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Metastasis-associated lung adenocarcinoma transcript 1 (MALAT1), a long non-coding RNA (lncRNA), has been confirmed to recruit enhancer of zeste 2 polycomb repressive complex 2 subunit (EZH2) to regulate cardiomyocyte apoptosis in diabetic cardiomyopathy. However, whether the similar regulatory axis exists in sepsis-induced myocardial dysfunction (SIMD) has not been clearly established. The current study sought to define the mechanism governing MALAT1-mediated EZH2 in SIMD. MALAT1 was significantly upregulated in lipopolysaccharide-induced cardiomyocytes. Depletion of MALAT1 by caudal vein injection of small interfering RNA targeting MALAT1 alleviated myocardial injury in SIMD rats, restored cardiac function, reduced oxidative stress production and fibrosis, and inhibited inflammatory factors and apoptosis in myocardial tissues. Moreover, MALAT1 bound to EZH2 and promoted EZH2 activity in the nucleus of cardiomyocytes. EZH2 repressed ubiquitin-specific peptidase 22 (USP22) expression through H3K27me3 modification. EZH2 elevation aggravated the cardiac injury in SIMD rats, while USP22 upregulation inhibited the effect of EZH2, which reduced the cardiac injury in SIMD rats. Taken together, MALAT1 decreased USP22 expression by interacting with EZH2, thereby worsening SIMD, highlighting an attractive therapeutic strategy for SIMD.

The discrete unified gas kinetic scheme (DUGKS) combines the advantages of both the unified gas kinetic scheme (UGKS) and the lattice Boltzmann method. It can adopt the flexible meshes, meanwhile, the flux calculation is simple. However, the original DUGKS is proposed for the compressible flows. When we try to solve a problem governed by the incompressible Navier-Stokes (N-S) equations, the original DUGKS may bring some undesirable errors because of the compressible effect. To eliminate the compressible effect, the DUGKS for incompressible N-S equations is developed in this work. In addition, the Chapman-Enskog analysis ensures that the present DUGKS can solve the incompressible N-S equations exactly, meanwhile, a new non-extrapolation scheme is adopted to treat the Dirichlet boundary conditions. To test the present DUGKS for incompressible N-S equations, four problems are adopted. The first one is a periodic problem driven by an external force, which is used to test the influences of Courant–Friedrichs–Lewy condition number and the M a c h number (Ma). Besides, some comparisons between the present DUGKS and some available results are also conducted. The second problem is Womersley flow, it is also used to test the influence of Ma, and the results show that the compressible effect is reduced obviously. Then, the two-dimensional lid-driven cavity flow is considered. In these simulations, the Reynolds number is varied from 400 to 1000000 to illustrate the accuracy, stability and efficiency of the present DUGKS. Finally, the numerical solutions of the three-dimensional lid-driven cavity flow suggest that the present DUGKS is suitable for the three-dimensional problems.

In this work, we propose a lattice Boltzmann method to model the electrohydrodynamic (EHD) conduction phenomenon in dielectric liquids. To incorporate the effect of the source term appeared in the charge conservation equations, some unified force terms that can be implemented locally are delicately designed in the charge evolution equations. We further validate the present model against some analytical solutions as well as experimental data for several problems, and a good agreement is obtained. Then it is used to study the fluid circulation in a square cavity generated by EHD conduction phenomenon, and the numerical results indicate that the geometric scale and the difference in the mobilities of the positive and negative charges could play an important role in studying EHD conduction phenomenon. Further, to analyze the characteristics of the EHD conduction mechanisms in the ohmic and saturation regimes, the influences of the conduction number W 0 as well as the parameter β , which is related to Onsager effect, are also investigated, and the numerical results fit well with previous work.

In this paper, a kind of finite-difference lattice Boltzmann method with the second-order accuracy of time and space (T2S2-FDLBM) is proposed. In this method, a two-stage time-accurate discretization approach is applied to construct time marching scheme, and the spatial gradient operator is discretized by a mixed difference scheme to maintain a second-order accuracy in space. It is shown that the previous finite-difference lattice Boltzmann method (FDLBM) (Guo and Zhao, 2003) is a special case of the T2S2-FDLBM. Through the von Neumann analysis, the stability of the method is analyzed and two specific T2S2-FDLBMs are discussed. The two T2S2-FDLBMs are applied to simulate some incompressible flows with the non-uniform grids. Compared with other high order FDLBMs, this two T2S2-FDLBMs keep the simplicity of standard lattice Boltzmann method. Compared with the previous FDLBM and lattice Boltzmann method, the two T2S2-FDLBMs are more accurate and more stable. The value of the Courant–Friedrichs–Lewy condition number in our method can be up to 0.9, which also significantly improves the computational efficiency.

The phenomena of capillary rise with the effect of displaced fluids are ubiquitous in both science and engineering, and the mathematical models and their analytical solutions of this problem have also received increasing attention. In this paper, a theoretical study on the rising dynamics of non-Newtonian power-law fluids in a capillary is performed, and the classical Lucas–Washburn equation is generalized to a nonlinear second-order differential equation in which the effects of the power-law index and displaced fluids are included. We analyze the imbibition behaviors of power-law fluids under the influence of displaced fluids in details, and also present some analytical solutions of different special cases and different time stages. The results show that for different special cases, it takes shorter time for shear thickening fluid to reach the equilibrium height. For different time stages, however, the rising phenomena of power-law fluids are more complex, and the average time for the shear-thinning fluid to reach the equilibrium height is longer compared to shear-thickening fluid, but an opposite phenomenon is observed for the case of μ 1 , 0 / μ 2 , 0 = 100 (here μ1,0/μ2,0 is the viscosity ratio of two power-law fluids) in viscous time stage. In inertial time stages, the density ratio of the absorbed fluid to the displaced fluid also has a significant effect on the rising dynamics of imbibing fluid. Furthermore, the effect of dynamic contact angle is also included in the governing equation and analytical solutions. Through a comparison between the theoretical results and experimental data, a good agreement is observed. These results can be used as a priori for liquid absorption in industrial applications, including oil recovery, fuel cells and water collection of artificial silk.

In this paper, a generalized lattice Boltzmann (LB) model with a source term in the continuity equation is proposed to solve both incompressible and nearly incompressible Navier–Stokes (N–S) equations. This model can be used to deal with single-phase and two-phase flows problems with a source term in the continuity equation. From this generalized model, we can not only get some existing models, but also derive new models. Moreover, for the incompressible model derived, a modified pressure scheme is introduced to calculate the pressure, and then to ensure the accuracy of the model. In this work, we will focus on a two-phase flow system, and in the frame work of our generalized LB model, a new phase-field-based LB model is developed for incompressible and quasi-incompressible two-phase flows. A series of numerical simulations of some classic physical problems, including a spinodal decomposition, a static droplet, a layered Poiseuille flow, and a bubble rising flow under buoyancy, are performed to validate the developed model. Besides, some comparisons with previous quasi-incompressible and incompressible LB models are also carried out, and the results show that the present model is accurate in the study of two-phase flows. Finally, we also conduct a comparison between quasi-incompressible and incompressible LB models for two-phase flow problems, and find that in some cases, the proposed quasi-incompressible LB model performs better than incompressible LB models.

In this paper, a multiple-relaxation-time finite-difference lattice Boltzmann method (MRT-FDLBM) is developed for the nonlinear convection-diffusion equation (NCDE). Through designing the equilibrium distribution function and the source term properly, the NCDE can be recovered exactly from MRT-FDLBM. We also conduct the von Neumann stability analysis on the present MRT-FDLBM and its special case, i.e., single-relaxation-time finite-difference lattice Boltzmann method (SRT-FDLBM). Then, a simplified version of MRT-FDLBM (SMRT-FDLBM) is also proposed, which can save about 15% computational cost. In addition, a series of real and complex-value NCDEs, including the isotropic convection-diffusion equation, Burgers-Fisher equation, sine-Gordon equation, heat-conduction equation, and Schrödinger equation, are used to test the performance of MRT-FDLBM. The results show that both MRT-FDLBM and SMRT-FDLBM have second-order convergence rates in space and time. Finally, the stability and accuracy of five different models are compared, including the MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method [H. Wang, B. Shi et al., Appl. Math. Comput. 309, 334 (2017)10.1016/j.amc.2017.04.015], and the lattice Boltzmann method (LBM). The stability tests show that the sequence of stability from high to low is as follows: MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method, and LBM. In most of the precision test results, it is found that the order from high to low of precision is MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, and the previous finite-difference lattice Boltzmann method.

In this paper, we present an improved phase-field-based lattice Boltzmann (LB) method for thermocapillary flows with large density, viscosity, and thermal conductivity ratios. The present method uses three LB models to solve the conservative Allen-Cahn equation, the incompressible Navier-Stokes equations, and the temperature equation. To overcome the difficulty caused by the convection term in solving the convection-diffusion equation for the temperature field, we first rewrite the temperature equation as a diffuse equation where the convection term is regarded as the source term and then construct an improved LB model for the diffusion equation. The macroscopic governing equations can be recovered correctly from the present LB method; moreover, the present LB method is much simpler and more efficient. In order to test the accuracy of this LB method, several numerical examples are considered, including the planar thermal Poiseuille flow of two immiscible fluids, the two-phase thermocapillary flow in a nonuniformly heated channel, and the thermocapillary Marangoni flow of a deformable bubble. It is found that the numerical results obtained from the present LB method are consistent with the theoretical prediction and available numerical data, which indicates that the present LB method is an effective approach for the thermocapillary flows.

In this paper, we present a brief overview of the phase-field-based lattice Boltzmann method (LBM) that is a distinct and efficient numerical algorithm for multiphase flow problems. We first give an introduction to the mathematical theory of phase-field models for multiphase flows, and then present some recent progress on the LBM for the phase-field models which are composed of the classic Navier-Stokes equations and the Cahn-Hilliard or Allen-Cahn equation. Finally, some applications of the phase-field-based LBM are also discussed. Cited as : Wang, H., Yuan, X., Liang, H., Chai, Z., Shi, B. A brief review of the phase-field-based lattice Boltzmann method for multiphase flows. Capillarity, 2019, 2(3): 33-52, doi: 10.26804/capi.2019.03.01

In this paper, an improved lattice Boltzmann model that incorporates the effect of heat capacity is adopted to simulate the conjugate natural convection in a square enclosure with a circular cylinder, and the square cavity considered is bounded by four finite thickness and conductive walls. Influences of Rayleigh number, wall thermophysical properties, and wall thickness on streamlines, isotherms and Nusselt number are analyzed. It is found that the heat transfer rate obtained for different wall thicknesses is usually smaller than that for the zero wall thickness, while this difference is not apparent for a larger thermal conductivity ration. Additionally, the study also shown that the increase of the outer wall thickness tends to decrease the heat transfer rate. The effects of the geometric parameter, including aspect ratio and vertical cylinder location, on heat transfer are also studied.

The motion of gravity-driven deformable droplets passing through a confining orifice in two-dimensional ( 2 D ) space is numerically studied by the phase-field-based multiple-relaxation-time (MRT) lattice Boltzmann (LB) model, and the ratio of orifice-to-droplet diameter is less than 1. Droplets are placed just above a sink with an orifice in the middle, accelerate under gravity and encounter the orifice plate. In this work, we mainly consider the effects of the Bond number ( B o ), orifice-to-droplet diameter ratio ( r = d ∕ D ), plate thickness ( H t ), wettability (or contact angle) and the diameter ratio of two droplets ( r d = D 1 ∕ D 2 ) on the dynamic behavior of droplet through the orifice. The results show that these issues have great influences on the typical flow patterns (i.e., release and capture). With the decrease of contact angle, the droplet is more easily captured, and there exists a critical equilibrium contact angle θ e q when the Bond number and the orifice-to-droplet diameter ratio as well as the thickness of the plate are specified. For the case with θ > θ e q , the droplet can finally pass through the orifice, otherwise, the droplet cannot pass through the orifice. In addition, the droplet is more likely to pass through the orifice as the thickness of the obstacle increases. Actually, when the obstacle thickness is large enough, droplet breaks into three segments and a liquid slug is formed in a hydrophilic orifice. Finally, for the evolution of two droplets with a larger diameter ratio ( r d = 1 . 0 ), the combined droplet finally passes through the orifice due to greater inertia than the cases with r d = 0 and r d = 0 . 43 . Besides, we also establish the relation r = 0 . 5 7 2 3 B o − 1 3 which can be used to separate droplet release from capture at H t = 1 . 2 m m .

In this paper, the Buongiorno’s two-phase model is adopted to study natural convection in a partially heated enclosure filled with nanofluids having temperature-dependent properties. In order to solve the governing equations, a hybrid lattice Boltzmann (LB) and total variation diminishing (TVD) scheme is proposed, in which the traditional LB method is employed to handle the flow and temperature fields, while the volume fraction equation is solved by a TVD method due to the fact that the corresponding equation is a type of convection-dominated equation. Additionally, to improve the computational efficiency, the proposed algorithm is executed on the Graphics Processing Unit (GPU) by using “Compute Unified Device Architecture (CUDA)” programming. The effect of several parameters, such as Rayleigh number, nanoparticle diameter, temperature difference between the sidewalls, heating location and heater length on heat transfer rate and nanoparticle distribution are analyzed. It is observed that at low Rayleigh numbers, the heat transfer enhancement increases in nanoparticle volume fraction, while at high Rayleigh numbers, there exists an optimal volume fraction at which the heat transfer performance has a peak. Moreover, the average Nusselt number and the heat transfer enhancement are found to decrease with increasing heater length.

In this paper, a lattice Boltzmann (LB) model with double distribution functions is proposed for two-phase flow in porous media where one distribution function is used for pressure governed by the Poisson equation, and the other is applied for saturation evolution described by the convection-diffusion equation with a source term. We first performed a Chapman-Enskog analysis, and show that the macroscopic nonlinear equations for pressure and saturation can be recovered correctly from present LB model. Then in the framework of LB method, we develop a local scheme for pressure gradient or equivalently velocity, which may be more efficient than the nonlocal second-order finite-difference schemes. We also perform some numerical simulations, and the results show that the developed LB model and local scheme for velocity are accurate and also have a second-order convergence rate in space. Finally, compared to the available pore-scale LB models for two-phase flow in porous media, the present LB model has more potential in the study of the large-scale problems., Comment: 40 pages, 18 figures

In this paper, the effects of temperature-dependent properties on natural convection of nanofluids in rectangular cavities with sinusoidal temperature distribution are investigated in detail with lattice Boltzmann method. To improve the computational efficiency, all simulations are performed on the Graphics Processing Unit (GPU) using NVIDIA’s CUDA. The fluid in the enclosure is a water-based nanofluid containing Al 2 O 3 nanoparticles. The effects of power-law index ( 0.5 ⩽ n ⩽ 1.5 ), thermal Rayleigh number ( 10 4 ⩽ Ra f ⩽ 10 6 ), diameter of nanoparticle ( 25 nm ⩽ d s ⩽ 100 nm ), nanoparticle volume fraction ( 0.0 ⩽ ϕ ⩽ 0.04 ), temperature of the cooled sidewall ( 315 K ⩽ T c ⩽ 335 K ), temperature difference between the sidewalls ( 10 K ⩽ Δ T ⩽ 50 K ), amplitude ratio ( 0.0 ⩽ A ⩽ 1.0 ), wave number ( 0.0 ⩽ ω ⩽ 6.0 ), phase deviation ( 0.0 ⩽ θ ⩽ π ) and aspect ratio ( 0.250 ⩽ AR ⩽ 4.00 ) on heat and fluid flows are investigated. The results reveal that there is an optimal volume fraction ϕ opt at which the maximum heat transfer enhancement is obtained, and the value of ϕ opt is found to increase slightly with decreasing the nanoparticle diameter, and to increase remarkably with increasing the temperature of T c or Δ T . In addition, the average Nusselt number is generally decreased with increasing power-law index, while increased with increasing A and ω . Further, we found that the average Nusselt number behaves nonlinearly with the phase deviation parameter. Moreover, the present results also indicate that there is an optimal value of aspect ratio at which the impact of AR on heat transfer is the most pronounced.

A three-dimensional numerical study is conducted to investigate the effect of heat-flux source location on the melting of phase change material (PCM) in a cubical cavity. An enthalpy-based incompressible thermal lattice Boltzmann model (iTLBM) has been established and validated by two benchmark problems. In order to reveal the effect of heat-flux source location in melting rate in three-dimensional space, both horizontal and vertical locations are considered. The numerical results show that the size of local heater and its heat flux play an essential role in melting rate. Furthermore, the variation of the location of the heat-flux source from both sides to the middle region of the cavity leads to the increase of the total heat flux on the interface, which also results in the increase of melting rate. Finally, we also find that the melting rate increases as the location of the heat-flux source is changed from top to the bottom region of the cavity, which is attributed to the enhancement of convection heat transfer.

In this paper, a multiple-distribution-function lattice Boltzmann method (MDF-LBM) with multiple-relaxation-time model is proposed for incompressible Navier-Stokes equations (NSEs) which are considered as the coupled convection-diffusion equations (CDEs). Through direct Taylor expansion analysis, we show that the Navier-Stokes equations can be recovered correctly from the present MDF-LBM, and additionally, it is also found that the velocity and pressure can be directly computed through the zero and first-order moments of distribution function. Then in the framework of present MDF-LBM, we develop a locally computational scheme for the velocity gradient where the first-order moment of the non-equilibrium distribution is used, this scheme is also extended to calculate the velocity divergence, strain rate tensor, shear stress and vorticity. Finally, we also conduct some simulations to test the MDF-LBM, and find that the numerical results not only agree with some available analytical and numerical solutions, but also have a second-order convergence rate in space., Comment: 32 pages, 18 figures

In this work, we consider a general consistent and conservative phase-field model for the incompressible two-phase flows. In this model, not only the Cahn-Hilliard or Allen-Cahn equation can be adopted, but also the mass and the momentum fluxes in the Navier-Stokes equations are reformulated such that the consistency of reduction, consistency of mass and momentum transport, and the consistency of mass conservation are satisfied. We further develop a lattice Boltzmann (LB) method, and show that through the direct Taylor expansion, the present LB method can correctly recover the consistent and conservative phase-field model. Additionally, if the divergence of the extra momentum flux is seen as a force term, the extra force in the present LB method would include another term which has not been considered in the previous LB methods. To quantitatively evaluate the incompressibility and the consistency of the mass conservation, two statistical variables are introduced in the study of the deformation of a square droplet, and the results show that the present LB method is more accurate. The layered Poiseuille flow and a droplet spreading on an ideal wall are further investigated, and the numerical results are in good agreement with the analytical solutions. Finally, the problems of the Rayleigh-Taylor instability, a single rising bubble, and the dam break with the high Reynolds numbers and/or large density ratios are studied, and it is found that the present consistent and conservative LB method is robust for such complex two-phase flows.

In this paper, we first present a multiple-relaxation-time lattice Boltzmann (MRT-LB) model for one-dimensional diffusion equation where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is considered. Then through the theoretical analysis, we derive an explicit four-level finite-difference scheme from this MRT-LB model. The results show that the four-level finite-difference scheme is unconditionally stable, and through adjusting the weight coefficient $\omega_{0}$ and the relaxation parameters $s_1$ and $s_2$ corresponding to the first and second moments, it can also have a sixth-order accuracy in space. Finally, we also test the four-level finite-difference scheme through some numerical simulations, and find that the numerical results are consistent with our theoretical analysis., Comment: 20 pages, 5 figures

In this paper, we develop an efficient lattice Boltzmann (LB) model for simulating immiscible incompressible $N$-phase flows $(N \geq 2)$ based on the Cahn-Hilliard phase field theory. In order to facilitate the design of LB model and reduce the calculation of the gradient term, the governing equations of the $N$-phase system are reformulated, and they satisfy the conservation of mass, momentum and the second law of thermodynamics. In the present model, $(N-1)$ LB equations are employed to capture the interface, and another LB equation is used to solve the Navier-Stokes (N-S) equations, where a new distribution function for the total force is delicately designed to reduce the calculation of the gradient term. The developed model is first validated by two classical benchmark problems, including the tests of static droplets and the spreading of a liquid lens, the simulation results show that the current LB model is accurate and efficient for simulating incompressible $N$-phase fluid flows. To further demonstrate the capability of the LB model, two numerical simulations, including dynamics of droplet collision for four fluid phases and dynamics of droplets and interfaces for five fluid phases, are performed to test the developed model. The results show that the present model can successfully handle complex interactions among $N$ ($N \geq 2$) immiscible incompressible flows.

In this paper, a diffuse-interface lattice Boltzmann method (DI-LBM) is developed for fluid-particle interaction problems. In this method, the sharp interface between the fluid and solid is replaced by a thin but nonzero thickness transition region named diffuse interface, where the physical variables change continuously. In order to describe the diffuse interface, we introduce a smooth function, which is similar to the order parameter in the phase-field model or the volume fraction of solid phase in the partially saturated lattice Boltzmann method (PS-LBM). In addition, to depict the fluid-particle interaction more accurately, a modified discrete force term is also proposed and included in the evolution equation of the DI-LBM. Some classical problems are used to test the DI-LBM, and the results are in good agreement with some available theoretical and numerical data. Finally, it is also found that the DI-LBM is more efficient and accurate than the PS-LBM with the superposition model.

The local and nonlocal Allen-Cahn equations (ACEs) have received increasing attention in the study of the complicated interfacial problems. In this paper, we conduct a comparison between local and nonlocal ACEs with the property of mass conservation in the framework of lattice Boltzmann (LB) method. To this end, we first propose two simple multiple-relaxation-time LB models for local and nonlocal ACEs, and through the Chapman-Enskog expansion, the local and nonlocal ACEs can be recovered correctly from the developed LB models. Then we test these two LB models with several examples, and the numerical results show that the developed LB models are accurate and also have a second-order convergence rate in space. Finally, a comparison between the local and nonlocal ACEs is also performed in terms of mass conservation, stability and accuracy. The results show that both local and nonlocal ACEs can preserve mass conservation of system and each phase. And additionally, it is also found that the local ACE is more accurate than nonlocal ACE in capturing the interface profile, but the latter is more stable than the former.

In this paper, two lattice Boltzmann models for two-dimensional coupled Burgers’ equations are proposed through treating the part or all of convection items as the source term, where the spatial gradient can be calculated by the distribution function. The models can exactly recover the Burgers’ equations without any assumptions. Some numerical tests are also performed to validate the present models. It is found that the proposed models are more accurate and efficient in solving two-dimensional coupled Burgers’ equations.

In this paper, the effects of temperature-dependent properties on natural convection of nanofluids in a partially heated cubic enclosure are investigated in detail with lattice Boltzmann method. To improve the computational efficiency, all simulations are performed on the Graphics Processing Unit (GPU) using NVIDIA’s CUDA. The fluid in the cuibc cavity is a water-based nanofluid containing Al 2 O 3 nanoparticles. The effects of thermal Rayleigh number ( 10 4 ⩽ Ra f ⩽ 10 6 ), diameter of nanoparticle ( 25 nm ⩽ d s ⩽ 100 nm ), nanoparticle volume fraction ( 0.0 ⩽ ϕ ⩽ 0.04 ), temperature of the cooled sidewall ( 315 K ⩽ T c ⩽ 335 K ), temperature difference between the sidewalls ( 10 K ⩽ Δ T ⩽ 50 K ), aspect ratio ( 0.50 ⩽ AR ⩽ 1.00 ) and heating location on temperature field and fluid flows are investigated. The results reveal that the average Nusselt number is decreased with the increase of nanoparticle volume fraction. In addition, it is also observed that there is an optimal volume fraction ϕ max at which the maximum heat transfer enhancement is obtained, and the value of ϕ max is found to increase slightly with decreasing the nanoparticle diameter, and to increase remarkably with increasing the temperature of T c or Δ T . Moreover, we also find that the average Nusselt number and the heat transfer enhancement are decreased with the increase of aspect ratio.

In this paper, we propose a lattice Boltzmann (LB) model for the generalized coupled cross-diffusion-fluid system. Through the direct Taylor expansion method, the proposed LB model can correctly recover the macroscopic equations. The cross diffusion terms in the coupled system are modeled by introducing additional collision operators, which can be used to avoid special treatments for the gradient terms. In addition, the auxiliary source terms are constructed properly such that the numerical diffusion caused by the convection can be eliminated. We adopt the developed LB model to study two important systems, i.e., the coupled chemotaxis-fluid system and the double-diffusive convection system with Soret and Dufour effects. We first test the present LB model through considering a steady-state case of coupled chemotaxis-fluid system, then we analyze the influences of some physical parameters on the formation of sinking plumes. Finally, the double-diffusive natural convection system with Soret and Dufour effects is also studied, and the numerical results agree well with some previous works.

Suppression of spiral wave attracts more and more attention in nonlinear systems. In this paper, a spiral wave local feedback control approach based on the FitzHugh–Nagumo (FHN) model is studied with lattice Boltzmann method. Numerical simulations are performed to investigate the effects of initial conditions for the spiral wave formation, model parameters, size and position of the feedback control region, and feedback control parameters on the behavior of spiral wave. The results show that there are three characteristics of spiral wave elimination. The first is that initial conditions of the spiral wave formation have little influence on feedback control of spiral wave. Secondly, the model parameters are related to the time needed for the elimination of spiral wave, for example, the larger the mutual time scales, the faster the elimination of spiral wave. Finally, through selecting the size and position of the feedback region, spiral wave can be effectively removed with weak feedback signal.

In this paper, a finite-difference lattice Boltzmann (LB) model for nonlinear isotropic and anisotropic convection-diffusion equations is proposed. In this model, the equilibrium distribution function is delicately designed in order to recover the convection-diffusion equation exactly. Different from the standard LB model, the temporal and spatial steps in this model are decoupled such that it is convenient to study convection-diffusion problem with the non-uniform grid. In addition, it also preserves the advantage of standard LB model that the complex-valued convection-diffusion equation can be solved directly. The von Neumann stability analysis is conducted to discuss the stability region which can be used to determine the free parameters appeared in the model. To test the performance of the model, a series of numerical simulations of some classical problems, including the diffusion equation, the nonlinear heat conduction equation, the Sine-Gordon equation, the Gaussian hill problem, the BurgersFisher equation, and the nonlinear Schrdinger equation, have also been carried out. The results show that the present model has a second-order convergence rate in space, and generally it is also more accurate than the standard LB model.

In this paper, the magnetic field effects on natural convection of power-law nanofluids in rectangular enclosures are investigated numerically with the lattice Boltzmann method. The fluid in the cavity is a water-based nanofluid containing Cu nanoparticles and the investigations are carried out for different governing parameters including Hartmann number (0.0≤Ha≤20.0), Rayleigh number (104≤Ra≤106), power-law index (0.5≤n≤1.0), nanopartical volume fraction (0.0≤ϕ≤0.1) and aspect ratio (0.125≤AR≤8.0). The results reveal that the flow oscillations can be suppressed effectively by imposing an external magnetic field and the augmentation of Hartmann number and power-law index generally decreases the heat transfer rate. Additionally, it is observed that the average Nusselt number is increased with the increase of Rayleigh number and nanoparticle volume fraction. Moreover, the present results also indicate that there is a critical value for aspect ratio at which the impact on heat transfer is the most pronounced.

In this paper, an efficient phase-field-based lattice Boltzmann (LB) model with multiple-relaxation-time (MRT) collision operator is developed for the simulation of three-dimensional multiphase flows. This model is an extension of our previous two-dimensional model (Liang etal., 2014) to the three dimensions using the D3Q7 (seven discrete velocities in three dimensions) lattice for the CahnHilliard equation and the D3Q15 lattice for the NavierStokes equations. Due to the less lattice-velocity directions used, the computational efficiency can be significantly improved in the study of three-dimensional multiphase flows, and simultaneously the present model can recover the CahnHilliard equation and the NavierStokes equations correctly through the ChapmanEnskog procedure. We compare the present MRT model with its single-relaxation-time version and the previous LB model by using two benchmark interface-tracking problems, and numerical results show that the present MRT model can achieve a significant improvement in the accuracy and stability of the interface capturing. The developed model can also be able to deal with multiphase fluids with high viscosity ratio, which is demonstrated by the simulation of the layered Poiseuille flow and RayleighTaylor instability at various viscosity ratios. The numerical results are found to be in good agreement with the analytical solutions or some available results. In addition, it is also found that the instability induces a more complex structure of the interface at a low viscosity.

The process of the droplet impact onto the liquid film on the rough solid surface, as one of the basic multiphase problems, is very important in many fields of science and engineering. On the other hand, the problem is also very complicated since there are many parameters that may influence the process of the droplet impact on the rough solid surface with a liquid film. Up to now, there are still little research on this problem, and to gain a better understanding on the physical mechanics of the droplet impact onto the film on the rough solid surface, it is desirable to conduct a detailed study. To clearly understand the physical phenomena appearing in the process of droplet impact on the liquid film, a parametric study on this problem is also carried out based on a recently developed lattice Boltzmann method in which a MRT lattice Boltzmann model is used to solve the Navier-Stokes equations, and the other is adopted to solve the Cahn-Hilliard equation that is used to depict the interface between different phases. In this paper, the effects of the relative thickness of film ( h ), the relative width of cavity ( d *) and the relative depth of cavity ( L *) on the dynamic behavior of interface are investigated in detail, and the velocity and pressure fields are also presented. In order to reduce the influence of lattice, we fix the lattice to be 600×120 for gas, which is fine enough to give accurate results. In addition, in our simulations, We =500, Re =480, viscosity ratio and density ratio are set to be 2:1. The numerical results first show that, the phenomena of crown and entrainment can be observed obviously during the process of droplet impact onto the liquid film on the rough interface when We and Re are large. The radius of spray ( r ), which is formed by the droplet impact onto liquid film, is related to time through the relation r / 2 R ≈ α U t / 2 R when h is small, which is coincident with the result of droplet impact onto the liquid film on smooth surface, and additionally the coefficient α would decrease with the increase of h . However, this relation seems not accurate for the case with a large h , and simultaneously, the splashing phenomenon has not been observed. Secondly, the relative width of cavity d * plays an important role on the phenomena of splashing. When d *=1, there will be two small droplets through the splashing phenomenon (left half part), then with this parameter increase, the number of small droplet and the point where the splashing occur will also change, and there also are much difference in relation of spray radius and time. Actually, if d * is small, the coefficient α would first decrease and then increase with the increase of d *, while if d *>8, the cavity width would only have a little influence on the behavior of spray. Finally, it is also found that the pressure change near the cavity bottom is small at different L *, that is to say, the relative depth of cavity L * seems to has no apparent effect on the formation of spray, but it brings a great influence on the splashing of spray and the movement of the droplet which is produced in the process of splashing.

In this work, a mixed bounce-back boundary scheme of general propagation lattice Boltzmann (GPLB) model is proposed for isotropic advection-diffusion equations (ADEs) with Robin boundary condition, and a detailed asymptotic analysis is also conducted to show that the present boundary scheme for the straight walls has a second-order accuracy in space. In addition, several numerical examples, including the Helmholtz equation in a square domain, the diffusion equation with sinusoidal concentration gradient, one-dimensional transient ADE with Robin boundary and an ADE with a source term, are also considered. The results indicate that the numerical solutions agree well with the analytical ones, and the convergence rate is close to 2.0. Furthermore, through adjusting the two parameters in the GPLB model properly, the present boundary scheme can be more accurate than some existing lattice Boltzmann boundary schemes.