Stabilizing effect of diffusion in enhanced oil recovery and three-layer Hele-Shaw flows with viscosity gradient

In the presence of diffusion, stability of three-layer Hele-Shaw flows which models enhanced oil recovery processes by polymer flooding is studied for the case of variable viscosity in the middle layer. This leads to the coupling of the momentum equation and the species advection-diffusion equation the hydrodynamic stability study of which is presented in this paper. Linear stability analysis of a potentially unstable three-layer rectilinear Hele-Shaw flow is used to examine the effects of species diffusion on the stability of the flow. Using a weak formulation of the disturbance equations, upper bounds on the growth rate of individual disturbances and on the maximal growth rate over all possible disturbances are found. Analytically, it is shown that a short-wave disturbance if unstable can be stabilized by mild diffusion of species, where as an unstable long-wave disturbance can always be stabilized by strong diffusion of species. Thus, an otherwise unstable three-layer Hele-Shaw flow can be completely stabilized by a large enough diffusion, i.e., by increasing enough the magnitude of the species diffusion coefficient. The magnitude of this diffusion coefficient required to completely stabilize the flow will depend on the magnitude of interfacial viscosity jumps and the viscosity gradient of the basic viscous profile of the middle layer.     

The instability mechanism of single and multilayer Newtonian and viscoelastic flows down an inclined plane

The instability mechanism of single and multilayer flow of Newtonian and viscoelastic fluids down an inclined plane has been examined based on a rigorous energy analysis as well as careful examination of the eigenfunctions. These analyses demonstrate that the free surface instability in single and multilayer flows in the limit of longwave disturbances (i.e., the most dangerous disturbances) arise due to the perturbation shear stresses at the free surface. Specifically, for viscoelastic flows, the elastic forces are destabilizing and the main driving force for the instability is the coupling between the base flow and the perturbation velocity and stresses and their gradient at the free surface. For Newtonian flows at finite Re, the driving force for the interfacial instability in the limit of longwaves depends on the placement of the less viscous fluid. If the less viscous fluid is adjacent to the solid surface then the main driving force for the instability is interfacial friction, otherwise the bulk contribution of Reynolds stresses drives the instability. For viscoelastic fluids in the limit of vanishingly small Re, the driving force for the instability is the coupling of the base flow and perturbation velocity and stresses and their gradients across the interface. In the limit of shortwaves the interfacial stability mechanism of flow down inclined plane is the same as plane Poiseuille flows (Ganpule and Khomami 1998, 1999a, b). Received: 20 October 2000/Accepted: 11 January 2001     

The effect of transient viscoelastic properties on interfacial instabilities in superposed pressure driven channel flows

In a recent study, Ganpule and Khomami (submitted to J. Non-Newtonian Fluid Mech.) have shown that in order to accurately describe the experimentally observed interfacial instability phenomenon in superposed channel flow of viscoelastic fluids, a constitutive equation that can accurately depict not only the steady viscometric properties of the experimental test fluids, but also their transient viscoelastic properties must be used in the analysis. In the present study, the effect of differences in transient viscoelastic properties which can arise either due to the differences in the predictive capabilities of various constitutive models or from the presence of multiple modes of relaxation on the interfacial instabilities of the superposed pressure driven channel flows has been investigated. Specifically, a linear stability analysis is performed using nonlinear constitutive equations which predict identical steady viscometric properties but different transient viscoelastic properties. It is shown that different nonlinear constitutive equations give rise to the same mechanism of interfacial instability, but the boundaries of the neutral stability contours and the magnitudes of the growth/decay rates, especially at intermediate and shortwaves, are shifted due to the overshoots in the transient viscoelastic responses predicted by the constitutive equations. In addition, the effect of the presence of multiple modes of relaxation on interfacial stability is studied using single and multiple mode upper convected Maxwell (UCM) fluids and it is shown that pronounced differences in the intermediate and shortwave linear stability predictions arise due to the fact that the increase in the number of modes gives rise to additional fast as well as slow relaxation modes and the presence of these additional relaxation modes gives rise to differences in the transient viscoelastic response of the fluids in the absence of any overshoots. The effect of fluid inertia on the interfacial stability of viscoelastic liquids is examined and it is shown that at longwaves, inertia has a pronounced effect on the stability of the interface, whereas at shortwaves, elastic and viscous effects dominate. Furthermore, the mechanism of viscoelastic interfacial instabilities is studied by a careful examination of disturbance eigenfunctions as well as performing a disturbance energy analysis. The results indicate that the mechanism of viscoelastic interfacial instabilities can be described in terms of interaction of mechanisms of purely viscous and purely elastic instabilities. However, since more than one mechanism for the instability is at work, the disturbance energy analysis can not clearly distinguish between them due to the fact that the eigenfunctions used in the energy analysis contain the information regarding both viscous and elastic effects. Hence, the mechanism of the instability must be determined by a careful examination of disturbance eigenfunctions.     

Irreversible thermodynamics of liquid crystal interfaces

A macroscopic theory for the dynamics of isothermal compressible interfaces between nematic liquid crystalline polymers and isotropic viscous fluids has been formulated using classical irreversible thermodynamics. The theory is based on the derivation of the interfacial rate of entropy production for ordered interfaces, that takes into account interfacial anisotropic viscous dissipation as well as interfacial anisotropic elastic storage. The symmetry breaking of the interface provides a natural decomposition of the forces and fluxes appearing in the entropy production, and singles out the symmetry properties and tensorial dimensionality of the forces and fluxes. Constitutive equations for the surface extra stress tensor and for surface molecular field are derived, and their use in interfacial balance equations for ordered interfaces is identified. It is found that the surface extra stress tensor is asymmetric, since the anisotropic viscoelasticity of the nematic phase is imprinted onto the surface. Consistency of the proposed surface extra stress tensor with the classical Boussinesq constitutive equation appropriate to Newtonian interfaces is demonstrated. The anisotropic viscoelastic nature of the interface between nematic polymers (NPs) and isotropic viscous fluids is demonstrated by deriving and characterizing the dynamic interfacial tension. The theory provides for the necessary theoretical tools needed to describe the interfacial dynamics of NP interfaces, such as capillary instabilities, Marangoni flows, wetting and spreading phenomena.     

Instability of the interface in two-layer flows with large viscosity contrast at small Reynolds numbers

The Kelvin–Helmholtz instability is believed to be the dominant instability mechanism for free shear flows at large Reynolds numbers. At small Reynolds numbers, a new instability mode is identified when the temporal instability of parallel viscous two fluid mixing layers is extended to current-fluid mud systems by considering a composite error function velocity profile. The new mode is caused by the large viscosity difference between the two fluids. This interfacial mode exists when the fluid mud boundary layer is sufficiently thin. Its performance is different from that of the Kelvin–Helmholtz mode. This mode has not yet been reported for interface instability problems with large viscosity contrasts.These results are essential for further stability analysis of flows relevant to the breaking up of this type of interface.     

On Stabilization of Multi-Layer Hele-Shaw and Porous Media Flows in the Presence of Gravity

Stabilization of multi-layer Hele-Shaw flows is studied here by including the influence of Rayleigh?CTaylor instability in our earlier work (Daripa, J. Stat. Mech. 12:28, 2008a) on stabilization of multi-layer Saffman?CTaylor instability. Furthermore, this article goes beyond our previous work with few extensions, improvements, new interpretations, and clarifications on the use of some terminologies. Results of two complete studies have been presented: the first investigates the effect of individually unstable interfaces on the overall stability of the flow, and the second studies the cumulative effect of unstable interfaces as well as unstable internal viscous layers. In each case, modal and absolute upper bounds on the growth rate are reported. Next, these bounds are used to investigate (i) stabilization of long waves on various interfaces; (ii) stabilization of all waves on all interfaces in comparison to pure Taylor instability; (iii) stabilization of disturbances on interior interfaces instead of exterior interfaces. In the first study, notions of partial and total stabilization with respect to the pure Taylor growth rate are introduced. Then necessary and sufficient conditions for partial and total stabilizations are found. Proof of stabilization of long waves on one of the two external interfaces in multi-layer flows is also proved. In the second study, an absolute upper bound is obtained in the presence of stabilizing density stratification across each internal interface even though all interfaces and layers have unstable viscous profiles. Exact results on the upper bounds, and necessary and sufficient conditions for control of instabilities driven by stable/unstable density stratification, unstable viscous layers and unstable interfaces are new and may be relevant to explain observed phenomena in many complex flows generating these kinds of viscous profiles and density stratification as they evolve. The present work builds upon and goes much further in details and new results than our previous work. The gravity effect included here brings with it restrictions which have not been addressed before in this multi-layer context.     

Instability through porous media of three layers superposed conducting fluids

The instability properties of streaming superposed conducting fluids through porous media under the influence of uniform magnetic field have been investigated. The system is composed of a middle fluid sheet of finite thickness embedded between two semi-infinite fluids. The fluids are assumed to be incompressible, perfectly conducting and there are weak viscous stresses on the interfaces. The Rayleigh–Taylor and Kelvin–Helmholtz problems have been studied. Such configurations are of relevance in a variety of astrophysical and space configurations. The solutions of the linearized equations of motion together with the boundary conditions lead to deriving the dispersion equation with complex coefficients. The limiting case of the stability of one interface between two fluids has been discussed. The stability criteria are discussed theoretically and numerically in which stability diagrams are obtained. It has been found that the increase of the viscosity coefficient as well as the porosity plays a regular stabilizing role in the stability behavior, while the increase of the fluid velocity plays a destabilizing influence in the stability criteria.     

Stability Analysis of a Falling Film Flow Down a Plane with Sinusoidal Corrugations

The liquid viscous film falling down a vertical wall with sinusoidal relief is considered. The linear stability of steady-state flow with respect to time-periodic disturbances is studied using the Floquet theory. It is shown that in the case of applying corrugations the variation in the disturbance growth rate is proportional to the second power of their undulations. Depending on the relief parameters there exist two possibilities: the instability domain can expand or certain disturbances can be stabilized. The growth rates are obtained numerically and analytically in the approximation of low-amplitude corrugations. The development of waves from small disturbances is simulated within the framework of nonlinear equations and the formation of structures whose wavelength is significantly greater than the space relief period is found out.     

Viscosity-stratified flow in a Hele–Shaw cell

A hierarchy of mathematical models describing viscosity-stratified flow in a Hele-Shaw cell is constructed. Numerical modelling of jet flow and development of viscous fingers with the influence of inertia and friction is carried out. One-dimensional multi-layer flows are studied. In the framework of three-layer flow the interpretation of the Saffman–Taylor instability is given. A kinematic-wave model of viscous fingering taking into account friction between the fluid layers is proposed. Comparison with calculations on the basis of two-dimensional equations shows that this model allows to determine the propagation velocity of the viscous fingers.     

Viscous and Inviscid Instabilities of Flow Along a Streamwise Corner

Here we consider the stability of flow along a streamwise corner formed by the intersection of two large flat plates held perpendicular to each other. Self-similar solutions for the steady laminar mean flow in the corner region have been obtained by solving the boundary layer equations for zero and nonzero streamwise pressure gradients. The stability of the mean flow is investigated using linear stability analysis. An eigensolver has been developed to solve the resulting linear eigenvalue problem either in a global mode to obtain an approximation to all the dominant eigenmodes or in a local mode to refine a particular eigenmode. The stability results indicate that the entire spectrum of two-dimensional and oblique viscous modes of a two-dimensional Blasius boundary layer is active in the case of a corner layer as well, but away from the cornerline. In a corner region of finite spanwise extent, the continuous spectrum of oblique modes degenerates to a discrete spectrum of modes of increasing spanwise wave number. The effect of the corner on the two-dimensional viscous instability is small and decreases the growth rate. The growth rate of outgoing oblique disturbances is observed to decrease, while the growth rate of incoming oblique disturbances is enhanced by the corner. This asymmetry between the outgoing and incoming viscous modes increases with increasing obliqueness of the disturbance. The instability of a zero pressure gradient corner layer is dominated by the viscous modes; however, an inviscid corner mode is also observed. The critical Reynolds number of the inviscid mode rapidly decreases with even a small adverse streamwise pressure gradient and the inviscid mode becomes the dominant one. Received 17 March 1998 and accepted 28 April 1999     

An integrated smoothed particle hydrodynamics model for complex interfacial flows with large density ratios

In this paper, an integrated smoothed particle hydrodynamics (SPH) model for complex interfacial flows with large density ratios is developed. The discrete continuity equation and acceleration equation are obtained by considering the time derivative of the volume of particle and Eckart's continuum Lagrangian equation. A continuum surface force model is used to meet the fact that surface force may not be distributed uniformly on each side of the interface. An improved boundary condition is imposed to model wall free-slip and no-slip condition for interfacial flows with large density ratios. Particle shifting algorithm (PSA) is added for interfacial flows by imposing the normal correction near the interface, called as Interface-PSA. Then four representative numerical examples, including droplet deformation, Rayleigh-Taylor instability, dam breaking, and bubble rising, are presented and compared well with reference data. It is demonstrated that inherent interfacial flow physics can be well captured, including surface tension and the dynamic evolution of the complex interfaces.     

Effect of small-amplitude vibrations on viscoelastic flows in a plane slider bearing

The qualitative changes of dynamic lift and friction forces caused by small-amplitude harmonic vibrations superimposed on flows in a plane slider bearing are considered for simple viscous and viscoelastic lubricating fluids. Low- and high-frequency disturbances are analysed in greater detail and the most beneficial situations discussed.     

Experiments on the linear instability of flow in a wavy channel

The effects of wall corrugation on the stability of wall-bounded shear flows have been examined experimentally in plane channel flows. One of the channel walls has been modified by introduction of the wavy wall model with the amplitude of 4% of the channel half height and the wave number of 1.02. The experiment is focused on the two-dimensional travelling wave instability and the results are compared with the theory [J.M. Floryan, Two-dimensional instability of flow in a rough channel, Phys. Fluids 17 (2005) 044101 (also: Rept. ESFD-1/2003, Dept. of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario, Canada, 2003)]. It is shown that the flow is destabilized by the wall corrugation at subcritical Reynolds numbers below 5772, as predicted by the theory. For the present corrugation geometry, the critical Reynolds number is decreased down to about 4000. The spatial growth rates, the disturbance wave numbers and the distribution of disturbance amplitude measured over such wavy wall also agree well with the theoretical results.     

Taylor instability of a fluid film on a body

The Taylor instability develops in a parallel flows when the body force acts in the direction from the heavier fluid toward the lighter [1]. It has been suggested that an increase in flow vorticity may have a stabilizing influence on the Taylor instability [2]. In studying the hydrodynamic stability of a viscous film on a body in a flow of a low-viscosity fluid [3], the author noted some stabilization of the Taylor instability with increase in Reynolds number, and suggested that cases of complete stabilization of the flow with respect to two-dimensional disturbances are possible with some increase in Reynolds number. In the present investigation, calculations revealed cases in which with increase in Reynolds number the Taylor instability goes over into a Helmholtz instability, which increases with increase in Reynolds number, and also cases in which the Taylor instability completely disappears at some value of the Reynolds number before a Helmholtz instability has developed, i.e., cases of complete stabilization of the flow with respect to two-dimensional disturbances as a result of an increase in Reynolds number.     

Surface instability and primary atomization characteristics of straight liquid jet sprays

Using the detailed numerical simulation data of primary atomization, the liquid surface instability development that leads to atomization is characterized. The numerical results are compared with a theoretical analysis of liquid–gas layer for a parameter range close to high-speed Diesel jet fuel injection. For intermittent and short-duration Diesel injection, the aerodynamic surface interaction and transient head formation play an important role. The present numerical setting excludes nozzle disturbances to primarily investigate this interfacial instability mechanism and the role of jet head. The first disturbed area is the jet head region, and the generated disturbances are fed into the upstream region through the gas phase. This leads to the viscous boundary layer instability development on the liquid jet core. By temporal tracking of surface pattern development including the phase velocity and stability regime and by the visualization of vortex structures near the boundary layer region, it is suggested that the instability mode is the Tollmien–Schlichting (TS) mode similar to the turbulent transition of solid-wall boundary layer. It is also demonstrated that the jet head and the liquid core play an interacting role, thus the jet head cannot be neglected in Diesel injection. In this study, this type of boundary layer instability has been demonstrated as a possible mechanism of primary atomization, especially for high-speed straight liquid jets. The effect of nozzle turbulence is a challenging but important issue, and it should be examined in the future.     

Simulation of three-dimensional polymer mould filling processes using a pseudo-concentration method

Mould filling processes, in which a material flow front advances through a mould, are typical examples of moving boundary problems. The moving boundary is accompanied by a moving contact line at the mould walls causing, from a macroscopic modelling viewpoint, a stress singularity. In order to be able to simulate such processes, the moving boundary and moving contact line problem must be overcome. A numerical model for both two- and three-dimensional mould filling simulations has been developed. It employs a pseudo-concentration method in order to avoid elaborate three-dimensional remeshing, and has been implemented in a finite element program. The moving contact line problem has been overcome by employing a Robin boundary condition at the mould walls, which can be turned into a Dirichlet (no-slip) or a Neumann (free-slip) boundary condition depending on the local pseudo-concentration. Simulation results for two-dimensional test cases demonstrate the model's ability to deal with flow phenomena such as fountain flow and flow in bifurcations. The method is by no means limited to two-dimensional flows, as is shown by a pilot simulation for a simple three-dimensional mould. The reverse problem of mould filling is the displacement of a viscous fluid in a tube by a less viscous fluid, which has had considerable attention since the 1960's. Simulation results for this problem are in good agreement with results from the literature. © 1998 John Wiley & Sons, Ltd.     

Stability of viscoelastic flow around periodic arrays of cylinders

Low Reynolds number flow of Newtonian and viscoelastic Boger fluids past periodic square arrays of cylinders with a porosity of 0.45 and 0.86 has been studied. Pressure drop measurements along the flow direction as a function of flow rate as well as flow visualization has been performed to investigate the effect of fluid elasticity on stability of this class of flows. It has been shown that below a critical Weissenberg number (We_c), the flow in both porosity cells is a two-dimensional steady flow, however, pressure fluctuations appear above We_c which is 2.95±0.25 for the 0.45 porosity cell and 0.95±0.08 for the higher porosity cell. Specifically, in the low porosity cell as the Weissenberg number is increased above We_c a transition between a steady two-dimensional to a transient three-dimensional flow occurs. However, in the high porosity cell a transition between a steady two-dimensional to a steady three-dimensional flow consisting of periodic cellular structures along the length of the cylinder in the space between the first and the second cylinder occurs while past the second cylinder another transition to a transient three-dimensional flow occurs giving rise to time- dependent cellular structures of various wavelengths along the length of the cylinder. Overall, the experiments indicate that viscoelastic flow past periodic arrays of cylinders of various porosities is susceptible to purely elastic instabilities. Moreover, the instability observed in lower porosity cells where a vortex is present between the cylinders in the base flow is amplifieds spatially, that is energy from the mean flow is continuously transferred to the disturbance flow along the flow direction. This instability gives rise to a rapid increase in flow resistance. In higher porosity cells where a vortex between the cylinders is not present in the base flow, the energy associated with the disturbance flow is not greatly changed along the flow direction past the second cylinder. In addition, it has been shown that in both flow cells the instability is a sensitive function of the relaxation time of the fluid. Hence, the instability in this class of flows is a strong function of the base flow kinematics (i.e., curvature of streamlines near solid surfaces), We and the relaxation time of the fluid.     

A general theory for two- and three-dimensional wall-mode instabilities in boundary layers over isotropic and anisotropic compliant walls

An asymptotic theory is developed for two- and three-dimensional disturbances growing in a two-dimensional boundary layer over a compliant wall. The theory exploits the multideck structure of the boundary layer to derive asymptotic approximations at a high Reynolds number for the perturbation wall pressure and viscous stresses. These quantities can be regarded as driving the wall and, accordingly, the equation(s) of motion for the wall is (are) used as the characteristic equation(s) for finding the eigenvalue(s). The main assumptions are that the amplitude of the disturbance is sufficiently small for linear theory to hold, the Reynolds number is large, the disturbance wavelength is long compared with the boundary-layer thickness, and the critical and viscous wall layers are well separated. The theory was developed to study the travelling-wave flutter instability discussed by Carpenter and Garrad, i.e., the Class B instability of Benjamin and Landahl. Under certain limiting processes both the upper-branch and conventional triple-deck scalings for the Tollmien-Schlichting instability can be obtained with the present approach. Accordingly, the theory also gives a reliable qualitative guide to the effect of anisotropic wall compliance on the Tollmien-Schlichting instability.The theory is applied to various cases including two- and three-dimensional disturbances, developing in boundary layers over isotropic and anisotropic compliant walls. The disturbances can be treated as either temporally or spatially growing. Eigenvalues are very accurately predicted by means of the theory, especially near points of neutral stability. The computational requirements are trivial compared with those required for full numerical solution of the Orr-Sommerfeld equation. For isotropic compliant walls the theory confirms the earlier result of Miles and Benjamin that the phase shift in the disturbance velocity across the critical layer plays a dominant role in destabilization of the Class B travelling-wave flutter through making irreversible energy transfer possible due to the work done by the fluctuating pressure at the wall. The theory elucidates the secondary role played by the phase shift occurring across the wall layer. Viscous effects are much more important for anisotropic compliant walls which admit substantial horizontal, as well as vertical, displacement. For these walls an important mechanism for irreversible energy transfer is the work done by fluctuating shear stress. This almost invariably has a stabilizing effect on the travelling-wave flutter. In addition there is a weaker effect arising from the effect of anisotropic wall compliance on the phase shift across the wall layer. This may be stabilizing or destabilizing.This work was carried out with the support of the Ministry of Defence (Procurement Executive) and the Office of Naval Research and was completed while P.W.C. and J.S.B.G. were on study leave at the Department of Aerospace Engineering, The Pennsylvania State University, and the Department of Mathematics, Iowa State University, Ames, respectively. They would like to express their gratitude to those institutions and the Office of Naval Research for financial support during their study leaves.     

Weakly nonlinear stability analysis of a falling film with countercurrent gas flow

Weakly nonlinear stability analysis of a falling film with countercurrent gas–liquid flow has been investigated. A normal mode approach and the method of multiple scales are employed to carry out the linear and nonlinear stability solutions for the film flow system. The results show that both supercritical stability and subcritical instability are possible for a film flow system when the gas flows in the countercurrent direction. The stability characteristics of the film flow system are strongly influenced by the effects of interfacial shear stress when the gas flows in the countercurrent direction. The effect of countercurrent gas flow in a falling film is to stabilize the film flow system.