liquid bridges between fixed solid surfaces (parallel plates, spheres, ...) under zero gravity. A general investigation method for any kind of axisymmetric liquid bridge is exposed but the author focus his work on the spherical liquid bridge cases. In particular, he exposes a full theoretical study of spherical liquid bridges between two spheres, plates and cones.     

A blow-up criterion for a 2D viscous liquid-gas two-phase flow model

In this paper, we prove a blow-up criterion in terms of the upper bound of the liquid mass for the strong solution to the two-dimensional (2D) viscous liquid-gas two-phase flow model in a smooth bounded domain. The result also applies to three-dimensional (3D) case.     

A boundary layer problem arising in gravity-driven laminar film flow of power-law fluids along vertical walls

A rigorous mathematical analysis is given for a boundary layer problem for a third-order nonlinear ordinary differential equation, which arises in gravity-driven laminar film flow of power-law fluids along vertical walls. Firstly, the problem is transformed into a singular nonlinear two-point boundary value problem of second order. Next, the latter is proved to have a unique positive solution, for which some estimates are established. Finally, the result above-mentioned is turned over to the original problem. The conclusion of this paper is that the boundary layer problem has a unique normal solution if the power-law index is less than or equal to one and a generalized normal solution if the power-law index is greater than one. Also the asymptotic behavior of the normal solution at the infinity is displayed.The work was supported by NNSF of China.     

Velocity of the fourth sound in liquid helium II via extended thermodynamics

This work continues a study begun in previous works, where, using Extended Thermodynamics, a monofluid model of liquid helium II is formulated. The wave propagation in bulk liquid helium II is studied in the hypothesis that the thermal dilatation is not zero. The propagation of fourth sound, studied previously neglecting both the thermal dilatation and finite volume of the powder, is studied without these simplified hypotheses: a scattering correction n is introduced to take into account the porosity. The model is more general than the standard two-fluid model because it allows that a small amount of entropy is associated with helium when it flows through a very thin capillary or a porous medium. A comparison with experimental data is performed. From experimental values for velocities and attenuations of the two sounds in bulk liquid helium, the model provides the velocity of fourth sound in a porous medium. These values are determined at various temperatures and pressures and compared with fourth sound measurements in a packed powder.     

On Stokes flow with variable and degenerate surface tension coefficient

Short-time existence, uniqueness, and regularity results are shown for the moving boundary problem of a free drop of liquid governed by the Stokes equations and driven by surface tension. The value of the surface tension coefficient is variable, not necessarily strictly positive, and transported with the flow on the moving surface.By a perturbation of identity approach, the problem is transformed into a nonlinear, nonlocal first order degenerate parabolic evolution equation on a fixed reference manifold. Its solvability is proved by deriving a priori estimates and using Galerkin approximations.     

Global weak solutions for a viscous liquid–gas model with transition to single-phase gas flow and vacuum

This work deals with a viscous two-phase liquid–gas model relevant to the flow in wells and pipelines. The liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic. The model is rewritten in terms of Lagrangian coordinates and is studied in a free boundary setting where the liquid and gas masses are of compact support initially, and continuous at the boundary. Consequently, the initial masses involve a transition to single-phase gas flow and vacuum at the boundary. An appropriate balance between pressure and viscous forces is identified which allows obtaining pointwise upper and lower estimates of masses. These estimates rely on the assumption of a certain relation between the rate of degeneracy of the viscosity coefficient and the rate that determines how fast the initial masses are vanishing at the boundary. By combining these estimates with basic energy type of estimates, higher order regularity estimates are obtained. The existence of global weak solutions is then proved by showing compactness for a class of semi-discrete approximations.     

Flow past a porous approximate spherical shell

In this paper, the creeping flow of an incompressible viscous liquid past a porous approximate spherical shell is considered. The flow in the free fluid region outside the shell and in the cavity region of the shell is governed by the Navier–Stokes equation. The flow within the porous annulus region of the shell is governed by Darcy’s Law. The boundary conditions used at the interface are continuity of the normal velocity, continuity of the pressure and Beavers and Joseph slip condition. An exact solution for the problem is obtained. An expression for the drag on the porous approximate spherical shell is obtained. The drag experienced by the shell is evaluated numerically for several values of the parameters governing the flow.     

固体平板上超薄水膜去湿润过程的分子动力学模拟

运用分子动力学模拟研究固体表面超薄水膜的失稳和破裂过程．结果表明薄膜中小扰动将失稳,并在初始阶段线性增长．但固体和液体的相互作用对扰动的初期增长影响较小．最小厚度的下降导致薄膜发生破裂．此后破裂边缘以一定的动态接触角后退．与宏观理论预测一致,边缘半径随时间的变化与时间平方根成正比．若固液相互作用较强,将引起破裂时间延迟,动态接触角减小,固体表面附近的液体密度增加．     

A justification for the thin film approximation of Stokes flow with surface tension

In the free boundary problem of Stokes flow driven by surface tension, we pass to the limit of small layer thickness. It is rigorously shown that in this limit the evolution is given by the well-known thin film equation. The main techniques are appropriate scaling and uniform energy estimates in Sobolev spaces of sufficiently high order, based on parabolicity.     

Stokes flow inside a porous spherical shell: Stress jump boundary condition

An arbitrary Stokes flow of a viscous, incompressible fluid inside a sphere with internal singularities, enclosed by a porous spherical shell, using Brinkmans equation for the flow in the porous region is discussed. At the interface of the clear fluid and porous region stress jump boundary condition for tangential stresses is used. The drag and torque are found by deriving the corresponding Faxens laws. It is found that drag and torque not only change with the varying permeability, but also change for different values of stress jump coefficient. Critical permeability is found for which drag and torque change their behavior. As a limiting case the corresponding Faxens laws for the rigid spherical shell with internal singularities has been obtained.Received: December 17, 2002; revised: February 3, 2004     

Predictions of the gas–liquid flow in wet electrostatic precipitators

Based on Computational Fluid Dynamics (CFD), the present paper aims to simulate several important phenomena in a wet type ESP from the liquid spray generation to gas-droplet flow in electric field. A single passage between the adjacent plates is considered for the simulation domain. Firstly, the electric field intensity and ion charge density are solved locally around a corona emitter of a barbed wire electrode, which are applied to the entire ESP using periodic conditions. Next, the Euler–Lagrange method is used to simulate the gas-droplet flow. Water droplets are tracked statistically along their trajectories, together with evaporation and particle charging. Finally, the deposition density on the plate is taken as the input for the liquid film model. The liquid film is simulated separately using the homogenous Eulerian approach in ANSYS-CFX. In the current case, since the free surface of the thin water film is difficult to resolve, a special method is devised to determine the film thickness.As parametric study, the variables considered include the nozzle pressure, initial spray spreading patterns (solid versus hollow spray) and plate wettability. The droplet emission rate and film thickness distribution are the results of interest. Main findings: electric field has strong effect on the droplet trajectories. Hollow spray is preferred to solid spray for its lower droplet emission. The liquid film uniformity is sensitive to the plate wettability.     

Global well-posedness of incompressible flow in porous media with critical diffusion in Besov spaces

In this paper we study the model of heat transfer in a porous medium with a critical diffusion. We obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in Besov spaces with 1?p?∞ by the method of modulus of continuity and Fourier localization technique.     

Modeling of van der Waals force with smoothed particle hydrodynamics: Application to the rupture of thin liquid films

The rupture of thin liquid films driven by the van der Waals force is of significance in many engineering processes, and most previous studies have relied on the lubrication approximation. In this paper, we develop a smoothed particle hydrodynamics (SPH) representation for the van der Waals force and simulate the rupture of thin liquid films without resort to lubrication theory. The van der Waals force in SPH is only imposed on one layer, i.e., the outermost layer of fluid particles, where a weighting function is deployed to evaluate the contributions of particles on or near the interface. However, to obtain an accurate hydrostatic pressure in reaction to the van der Waals force, a smaller smoothing length is used for the calculation of the weighting function than that used for SPH discretizations of the bulk fluid. The same surface particles are also used to model the surface tension. To deal with the rupture of a thin liquid film with a very small aspect ratio ε (ε = thickness/length), a coordinate transformation is introduced to shrink the length of the liquid film to achieve accurate numerical resolution with a manageable number of particles. As verifications of our physical model and numerical algorithm, we simulate the hydrostatic pressure in a stationary film and the relaxation of an initially square droplet and compare the SPH results with the analytical solutions. The method is then applied to simulate the rupture of thin liquid films with moderate and small aspect ratios (ε = 0.5 and 0.005). The convergence of the method is verified by refining particle spacing to four different levels. The effect of the capillary number on the rupture process is analyzed.     

The asymptotics of neutral curve crossing in Taylor–Dean flow

The fluid flow between a pair of coaxial circular cylinders generated by the uniform rotation of the inner cylinder and an azimuthal pressure gradient is susceptible to both Taylor and Dean type instabilities. The flow can be characterised by two parameters: a measure of the relative magnitude of the rotation and pressure effects and a non-dimensional Taylor number. This work considers the small gap, large wavenumber limit for linear perturbations when the onset of the Taylor and Dean instabilities is concurrent. A consistent, matched asymptotic solution is found across the whole annular domain and identifies five regions of interest: two boundary adjustment regions and three internal critical points. Necessary conditions for the Taylor number and wavenumber are found and interpreted with reference to the suggestion of neutral curve kinks occurring at moderate wavelengths. Received: October 21, 2003; revised: November 11, 2004