Binary Fluid Flow Research Articles

Wetting and pressure gradient performance in a lattice Boltzmann color gradient model

An accurate implementation of wetting and pressure drop is crucial to correctly reproduce fluid displacement processes in porous media. Although several strategies have been proposed in the literature, a systematic comparison of them is needed to determine the most suitable for practical applications. Here, we carried out numerical simulations to investigate the performance of two widely used wettability schemes in the lattice Boltzmann color gradient model, namely, the geometrical wetting scheme by Leclaire et al. [Phys. Rev. E 95(3), 033306 (2017)](scheme-I) and the modified direction of the color gradient scheme by Akai et al. [Adv. Water Resour. 116, 56–66 (2018)] (scheme-II). We showed that scheme-II was more accurate in simulating static contact angles of a fluid droplet on a solid surface. However, scheme-I was more accurate in simulating a dynamic case of a binary fluid flow in a horizontal capillary tube described by the Washburn equation. Moreover, we investigated the performance of two popular pressure gradient implementation types. Type-I used the so-called Zou–He pressure boundary conditions at the inlet and the outlet of the domain, while type-II used an external body force as a pressure gradient. We showed that the type-I implementation was slightly more accurate in simulating a neutrally wetting fluid in a horizontal capillary tube described by the Washburn equation. We also investigated the differences between the two types of pressure gradient implementation in simulating two fluid displacement processes in a Bentheimer sandstone rock sample: the primary drainage and the imbibition displacement processes.

Color-gradient-based phase-field equationfor multiphase flow.

In this paper, the underlying problem with the color-gradient (CG) method in handling density-contrast fluids is explored. It is shown that the CG method is not fluid invariant. Based on nondimensionalizing the CG method, a phase-field interface-capturing model is proposed which tackles the difficulty of handling density-contrast fluids. The proposed formulation is developed for incompressible, immiscible two-fluid flows without phase-change phenomena, and a solver based on the lattice Boltzmann method is proposed. Coupled with an available robust hydrodynamic solver, a binary fluid flow package that handles fluid flows with high density and viscosity contrasts is presented. The macroscopic and lattice Boltzmann equivalents of the formulation, which make the physical interpretation of it easier, are presented. In contrast to existing color-gradient models where the interface-capturing equationsare coupled with the hydrodynamic ones and include the surface tension forces, the proposed formulation is in the same spirit as the other phase-field models such as the Cahn-Hilliard and the Allen-Cahn equationsand is solely employed to capture the interface advected due to a flow velocity. As such, similarly to other phase-field models, a so-called mobility parameter comes into play. In contrast, the mobility is not related to the density field but a constant coefficient. This leads to a formulation that avoids individual speed of sound for the different fluids. On the lattice Boltzmann solver side, two separate distribution functions are adopted to solve the formulation, and another one is employed to solve the Navier-Stokes equations, yielding a total of three equations. Two series of numerical tests are conducted to validate the accuracy and stability of the model, where we compare simulated results with available analytical and numerical solutions, and good agreement is observed. In the first set the interfacial evolution equationsare assessed, while in the second set the hydrodynamic effects are taken into account.

Fluid–structure-interactive elasto- and thermo-hydrodynamics of electrokinetic binary fluid flows in compliant micro-confinements

We explore the intricate two-way fluid structure interaction arising due to the flow of a binary system of immiscible Newtonian fluids, composed of upper electrically conducting and lower electrically insulating fluids, flowing within a compliant microchannel, whose walls behave as linear elastic solids. The transport of the fluids along the domain occurs due to the collective impact of pressure gradient and applied electric field. We solve the closed-form system of equations and obtain semi-analytical expressions for the velocity fields and channel wall deformation from the coupled elasto-hydrodynamic problem. We then delineate the effect of four pivotal parameters: (a) Debye–Hückel parameter κ¯, (b) upper wall slip length, ls¯, (c) viscosity ratio, μr, and (d) elasticity ratio, Nr, on the morphological evolution of the wall deformation characteristics and the spatial distribution of the velocity profile of the fluids. Observations establish a positive co-relationship of wall deformation with fluid pressure, showcasing an increased collapsibility with augmented pressure gradients. Consequently, the channel walls show enhanced deformation with a decrease in κ¯, ls¯, μr and with an increase in Nr. We also demonstrate from our model that by properly tuning the applied pressure gradient and electric field, it is possible to achieve counterflow of the two fluids. We also consider the effect of heat generation in the fluids due to viscous dissipation and Joule heating, which dissipates to the surrounding by the mechanism of conjugate heat transfer. Accordingly, we provide semi-analytical expressions for the temperature profile distribution within the channel, and discuss their variation with three thermo-physical parameters: (a) Biot number of the top wall (Bi1), (b) Peclet number of the top fluid (Pe1), and (c) ratio of the thermal conductivities of the upper conducting fluid to that of the upper solid wall (kr2). We establish from our investigation that with the increase in Pe1 and with the decrease in Bi1 and kr2, the overall system temperature enhances. Finally, in order to design a thermally efficient system, we compute the global entropy generation rate in the system and evaluate optimum values of, Pe1, Bi1, and kr2 for which the system exhibits highest second law efficiency. We expect our findings to contribute toward the development of optimized microfluidic devices fabricated from deformable elastic materials.

Binary fluid flow simulations with free energy lattice Boltzmann methods

We use free energy lattice Boltzmann methods to simulate shear and extensional flows of a binary fluid in two and three dimensions. To this end, two classical configurations are digitally twinned, namely a parallel-band device for binary shear flow and a four-roller apparatus for binary extensional flow. The free energy lattice Boltzmann method and the test cases are implemented in the open-source parallel C++ framework OpenLB and evaluated for several non-dimensional numbers. Characteristic deformations are captured, where breakup mechanisms occur for critical capillary regimes. Though the known mass leakage for small droplet-domain ratios and large Cahn numbers is observed, suitable mesh sizes show good agreement to analytical predictions and reference results.

Exact Solutions to the Navier–Stokes Equations with the Boussinesq Approximation for Describing Binary Fluid Flows

Exact Solutions to the Navier–Stokes Equations with the Boussinesq Approximation for Describing Binary Fluid Flows

Modified diffuse interface fluid model and its consistent energy-stable computation in arbitrary domains

Binary fluid flows in irregular domains are common phenomena in natural and industrial fields. To describe these physical processes, we herein present a novel fluid flow-coupled two-material model reflecting different wetting conditions on an arbitrary fluid-solid interface. A modified ternary Cahn–Hilliard (CH) diffuse interface model is adopted to capture the fluid-fluid interface. One of three components is fixed for all time to represent the profile of complex domains. By considering the contact angle condition on solid-fluid interface and equilibrium interface assumption, an extra wetting term is derived. To treat the fluid flow in domains with solid obstacles, we consider the entire system as a porous medium with variable permeability and add a large penalty term into the incompressible Navier–Stokes (NS) equations. It can be proven that the proposed modified Cahn–Hilliard–Navier–Stokes (CHNS) system satisfies the energy dissipation law (energy stability). Based on the scalar auxiliary variable (SAV) approach with appropriate correction techniques, the linear and consistent energy-stable scheme is developed. We introduce the numerical implementation and estimate the discrete version of energy stability in detail. The proposed method can be implemented on regular Cartesian grids with the absence of explicit boundary treatment. Moreover, the calculations are totally decoupled in each time step. Extensive numerical experiments not only indicate the expected accuracy and stability but also show the superior performance in arbitrary domains with different wetting conditions.

Quasi-incompressible models for binary fluid flows in porous media

We derive a quasi-incompressible hydrodynamic phase field model and consistent physical boundary conditions for flows of binary incompressible fluids of distinct intrinsic densities in porous media subject to an external force using the generalized Onsager principle (GOP). When the external force is conservative, the model not only conserves mass and volume fraction but also dissipates the total mechanical energy with respect to the consistent boundary conditions in a fixed domain. In the case of gravity, an extended family of hydrodynamical phase field models parametrized by a specific volume fraction parameter, ϕˆ, is derived via the GOP in the context of the buoyancy force, which reduces to the previous model at ϕˆ=0. In the case of a constant mobility and equal intrinsic densities in the binary fluid system, the extended model is identical to the original one for any ϕˆ. While ϕˆ is chosen as the spatially averaged volume fraction in the general case, the extended model may not be dissipative and thus different from the model at ϕˆ=0. Numerical examples are presented to highlight the difference between the two models and interfacial instability with respect to distinct densities when the two models are distinct.

A phase separation model for binary fluids with hereditary viscosity

We consider a diffuse interface model for an incompressible binary viscoelastic fluid flow. The model consists of the Navier–Stokes–Voigt equations where the instantaneous kinematic viscosity has been replaced by a memory term incorporating hereditary effects coupled with the Cahn–Hilliard equation with Flory–Huggins potential. The resulting system is subject to no‐slip condition for the (volume averaged) fluid velocity and to no‐flux boundary conditions for the order parameter and the chemical potential . We first establish the well‐posedness of the initial and boundary value problem. Then, taking advantage of an Ekman‐type damping, we obtain a dissipative estimate. Thus, we can define a dissipative dynamical system on a suitable phase space. Also, we show that any weak solution is such that regularizes in finite time, and in dimension two, it stays uniformly away from pure phases; that is, the instantaneous strict separation property holds. Finally, we prove the existence of the global attractor, and exploiting the strict separation property, we demonstrate that an exponential attractor exists in dimension two.

Effect of chemical reaction, heat source/sink, and thermal diffusion on transient natural convection through a tube

AbstractA semi‐analytical investigation of the effect of a chemical reaction and thermal diffusion on transient natural convection flow of a viscous incompressible binary fluid with heat source/sink in a vertical and infinite tube is presented. The equations for the model are simplified with the help of the Laplace transform, solved analytically to obtain solutions whose expressions are given in modified Bessel functions. These solutions are then inverted numerically using the Riemann sum approximation method. An exhaustive model analysis is achieved by exhibiting the temperature, concentration, and velocity profiles graphically. Also, numerical values are displayed in tabular form for the Nusselt number, Sherwood number, and the shear stress. Furthermore, a version of the governing equations which are independent of time is solved analytically. The obtained results are found to coincide with the results of the time‐dependent problem at an appreciable value of time. A close examination of the results reveals that raising the generative chemical reaction parameter strengthens both the concentration and momentum of the flow while they are weakened with a surge of the destructive chemical reaction. As the heat source intensifies, both fluid temperature and velocity are elevated while the species concentration is abated with the reversed phenomenon occurring as the heat sink intensifies. An improvement is observed in the concentration and velocity distribution as the Soret parameter increases. This model is useful in petroleum reservoirs, the mining industry, and many industrial processes.

Double-flux model for supercritical multicomponent flows at low Mach numbers with preconditioning method

A double-flux model for preconditioned systems was developed for supercritical multi-component flows under low Mach-number conditions. The mechanism of spurious oscillations in a preconditioned system was classified based on the temporal variation of binary and different temperature fluids in a one-dimensional Euler system. Double-flux models with the original and simplified preconditioning matrices were demonstrated in one-dimensional advection problems, where the latter ignored the partial derivatives of density with respect to the mass fraction for simplicity. The assumption of uniform mass fraction and temperature in the double-flux approach improved the pressure and temperature equilibrium at the material interface, if the simplified preconditioning matrix was used, while the original preconditioning matrix satisfied them by the uniform temperature assumption. A conservation-error analysis was conducted for the one-dimensional advection of the n-dodecane interface using the double-flux model coupled with the preconditioning method. The double flux treatment affected the conservation of not only the total energy but also mass and momentum, because of the preconditioning matrix. The present method enabled the simulation of the advection of methane in an n-dodecane atmosphere and the natural convection of n-dodecane without spurious oscillations, even though these are the binary fluid flow condition or the transcritical conditions. Finally, as a practical application for reactive flows under transcritical conditions, a simulation of n-dodecane with the pyrolysis reaction in a heated circular tube was carried out. Pyrolysis occurred near the heating wall, and the density rapidly changed because of the transition to the supercritical state and the decomposed lower hydrocarbon. When applying the double-flux model, converged results without any spurious oscillations were obtained. The outlet conversion rates were compared with the experimental data, and the present method could reproduce the pyrolysis reaction of n-dodecane in a heated circular pipe.

Electromagnetohydrodynamic bioconvective flow of binary fluid containing nanoparticles and gyrotactic microorganisms through a stratified stretching sheet

Bioconvection has recently been the subject of dispute in a number of biotechnological fields that depend on fluids and their physical properties. When mixed nanofluids are subjected to heat and mass transmission, the process of bioconvection occurs. This attempt conveys the theoretical analysis of two-dimensional electrically conducting and magnetically susceptible binary fluid containing nanoparticles and gyrotactic microorganisms past a stratified stretching surface. Furthermore binary chemical reaction, thermal radiation, and activation energy are taken into assumptions. The analytical solution based on HAM has been performed. The convergence of HAM is presented with the help of figures. The present study is compared with previously published results and has established an excessive agreement which validate the present study. It is perceived that the presence and absence of an electric field influences the variations in fluid velocities due to presence of magnetic field. The micropolar constant heightens the velocity and microrotation of the fluid flow. The buoyancy parameter and bioconvection Rayleigh number diminish the velocity function while these parameters show dual impact on microrotation function. The skin friction and couple stress escalates with the increasing buoyancy ratio parameter and bioconvection Rayleigh number.

Second order linear thermodynamically consistent approximations to nonlocal phase field porous media models

We present a weakly compressible, thermodynamically consistent, hydrodynamical model for the binary fluid flow in porous media using the one-fluid multiple component formulation, consisting of a force balance equation with a weak inertia, weakly perturbed continuity equation, and a volume-preserving, nonlocal Allen–Cahn equation for the phase field variable. This model is called the nonlocal Allen–Cahn-Extended-Darcy (NACED) model. In the incompressible and/or the inertialess limit, it reduces to two limiting models that we focus on in this study: the incompressible, nonlocal Allen–Cahn-Extended-Darcy (NACED) model and nonlocal Allen–Cahn–Darcy (NACD) model, respectively. The weakly compressible model provides a pathway for us to develop thermodynamically consistent numerical algorithms for the incompressible models. Guided by the energy variational formulation of the models, we use the energy quadratization method in time and finite difference method in space on staggered grids to derive second-order, linear, coupled and decoupled, energy dissipation rate preserving schemes for the incompressible models. The decoupled schemes for NACED model are obtained by exploiting the intrinsic relation between the incompressible and weakly compressible model. We then prove that the linear systems resulted from the linear schemes are uniquely solvable. Mesh refinement tests are conducted to confirm the convergence rates and benchmark examples of binary fluid flow motion in porous media with and without gravity are presented to showcase the schemes’ accuracy and usefulness.

Second order linear decoupled energy dissipation rate preserving schemes for the Cahn-Hilliard-extended-Darcy model

The thermodynamically consistent Cahn-Hilliard-Extended-Darcy (CHED) model has been used to describe transient motion of a binary incompressible fluid flow in porous media. In this paper, we develop a series of linear, second-order, energy-dissipation-rate preserving numerical algorithms for the CHED model based on the energy quadratization strategy. We first extend the incompressible CHED model into a weakly compressible, thermodynamically consistent one using the generalized Onsager principle. Guided by the weakly compressible model, we then devise a couple of linear, second-order, decoupled, semi-discrete, temporal algorithms in the form of projection and the energy quadratization (EQ) method. The fully discrete algorithms are obtained by the use of the second-order finite difference method on staggered grids in space. We show theoretically that the obtained numerical algorithms respect the energy-dissipation-rate and the volume conservation property at the discrete level for any time steps, making them unconditionally energy stable. Mesh refinement tests, coarsening dynamics of binary fluids, and the buoyancy-driven binary fluid motion in porous media are investigated numerically. In buoyancy-driven flow simulations, a new set of inflow and outflow boundary conditions are devised using the model. The numerical results compare well with the results in the literature.

A second-order accurate, unconditionally energy stable numerical scheme for binary fluid flows on arbitrarily curved surfaces

In this paper, a second-order temporal and spatial accurate, unconditionally energy stable scheme for the binary fluid flows model on arbitrarily curved surfaces is proposed. We construct a novel surface discrete finite volume method for the surface computation with second-order spatial accuracy. The discretization can be obtained based on the surface mesh consisting of triangular grids. In order to obtain second order temporal accuracy, we apply a Crank–Nicolson-type method to the Cahn–Hilliard–Navier–Stokes system under the projection framework. The resulting system is solved by the Jacobi-type iteration method and bi-conjugate gradient stabilized method. The proposed scheme is proved to be unconditionally energy stable, which implies that a larger time step can be used. Additionally, our scheme has been proved to satisfy mass conservation property. Various numerical experiments are presented to demonstrate the efficiency and robustness of the proposed method.

A phase-field model and its efficient numerical method for two-phase flows on arbitrarily curved surfaces in 3D space

Herein, we present a phase-field model and its efficient numerical method for incompressible single and binary fluid flows on arbitrarily curved surfaces in a three-dimensional (3D) space. An incompressible single fluid flow is governed by the Navier–Stokes (NS) equation and the binary fluid flow is governed by the two-phase Navier–Stokes–Cahn–Hilliard (NSCH) system. In the proposed method, we use a narrow band domain to embed the arbitrarily curved surface and extend the NSCH system and apply a pseudo-Neumann boundary condition that enforces constancy of the dependent variables along the normal direction of the points on the surface. Therefore, we can use the standard discrete Laplace operator instead of the discrete Laplace–Beltrami operator. Within the narrow band domain, the Chorin’s projection method is applied to solve the NS equation, and a convex splitting method is employed to solve the Cahn–Hilliard equation with an advection term. To keep the velocity field tangential to the surface, a velocity correction procedure is applied. An effective mass correction step is adopted to preserve the phase concentration. Computational results such as convergence test, Kevin–Helmholtz instability, and Rayleigh–Taylor instability on curved surfaces demonstrate the accuracy and efficiency of the proposed method.

Structure-Preserving Numerical Approximations to a Non-isothermal Hydrodynamic Model of Binary Fluid Flows

We present two second order, structure-preserving numerical schemes for a newly derived thermodynamically consistent, non-isothermal hydrodynamical phase field model for incompressible binary viscous fluid flows. The schemes preserve the volume of each fluid phase, the total energy and the positive entropy production rate. The entropy quadratization approach is employed to devise the two semi-discrete numerical schemes in time, preserving both the total energy and the positive entropy production rate. The first scheme is weakly nonlinear, which is solved using iterative methods aided by fast Fourier algorithms. The second scheme is linear, in which a time-dependent supplementary variable is added to preserve the positive entropy production rate. The semi-discrete schemes are discretized in space by a finite difference method on staggered grids subsequently to yield two fully discrete schemes. Mesh refinement is carried out to confirm the order of the schemes and several numerical examples are provided to show hydrodynamic as well as thermal effects in resolving thermocapillary convection near the fluid interface in the incompressible binary viscous fluid flow.

Numerical Investigation of Binary Fluid Flow through Propped Fractures by Lattice Boltzmann Method

Abstract In this paper, a pore-scale multiphase, multicomponent Shan–Chen model with proper pressure boundary scheme is developed and implemented to model binary flows in porous media, which enable...

Mixed Oscillation Flow of Binary Fluid with Minus One Capillary Ratio in the Czochralski Crystal Growth Model

This work presented a series of three-dimensional unsteady numerical simulations on the characteristics of the mixed oscillation flows of binary mixture in a Czochralski crystal growth model. The silicon-germanium melt is investigated and the capillary ratio is minus one. The simulation results showed that, for the special capillary ratio, the thermal and solutocapillary forces are imposed in opposite directions and counteract each other. With the effect of buoyancy, the balance between the capillary forces is disturbed. Mixed with the forced convection driven by rotation, the capillary-buoyancy convection is complex. The basic mixed flow streamlines are presented as various rolling cells. The directions of the rolls are dependent on the combinations of surface and body forces. With the increase of temperature gradient, the basic flow stability is broken, and the oscillations occur. The crucible rotation has an effective influence on the stability enhancement. However, affected by the crystal rotation, the critical condition experiences an increase to a turning point, and then undergoes a sharp reduction to zero. Once the instability is incubated, the surface oscillations are analyzed. For the three-dimensional steady flow, only spatial oscillations are observed circumferentially, and the surface patterns of spokes, rosebud, and pulsating ring are obtained. For the unsteady oscillation flow, the spiral hydrosoultal waves, rotating waves, and superimposition of spirals and spokes are observed, and the oscillation behaviors are also discussed.

Thermodynamically consistent spatial discretization of the one-dimensional regularized system of the Navier–Stokes–Cahn–Hilliard equations

This paper is devoted to the construction of a thermodynamically consistent (with non-increasing total energy on the discrete level) semi-discrete approximation (continuous in time and discrete in space) for the one-dimensional regularized system of the Navier–Stokes–Cahn–Hilliard equations that describes a compressible isothermal flow of two-phase binary fluid with surface effects. Thermodynamical consistency is achieved using special form of momentum equation with surface force term written in so called potential form. Results of numerical study are presented.

Stability of Stationary Plane-Parallel Flow of Binary Fluid with the Soret Effect in Vertical Layer with Differentially Heated Boundaries

Stability of stationary plane-parallel flow of binary mixture with the Soret effect in vertical layer with differentially heated boundaries is studied. Main attention is paid to the thermal wave instability mode. The dependence of the threshold value of Prandtl number Pr∗, at which the growing thermal wave appear, on the separation ratio e is obtained. Complex scenario of stability change with respect to thermal waves is discovered for the case of positive Soret effect: it is shown that for e Pr∗, with the increase in separation ratio, additional range of growing thermal waves appears near Pr = 0, for e > 0.2634 the thermal wave instability exists at $$ 0\le \mathit{\Pr}\le {\mathit{\Pr}}_1^{\ast } $$ and at $$ \mathit{\Pr}>{\mathit{\Pr}}_2^{\ast } $$. For e > 0.278 the thermal wave instability is possible at any values of Prandtl number.