Effect of pulsations on two-layer channel flow

The effect of vertical wall vibrations on two-phase channel flow is examined. The basic flow consists of two superposed fluid layers in a channel whose walls oscillate perpendicular to themselves in a prescribed, time-periodic manner. The solution for the basic flow is presented in closed form for Stokes flow, and its stability to small periodic perturbations is assessed by means of a Floquet analysis. It is found that the pulsations have a generally destabilizing influence on the flow. They tend to worsen the Rayleigh–Taylor instability present for unstably stratified fluids; the larger the amplitude of the pulsations, the greater the range of unstable wave numbers. For stably stratified fluids, the pulsations raise the growth rate of small perturbations, but are not sufficient to destabilize the flow. In the latter part of the paper, the basic flow for arbitrary Reynolds number is computed numerically assuming a flat interface, and the motion of the interface in time is predicted. The existence of a time-periodic flow is demonstrated in which the ratio of the layer thicknesses remains constant throughout the motion.     

Stretching of a liquid layer underneath a semi-infinite fluid

Exact similarity solutions of the Navier–Stokes equation are derived describing the flow of a liquid layer coated on a stretching surface underneath another semi-infinite fluid. In the absence of hydrodynamic instability, the interface remains flat as the layer thickness decreases in time. When the physical properties of the fluids are matched, we obtain Crane’s analytical solution for two-dimensional (2D) flow and a corresponding numerical solution for axisymmetric flow. When the rate of stretching of the surface is constant in time, the temporal evolution of the interface between the layer and the overlying fluid is computed by integrating in time a system of coupled partial differential equations for the velocity in each fluid together with an ordinary differential equation expressing kinematic compatibility at the interface, subject to appropriate boundary, interfacial, and far-field conditions. Multiple solutions are found in certain ranges of the density and viscosity ratios. Additional similarity solutions are presented for accelerated 2D and axisymmetric stretching. The numerical prefactors that appear in the analytical expressions for the interface location and wall shear stress are presented for different ratios of the densities and viscosities of the two fluids.     

Explosive collapse of vapor film under conditions of intensive heat fluxes

The stability is investigated (linear and nonlinear analysis) of the interface between a thin vapor film and a layer of liquid in the presence of a steady heat flux from a metal surface heated to a high temperature to the vapor film and then from vapor to subcooled liquid. In view of thermal disequilibrium which takes into account the temperature dependence of saturation pressure, boundary conditions on the vapor-liquid interface are derived, which generalize the known correlations on the free surface of liquid in the gravity field. A number of new effects arise in the problem under consideration, as distinct from the classical problem. The thermal processes, which occur on the phase boundary and are possible in the absence of the force of gravity as well, lead to the generation of weakly decaying periodic waves of low amplitude, whose velocity may exceed significantly that of gravity waves. The heat flux through the interface may cause on this surface periodic waves of small length (ripple) which are not capillary. The processes of phase transition on the interface are capable of providing for the stability of vapor film under the layer of liquid in the gravity field. Along with periodic waves and solitons, the mode of explosive instability may arise in the nonlinear stage because of a weak variation of the film thickness, where the amplitude of an initially low-amplitude plane wave rises to infinity during a finite period of time.     

Surface tension effects on the nonlinear behavior of long waves in a two layer flow

 The propagation of long waves of finite amplitude at the interface of two viscous fluids in the presence of interfacial tension is examined. The effect of capillarity on the shape of the waves at the interface of two superposed fluids is investigated for a wide range of density differences, viscosity ratios and imposed pressure gradients. It is found that in planar geometry surface tension stabilizes the interfacial disturbances. Attention is given to the case in which the upper fluid is more dense and comprises a thin film above the lower fluid. With the heavier fluid on the top the flow pattern is always unstable when surface tension effects are neglected. In this case the interfacial waves do not grow forever and reach a finite amplitude only when the interfacial tension is greater than a critical value.     

Stability of Plane-Parallel Pulsational Flow of Two Miscible Fluids under High Frequency Horizontal Vibrations

Linear stability of plane-parallel pulsational flow of two miscible fluids in a horizontal layer subjected to high frequency horizontal vibrations is investigated neglecting viscosity and diffusion. Long-wave instability is studied analytically and instability to the perturbations with finite wavelength—numerically.     

Frequency-Dependent Elongational Viscosity by Nonequilibrium Molecular Dynamics

The elongational viscosity of a liquid describes the response of the liquid to simultaneous stretching and compression in various directions, subject to the restriction that the trace of the rate of the strain tensor is zero (or the density is constant). Despite the growing popularity and usefulness of nonequilibrium molecular dynamics methods in studies of the shear viscosity of simple and complex fluids, the elongational viscosity remains a relatively neglected property in computer simulation studies. This stems from some significant technical difficulties that arise when standard methods such as the constant strain rate SLLOD algorithm are applied to elongational flow. For example, if planar elongational flow with a constant elongation rate is applied in a computer simulation with periodic boundary conditions, the box size in the contracting direction quickly becomes smaller than twice the range of the potential, violating the minimum image convention. The time at which this occurs may be less than the time required for the system to reach a steady state, making it impossible to compute the steady-state elongational viscosity. This difficulty can be avoided by applying an oscillating elongational strain rate to the liquid, and computing frequency dependent elements of the stress tensor, which can then be extrapolated to zero frequency to evaluate the steady-state elongational viscosity. We have used this method to compute the elongational viscosity of a simple atomic liquid and discuss its possible application to a model polymeric liquid.     

Interfacial capillary–gravity waves due to a fundamental singularity in a system of two semi-infinite fluids

The interfacial capillary–gravity waves due to a transient fundamental singularity immersed in a system of two semi-infinite immiscible fluids of different densities are investigated analytically for two- and three- dimensional cases. The two-fluid system, which consists of an inviscid fluid overlying a viscous fluid, is assumed to be incompressible and initially quiescent. The two fluids are each homogeneous, and separated by a sharp and stable interface. The Laplace equation is taken as the governing equation for the inviscid flow, while the linearized unsteady Navier–Stokes equations are used for the viscous flow. With surface tension taken into consideration, the kinematic and dynamic conditions on the interface are linearized for small-amplitude waves. The singularity is modeled as a simple mass source when immersed in the inviscid fluid above the interface, or as a vertical point force when immersed in the viscous fluid beneath the interface. Based on the integral solutions for the interfacial waves, the asymptotic wave profiles are derived for large times with a fixed distance-to-time ratio by means of the generalized method of stationary phase. It is found that there exists a minimum group velocity, and the wave system observed will depend on the moving speed of the observer. Two schemes of expansion of the phase function are proposed for the two cases when the moving speed of an observer is larger than, or close to the minimum group velocity. Explicit analytical solutions are presented for the long gravity-dominant and the short capillary-dominant wave systems, incorporating the effects of density ratio, surface tension, viscosity and immersion depth of the singularity.     

Flows induced by the surface motions of a cylindrical sheet

Fluid flows induced by the surface stretching or shearing motion of cylindrical sheets are investigated. Steady and unsteady exact solutions of the Navier–Stokes equations are found for periodic axial shearing of an impermeable sheet and for periodic azimuthal stretching of a permeable sheet. Steady Stokes-flow solutions induced by the periodic axial stretching and the periodic azimuthal stretching of impermeable cylindrical sheets are also reported. In each case flows interior and exterior to a cylinder are considered, as well as the flow in the annulus between concentric cylinders.     

Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis

The effect of inertia on the Yih–Marangoni instability of the interface between two liquid layers in the presence of an insoluble surfactant is assessed for shear-driven channel flow by a normal-mode linear stability analysis. The Orr–Sommerfeld equation describing the growth of small perturbations is solved numerically subject to interfacial conditions that allow for the Marangoni traction. For general Reynolds numbers and arbitrary wave numbers, the surfactant is found to either provoke instability or significantly lower the rate of decay of infinitesimal perturbations, while inertial effects act to widen the range of unstable wave numbers. The nonlinear evolution of growing interfacial waves consisting of a special pair of normal modes yielding an initially flat interface is analysed numerically by a finite-difference method. The results of the simulations are consistent with the predictions of the linear theory and reveal that the interfacial waves steepen and eventually overturn under the influence of the shear flow.     

Nonstationary effect of artificial perturbations on a nonisothermal falling liquid film

The effect of nonstationary artificial perturbations on the formation of rivulets on the surface of a nonisothermal liquid film flowing down a vertical plate with a heater has been experimentally studied. The film thickness was measured using a fluorescent technique. It is demonstrated that, using periodic perturbations, it is possible to change the distance between rivulets with time. An increase in the wave amplitude at the front of a propagating perturbation has been observed. It is established that the wave amplitude growth and the liquid mass transfer across the flow lead to the washing of dry spots downstream the heater.     

Effects of an unsteady rotation on the electrohydrodynamic stability of a cylindrical interface

The effect of a periodic rotation on two dielectric inviscid fluids separated by a cylindrical interface is studied. The system is influenced by a constant perpendicular electric field. The model allows for general forms of deformations of the interface. The standard normal modes approach is utilized. The amplitude of the periodic rotation is considered as a smallness parameter. The method of multiple time scales is used to achieve the stability of the problem. Fourier series approach is used to solve the equations of motion. The solutions are obtained in terms of modified Bessel functions. A transcendental dispersion relation is obtained in zero-order perturbations. Several special cases are discussed. A generalization to Rayleigh's theorem is obtained. The solvability condition, in first order perturbations, is derived. It is found that the resonance regions may appear due to the periodicity of the rotation. According to the periodicity the electric field has change its mechanism at the resonance case.     

Effect of artificial perturbations on the formation of structures in a nonisothermal liquid film

The effect of artificial perturbations on the formation of structures in a nonisothermal liquid film flowing down a vertical plate with a 150 × 150 mm heater has been experimentally studied. The action of heat flux on the wave flow leads to the formation of periodic flowing rivulets separated by thin film regions. Artificial perturbations in a certain interval of wavelengths produce a change in the number of rivulets formed on the heater surface. The “most dangerous” wavelength of artificial perturbations altering the flow structure is determined.     

Mechanism of stabilization of location of a droplet cluster above the liquid-gas interface

We studied the influence of sizes of droplets, forming the ??droplet cluster?? dissipative structure, on their levitation height in the vapor-air flow, which appears when free surface of horizontal water layer is locally heated. A sharp decrease in the velocity of the vapor-air flow takes place at a distance from the surface comparable with the droplet diameter. Allowing for the aerodynamic nature of the droplet levitation, this peculiarity of the flow determines the high stability of location of the droplet cluster above the interface. Existence of droplets that are anomalously heavy in the slope of the Stokes levitation mechanism is described.     

Stability of sessile and pendant liquid drops

The solution space of axisymmetric liquid drops attached to a horizontal plane is investigated, and the stability of hydrostatic shapes is assessed by a novel numerical linear stability analysis involving discrete perturbations. For a given contact angle and Bond number, multiple interfacial shapes exist with compact, lightbulb, hourglass, and more convoluted pearly shapes. It is found that more than one solution branch can be stable, and that negative curvature at the contact line of a pendant drop is not a prerequisite for instability. Numerical simulations based on the boundary-integral method for Stokes flow illustrate the process of unstable drop detachment. Unstable drops transform into elongated threads with a spherical head whose volume is determined by a Bond number expressing the significance of surface tension. A complementary investigation of the shape and stability of two-dimensional drops attached to a horizontal or inclined plane reveals that hydrostatic shapes are least stable in the inclined configuration and most stable in the pendant or sessile configuration.     

Effect of long undulated bottoms on thin gravity-driven films

Summary We carry out a perturbation analysis for steady gravity-driven film flow over undulations of moderate steepness that are long compared to the film thickness and study the linear stability of the flow in the framework of Floquet analysis. The effect of geometric nonlinearities on the instability becomes relevant for moderate bottom variations. We find that the critical Reynolds number for the onset of surface waves is higher than that for a flat bottom. At higher inclination angles, the theoretical results are in good quantitative agreement with experiment. At inclination angles where the flat part of the undulation is close to being horizontal, the basic solution for the steady flow fails to describe the flow in the flat part, and the linear stability analysis overestimates the critical Reynolds number.     

Gravity-driven film flow down an inclined wall with three-dimensional corrugations

Summary The gravity-driven flow of a liquid film down an inclined wall with three-dimensional doubly periodic corrugations is investigated in the limit of vanishing Reynolds number. The film surface may exhibit constant or variable surface tension due to an insoluble surfactant. A perturbation analysis for small-amplitude corrugations is performed, wherein the wall geometry is expressed as a Fourier series consisting of a linear superposition of two-dimensional oblique waves defined by two base vectors. Each of the constituent perturbation flows over the individual oblique waves is further decomposed into a two-dimensional flow transverse to the oblique waves and a unidirectional flow parallel to the waves. Both the transverse and the parallel flow are calculated by carrying out an analysis in oblique coordinates, similar to that conducted for two-dimensional flow. The particular cases of flow down a wall with oblique two-dimensional, orthogonal three-dimensional, and hexagonal three-dimensional corrugations are considered. The results illustrate the surface velocity field and the distribution of the surfactant. The three-dimensional wall geometry is found to reduce the surface deformation with respect to its two-dimensional counterpart by increasing the effective wave numbers and decreasing the effective capillary number encapsulating the effect of surface tension.     

Development of artificial perturbations in a nonisothermal liquid film

The effect of artificial perturbations on the formation of structures in a water film flowing over a vertical plate with a heater has been experimentally investigated. Attention has been focused on the study of the propagation of nonstationary perturbations in time immediately after their creation on the surface of the liquid film. The fluorescence method has been used to measure the film thickness field. The temperature field on the film surface has been measured by a high-speed infrared scanner. It has been shown that the passage of the wave front after the action of a perturbation system with the “most dangerous” distance between the cylinders leads to a change in the distance between the rivulets. It has been found that the intensity of introduced perturbations (cylinder diameters) and the characteristics of waves flowing to the heater significantly affect the change in the distance between the rivulets. An increase in the amplitude of the waves at the front of propagating perturbation has been detected. The coalescence of the rivulets in the lower part of the heater has been observed. It has been shown that the passage of several wave fronts is necessary for the complete rearrangement of the flow even when the intensity of perturbations is high.     

An adaptive fixed mesh method for modelling and simulating of unsteady two‐phase flows with sharp physical interfaces

We present in this paper a new computational method for simulation of two‐phase flow problems with moving boundaries and sharp physical interfaces. An adaptive interface‐capturing technique (ICT) of the Eulerian type is developed for capturing the motion of the interfaces (free surfaces) in an unsteady flow state. The adaptive method is mainly based on the relative boundary conditions of the zero pressure head, at which the interface is corresponding to a free surface boundary. The definition of the free surface boundary condition is used as a marker for identifying the position of the interface (free surface) in the two‐phase flow problems. An initial‐value‐problem (IVP) partial differential equation (PDE) is derived from the dynamic conditions of the interface, and it is designed to govern the motion of the interface in time. In this adaptive technique, the Navier–Stokes equations written for two incompressible fluids together with the IVP are solved numerically over the flow domain. An adaptive mass conservation algorithm is constructed to govern the continuum of the fluid. The finite element method (FEM) is used for the spatial discretization and a fully coupled implicit time integration method is applied for the advancement in time. FE‐stabilization techniques are added to the standard formulation of the discretization, which possess good stability and accuracy properties for the numerical solution. The adaptive technique is tested in simulation of some numerical examples. With the test problems presented here, we demonstrated that the adaptive technique is a simple tool for modelling and computation of complex motion of sharp physical interfaces in convection–advection‐dominated flow problems. We also demonstrated that the IVP and the evolution of the interface function are coupled explicitly and implicitly to the system of the computed unknowns in the flow domain. Copyright © 2005 John Wiley & Sons, Ltd.     

General slow viscous flows in a two-fluid system

Summary A general solution of the creeping flow equations suitable for a flow that is bounded by a nondeforming planar interface is presented. New compact representations for the velocity and pressure fields are given in terms of two scalar functions which describe arbitrary Stokes flow. A general reflection theorem is derived for a fluid-fluid interface problem containing Lorentz reflection formula as a particular case. The theorem allows a better interpretation of the image system for various singularities in the presence of a planar interface. The general solution is further used to describe the first-order approximation of the deformed interface by performing normal stress balance. It is found that the normal stress imbalance and the interface displacement are independent of the viscosity ratio of two fluids (!) and only depend on the location of initial singularity.