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Three-phase flows through a pore network of Berea sandstone are studied numerically under critical interfacial tension condition. Results show that the relative permeability of each fluid increases as its own saturation increases. The specific interfacial length between wetting and non-wetting fluids monotonously decreases with increasing the saturation of intermediate-wetting fluid, while the other two specific interfacial lengths exhibit a non-monotonous variation. As the wetting (non-wetting) fluid becomes less wetting (non-wetting), the relative permeability of wetting fluid monotonously increases, while the other two relative permeabilities show a non-monotonous trend. Due to the presence of spreading layer, the specific interfacial length between wetting and non-wetting fluids always stabilizes at a low level. As the viscosity ratio of wetting (non-wetting) to intermediate-wetting fluids increases, the relative permeability of wetting (non-wetting) fluid increases. With the viscosity ratio deviating from unity, the phase interfaces become increasingly unstable, leading to an increased specific interfacial length.

A numerical method for simulating three-phase flows with moving contact lines on arbitrarily complex surfaces is developed in the framework of lattice Boltzmann method. In this method, the immiscible three-phase flow is modeled through a multiple-relaxation-time color-gradient model, which not only allows for a full range of interfacial tensions but also produces stable outcomes for a wide range of viscosity ratios. A characteristic line model is introduced to implement the wetting boundary condition, which is not only easy to implement but also able to handle arbitrarily complex boundaries with prescribed contact angles. The developed method is first validated by the simulation of a Janus droplet resting on a flat surface, a perfect Janus droplet deposited on a cylinder, and the capillary intrusion of ternary fluids for various viscosity ratios. It is then used to study a compound droplet subject to a uniform incoming flow passing through a multi-pillar structure, where three different values of surface wettability are considered. The simulated results show that the surface wettability has significant impact on the droplet dynamic behavior and final fluid distribution.

A deep understanding of two-phase displacement in porous media with permeability contrast is essential for the design and optimisation of enhanced oil recovery processes. In this paper, we investigate the forced imbibition behaviour in two dual-permeability geometries that are of equal permeability contrast. First, a mathematical model is developed for the imbibition in a pore doublet, which shows that the imbibition dynamics can be fully described by the viscosity ratio $\lambda$ and capillary number $Ca_m$ which creatively incorporates the influence of channel width and length. Through the finite difference solution of the mathematical model, a $\lambda-Ca_m$ phase diagram is established to characterise the imbibition preference in the pore doublet. We then investigate the imbibition process in a dual-permeability pore network using a well-established lattice Boltzmann method, focusing on the competition between the viscous and capillary forces. Like in the pore doublet, the preferential imbibition occurs in high permeability zone at high $Ca_{m}$ but in low permeability zone at low $Ca_{m}$. When $Ca_m$ is not sufficiently high, an oblique advancing pattern is observed which is attributed to non-trivial interfacial tension. Thanks to the newly defined capillary number, the critical $Ca_{m}$ curve on which the breakthrough simultaneously occurs in both permeability zones, is found to match perfectly with that from the pore doublet and it is the optimal condition for maximising the imbibition efficiency in the entire pore network., Comment: 11 figures, 21 pages

Although the thermal conductivity of molecular gases can be measured straightforwardly and accurately, it is difficult to experimentally determine its separate contributions from the translational and internal motions of gas molecules. Yet this information is critical in rarefied gas dynamics as the rarefaction effects corresponding to these motions are different. In this paper, we propose a novel methodology to extract the translational thermal conductivity (or equivalently, the translational Eucken factor) of molecular gases from the Rayleigh-Brillouin scattering (RBS) experimental data. From the numerical simulation of the \cite{LeiJFM2015} model we find that, in the kinetic regime, in addition to bulk viscosity, the RBS spectrum is sensitive to the translational Eucken factor, even when the total thermal conductivity is fixed. Thus it is not only possible to extract the bulk viscosity, but also the translational Eucken factor of molecular gases from RBS light scattering spectra measurements. Such experiments bear the additional advantage that gas-surface interactions do not affect the measurements. For the first time, bulk viscosities (due to the rotational relaxation of gas molecules only) and translational Eucken factors of N2, CO2 and SF6 are simultaneously extracted from RBS experiments.

Double emulsion formation in a hierarchical flow-focusing channel is systematically investigated using a free energy ternary lattice Boltzmann model. A three dimensional formation regime diagram is constructed based on the capillary numbers of the inner ($Ca_i$), middle ($Ca_m$) and outer ($Ca_o$) phase fluids. The results show that the formation diagram can be classified into periodic two-step region, periodic one-step region, and non-periodic region. By varying $Ca_i$ and $Ca_m$ in the two-step formation region, different morphologies are obtained, including the regular double emulsions, decussate regimes with one or two alternate empty droplets, and structures with multiple inner droplets contained in the continuous middle phase thread. Bidisperse behaviors are also frequently encountered in the two-step formation region. In the periodic one-step formation region, scaling laws are proposed for the double emulsion size and for the size ratio between the inner droplet and the overall double emulsion. Furthermore, we show that the interfacial tension ratio can greatly change the morphologies of the obtained emulsion droplets, and the channel geometry plays an important role in changing the formation regimes and the double emulsion sizes. In particular, narrowing the side inlets or the distance between the two side inlets promotes the conversion from the two-step formation regime to the one-step formation regime., Comment: 14 figures; Accepted for publication in the Journal of Fluid Mechanics

In this work, a novel synthetic iteration scheme (SIS) is developed for the LBE to find solutions to Kramer's problem accurately and efficiently: the velocity distribution function is first solved by the conventional iterative scheme, then it is modified such that in each iteration i) the flow velocity is guided by an ordinary differential equation that is asymptotic-preserving at the Navier-Stokes limit and ii) the shear stress is equal to the average shear stress. Based on the Bhatnagar-Gross-Krook model, the SIS is assessed to be efficient and accurate. Then we investigate the Kramer's problem for gases interacting through the inverse power-law, shielded Coulomb, and Lennard-Jones potentials, subject to diffuse-specular and Cercignani-Lampis gas-surface boundary conditions. When the tangential momentum accommodation coefficient (TMAC) is not larger than one, the Knudsen layer function is strongly affected by the potential, where its value and width increase with the effective viscosity index of gas molecules. Moreover, the Knudsen layer function exhibits similarities among different values of TMAC when the intermolecular potential is fixed. For Cercignani-Lampis boundary condition with TMAC larger than one, both the viscous slip coefficient and Knudsen layer function are affected by the intermolecular potential, especially when the "backward" scattering limit is approached. With the asymptotic theory by Jiang and Luo for the singular behavior of the velocity gradient in the vicinity of the solid surface, we find that the whole Knudsen layer function can be well fitted by power series., Comment: 28pages, 10 figures

We propose a lattice Boltzmann color-gradient model for immiscible ternary fluid flows, which is applicable to the fluids with a full range of interfacial tensions, especially in near-critical and critical states. An interfacial force for N-phase systems is derived based on the previously developed perturbation operator and is then introduced into the model using a body force scheme, which helps reduce spurious velocities. A generalized recoloring algorithm is applied to produce phase segregation and ensure immiscibility of three different fluids, where a novel form of segregation parameters is proposed by considering the existence of Neumann's triangle and the effect of equilibrium contact angle in three-phase junction. The proposed model is first validated with three typical examples, namely the interface capturing for two separate static droplets, the Young-Laplace test for a compound droplet, and the spreading of a droplet between two stratified fluids. This model is then used to study the structure and stability of double droplets in a static matrix. Consistent with the theoretical stability diagram, seven possible equilibrium morphologies are successfully reproduced by adjusting two ratios of the interfacial tensions. By simulating Janus droplets in various geometric configurations, the model is shown to be accurate when three interfacial tensions satisfy a Neumann's triangle. In addition, we also simulate the near-critical and critical states of double droplets where the outcomes are very sensitive to the model accuracy. Our results show that the present model is advantageous to three-phase flow simulations, and allows for accurate simulation of near-critical and critical states., Comment: 24 pages, 11 figures

An improved wetting boundary implementation strategy is proposed based on lattice Boltzmann color-gradient model in this paper. In this strategy, an extra interface force condition is demonstrated based on the diffuse interface assumption and is employed in contact line region. It has been validated by three benchmark problems: static droplet wetting on a flat surface and a curved surface, and dynamic capillary filling. Good performances are shown in all three cases. Relied on the strict validation to our scheme, the viscous fingering phenomenon of immiscible fluids displacement in a two-dimensional channel has been restudied in this paper. High viscosity ratio, wide range contact angle, accurate moving contact line and mutual independence between surface tension and viscosity are the obvious advantages of our model. We find the linear relationship between the contact angle and displacement velocity or variation of finger length. When the viscosity ratio is smaller than 20, the displacement velocity is increasing with increasing viscosity ratio and reducing capillary number, and when the viscosity ratio is larger than 20, the displacement velocity tends to a specific constant. A similar conclusion is obtained on the variation of finger length., Comment: 24 pages, 11 figures

Rooted from the gas kinetics, the lattice Boltzmann method is a powerful tool in modeling hydrodynamics. In the past decade, it has been extended to simulate the rarefied gas flow beyond the Navier-Stokes level, either by using the high-order Gauss-Hermite quadrature, or by introducing the relaxation time that is a function of the gas-wall distance. While the former method, with a limited number of discrete velocities (i.e. D2Q36), is accurate up to the early transition flow regime, the latter method, with the same discrete velocities as that used in simulating hydrodynamics (i.e. D2Q9), is accurate up to the free-molecular flow regime in the Poiseuille flow between two parallel plates. This is quite astonishing in the sense that more discrete velocities are less accurate. In this paper, by solving the Bhatnagar-Gross-Krook kinetic equation accurately via the discrete velocity method, we find that the accuracy of the lattice Boltzmann method is reduced significantly in the simulation of rarefied gas flows through the rough surface and porous media. Our simulation results could serve as benchmarking cases for future development of the lattice Boltzmann method for modeling and simulation of rarefied gas flows in complex geometries.

An iterative scheme can be used to find a steady-state solution to the Boltzmann equation, however, it is very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion equation which is asymptotic-preserving in the Navier-Stokes limit. The efficiency of the new scheme is verified by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum flow regimes. Moreover, due to the asymptotic-preserving properties, the SIS needs less spatial resolution in the near-continuum flow regimes, which makes it even faster than the conventional iterative scheme. Using this synthetic iterative scheme, and the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections, and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard-Jones potential for the first time, and the difference between these results and those using hard-sphere intermolecular potential is discussed., Comment: 13 figs, 5 tables

Over the last two decades, lattice Boltzmann methods have become an increasingly popular tool to compute the flow in complex geometries such as porous media. In addition to single phase simulations allowing, for example, a precise quantification of the permeability of a porous sample, a number of extensions to the lattice Boltzmann method are available which allow to study multiphase and multicomponent flows on a pore scale level. In this article we give an extensive overview on a number of these diffuse interface models and discuss their advantages and disadvantages. Furthermore, we shortly report on multiphase flows containing solid particles, as well as implementation details and optimization issues., Comment: 50 pages, 13 figures in Computational Geosciences (2016)