Unstable Displacement of Non-aqueous Phase Liquids with Surfactant and Polymer

In this paper, we study two-phase multicomponent displacement of two immiscible fluids in both homogeneous and heterogeneous porous media. In many applications such as enhanced oil recovery, fluid mixing and spreading can be detrimental to the efficacy of the process. Here, we show that when an initially immobile phase is being displaced by a finite-size slug of solvents (surfactant and polymer), viscous fingering significantly enhances mixing and spreading of solvents. These effects are similar to those caused by medium heterogeneity and lead to poor displacement efficiency. We first quantify the displacement efficiency subject to different mobility ratios, Peclet numbers, and levels of medium heterogeneity. We observe a non-monotonic behavior in displacement efficiency as a function of mobility ratio, indicating that although stable frontal interface is desirable, miscible viscous fingering on the rear interface will eventually disintegrate the solvents slugs and reduce the displacement efficiency. Then, we show that miscible viscous fingering developing on the rear interface of the chemical slug could be greatly suppressed when viscosity contrast is gradually decreased using exponential or linear functions, leading to 10% increase in displacement efficiency while using the same amount of chemicals. To elucidate this low displacement efficiency, we study the evolution of mixing, spreading, and interfacial length and show that while higher viscosity ratios are quite effective in mobilizing the initially immobile phase in 1D displacements, they are in fact detrimental in 2D unstable displacements since they enhance mixing and spreading of solvents.

     

Experimental studies of interfacial instabilities in multilayer pressure-driven flow of polymeric melts

The interfacial deformation and stability of two-(A-B) as well as three-layer symmetric (A-B-A) and asymmetric (A-B-C) pressure-driven flow of viscoelastic fluids has been investigated. Flow visualization in conjunction with digital image processing has been used to observe and measure the rate of encapsulation and interfacial stability/instability of the flow. Specifically, the encapsulation behavior as well as stability/instability of the interface and the corresponding growth or decay rate of disturbances as a function of various important parameters, namely, number of layers and their arrangement, layer depth ratio, viscosity and elasticity ratio as well as disturbance frequency, have been investigated. Based on these experiments, we have shown that the encapsulation phenomena occurs irrespective of the stability/instability of the interface and in cases when both encapsulation and instability occur simultaneously their coupling leads to highly complex and three-dimensional interfacial wave patterns. Moreover, it has been shown that the simple notion that less viscous fluids encapsulate more viscous fluids is incorrect and depending on the wetting properties of the fluid as well as their first and second normal stresses the reverse could occur. Additionally, in two- and three-layer flows it has been shown that by placing a thin, less viscous layer adjacent to the wall longwave disturbances can be stabilized while short and intermediate wavelength disturbances are stabilized when the more elastic fluid is the majority component. Furthermore, in three-layer flows it has been demonstrated that in the linear instability regime no dynamic interaction between the two interfaces is possible for short and intermediate wavenumber disturbances. However, in the nonlinear stability regime dynamic interactions between interfaces have been observed in this range of disturbance wavenumbers leading to highly chaotic flows. Finally, in the parameter space of this study no subcritical bifurcations were observed while supercritical bifurcations resulting in waves with a pointed front and a gradual tail were observed.     

Suppression or enhancement of interfacial instability in two-layer plane Couette flow of FENE-P fluids past a deformable solid layer

The linear stability of two-layer plane Couette flow of FENE-P fluids past a deformable solid layer is analyzed in order to examine the effect of solid deformability on the interfacial instability due to elasticity and viscosity stratification at the two-fluid interface. The solid layer is modeled using both linear viscoelastic and neo-Hookean constitutive equations. The limiting case of two-layer flow of upper-convected Maxwell (UCM) fluids is used as a starting point, and results for the FENE-P case are obtained by numerically continuing the UCM results for the interfacial mode to finite values of the chain extensibility parameter. For the case of two-layer plane Couette flow past a rigid solid surface, our results show that the finite extensibility of the polymer chain significantly alters the neutral stability boundaries of the interfacial instability. In particular, the two-layer Couette flow of FENE-P fluids is found to be unstable in a larger range of nondimensional parameters when compared to two-layer flow of UCM fluids. The presence of the deformable solid layer is shown to completely suppress the interfacial instability in most of the parameter regimes where the interfacial mode is unstable, while it could have a completely destabilizing effect in other parameter regimes even when the interfacial mode is stable in rigid channels. When compared with two-layer UCM flow, the two-layer FENE-P case is found in general to require solid layers with relatively lower shear modulii in order to suppress the interfacial instability. The results from the linear elastic solid model are compared with those obtained using the (more rigorous) neo-Hookean model for the solid, and good agreement is found between the two models for neutral stability curves pertaining to the two-fluid interfacial mode. The present study thus provides an important extension of the earlier analysis of two-layer UCM flow [V. Shankar, Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer: implications of fluid viscosity stratification, J. Non-Newtonian Fluid Mech. 125 (2005) 143–158] to more accurate constitutive models for the fluid and solid layers, and reaffirms the central conclusion of instability suppression in two-layer flows of viscoelastic fluids by soft elastomeric coatings in more realistic settings.     

Displacement of a viscous fluid from a Hele-Shaw cell with a sink

It is found that the radial geometry does not stabilize the evolution of instability in the displacement of a more viscous fluid by a less viscous fluid from a circular Hele-Shaw cell with a sink. A linear analysis shows the absolute instability of the radial displacement front. The appearance of isolated fingers is observed during numerical simulations.     

Effect of small parameters on the structure of the front formed by unstable viscous-fluid displacement from a Hele-Shaw cell

The process of displacement of a viscous fluid from a Hele-Shaw cell consisting of two plates separated by a small gap is investigated. The front formed when the fluid is displaced from the cell by another, lower-viscosity fluid is unstable. The lower-viscosity fluid breaks through the layer of displaced fluid and forms channels called viscous fingers. As a result, a mixing zone occupied by both displaced and displacing fluids is formed. The structure of the unstable displacement front is investigated when the surface tension forces have no effect on the shape of the fingers. This situation is realized when a water-glycerin mixture is rapidly displaced from the cell by water. Equations taking the inertial and viscous forces acting in the plane of the plates into account are obtained by averaging the Navier-Stokes equations over the cell gap. Using the equations obtained the stability of a plane displacement front traveling in the direction of its normal and the stability of the lateral surfaces of the viscous fingers is investigated when the fluid velocities are parallel to the interface. From the solution for stability of the transverse displacement front it follows that the viscous forces acting in the plane of the plates determine the finger width (when there is no surface tension). Instability also develops in the flow on the longitudinal fluid interface. In this case the destabilizing factor is the inertial forces. Under the action of this instability the fingers, in their turn, lose stability and disintegrate into viscous bubbles.     

Bénard-Marangoni Instability of a Two-Layer System with Allowance for Variations in the Interfacial Internal Energy

The effect of variations of the internal surface energy due to local increments in the interfacial area on the conditions of onset of thermocapillary Marangoni instability in a two-layer system of reduced-viscosity fluids is studied. It is shown that in the linear approximation the effect considered leads to stabilization of the development of the monotonic instability mode.     

Magnetohydrodynamic laminar boundary-layer flow over a porous wedge with thermophoresis particle deposition in the presence of variable stream conditions

An analysis is presented to investigate the effects of thermophoresis variable viscosity on MHD mixed convective heat and mass transfer of a viscous, incompressible and electrically conducting fluid past a porous wedge in the presence of chemical reaction. The wall of the wedge is embedded in a uniform porous medium in order to allow for possible fluid wall suction or injection. The governing boundary layer equations are written into a dimensionless form by local non-similarity transformations. The transformed coupled nonlinear ordinary differential equations are solved numerically by using the R.K. Gill and shooting methods. Favorable comparison with previously published work is performed. Numerical results for the dimensionless velocity, temperature and concentration profiles are obtained and displayed graphically for pertinent parameters to show interesting aspects of the solution.     

Numerical study on a vertical plate with variable viscosity and thermal conductivity

Free convection over an isothermal vertical plate immersed in a fluid with variable viscosity and thermal conductivity is studied in this paper. We consider the two-dimensional, laminar and unsteady boundary layer equations. Using the appropriate variables, the basic governing equations are transformed to non-dimensional governing equations. These equations are then solved numerically using a very efficient implicit finite difference scheme known as Crank–Nicolson scheme. The fluid considered in this study is of viscous incompressible fluid of temperature dependent viscosity and thermal conductivity. The effect of varying viscosity and thermal conductivity on velocity, temperature, shear stress and heat transfer rate are discussed. The velocity and temperature profiles are compared with previously published works and are found to be in good agreement.     

Frontal collision of internal solitary waves of first mode

The dynamics and energetics of a frontal collision of internal solitary waves (ISW) of first mode in a fluid with two homogeneous layers separated by a thin interfacial layer are studied numerically within the framework of the Navier–Stokes equations for stratified fluid. It was shown that the head-on collision of internal solitary waves of small and moderate amplitude results in a small phase shift and in the generation of dispersive wave train travelling behind the transmitted solitary wave. The phase shift grows as amplitudes of the interacting waves increase. The maximum run-up amplitude during the wave collision reaches a value larger than the sum of the amplitudes of the incident solitary waves. The excess of the maximum run-up amplitude over the sum of the amplitudes of the colliding waves grows with the increasing amplitude of interacting waves of small and moderate amplitudes whereas it decreases for colliding waves of large amplitude. Unlike the waves of small and moderate amplitudes collision of ISWs of large amplitude was accompanied by shear instability and the formation of Kelvin–Helmholtz (KH) vortices in the interface layer, however, subsequently waves again become stable. The loss of energy due to the KH instability does not exceed 5%–6%. An interaction of large amplitude ISW with even small amplitude ISW can trigger instability of larger wave and development of KH billows in larger wave. When smaller wave amplitude increases the wave interaction was accompanied by KH instability of both waves.     

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.     

Nonlinear effects in absolute and convective instabilities of a near-wake

Linear and nonlinear initial-value problems are discussed for planar inviscid disturbances in streamlined near-wakes. This is mostly for those areas of near-wake flow where the basic motion comprises nearly uniform shear with or without normal influx into the accompanying viscous interfacial layer, although agreement is found with linear properties for full velocity profiles of double-Blasius, double-Jobe–Burggraf, Hakkinen–Rott and Goldstein form. With nonlinear disturbances, wavelike initial conditions yield a known critical-layer development, whereas more general, non-wave, initial conditions lead to a new integro-partial-differential amplitude equation which is studied analytically and numerically. The solutions show decay, finite-time blowup or nonlinear upstream-travelling disturbances. The normal influx proves crucial. Absolute and upstream- or downstream-convective instability is encountered (depending on the profiles, and flow reversal, for example); and in generic cases (for any thin airfoil) nonlinearity is shown analytically to provoke upstream convection. Increased nonlinearity drives the typical transition point extremely close to the trailing edge. Comparisons are made with three-dimensional behaviour in the linear case and with a direct simulation in the nonlinear regime.     

Boundary conditions for contacts lines in coextrusion flows

A bicomponent coextrusion process is modelled using a 3-D finite element formulation. The layer uniformity problem in coextrusion is addressed by examining the effects of the polymer melt/polymer melt/die wall contact line boundary condition. It has been observed that the less viscous polymer layer will tend to displace the more viscous polymer layer near the die wall. The behaviour of the contact lisle is considered to be either a stick or slip boundary condition. In the stick boundary condition, the contact line does not move from its original position after the two polymer layers meet, A slip boundary condition allows the contact line to move along the die wall. The calculated interfaces which result from different contact line assumptions are determined. Results show that if a stick boundary condition is appropriate for a given fluid/fluid/solid contact line, then a very thin entrained layer of the more viscous polymer melt will be trapped between the less viscous polymer melt and the die wall. Slip boundary conditions would allow complete displacement of the contact line along the die wall. Both slip and stick boundary conditions produce similar interface profiles far away from the die wall for small viscosity ratios. In certain eases, the displacement of the more viscous material by the less viscous material will cease and a static interface structure is produced regardless of die length. Experimental work with polycarbonate melts is compared with the numerical simulations.A. Torres on leave from Investigación y Desarrollo,, C.A. (INDESCA), P.O. Box 10319, Complejo Petroquímico El Tablazo, Maracaibo, 4001, Venezuela.     

Thermophoresis and variable viscosity effects on MHD mixed convective heat and mass transfer past a porous wedge in the presence of chemical reaction

An analysis is presented to investigate the effects of thermophoresis and variable viscosity on MHD mixed convective heat and mass transfer of a viscous, incompressible and electrically conducting fluid past a porous wedge in the presence of chemical reaction. The wall of the wedge is embedded in a uniform porous medium in order to allow for possible fluid wall suction or injection. The governing boundary layer equations are written into a dimensionless form by similarity transformations. The transformed coupled nonlinear ordinary differential equations are solved numerically by using the R.K. Gill and shooting methods. Favorable comparison with previously published work is performed. Numerical results for the dimensionless velocity, temperature and concentration profiles as well as for the skin friction, heat and mass transfer and deposition rate are obtained and displayed graphically for pertinent parameters to show interesting aspects of the solution.     

Convection in a two-layer system due to the combined action of the Rayleigh and thermocapillary instability mechanisms

In a two-layer system loss of stability may be monotonic or oscillatory in character. Increasing oscillatory perturbations have been detected in the case of both Rayleigh [1, 2] and thermocapillary convection [3–5]; however, for many systems the minimum of the neutral curve corresponds to monotonic perturbations. In [5] an example was given of a system for which oscillatory instability is most dangerous when the thermogravitational and thermocapillary instability mechanisms are simultaneously operative. In this paper the occurrence of convection in a two-layer system due to the combined action of the Rayleigh (volume) and thermocapillary (surface) instability mechanisms is systematically investigated. It is shown that when the Rayleigh mechanism operates primarily in the upper layer of fluid, in the presence of a thermocapillary effect oscillatory instability may be the more dangerous. If thermogravitational convection is excited in the lower layer of fluid, the instability will be monotonic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 166–170, January–February, 1987.     

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     

Convective stability of a horizontal rotating fluid layer with distributed heat sources

The results of investigating the convective instability of a horizontal layer of rotating fluid, created by a temperature difference applied at the boundaries of the layer and by heat sources distributed according to various laws, are presented. It is shown that, when the other parameters of the problem are fixed, an increase in the internal heat release lowers the limits of both monotonic and oscillatory stability of the layer, increases the wave number and reduces the neutral oscillation frequency. An increase in source concentration towards the center of the layer intensifies the effect. As the strength of the internal heat sources and their concentration towards the center of the layer increase, the oscillating convection that develops at the stability limit when the Prandtl number is low and the rotation fairly fast is first replaced by monotonic convection and then ceases altogether.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–28, January–February, 1989.     

Coupled frequencies of a hydroelastic viscous liquid system

The coupled frequencies of a hydroelastic system consisting of an elastic shell and a viscous liquid layer with a free surface have been treated. The system exhibits no z-dependency and may be either an annular liquid layer around an elastic center shell or a liquid layer inside an elastic container. The first case has been evaluated numerically, where the influence of the liquid surface tension parameter, the elasticity parameter of the shell and the thickness of the layer have been determined. In contrast to the hydroelastic system with an ideal liquid, the system with viscous liquid exhibits instability of the liquid surface as well as the shell.     

The Onset of Double-Diffusive Convection in a Superposed Fluid and Porous Layer Under High-Frequency and Small-Amplitude Vibrations

We numerically simulate the initiation of an average convective flow in a system composed of a horizontal binary fluid layer overlying a homogeneous porous layer saturated with the same fluid under gravitational field and vibration. In the layers, fixed equilibrium temperature and concentration gradients are set. The layers execute high-frequency oscillations in the vertical direction. The vibration period is small compared with characteristic timescales of the problem. The averaging method is applied to obtain vibrational convection equations. Using for computation the shooting method, a numerical investigation is carried out for an aqueous ammonium chloride solution and packed glass spheres saturated with the solution. The instability threshold is determined under two heating conditions—on heating from below and from above. When the solution is heated from below, the instability character changes abruptly with increasing solutal Rayleigh number, i.e., there is a jump-wise transition from the most dangerous shortwave perturbations localized in the fluid layer to the long-wave perturbations covering both layers. The perturbation wavelength increases by almost 10 times. Vibrations significantly stabilize the fluid equilibrium state and lead to an increase in the wavelength of its perturbations. When the fluid with the stabilizing concentration gradient is heated from below, convection can occur not only in a monotonous manner but also in an oscillatory manner. The frequency of critical oscillatory perturbations decreases by 10 times, when the long-wave instability replaces the shortwave instability. When the fluid is heated from above, only stationary convection is excited over the entire range of the examined parameters. A lower monotonic instability level is associated with the development of perturbations with longer wavelength even at a relatively large fluid layer thickness. Vibrations speed up the stationary convection onset and lead to a decrease in the wavelength of most dangerous perturbations of the motionless equilibrium state. In this case, high enough amplitudes of vibration are needed for a remarkable change in the stability threshold. The results of numerical simulation show good agreement with the data of earlier works in the limiting case of zero fluid layer thickness.     

Effect of disjoining pressure on the instability of a charged thin liquid film on a spherical rigid core

It is shown that the critical Rayleigh number which characterizes the stability of a thin charged viscous fluid film on the surface of a rigid spherical core develops rapidly with decrease in the film thickness to 100 nm when the effect of the disjoining pressure becomes significant. The dependence of the instability growth rate on the thickness of the fluid layer is obtained by analyzing the dispersion relation numerically. Yaroslavl’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 102–106, January–February, 1999.     

Mixed convection boundary layer flow past a wedge with permeable walls

The effects of suction/injection on steady laminar mixed convection boundary layer flow over a permeable horizontal surface of a wedge in a viscous and incompressible fluid is considered in this paper. The similarity solutions of the governing boundary layer equations are obtained for some values of the suction/injection parameter f ₀, the constant exponent m of the wall temperature as well as the mixed convection parameter λ. The resulting system of nonlinear ordinary differential equations is solved numerically for both assisting and opposing flow regimes using an implicit finite-difference scheme known as the Keller-box method. Numerical results for the reduced skin friction coefficient, the local Nusselt number, and the velocity and temperature profiles are obtained for various values of parameters considered. Dual solutions are found to exist for the case of opposing flow.