Flow Of Immiscible Fluids Research Articles

Multilayer analysis of immiscible power-law fluids under magnetohydrodynamic and pressure-driven effects in a microchannel

In the present study, the combined magnetohydrodynamic and pressure-driven flow of multilayer immiscible fluids into a parallel flat plate microchannel is semi-analytically solved. Due to the handling of complex fluids in various microfluidic platform applications, the fluid transport reviewed here considers the power-law model. The movement of electrically conductive fluid layers is due to Lorentz forces that arise from the interaction between an electric current and a magnetic field. To find a solution for the flow field, the momentum equation and the rheological model for each fluid layer, together with the corresponding boundary conditions at the liquid-liquid and solid-liquid interfaces, are solved simultaneously through a closed system of nonlinear equations. The graphical results show the influence of the dimensionless parameters that arise from the mathematical modeling on the velocity profiles and flow rate. These are the magnetic parameters, the fluid layers thickness, the viscosity coefficients, the ratios between pressure forces and magnetic forces, and the flow behavior indexes. This theoretical work contributes to the design of microfluidic devices for flow-focusing tasks in chemical, clinical, and biological areas.

Numerical Solution of the Time-Depending Flow of Immiscible Fluids with Fuzzy Boundary Conditions

Fluid flow modeling using fuzzy boundary conditions is one of the viable areas in biofluid mechanics, drug suspension in pharmacology, as well as in the cytology and electrohydrodynamic analysis of cerebrospinal fluid data. In this article, a fuzzy solution for the two immiscible fluid flow problems is developed, which is motivated by biomechanical flow engineering. Two immiscible fluids, namely micropolar and Newtonian fluid, are considered with fuzzy boundary conditions in the horizontal channel. The flow is considered unsteady and carried out by applying a constant pressure gradient in the X-direction of the channel. The coupled partial differential equations are modeled for fuzzy profiles of velocity and micro-rotation vectors then the numerical results are obtained by the modified cubic B - spline differential quadrature method. The evolution of membership grades for velocity and microrotation profiles has been depicted with the fuzzy boundaries at the channel wall. It is observed that Micropolar fluid has a higher velocity change than Newtonian fluid, and both profiles indicate a declining nature toward the interface.

Dispersion tensor in a two-phase flow in a slit

Taylor's approach on the dispersion phenomenon is generalized for solute transport in a two-phase laminar flow of immiscible fluids in a slit. The reduced-order models for solute transport are derived using Reynolds decomposition and averaging techniques from which the exact analytical expressions for all elements of the dispersion tensor and the matrix of coefficients of the advection term are derived. It is shown that the dispersion tensor is generally not symmetric, and the asymmetry originates from the presence of an interface between the two fluids. We also discussed conditions at which the solute transport in a two-phase laminar flow in a slit lead to dispersion barrier, osmotic dispersion, and reverse dispersion. The results provide a thorough insight into modeling solute transport across an interface/film in two-phase stratified flows and find applications in the design and optimization of microfluidic devices where two fluids flow in laminar contact.

Capillary fluctuations and energy dynamics for flow in porous media

Capillary energy barriers have important consequences for immiscible fluid flow in porous media. We derive a time-and-space averaging theory to account for the non-equilibrium behavior and understand the role of athermal capillary fluctuations in the context of their relationship to larger scale phenomenological equations. The formulation resolves several key challenges associated with two-fluid flow in porous media: (1) geometric and thermodynamic quantities are constructed as smooth functions of time based on time-and-space averages; (2) averaged thermodynamics are developed for films; (3) multi-scale fluctuation terms are identified, which account for transient behaviors of interfaces and films that occur due to pore-scale events; (4) geometric constraints are derived and imposed on the averaged thermodynamics; (5) a new constitutive model is proposed for capillary pressure dynamics that includes contributions from films; and (6) a time-and-space criterion for representative elementary volume is established based on capillary fluctuations. Capillary fluctuations are assessed quantitatively based on pore-scale simulations and experimental core-flooding data.

Pore-scale numerical prediction of three-phase relative permeability in porous media using the lattice Boltzmann method

Three immiscible fluids systems are often encountered in energy and environmental disciplines, especially in the oil exploration field. It is essential to enhance oil recovery (EOR) by identifying the three-phase fluid flow mechanism of oil-gas-water system in complex geological structures. In the present study, a lattice Boltzmann (LB) color gradient model for three immiscible fluid flow in three-dimensional porous media is employed and validated using five benchmarks (i.e., the multiphase spinodal decomposition; the multiphase Young-Laplace test; the liquid lens; contact angles and 3D channel layered three-phase fluid flow). The effects of wettability, viscosity ratio (M) and capillary number (Ca) on the relative permeability curves are analyzed in detail. When the wetting condition is enhanced, the results indicate that the relative permeability of non-wetting phase raises while that for the wetting phase exhibits a decline. When M is not equal 1, the relative permeability of high viscosity phase is significantly increased, while the other two phases are less affected. Meanwhile, the maximal value of relative permeability does not exceed unity because of the low porosity in the porous medium. For the influence of Ca, simulation results indicate the relative permeability shows weak sensitivity to Ca.

Advected phase-field method for bounded solution of the Cahn–Hilliard Navier–Stokes equations

Phase-field methods based on the Cahn–Hilliard (CH) equation coupled to the incompressible Navier–Stokes equation are becoming increasingly popular for interface resolving numerical simulations of two-phase flows of immiscible fluids. One major limitation of this approach, however, is that the volume of each phase is not inherently preserved. This is associated with the phase-discriminating order parameter, which in the course of the simulation remains in general not within its initial physical bounds. This shortcoming relates to the fact that the CH equation with standard Ginzburg–Landau chemical potential has no volume-preserving stationary solution for interfaces with uniform (non-zero) curvature. In this paper, a curvature-dependent chemical potential is proposed which allows for bounded stationary solutions of the CH equation for drops/bubbles exhibiting uniform curvature. Numerical solutions of the coupled Cahn–Hilliard Navier–Stokes equations show that the proposed chemical potential significantly improves boundedness and phase volume conservation over the standard one.

High-resolution shock-capturing numerical simulations of three-phase immiscible fluids from the unsaturated to the saturated zone

Numerical modeling of immiscible contaminant fluid flow in unsaturated and saturated porous aquifers is of great importance in many scientific fields to properly manage groundwater resources. We present a high-resolution numerical model that simulates three-phase immiscible fluid flow in both unsaturated and saturated zone in a porous aquifer. We use coupled conserved mass equations for each phase and study the dynamics of a multiphase fluid flow as a function of saturation, capillary pressure, permeability, and porosity of the different phases, initial and boundary conditions. To deal with the sharp front originated from the partial differential equations’ nonlinearity and accurately propagate the sharp front of the fluid component, we use a high-resolution shock-capturing method to treat discontinuities due to capillary pressure and permeabilities that depend on the saturation of the three different phases. The main approach to the problem’s numerical solution is based on (full) explicit evolution of the discretized (in-space) variables. Since explicit methods require the time step to be sufficiently small, this condition is very restrictive, particularly for long-time integrations. With the increased computational speed and capacity of today’s multicore computer, it is possible to simulate in detail contaminants’ fate flow using high-performance computing.

Slip effect on the unsteady electroosmotic and pressure-driven flows of two-layer fluids in a rectangular microchannel

Abstract of the paper: Electroosmotic flows of two-layer immiscible Newtonian fluids under the influence of time-dependent pressure gradient in the flow direction and different zeta potentials on the walls have been investigated. The slippage on channel walls is, also, considered in the mathematical model. Solutions to fluid velocities in the transformed domain are determined by using the Laplace transform with respect to the time variable and the classical method of the ordinary differential equations. The inverse Laplace transforms are obtained numerically by using Talbot’s algorithm and the improved Talbot’s algorithm. Numerical results corresponding to a time-exponential pressure gradient and translational motion with the oscillating velocity of the channel walls have been presented in graphical illustrations in order to study the fluid behaviour. It has been found that the ratio of the dielectric constant of fluid layers and the interface zeta potential difference have a significant influence on the fluid velocities.

Computational modeling of three-dimensional thermocapillary flow of recalcitrant bubbles using a coupled lattice Boltzmann-finite difference method

This study analyzes the thermocapillary flow of recalcitrant bubbles within thin channels using a hybrid finite difference lattice Boltzmann method (LBM). It extends a recently developed phase-field LBM to account for temperature effects by coupling the scheme with a fourth-order Runge–Kutta algorithm to solve the governing energy equation. The LBM makes use of a weighted-multiple relaxation-time collision scheme, which has been previously shown to capture high density and viscosity contrasts. This paper makes contributions in two fundamental areas relating to thermocapillary flow. First, it presents and verifies a novel, three-dimensional model to resolve thermocapillary dynamics for practical applications. The verification was undertaken via comparison with analytical solutions for the flow of immiscible fluids in a heated microchannel and for the migration of a droplet in a temperature field. Second, it provides new insight into the inherently three-dimensional nature of recalcitrant bubbles. It was found that the competing inertial and thermal effects allow these bubbles to propagate against the bulk motion of the liquid toward regions of low surface tension.

An arbitrary Lagrangian–Eulerian‐based finite element strategy for modeling incompressible two‐phase flows

AbstractIn this work, we develop a numerical strategy for solving two‐phase immiscible incompressible fluid flows in a general arbitrary Lagrangian–Eulerian framework. As we use a conforming mesh moving with one of the fluids, there is no need to track or reconstruct the interface explicitly. A sharp interface, discontinuity in the fluid properties, and the jump in the pressure field are all accurately modeled using the dummy‐node technique. One of the challenges that this work addresses is to obtain a C0‐continuous approximation for the surface tension force term on the reference configuration, and to obtain its consistent linearization within the context of a Newton–Raphson strategy. It is shown by means of various numerical examples that the strategy is computationally efficient and robust, and circumvents the need for remeshing upto significantly large deformations.

Computational homogenization of fully coupled multiphase flow in deformable porous media

In this paper, a computational modeling tool is developed for fully coupled multiphase flow in deformable heterogeneous porous medium that consists of complex and non-uniform micro-structures using the dual continuum scales based on the computational homogenization approach. The first-order homogenization technique is employed to perform the multi-scale analysis. The governing equations of two-phase flow of immiscible fluids, including an equilibrium equation and two mass continuity equations, are considered based on the appropriate main variables. According to the well-known Hill–Mandel principle of macro-homogeneity, the proper energy types are defined instead of conventional stress power for linking micro- and macro-scales, which plays a significant role in determination of consistent microscopic fields. The finite element squared strategy is utilized to resolve the two scales simultaneously. The periodic and linear boundary conditions are exploited in the micro-scale analysis, and the macroscopic quantities such as stress tensor, inertial force vector, flux vectors and fluid contents are determined from the boundary information of microscopic domain. Moreover, a general approach is defined depending on the type of boundary condition in which the macroscopic tangent operators can be extracted directly from the converged microscopic Jacobian matrix. Finally, in order to illustrate the efficiency and accuracy of the proposed computational algorithm, several numerical examples are solved, and the effects of various parameters, such as boundary conditions, RVE types, RVE length scale, and volume fraction of heterogeneities are investigated.

INFLUENCE OF HEAT TRANSFER ON THE FLOW OF IMMISCIBLE FLUIDS THROUGH PIPES: AN ANALYTICAL STUDY

INFLUENCE OF HEAT TRANSFER ON THE FLOW OF IMMISCIBLE FLUIDS THROUGH PIPES: AN ANALYTICAL STUDY

Role of Pore‐Scale Disorder in Fluid Displacement: Experiments and Theoretical Model

AbstractThe flow of multiple immiscible fluids in disordered porous media is important in many natural processes and subsurface applications. The pore‐scale disorder affects the fluid invasion pathways significantly and induces the transitions of displacement patterns in porous media. Extensive studies focus on pattern transitions affected by disorder under quasistatic or dynamic conditions, but how the disorder controls the pattern transitions from capillary‐dominated regime to viscous‐dominated regime is not well understood. Here, we combine microfluidic experiments and theoretical analysis to investigate the role of disorder in fluid displacement. We perform drainage experiments with four different disorders under six flow rate conditions and show that increasing disorder destabilizes displacement fronts for all flow rates considered. Based on the scaling analysis of pore‐filling events, we propose a theoretical model that describes the pattern transitions from compact displacement to capillary to viscous fingering as functions of disorder and capillary number. The effects of disorder on both capillary and viscous forces are quantified within the theoretical model. The phase diagram predicted by this model agrees well with our experimental results. We further elucidate the role of disorder in fluid displacement via energy conversion and dissipation. We find that increasing disorder enhances the capillary instabilities and induces more energy dissipated in a capillary‐dominated regime, with the dissipation ratio increasing from 28.3% to 56.7%. Our work extends the classic phase diagram to consider the effect of disorder and provides a better understanding of the impact of the disorder on flow behaviors by energy dissipation.

Two-layer nanofluid flow and heat transfer in a horizontal microchannel with electric double layer effects and magnetic field

Purpose The purpose of this paper is to investigate the immiscible two-layer heat fluid flows in the presence of the electric double layer (EDL) and magnetic field. The effects of EDL, magnetic field and the viscous dissipative term on fluid velocity and temperature, as well as the important physical quantities, are examined and discussed. Design/methodology/approach The upper and lower regions in a horizontal microchannel with one layer being filled with a nanofluid and the other with a viscous Newtonian fluid. The nanofluid flow in the lower layer is described by the Buongiorno’s nanofluid model with passively controlled model at the boundaries. An appropriate set of non-dimensional quantities are used to simplify the nonlinear systems. The resulting coupled nonlinear equations are solved by using homotopy analysis method. Findings The present work demonstrates that increasing the EDL thickness and Hartmann number can restrain the fluid flow. The Brinkmann number has a significant role in the enhancement of heat transfer. It is also identified that the influence of EDL effects on microflow cannot be ignored. Originality/value The effects of viscous dissipation involved in the heat transfer process and the body force because of the EDL and the magnetic field are considered in the thermal energy and momentum equations for both regions. The detailed derivation procedure of the analytical solution for electrostatic potential is provided. The analytical solutions can lead to improved understanding of the complex microfluidic systems.

Two-phase MHD Flow through Porous Medium with Heat Transfer in a Horizontal Channel

The present study deals with two layered MHD immiscible fluid flow through porous medium in presence of heat transfer through parallel plate channel. The fluids are incompressible, and flow is fully developed. The fluids are of different viscosities and thermal conductivities so flowing without mixing each other. Two different phases are accounted for study and are electrically conducting. Temperature of the walls of parallel plate channel is constant. Rheological properties of the immiscible fluids are constant in nature. The flow is governed by coupled partial differential equations which are converted to ordinary differential equations and exact solutions are obtained. The velocity profile and temperature distribution are evaluated and solved numerically for different heights and viscosity ratios for the two immiscible fluids. The effect of magnetic field parameter M and porosity parameter K is discussed for velocity profile and temperature distribution. Combined effects of porous medium and magnetic fields are accelerating the flow which, can be helpful in draining oil from oil wells.

Pore-Scale Study of the Thermochemical Process in Porous Media With Immiscible Phase by Lattice Boltzmann Method

Abstract Reactive flow happens in carbonate rocks which are porous media during acidification. In this study, a thermochemical dissolution model based on the lattice Boltzmann method (LBM) is established to investigate the complex thermochemical process in porous media with immiscible phase at pore scale. In the model, the immiscible fluid flow, solute transport, and heat transfer are solved by Shan–Chen multicomponent LB model, mass transport LB model, and multicomponent thermal LB model, respectively. The porous media is generated by the quartet structure generation set, and the evolution of solid phase is addressed by volume of pixel (VOP) method. The detailed thermochemical process in porous media with immiscible phase is revealed, and the effects of velocity, concentration, and temperature on mass and heat transfer are further analyzed. The results show that increasing inlet velocity, inlet concentration, and temperature accelerates acidizing process and influences the temperature evolution in porous media significantly.

Poroelastic theory of consolidation for a two-layer system with an upper unsaturated soil and a lower saturated soil under fully permeable boundary conditions

Soils are naturally present in the form of stratified layers, each being distinguishable from adjacent layers by a distinctive set of hydraulic and elasticity properties. Lo et al. (2020) have recently presented a theoretical model of poroelasticity that provides a detailed description of simultaneous solid framework deformation and immiscible fluid flow in a two-layer soil system that is made up of upper unsaturated and lower saturated zones caused by a vertical constant surface load under semi-permeable boundary drainage conditions. Due to the existence of different moisture contents (partial and full saturations) in the upper and lower layers, excess pore fluid pressure does not exhibit a symmetric distribution with respect to depth. Therefore, an exact, analytical solution for fully permeable boundary drainage conditions cannot be determined readily by applying scaling transformations and spatial translations from our recent result even though the coupled model equations possess an invariance feature.In this paper, a well-defined boundary-value problem based on the decoupled model equations of poroelasticity generalized for a two-fluid system is rigorously formulated for a double-layer (unsaturated-saturated) soil profile that allows water drainage at both top and bottom surfaces. Using Laplace time transformation, we derive the closed-form, exact-analytical solution accounting for the intricate interaction between deformable solid matrix and compressible interstitial fluids in such a two-layer system. The solution takes a more complicated form than that for the semi-permeable case. Boundary drainage conditions indeed affect the development of excess pore water and air pressures along with the time evolution of total settlement. We show that when the bottom boundary of the saturated layer overlain by the unsaturated layer with a lower hydraulic conductivity is impervious, the curve describing the relationship between the depth and excess water pressure at the interface has a negative slope, whereas the slope is positive when the boundary is pervious. The total settlement in the lower saturated layer would thus achieve an equilibrium state faster.

Buckley–Leverett Theory for Two-Phase Immiscible Fluids Flow Model with Explicit Phase-Coupling Terms

The theory of two-phase immiscible flow in porous media is based on the extension of single phase models through the concept of relative permeabilities. It mimics Darcy’s law for a fixed average saturation through the introduction of saturation-based permeabilities to model the momentum exchange between the phases. In this paper, we present a model of two-phase flow, based on the extension of Darcy’s law including the effect of capillary pressure, but considering in addition the coupling between the phases modeled through flow cross-terms. In this work, we extend the Buckley–Leverett theory to the subsequent model, and provide numerical experiments shading the light on the effect of the coupling cross-terms in comparison to the classical Darcy’s approach.

Multi-layer flows of immiscible fractional second grade fluids in a rectangular channel

Multi-layer laminar unsteady flows of immiscible fractional second grade fluids in a rectangular channel made by two parallel plates are studied. The fluid motion is produced by the motion of parallel walls in their plane and by the time-dependent pressure gradient in the presence of the linear fluid–fluid interface conditions. The mathematical model is based on the generalized constitutive equations for the shear stress described by the time-fractional Caputo derivative. Integral transforms (finite Fourier sine transform and Laplace transform) have been used to obtain analytical and semi-analytical solutions for velocity, shear stress and the temperature fields. In the case of semi-analytical solutions, the Talbot’s algorithms are used for the inverse Laplace transform. The numerical calculations are carried out with the help of Mathcad software, and the results are illustrated graphically. It has been found that the memory effects have a significant influence on the motion of the fluids.

A Coupled Artificial Compressibility Method for Free Surface Flows

Modeling free surface flows in a CFD context typically requires an incompressible approach along with a formulation to account for the air–water interface. Commonly, pressure-correction algorithms combined with the Volume of Fluid (VOF) method are used to describe these kinds of flows. Pressure-correction algorithms are segregated solvers, which means equations are solved in sequence until convergence is accomplished. On the contrary, the artificial compressibility (AC) method solves a single coupled system of equations. Solving at each timestep a single system of equations obviates the need for segregated algorithms, since all equations converge simultaneously. The goal of the present work is to combine the AC method with VOF formulation and prove its ability to account for unsteady flows of immiscible fluids. The presented system of equations has a hyperbolic nature in pseudo-time, thus the arsenal of the hyperbolic discretization process can be exploited. To this end, a thorough investigation of unsteady flows is presented to demonstrate the ability of the method to accurately describe unsteady flows. Problems of wave propagation on constant and variable bathymetry are considered, as well as a fluid structure interaction problem, where viscous effects have a significant impact on the motion of the structure. In all cases the results obtained are compared with theoretical or experimental data. The straightforward implementation of the method, as well as its accurate predictions, shows that AC method can be regarded as a suitable choice to account for free surface flows.