ONE- AND TWO-PHASE FLOW IN NETWORK MODELS OF POROUS MEDIA

We discuss the low Reynolds number flow of one or two immiscible Newtonian fluids in network models of microscopically random porous media. For the case of a single fluid, we reduce the flow problem to an analog random electrical resistor problem and use an 'effective medium theory' to express the permeability of such networks in terms of the pore space geometry. For the flow of two fluids we use the Washburn approximation to incorporate capillary pressure differences, and show that this problem may also be formulated as a random electrical network. In this case, the capillary menisci correspond to moving batteries, and we follow the motion of the fluid-fluid interface (the ensemble of analog batteries) by a time-step procedure. We study the time evolution of the interface and the dynamics of blobs of one fluid contained in the other, as a function of the network geometry.’     

A stochastic analysis of the flow of two immiscible fluids in porous media: The case when the viscosities of the fluids are equal

A new stochastic theory is developed to explain the flow of two immiscible fluids in porous medium when the viscosity difference between two fluids is zero. In an individual micropore the radius of curvature of the interface separating the fluids is assumed constant and flow is modeled by the random jumping of microscopic interfaces. A one dimensional model composed of an array of parallel capillary tubes of constant radius is analyzed in detail. For the case in which two fluids have equal viscosity an analytical solution is obtained. The fluid displacement process is Fickian and dispersion is described in terms of a diffusion or spreading constant.     

Electrohydrodynamic linear stability of two immiscible fluids in channel flow

The electrohydrodynamic instability of the interface between two viscous fluids with different electrical properties in plane Poiseuille flow has recently found applications in mixing and droplet formation in microfluidic devices. In this paper, we perform the stability analysis in the case where the fluids are assumed to be leaky dielectrics. The two-layer system is subjected to an electric field normal to the interface between the two fluids. We make no assumption on the magnitude of the ratio of fluid to electric time scales, and thus solve the full conservation equation for the interfacial charge. The electric field is found to be either stabilizing or destabilizing, and the influence of the various parameters of the problem on the interface stability is thoroughly analyzed.     

The crossflow piston bed

The moving piston bed is a device for producing washed crystals from a slurry feed. Necessary for predicting the quality of a conceptual bed geometry is an evaluation method which accurately reveals the flow patterns in the dynamic system. The electrical analog method serves as a versatile guide to an evolutionary development of improved designs.All beds with a submerged drainage port have, for a given geometry, a unique relationship between loss of displacement fluid and rate of bed advance. The hydraulic diagram is unchanged, by the magnitude of applied forces. For rating productive capacity the use of gravity forces as a reference standard is recommended.A crossflow design is developed that predicts a productive rate over three times that achievable by downflow washing.     

Wave regimes in two-layer microchannel flow

We consider the dynamics of an interface separating two immiscible fluids in a vertically oriented channel. We use boundary-layer theory together with the Kármán-Pohlhausen approximation to derive coupled nonlinear evolution equations for the interface shape and the flow rate in one of the layers. These equations reduce to those that govern the evolution of falling films at moderate flow rates. We use bifurcation theory to determine the ‘families’ of travelling waves for given flow rate and physical properties of the fluids; these waves are characterised by their speed and frequency. The solutions are obtained via continuation from those calculated for the falling film case.     

Improved parametric characterization of flow geometries

An improved parametric method is presented for characterization of the flow geometry in the rectilinear flow of non-Newtonian fluids in open and closed conduits of arbitrary crosssection and in purely viscous, inelastic flow of non-Newtonian fluids through packed beds and porous media. In the new formulation, an infinite number of geometric parameters characterize the flow geometry. The actual number required in a particular application is shown to be determined by the fluid model equation representing the rheological behavior of the fluid. For Ostwald-de-Waele and Ellis fluids with flow behavior indices s = 1/n and α integers, the number of geometric parameters required to represent the relationship between flow rate and pressure drop is given by s + 1 and α + 1, respectively. The efficacy of the present method is demonstrated by comparisons with available results for various fluid models and flow geometries.     

Pumping,spraying, and mixing of fluids by electric fields

The mechanism of electrostatic spraying of insulating fluids, such as air or organic solvents, into relatively conductive fluids, such as water, is investigated in this work. Experiments with air sprayed into water through an electrified capillary showed that the pressure inside the capillary increases, reaches a maximum, and then decreases as the applied voltage is increased. The initial pressure increase is due to the electric stress on the fluid interface, while the decrease is due to the Coulombic electrohydrodynamic flow generated near the end of the capillary. It is shown that electric fields can cause simultaneous pumping, spraying, and mixing of fluids. This phenomenon is demonstrated for air and kerosene in water.     

Velocity profiles of the exit region of molten polyethylene extrudates

The objective of the work described herein is the experimental investigation of the velocity field of polymer melts flowing through a capillary in the regons of flow prior to and after the capillary exit. The fluids studied are branched polyethylene melts in steady laminar isothermal flow. The technique employed for the determination of the Eulerian velocity profiles is one that utilizes phototomicrogroaphy of the reflected light from tracer particles dispersed in the flowing medium. Axial acceleration of the fluid elements just before the capillary exit was observed. It was found that this accelearation is more pronounced in melts of low bulk viscosity. This observation region, non-viscometric. The translation of the velocity profiles of the fluids studied, from one resembling a parabola to that of “plug” flow, involves inflection points with minima in the velocity vector v (r, z). These minima appear near the surface of the extrudates and can not be accounted for by an existing theory. It was also found that the density of the viscoelastic fluids studied is a function of the axial position, in the region of flow investigated. The density decreases before the exit and, before it reaches an equilibuiu value at an axial position downstream equal to one or two diameters, increases beyond that value upon exit. This phenomenon is attributed to an “overshoot” in the process fo elastic recoil of the high polymer melts fron a strained structure to a random one.     

Interface determination in bicomponent extrusion

The unidirectional flow of two immiscible fluids with different viscosities in a long die of arbitrary shape is considered. Mathematically, the problem has a continuum of solutions corresponding to arbitrarily prescribed interface shapes, but experimental evidence indicates the existence of a unique interface shape with the less viscous fluid encapsulating the more viscous fluid. With the introduction of the minimum viscous dissipation principle, which postulates that the amount of viscous dissipation is minimized for a given flow rate, the problem becomes a nonlinearly constrained optimization problem. A generalized reduced gradient/finite element method combination is used to predict the interface shape when two inelastic fluids are considered. The effect of the viscosity ratio and flow-rate ratio on the interface shape is examined for different die geometries. Inner layer breakup phenomena are predicted and explained for complex die geometries.     

BLASIUS AND SAKIADIS FLOW WITH UNIFORM BLOWING OR SUCTION IN NON-NEWTONIAN POWER-LAW FLUIDS

The Blasius and Sakiadis flows of a non-Newtonian power-law fluid are considered. The plate is porous and fluid can be either injected or sucked through it. The boundary layer equations are transformed into a nondimensional form and are solved with a finite difference method. For the case of uniform suction, new results have been found, although this problem has been investigated in the past. Among them are analytical solutions for dilatant fluids of the Blasius flow and analytical solutions of the Sakiadis flow for all values of the power-law index. For the case of uniform injection, the characteristics of the flow until a separation state are investigated and discussed.     

Hydrodynamics of Taylor flow in noncircular capillaries

In this work, volume of fluid (VOF) technique, one computational fluid dynamics (CFD) method, was used to investigate the upward Taylor flow in vertical square and equi-triangular capillaries. For saving computation time, the simulations were carried out in a moving frame of reference attached to Taylor bubbles. The main flow parameters, involving bubble size and shape, liquid film thickness, velocity field and two-phase relative velocity, were studied as functions of capillary number. The numerical simulations were in good agreement with previous reports and showed that the flow in the sides and corners of polygonal capillaries were different. A comparative study was also conducted on Taylor flow in square and equi-triangular capillaries and their circular counterparts, where the influence of capillary geometry on the characteristics of Taylor flow was illustrated clearly.     

Experimental studies on the flow of two immiscible fluids in porous media using X-ray computed tomography: The case when the viscosities of the fluids are equal

Experiments were carried out to demonstrate the dispersion that occurs at the interface between fluids when two immiscible fluids flow in porous structures. In this work the porous medium was cellulosic absorbent and the fluids, decyl alcohol and water, were modified so as to cover a range of flow rates and identical fluid viscosities. Computerized Tomography (CT) was used to generate dynamic three-dimensional images of two-phase saturations and provided quantitative information of time evolution of fluid saturation at each position. Thus, use of CT made characterization of two-phase displacement history in cellulosic porous media possible. This work could relate experimental fluid saturation to theoretical model of immiscible fluids flow.     

On scaling of diffuse-interface models

Application of diffuse-interface models (DIM) yields a system of partial differential equations (PDEs) that generally requires a numerical solution. In the analyses of multiphase flows with DIM usually an artificial enlargement of the interface thickness is required for numerical reasons. Replacing the real interface thickness with a numerically acceptable one, while keeping the surface tension constant, can be justified based on the analysis of the equilibrium planar interface, but demands a change in the local part of the free energy. In a non-equilibrium situation, where the interface position and shape evolve with time, we need to know how to change the mobility in order to still model the same physical problem.Here we approach this question by studying the mixing of two immiscible fluids in a lid-driven cavity flow where the interface between the two fluids is stretched roughly linearly with time, before break-up events start. Scaling based on heuristics, where the mobility is taken inversely proportional to the interface thickness, was found to give fairly well results over the period of linear interface stretching for the range of Péclet numbers and viscosity ratios considered when the capillary number is O(10). None of the scalings studied was, however, able to capture the break-up events accurately.     

Two-phase flow in porous media

Two-phase flow in porous media depends on many factors, such as displacement vs steady two-phase flow, saturation, wettability conditions, wetting fluid vs non-wetting fluid is displacing, the capillary number, interfacial tension, viscosity ratio, pressure gradient, uniformly wetted vs mixed-wet pore surface, uniform vs distributed pore throats, small vs large pores, well-connected pores vs pores connected by small throats, etc. These parameters determine how the two fluids are distributed in the pores, e.g. whether they flow in seperate channels or side-by-side in the same channels, either with both fluids being continous or only one fluid being continous and the other discontinuous. In displacement, the capillary number and the viscosity ratio determine whether the displacement front is sharp, or if there is either capillary or viscous fingering.     

Multiphase mass transport with partitioning and inter-phase transport in porous media

We derive the effective mass-transfer coefficient between two fluid phases in a porous medium, one of which is flowing and the other is immobile. A passive tracer is advected by the flowing phase, becomes partitioned at the fluid-fluid interface and diffuses in the immobile phase. We use traditional volume-averaging methods to obtain a unit-cell boundary-value problem for the calculation of the effective mass-transfer coefficient. The problem is controlled by the Peclet number of the flowing phase, by a second dimensionless parameter that captures diffusion and partition in the two phases and by the geometrical properties of the porous medium.We derive asymptotic results for the scaling of the mass-transfer coefficient under various limiting conditions. Then, we use numerical methods that solve for the flow velocity field under Stokes flow conditions, and for the transport problem. The numerical results verify the asymptotic scaling expressions and provide estimates of the coefficient for a number of special cases. In particular, we find that when the immobile phase is wetting the solid (in the form of films), the mass transfer coefficient is larger than in the non-wetting case (where the phase is distributed in the form of blobs). Shape factors for practical applications are also obtained.     

Experimental and theoretical study of the extrusion of two-phase molten polymer systems

An experimental and theoretical study of two-phase flow of molten polymers has been carried out. The theoretical analyses apply the theory of nonlinear viscoelastic fluids to consider stress and velocity profiles and interface shape in stratified flow between parallel plates and in a tube. The second normal stress difference is predicted to influence interface shape. Experimental studies have been made of stratified two-phase flow of a low viscosity but elastic low-density polyethylene and a high-viscosity polystyrene in a capillary rheometer. In the stratified flow experiment, velocity fields and interface shape in the reservoir approaching capillary die and the emerging extrudate were investigated, the former being observed through visual experiments. The emerging extrudates possessed convex polystyrene surfaces at the interface. A strong tendency toward the collection of bubbles near the capillary entry was found. We have made an experimental study of the extrusion of disperse mixtures of polystyrene and different polyolefins. The morphology of the disperse two-phase emerging extrudates has been investigated.     

Inertia and gravitational effects in extrusion dies for non-Newtonian fluids

An analysis is presented which allows the sheet or film die designer to estimate when inertial and gravitational effects are important. General theoretical equations are developed for end fed dies with arbitrary variation of the cavity cross sectional shape, cavity taper, slot length, and gap over the width. The method assumes viscous flow and a two dimensional approximation for the cavity flow. For fluid flow properties, it is assumed only that the apparent viscosity is a single valued function of the shear rate. In the important special case of constant die geometry and power law fluids, three dimensionless numbers plus the power law index are the parameters controlling the uniformity of flow from the die. Results are presented that illustrate when die orientations with respect to gravity and when fluid inertia are important. When they are not, simple expressions for die inlet pressure and uniformity index are given.     

Fractal analysis of invasion depth of extraneous fluids in porous media

Applications of the fractal theory to analyze transport properties of porous media in science and engineering have received steady attention in the past two decades. However, the theory was rarely used to analyze invasion by extraneous fluids into a permeable bed where there is initially no such fluid present. Spills and leaks of non-aqueous phase liquids (NAPLs) and formation damage in drilling and completion wells are two typical examples. In this work, a fractal capillary model is proposed to analyze the depth of extraneous fluid invasion, where the tortuosity of capillaries and capillary pressure effect are taken into account. The quantitative relationship between average flow velocity and average beeline velocity are discussed based on the fractal geometry theory. Based on the proposed model, the depth of extraneous fluid invasion can be determined when the operation conditions, extraneous fluid properties and formation structure parameters are available, and the model predictions are in good agreement with the available data.     

A theoretical analysis on stability of a moving interface in a porous medium for bingham displacing fluids and its application in oil displacement mechanism

This paper deals specifically with the illustration of the rheological effects of non-Newtonian behaviour of the displacing fluid in a radial displacement mechanism in a porous medium. As an example, a Bingham displacing fluid is considered. For this case the necessary and sufficient conditions for stability are derived and discussed. The results obtained are relevant for an adequate understanding of the rheological effects associated with Bingham displacing fluids on the moving interface stability. The existence of a critical value for the interface location from which the interface protuberances begin to damp out is shown.