Search

In this paper we first propose a phase-field model for the containerless freezing problems, in which the volume expansion or shrinkage of the liquid caused by the density change during the phase change process is considered by adding a mass source term to the continuum equation. Then a phase-field-based lattice Boltzmann (LB) method is further developed to simulate solid-liquid phase change phenomena in multiphase systems. We test the developed LB method by the problem of conduction-induced freezing in a semi-infinite space, the three-phase Stefan problem, and the droplet solidification on a cold surface, and the numerical results are in agreement with the analytical and experimental solutions. In addition, the LB method is also used to study the rising bubbles with solidification. The results of the present method not only accurately capture the effect of bubbles on the solidification process, but also are in agreement with the previous work. Finally, a parametric study is carried out to examine the influences of some physical parameters on the sessile droplet solidification, and it is found that the time of droplet solidification increases with the increase of droplet volume and contact angle.

In this work, we develop a phase-field-based lattice Boltzmann (LB) method for a two-scalar model of the two-phase flows with interfacial mass or heat transfer. Through the Chapman-Enskog analysis, we show that the present LB method can correctly recover the governing equations for phase field, flow field, and concentration or temperature field. In particular, to derive the two-scalar equations for the mass or heat transfer, we propose a new LB model with an auxiliary source distribution function to describe the extra flux terms, and the discretizations of some derivative terms can be avoided. The accuracy and efficiency of the present LB method are also tested through several benchmark problems, and the influence of mass or heat transfer on the fluid viscosity is further considered by introducing an exponential relation. The numerical results show that the present LB method is suitable for the two-phase flows with interfacial mass or heat transfer.

In this paper we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations (MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation (NACDE) and Navier-Stokes equations (NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation (CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, three benchmark problems, the Gauss hill problem, the CDE with nonlinear convection and diffusion terms, and the Taylor-Green vortex flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results not only are in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis.

Auto-ejection of liquid is an important process in engineering applications, and is also very complicated since it involves interface moving, deforming, and jet breaking up. In this work, a theoretical velocity of meniscus at nozzle exit is first derived, which can be used to analyze the critical condition for auto-ejection of liquid. Then a consistent and conservative axisymmetric lattice Boltzmann (LB) method is proposed to study the auto-ejection process of liquid jet from a nozzle. We test the LB model by conducting some simulations, and find that the numerical results agree well with the theoretical and experimental data. We further consider the effects of contraction ratio, length ratio, contact angle, and nozzle structure on the auto-ejection, and observe some distinct phenomena during the ejection process, including the deformation of meniscus, capillary necking, and droplet pinch off. Finally, the results reported in the present work may play an instructive role on the design of droplet ejectors and the understanding of jetting dynamics in microgravity environment.

Quantum systems usually feature a rich multilevel structure with promising resources for developing superior quantum technologies compared with their binary counterpart. Single-shot readout of these high-dimensional quantum systems is essential for exploiting their potential. Although there have been various high-spin systems, the single-shot readout of the overall state of high-spin systems remains a challenging issue. Here we demonstrate a reliable single-shot readout of spin qutrit state in a low-temperature solid-state system utilizing a binary readout scheme. We achieve a single-shot readout of an electron spin qutrit associated with a single nitrogen-vacancy center in diamond with an average fidelity of 87.80%. We use this spin qutrit system to verify quantum contextuality, a fundamental test of quantum mechanics. We observe a violation of the noncontextual hidden variable inequality with the developed single-shot readout in contrast to the conventional binary readout. These results pave the way for developing quantum information processing based on spin qutrits.

In this paper, we develop a macroscopic finite-difference scheme from the mesoscopic regularized lattice Boltzmann (RLB) method to solve the Navier-Stokes equations (NSEs) and convection-diffusion equation (CDE). Unlike the commonly used RLB method based on the evolution of a set of distribution functions, this macroscopic finite-difference scheme is constructed based on the hydrodynamic variables of NSEs (density, momentum, and strain rate tensor) or macroscopic variables of CDE (concentration and flux), and thus shares low memory requirement and high computational efficiency. Based on an accuracy analysis, it is shown that, the same as the mesoscopic RLB method, the macroscopic finite-difference scheme also has a second-order accuracy in space. In addition, we would like to point out that compared with the RLB method and its equivalent macroscopic numerical scheme, the present macroscopic finite-difference scheme is much simpler and more efficient since it is only a two-level system with macroscopic variables. Finally, we perform some simulations of several benchmark problems, and find that the numerical results are not only in agreement with analytical solutions, but also consistent with the theoretical analysis.

In this paper, we develop a general rectangular multiple-relaxation-time lattice Boltzmann (RMRT-LB) method for the Navier-Stokes equations (NSEs) and nonlinear convection-diffusion equation (NCDE) by extending our recent unified framework of the multiple-relaxation-time lattice Boltzmann (MRT-LB) method [Chai and Shi, Phys. Rev. E 102, 023306 (2020)10.1103/PhysRevE.102.023306], where an equilibrium distribution function (EDF) [Lu et al., Philos. Trans. R. Soc. A 369, 2311 (2011)10.1098/rsta.2011.0022] on a rectangular lattice is utilized. The anisotropy of the lattice tensor on a rectangular lattice leads to anisotropy of the third-order moment of the EDF, which is inconsistent with the isotropy of the viscous stress tensor of the NSEs. To eliminate this inconsistency, we extend the relaxation matrix related to the dynamic and bulk viscosities. As a result, the macroscopic NSEs can be recovered from the RMRT-LB method through the direct Taylor expansion method. Whereas the rectangular lattice does not lead to the change of the zero-, first- and second-order moments of the EDF, the unified framework of the MRT-LB method can be directly applied to the NCDE. It should be noted that the RMRT-LB model for NSEs can be derived on the rDdQq (q discrete velocities in d-dimensional space, d≥1) lattice, including rD2Q9, rD3Q19, and rD3Q27 lattices, while there are no rectangular D3Q13 and D3Q15 lattices within this framework of the RMRT-LB method. Thanks to the block-lower triangular relaxation matrix introduced in the unified framework, the RMRT-LB versions (if existing) of the previous MRT-LB models can be obtained, including those based on raw (natural) moment, central moment, Hermite moment, and central Hermite moment. It is also found that when the parameter c_{s} is an adjustable parameter in the standard or rectangular lattice, the present RMRT-LB method becomes a kind of MRT-LB method for the NSEs and NCDE, and the commonly used MRT-LB models on the DdQq lattice are only its special cases. We also perform some numerical simulations, and the results show that the present RMRT-LB method can give accurate results and also have a good numerical stability.

In this paper, we first develop a fourth-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the one-dimensional convection-diffusion equation (CDE) with the constant velocity and diffusion coefficient, where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is used. We also perform the Chapman-Enskog analysis to recover the CDE from the MRT-LB model. Then an explicit four-level finite-difference (FLFD) scheme is derived from the developed MRT-LB model for the CDE. Through the Taylor expansion, the truncation error of the FLFD scheme is obtained, and at the diffusive scaling, the FLFD scheme can achieve the fourth-order accuracy in space. After that, we present a stability analysis and derive the same stability condition for the MRT-LB model and FLFD scheme. Finally, we perform some numerical experiments to test the MRT-LB model and FLFD scheme, and the numerical results show that they have a fourth-order convergence rate in space, which is consistent with our theoretical analysis.

In this work we develop an improved phase-field based lattice Boltzmann (LB) method where a hybrid Allen-Cahn equation (ACE) with a flexible weight instead of a global weight is used to suppress the numerical dispersion and eliminate the coarsening phenomenon. Then two LB models are adopted to solve the hybrid ACE and the Navier-Stokes equations, respectively. Through the Chapman-Enskog analysis, the present LB model can correctly recover the hybrid ACE, and the macroscopic order parameter used to label different phases can be calculated explicitly. Finally, the present LB method is validated by five tests, including the diagonal translation of a circular interface, two stationary bubbles with different radii, a bubble rising under the gravity, the Rayleigh-Taylor instability in two-dimensional and three-dimensional cases, and the three-dimensional Plateau-Rayleigh instability. The numerical results show that the present LB method has a superior performance in reducing the numerical dispersion and the coarsening phenomenon.

In this paper, a multiple-distribution-function lattice Boltzmann method (MDF-LBM) with a multiple-relaxation-time model is proposed for incompressible Navier-Stokes equations which are considered as coupled convection-diffusion equations. 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 the distribution function. Then in the framework of the present MDF-LBM, we develop a locally computational scheme for the velocity gradient in which the first-order moment of the nonequilibrium 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.

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 developed a coupled diffuse-interface lattice Boltzmann method (DI-LBM) to study the transport of a charged particle in the Poiseuille flow, which is governed by the Navier-Stokes equations for fluid field and the Poisson-Boltzmann equation for electric potential field. We first validated the present DI-LBM through some classical benchmark problems, and then investigated the effect of electric field on the lateral migration of the particle in the Poiseuille flow. The numerical results show that the electric field has a significant influence on the particle migration. When an electric field in the vertical direction is applied to the charged particle initially located above the centerline of the channel, the equilibrium position of the particle would drop suddenly as the electric field is larger than a critical value. This is caused by the wall repulsion due to lubrication, the inertial lift related to shear slip, the lift owing to particle rotation, the lift due to the curvature of the undisturbed velocity profile, and the electric force. On the other hand, when an electric field in the horizontal direction is adopted, the equilibrium position of the particle would move toward the centerline of the channel with the increase of the electric field.

Following the rising interest in quantum information science, the extension of a heat engine to the quantum regime by exploring microscopic quantum systems has seen a boon of interest in the last decade. Although quantum coherence in the quantum system of the working medium has been investigated to play a nontrivial role, a complete understanding of the intrinsic quantum advantage of quantum heat engines remains elusive. We experimentally demonstrate that the quantum correlation between the working medium and the thermal bath is critical for the quantum advantage of a quantum Szilárd engine, where quantum coherence in the working medium is naturally excluded. By quantifying the nonclassical correlation through quantum steering, we reveal that the heat engine is quantum when the demon can truly steer the working medium. The average work obtained by taking different ways of work extraction on the working medium can be used to verify the real quantum Szilárd engine.

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, 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 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 ω_{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.

In this paper, we first present a unified framework of multiple-relaxation-time lattice Boltzmann (MRT-LB) method for the Navier-Stokes and nonlinear convection-diffusion equations where a block-lower-triangular-relaxation matrix and an auxiliary source distribution function are introduced. We then conduct a comparison of the four popular analysis methods (Chapman-Enskog analysis, Maxwell iteration, direct Taylor expansion, and recurrence equations approaches) that have been used to obtain the macroscopic Navier-Stokes and nonlinear convection-diffusion equations from the MRT-LB method and show that from mathematical point of view, these four analysis methods can give the same equations at the second-order of expansion parameters. Finally, we give some elements that are needed in the implementation of the MRT-LB method and also find that some available LB models can be obtained from this MRT-LB method.

Quantum simulation, as a state-of-the-art technique, provides a powerful way to explore topological quantum phases beyond natural limits. Nevertheless, it is usually hard to simulate both the bulk and surface topological physics at the same time to reveal their correspondence. Here we build up a quantum simulator using nitrogen-vacancy center to investigate a three-dimensional (3D) chiral topological insulator, and demonstrate the study of both the bulk and surface topological physics by quantum quenches. First, a dynamical bulk-surface correspondence in momentum space is observed, showing that the bulk topology of the 3D phase uniquely corresponds to the nontrivial quench dynamics emerging on 2D momentum hypersurfaces called band inversion surfaces (BISs). This is the momentum-space counterpart of the bulk-boundary correspondence in real space. Further, the symmetry protection of the 3D chiral phase is uncovered by measuring dynamical spin textures on BISs, which exhibit perfect (broken) topology when the chiral symmetry is preserved (broken). Finally, we measure the topological charges to characterize directly the bulk topology and identify an emergent dynamical topological transition when varying the quenches from deep to shallow regimes. This work demonstrates how a full study of topological phases can be achieved in quantum simulators.

In this paper, we develop an efficient and alternative lattice Boltzmann (LB) model for simulating immiscible incompressible N-phase flows (N≥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≥2) immiscible incompressible flows.

In this paper, we develop a discrete unified gas kinetic scheme (DUGKS) for a general nonlinear convection-diffusion equation (NCDE) and show that the NCDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the present DUGKS through some classic convection-diffusion equations, and we find that the numerical results are in good agreement with analytical solutions and that the DUGKS model has a second-order convergence rate. Finally, as a finite-volume method, the DUGKS can also adopt the nonuniform mesh. Besides, we perform some comparisons among the DUGKS, the finite-volume lattice Boltzmann model (FV-LBM), the single-relaxation-time lattice Boltzmann model (SLBM), and the multiple-relaxation-time lattice Boltzmann model (MRT-LBM). The results show that the present DUGKS is more accurate than the FV-LBM, more stable than the SLBM, and almost has the same accuracy as the MRT-LBM. Moreover, the use of nonuniform mesh may make the DUGKS model more flexible.

Within the phase-field framework, we present an accurate and robust lattice Boltzmann (LB) method for simulating contact-line motion of immiscible binary fluids on the solid substrate. The most striking advantage of this method lies in that it enables us to handle two-phase flows with mass conservation and a high density contrast of 1000, which is often unavailable in the existing multiphase LB models. To simulate binary fluid flows, the present method utilizes two LB evolution equations, which are respectively used to solve the conservative Allen-Cahn equation for interface capturing, and the incompressible Navier-Stokes equations for hydrodynamic properties. Besides, to account for the substrate wettability, two popular contact angle models including the cubic surface-energy model and the geometrical one are incorporated into the present method, and their performances are numerically evaluated over a wide range of contact angles. The contact-angle hysteresis effect, which is inherent to a rough or chemically inhomogeneous substrate, is also introduced in the present LB approach through the strategy proposed by Ding and Spelt [J. Fluid Mech. 599, 341 (2008)10.1017/S0022112008000190]. The present method is first validated by simulating droplet spreading and capillary intrusion on the ideal or smooth pipes. It is found that the cubic surface-energy and geometrical wetting schemes both offer considerable accuracy for predicting a static contact angle within its middle region, while the former is more stable at extremely small contact angles. Besides, it is shown that the geometrical wetting scheme enables us to obtain better accuracy for predicting dynamic contact points in capillary pipe. Then we use the present LB method to simulate the droplet shearing processes on a nonideal substrate with contact angle hysteresis. The geometrical wetting model is found to be capable of reproducing four typical motion modes of contact line, while the surface-energy wetting scheme fails to predict the hysteresis behaviors in some cases. At last, a complex contact-line dynamic problem of three-dimensional microscale droplet impact on a wettable solid is simulated, and it is found that the numerical results for droplet shapes agree well with the experimental data.

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.

The phenomena of diffusion in multicomponent (more than two components) mixtures are universal in both science and engineering, and from the mathematical point of view, they are usually described by the Maxwell-Stefan (MS)-theory-based diffusion equations where the molar average velocity is assumed to be zero. In this paper, we propose a multiple-relaxation-time lattice Boltzmann (LB) model for the mass diffusion in multicomponent mixtures and also perform a Chapman-Enskog analysis to show that the MS continuum equations can be correctly recovered from the developed LB model. In addition, considering the fact that the MS-theory-based diffusion equations are just a diffusion type of partial differential equations, we can also adopt much simpler lattice structures to reduce the computational cost of present LB model. We then conduct some simulations to test this model and find that the results are in good agreement with the previous work. Besides, the reverse diffusion, osmotic diffusion, and diffusion barrier phenomena are also captured. Finally, compared to the kinetic-theory-based LB models for multicomponent gas diffusion, the present model does not include any complicated interpolations, and its collision process can still be implemented locally. Therefore, the advantages of single-component LB method can also be preserved in present LB model.

In this paper, a general propagation lattice Boltzmann model is proposed for nonlinear advection-diffusion equations (NADEs), and the Chapman-Enskog analysis shows that the NADEs with variable coefficients can be recovered correctly from the present model. We also perform some simulations of the linear advection-diffusion equation, nonlinear heat conduction equation, NADEs with anisotropic diffusion, and variable coefficients to test the present model, and find that the numerical results agree well with the corresponding analytical solutions. Moreover, it is also shown that by properly adjusting the two free parameters introduced into the propagation step, the present model could be more stable and more accurate than the standard lattice Bhatnagar-Gross-Krook model.

In this paper, we present a simple and accurate lattice Boltzmann (LB) model for immiscible two-phase flows, which is able to deal with large density contrasts. This model utilizes two LB equations, one of which is used to solve the conservative Allen-Cahn equation, and the other is adopted to solve the incompressible Navier-Stokes equations. A forcing distribution function is elaborately designed in the LB equation for the Navier-Stokes equations, which make it much simpler than the existing LB models. In addition, the proposed model can achieve superior numerical accuracy compared with previous Allen-Cahn type of LB models. Several benchmark two-phase problems, including static droplet, layered Poiseuille flow, and spinodal decomposition are simulated to validate the present LB model. It is found that the present model can achieve relatively small spurious velocity in the LB community, and the obtained numerical results also show good agreement with the analytical solutions or some available results. Lastly, we use the present model to investigate the droplet impact on a thin liquid film with a large density ratio of 1000 and the Reynolds number ranging from 20 to 500. The fascinating phenomena of droplet splashing is successfully reproduced by the present model and the numerically predicted spreading radius exhibits to obey the power law reported in the literature.

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

In this paper, we will focus on the multiple-relaxation-time (MRT) lattice Boltzmann model for two-dimensional convection-diffusion equations (CDEs), and analyze the discrete effect on the halfway bounce-back (HBB) boundary condition (or sometimes called bounce-back boundary condition) of the MRT model where three different discrete velocity models are considered. We first present a theoretical analysis on the discrete effect of the HBB boundary condition for the simple problems with a parabolic distribution in the x or y direction, and a numerical slip proportional to the second-order of lattice spacing is observed at the boundary, which means that the MRT model has a second-order convergence rate in space. The theoretical analysis also shows that the numerical slip can be eliminated in the MRT model through tuning the free relaxation parameter corresponding to the second-order moment, while it cannot be removed in the single-relaxation-time model or the Bhatnagar-Gross-Krook model unless the relaxation parameter related to the diffusion coefficient is set to be a special value. We then perform some simulations to confirm our theoretical results, and find that the numerical results are consistent with our theoretical analysis. Finally, we would also like to point out the present analysis can be extended to other boundary conditions of lattice Boltzmann models for CDEs.

In this paper, a regularized lattice Boltzmann model for a class of nonlinear convection-diffusion equations with variable coefficients is proposed. The main idea of the present model is to introduce a set of precollision distribution functions that are defined only in terms of macroscopic moments. The Chapman-Enskog analysis shows that the nonlinear convection-diffusion equations can be recovered correctly. Numerical tests, including Fokker-Planck equations, Buckley-Leverett equation with discontinuous initial function, nonlinear convection-diffusion equation with anisotropic diffusion, are carried out to validate the present model, and the results show that the present model is more accurate than some available lattice Boltzmann models. It is also demonstrated that the present model is more stable than the traditional single-relaxation-time model for the nonlinear convection-diffusion equations.

In this paper, we present a numerical study on the deformation and breakup behavior of liquid droplet past a solid circular cylinder by using an improved interparticle-potential lattice Boltzmann method. The effects of the eccentric ratio β, viscosity ratio λ between the droplet and the surrounding fluid, surface wettability, and Bond number (Bo) on the dynamic behavior of the liquid droplet are considered. The parameter β represents the degree that the solid cylinder deviates from the center line, and Bo is the ratio between the inertial force and capillary force. Numerical results show that there are two typical patterns, i.e., breakup and no breakup, which are greatly influenced by the aforementioned parameters. When β increases to a critical value βc, the droplet can pass the circular cylinder without a breakup, otherwise, the breakup phenomenon occurs. The critical eccentric ratio βc increases significantly with increasing Bo for case with λ>1, while for the case with λ<1, the viscosity effects on the βc is not obvious when Bo is large. For the breakup case, the amount of deposited liquid on the tip of the circular cylinder is almost unaffected by β. In addition, the results also show that the viscosity ratio and wettability affect the deformation and breakup process of the droplet. For case with λ<1, the viscosity ratio plays a minor role in the thickness variations of the deposited liquid, which decreases to a nonzero constant eventually; while for λ>1, the increase of the viscosity ratio significantly accelerates the decrease of the deposited liquid, and finally no fluid deposits on the cylinder. In term of the wettability, there occurs continuous gas phase trapped by the wetting droplet, but this does not happen for nonwetting droplet. Besides, for λ<1, the time required to pass the cylinder (tp) decreases monotonically with decreasing contact angle, while a nonmonotonic decrease appears for λ>1. It is also found that tp decreases monotonically with increasing Bo and is less sensitive to λ at a large Bo.

In this paper, two modified lattice Boltzmann Bhatnagar-Gross-Krook (LBGK) models for incompressible Navier-Stokes equations and convection-diffusion equations are proposed via the addition of correction terms in the evolution equations. Utilizing this modification, the value of the dimensionless relaxation time in the LBGK model can be kept in a proper range, and thus the stability of the LBGK model can be improved. Although some gradient operators are included in the correction terms, they can be computed efficiently using local computational schemes such that the present LBGK models still retain the intrinsic parallelism characteristic of the lattice Boltzmann method. Numerical studies of the steady Poiseuille flow and unsteady Womersley flow show that the modified LBGK model has a second-order convergence rate in space, and the compressibility effect in the common LBGK model can be eliminated. In addition, to test the stability of the present models, we also performed some simulations of the natural convection in a square cavity, and we found that the results agree well with those reported in the previous work, even at a very high Rayleigh number (Ra = 10(12)).

In this paper, we propose a local nonequilibrium scheme for computing the flux of the convection-diffusion equation with a source term in the framework of the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). Both the Chapman-Enskog analysis and the numerical results show that, at the diffusive scaling, the present nonequilibrium scheme has a second-order convergence rate in space. A comparison between the nonequilibrium scheme and the conventional second-order central-difference scheme indicates that, although both schemes have a second-order convergence rate in space, the present nonequilibrium scheme is more accurate than the central-difference scheme. In addition, the flux computation rendered by the present scheme also preserves the parallel computation feature of the LBM, making the scheme more efficient than conventional finite-difference schemes in the study of large-scale problems. Finally, a comparison between the single-relaxation-time model and the MRT model is also conducted, and the results show that the MRT model is more accurate than the single-relaxation-time model, both in solving the convection-diffusion equation and in computing the flux.

We propose a lattice Boltzmann (LB) model for the convection-diffusion equation (CDE) and show that the CDE can be recovered correctly from the model by the Chapman-Enskog analysis. The most striking feature of the present LB model is that it enables the collision process to be implemented locally, making it possible to retain the advantage of the lattice Boltzmann method in the study of the heat and mass transfer in complex geometries. A local scheme for computing the heat and mass fluxes is then proposed to replace conventional nonlocal finite-difference schemes. We further validate the present model and the local scheme for computing the flux against analytical solutions to several classical problems, and we show that both the model for the CDE and the computational scheme for the flux have a second-order convergence rate in space. It is also demonstrated the present model is more accurate than existing LB models for the CDE.

In this work, the effect of the forcing term (or external force) in the multiple-relaxation-time lattice Boltzmann equation (MRTLBE) on the shear stress or the strain rate tensor is studied theoretically and numerically. Through a Chapman-Enskog analysis and numerical simulations, we show that the shear stress (or the strain rate tensor) derived from the MRTLBE is second-order accurate in space. We then examine the influence of the forcing term on the shear stress or the strain rate tensor, and demonstrate that the forcing term effect must be included when the shear stress or the strain rate tensor is computed with the nonequilibrium part of the distribution function.

In this paper, a general bounce-back scheme is proposed to implement concentration or thermal boundary conditions of convection-diffusion equation with the lattice Boltzmann method (LBM). Using this scheme, the general concentration boundary conditions, i.e., b1(∂Cw/∂n) + b2Cw = b3, can be easily implemented at boundaries with complex geometry structure like that in porous media. The numerical results obtained using the present scheme are in excellent agreement with the analytical solutions of flows with both stationary and moving interfaces. Furthermore, to better understand the halfway bounce-back scheme, an analytical study of the concentration jump is presented. The studies of theoretical analysis and numerical experiments demonstrate that the proposed scheme has second-order accuracy.

This Letter presents the first measurement of event-by-event fluctuations of the elliptic flow parameter v(2) in Au+Au collisions at square root(s(NN))=200 GeV as a function of collision centrality. The relative nonstatistical fluctuations of the v(2) parameter are found to be approximately 40%. The results, including contributions from event-by-event elliptic flow fluctuations and from azimuthal correlations that are unrelated to the reaction plane (nonflow correlations), establish an upper limit on the magnitude of underlying elliptic flow fluctuations. This limit is consistent with predictions based on spatial fluctuations of the participating nucleons in the initial nuclear overlap region. These results provide important constraints on models of the initial state and hydrodynamic evolution of relativistic heavy ion collisions.

A measurement of two-particle correlations with a high transverse momentum trigger particle (p(T)(trig) > 2.5 GeV/c) is presented for Au+Au collisions at square root(s(NN)) = 200 GeV over the uniquely broad longitudinal acceptance of the PHOBOS detector (-4 < Delta eta < 2). A broadening of the away-side azimuthal correlation compared to elementary collisions is observed at all Delta eta. As in p+p collisions, the near side is characterized by a peak of correlated partners at small angle relative to the trigger particle. However, in central Au+Au collisions an additional correlation extended in Delta eta and known as the "ridge" is found to reach at least |Delta eta| approximately = 4. The ridge yield is largely independent of Delta eta over the measured range, and it decreases towards more peripheral collisions. For the chosen (p(T)(trig) cut, the ridge yield is consistent with zero for events with less than roughly 100 participating nucleons.

We present the first measurements of the pseudorapidity distribution of primary charged particles in Cu+Cu collisions as a function of collision centrality and energy, sqrt[s&#95;{NN}]=22.4, 62.4, and 200 GeV, over a wide range of pseudorapidity, using the PHOBOS detector. A comparison of Cu+Cu and Au+Au results shows that the total number of produced charged particles and the rough shape (height and width) of the pseudorapidity distributions are determined by the number of nucleon participants. More detailed studies reveal that a more precise matching of the shape of the Cu+Cu and Au+Au pseudorapidity distributions over the full range of pseudorapidity occurs for the same N{part}/2A rather than the same N&#95;{part}. In other words, it is the collision geometry rather than just the number of nucleon participants that drives the detailed shape of the pseudorapidity distribution and its centrality dependence at RHIC energies.

We present a measurement of the spin-dependent cross sections for the 3He over -->(e over -->,e')X reaction in the quasielastic and resonance regions at a four-momentum transfer 0.1< or =Q2< or =0.9 GeV2. The spin-structure functions have been extracted and used to evaluate the nuclear Burkhardt-Cottingham and extended Gerasimov-Drell-Hearn sum rules for the first time. The data are also compared to an impulse approximation calculation and an exact three-body Faddeev calculation in the quasielastic region.

This Letter presents measurements of the elliptic flow of charged particles as a function of pseudorapidity and centrality from Cu-Cu collisions at 62.4 and 200 GeV using the PHOBOS detector at the Relativistic Heavy Ion Collider. The elliptic flow in Cu-Cu collisions is found to be significant even for the most central events. For comparison with the Au-Au results, it is found that the detailed way in which the collision geometry (eccentricity) is estimated is of critical importance when scaling out system-size effects. A new form of eccentricity, called the participant eccentricity, is introduced which yields a scaled elliptic flow in the Cu-Cu system that has the same relative magnitude and qualitative features as that in the Au-Au system.

We report on measurements of directed flow as a function of pseudorapidity in Au + Au collisions at energies of square root of SNN = 19.6, 62.4, 130 and 200 GeV as measured by the PHOBOS detector at the BNL Relativistic Heavy Ion Collider. These results are particularly valuable because of the extensive, continuous pseudorapidity coverage of the PHOBOS detector. There is no significant indication of structure near midrapidity and the data surprisingly exhibit extended longitudinal scaling similar to that seen for elliptic flow and charged particle pseudorapidity density.

We present transverse momentum distributions of charged hadrons produced in Cu + Cu collisions at square root of SNN = 62.4 and 200 GeV. The spectra are measured for transverse momenta of 0.25 < pT < 5.0 GeV/c at square root of SNN = 62.4 GeV and 0.25 < pT < 7.0 GeV/c at square root of SNN = 200 GeV, in a pseudorapidity range of 0.2 < eta < 1.4. The nuclear modification factor R(AA) is calculated relative to p + p data at both collision energies as a function of collision centrality. At a given collision energy and fractional cross section, R(AA) is observed to be systematically larger in Cu + Cu collisions compared to Au + Au. However, for the same number of participating nucleons, R(AA) is essentially the same in both systems over the measured range of pT, in spite of the significantly different geometries of the Cu + Cu and Au + Au systems.

We measured angular distributions of recoil-polarization response functions for neutral pion electroproduction for W = 1.23 GeV at Q(2) = 1.0 (GeV/c)(2), obtaining 14 separated response functions plus 2 Rosenbluth combinations; of these, 12 have been observed for the first time. Dynamical models do not describe quantities governed by imaginary parts of interference products well, indicating the need for adjusting magnitudes and phases for nonresonant amplitudes. We performed a nearly model-independent multipole analysis and obtained values for Re (S(1+)/M(1+)) = -(6.84 +/- 0.15)% and Re (E(1+)/M(1+)) = -(2.91 +/- 0.19)% that are distinctly different from those from the traditional Legendre analysis based upon M1+ dominance and ll(pi) < or = 1 truncation.

We have studied the quasielastic 3He(e,e(')p)2H reaction in perpendicular coplanar kinematics, with the energy and the momentum transferred by the electron fixed at 840 MeV and 1502 MeV/c, respectively. The 3He(e,e(')p)2H cross section was measured for missing momenta up to 1000 MeV/c, while the A(TL) asymmetry was extracted for missing momenta up to 660 MeV/c. For missing momenta up to 150 MeV/c, the cross section is described by variational calculations using modern 3He wave functions. For missing momenta from 150 to 750 MeV/c, strong final-state interaction effects are observed. Near 1000 MeV/c, the experimental cross section is more than an order of magnitude larger than predicted by available theories. The A(TL) asymmetry displays characteristic features of broken factorization with a structure that is similar to that generated by available models.

This Letter describes the measurement of the energy dependence of elliptic flow for charged particles in Au+Au collisions using the PHOBOS detector at the Relativistic Heavy Ion Collider. Data taken at collision energies of square root of s(NN)=19.6, 62.4, 130, and 200 GeV are shown over a wide range in pseudorapidity. These results, when plotted as a function of eta(')=|eta|-y(beam), scale with approximate linearity throughout eta('), implying no sharp changes in the dynamics of particle production as a function of pseudorapidity or increasing beam energy.

Results of the Jefferson Lab Hall A quasielastic 3He(e,e'p)pn measurements are presented. These measurements were performed at fixed transferred momentum and energy, q=1502 MeV/c and omega=840 MeV, respectively, for missing momenta p(m) up to 1 GeV/c and missing energies in the continuum region, up to pion threshold; this kinematic coverage is much more extensive than that of any previous experiment. The cross section data are presented along with the effective momentum density distribution and compared to theoretical models.

We have measured transverse momentum distributions of charged hadrons produced in Au+Au collisions at sqrt[s(NN)]=62.4 GeV. The spectra are presented for transverse momenta 0.25

The generalized forward spin polarizabilities gamma(0) and delta(LT) of the neutron have been extracted for the first time in a Q2 range from 0.1 to 0.9 GeV2. Since gamma(0) is sensitive to nucleon resonances and delta(LT) is insensitive to the Delta resonance, it is expected that the pair of forward spin polarizabilities should provide benchmark tests of the current understanding of the chiral dynamics of QCD. The new results on delta(LT) show significant disagreement with chiral perturbation theory calculations, while the data for gamma(0) at low Q2 are in good agreement with a next-to-leading-order relativistic baryon chiral perturbation theory calculation. The data show good agreement with the phenomenological MAID model.

We have measured the spin structure functions g(1) and g(2) of 3He in a double-spin experiment by inclusively scattering polarized electrons at energies ranging from 0.862 to 5.058 GeV off a polarized 3He target at a 15.5 degrees scattering angle. Excitation energies covered the resonance and the onset of the deep inelastic regions. We have determined for the first time the Q2 evolution of Gamma(1)(Q2)= integral (1)(0)g(1)(x,Q2)dx, Gamma(2)(Q2)= integral (1)(0)g(2)(x,Q2)dx, and d(2)(Q2)= integral (1)(0)x(2)[2g(1)(x,Q2)+3g(2)(x,Q2)]dx for the neutron in the range 0.1< or =Q2< or =0.9 GeV2 with good precision. Gamma(1)(Q2) displays a smooth variation from high to low Q2. The Burkhardt-Cottingham sum rule holds within uncertainties and d(2) is nonzero over the measured range.

We have measured the neutron spin asymmetry A(n)(1) with high precision at three kinematics in the deep inelastic region at x=0.33, 0.47, and 0.60, and Q(2)=2.7, 3.5, and 4.8 (GeV/c)(2), respectively. Our results unambiguously show, for the first time, that A(n)(1) crosses zero around x=0.47 and becomes significantly positive at x=0.60. Combined with the world proton data, polarized quark distributions were extracted. Our results, in general, agree with relativistic constituent quark models and with perturbative quantum chromodynamics (PQCD) analyses based on the earlier data. However they deviate from PQCD predictions based on hadron helicity conservation.