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1 This article was downloaded by: [Xi'an Jiaotong University] On: 28 February 2013, At: 06:00 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology Publication details, including instructions for authors and subscription information: A Molecular Dynamics and Lattice Boltzmann Multiscale Simulation for Dense Fluid Flows W. J. Zhou a, H. B. Luan a, J. Sun b, Y. L. He a & W. Q. Tao a a Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy & Power Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi, People's Republic of China b School of Engineering and Materials Science, Queen Mary, University of London, London, United Kingdom Version of record first published: 22 Jun To cite this article: W. J. Zhou, H. B. Luan, J. Sun, Y. L. He & W. Q. Tao (2012): A Molecular Dynamics and Lattice Boltzmann Multiscale Simulation for Dense Fluid Flows, Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 61:5, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Numerical Heat Transfer, Part B, 61: , 2012 Copyright # Taylor & Francis Group, LLC ISSN: print= online DOI: / A MOLECULAR DYNAMICS AND LATTICE BOLTZMANN MULTISCALE SIMULATION FOR DENSE FLUID FLOWS W. J. Zhou 1, H. B. Luan 1, J. Sun 2,Y.L.He 1, and W. Q. Tao 1 1 Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy & Power Engineering, Xi an Jiaotong University, Xi an, Shaanxi, People s Republic of China 2 School of Engineering and Materials Science, Queen Mary, University of London, London, United Kingdom A molecular dynamics (MD)-lattice Boltzmann (LB) hybrid scheme has been adopted to simulate dense fluid flows. Based on the domain decomposition method and the Schwarz alternating scheme, the Maxwell Demon approach is used to impose boundary conditions from the continuum to the atomistic region, while the reconstruction operator is implemented to construct the single-particle distribution function of the LB method from the results of the MD simulation. Couette flows and the flow of a dense fluid argon around a carbon nanotube (CNT) are solved to validate the hybrid method. When the mesh of the LB domain is refined and the size of corresponding sampling cells of the MD domain is reduced, the fluctuations of the results between two successive iterations of the hybrid method become more severe, although the results get closer to the MD reference solutions. To decrease the fluctuation due to the mesh refinement, a new weighting function is proposed for the sampling of MD simulation results. Numerical practice demonstrates its feasibility. 1. INTRODUCTION The Navier-Stokes (NS) equations are based on the conventional continuum assumption and can be used to solve macroscopic flow problems. However, when the system becomes small and reaches the nanometer scale, the continuum assumption breaks down. Fortunately, molecular dynamics (MD) simulations can be used to obtain details in small regions. Nevertheless, MD simulations are much more time-consuming than continuum models and are limited to really small size in both space and time. Thus the multiscale simulation, or hybrid model, emerges, which possesses the advantages of both the MD simulation and the continuum method and is becoming more and more popular. Microscopic details near the key positions Received 22 April 2011; accepted 3 February The authors wish to thank Dr. Hui Xu for very helpful technical discussion. This work was supported by the Key Projects of National Natural Science Foundation of China (No , U ). Address correspondence to W. Q. Tao, State Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi an Jiaotong University, 28 Xi an Ning Road, Xi an, Shaanxi , People s Republic of China. wqtao@mail.xjtu.edu.cn 369

3 370 W. J. ZHOU ET AL. NOMENCLATURE C lattice speed c s lattice sound speed C D Dirichlet compression operator C N Neumann compression operator d distance, velocity change rate D2Q9 2-dimension, 9-velocity lattice e error f lattice distribution function, force H height l length m mass of atom N number of atoms or grid points p pressure q property of atom Q macroscopic quantity, interface flux r position vector, radius r c cutoff radius r 0 radius of circle in MCIC scheme R D Dirichlet reconstruction operator t time u velocity vector u 1 freestream velocity w, W weighting factor x, y, z Cartesian coordinates X grid node Dx space step Dt time step e characteristic energy a, b coordinate direction indices C boundary dt time step n kinematic viscosity q density r characteristic length s nondimensional relaxation time / potential function Superscripts ArC related to an argon carbon pair eq equilibrium Subscripts c cutoff i ith j jth k kth such as interfaces between fluid and solid, where the continuum assumption breaks down, can be obtained by MD simulation, while the macro field of the remaining bulk region can be solved by continuum mechanics. As far as the ways for exchanging information at the interface in the multiscale simulation is concerned, generally it can be expressed as follows [1 4]. If the exchange of information at the interface ( hand-shaking region) is performed via Dirichlet type, then mathematically it can be expressed by U ¼ C D u u ¼ R D U ð1aþ where U and u are the macroscopic parameter, and microscopic=mesoscopic parameter, respectively. C D and R D are the Dirichlet compression and reconstruction operators, respectively. The information at mesoscale or microscale level may be transferred to the macroscale level via Neumann type, that is, by supplying flux at the interface; then we have [1 4] q ¼ C N u u ¼ R D U ð1bþ where q is the interface flux of the continuum region. As indicated in [1 4], at the interface between different regions there will be a mismatch in the kind and number of variables used by the different regions. The Dirichlet compression operator C D, which extracts the macroscopic parameters from a large amount of data at micro-or mesoscale level by some averaging or integrating

4 MULTISCALE SCALING FOR DENSE FLUID FLOWS 371 principles, is easy to define, but the reconstruction operator R D, which should extend a small amount of macroscopic parameters into a large amount of parameters at mesoscale or microscale, is quite difficult to construct. Here the problem of one-tomany is encountered, since the macroscopic (mesoscopic) variables have to be mapped to more MD variables. The design of the compression and reconstruction operators should abide by some basic physical laws or principles, such as mass, momentum, and energy conservation. In short, the exchange of information should be conducted in a way that is physically meaningful, mathematically stable, computationally efficient, and easy implement. It should be noted that the terminology operator means: (1) It is an actual mathematical formula for transferring (converting) results of different regions at the interface; or (2) it is a set of numerical treatments for transferring information which are developed from some fundamental considerations. At present, the second is the most frequently encountered. The pioneering work based on Eq. (1a), i.e., based on the state variable coupling, was done by O Connell and Thompson [5]. An unsteady shear flow was investigated, and the results from the hybrid method agreed well with the analytical solutions. Two main limitations of their coupling scheme are that the mass flux continuation between the atomistic and continuum regions was not dealt with, and the time scales of two different methods (micro versus macro) were not decoupled. Hadjiconstantinou and Patera [6] proposed a domain decomposition coupling scheme based on the Schwarz alternating method. They proposed a coupling scheme, called the Maxwell Demon method, to impose the continuum solutions to the boundary of the atomistic region. The implementation procedure of the Maxwell Demon will be presented later. The use of a particle reservoir ensures that there is no net mass flux into the region of interest, so that the mass conservation condition at the interface is guaranteed. Using the Schwarz alternating method, the time scales of the MD simulation and the continuum method were successfully decoupled. The results from the hybrid method agreed well with the solutions from the pure continuum and MD reference computations. Werder et al. [7] developed a hybrid scheme which can handle the nonperiodic velocity boundary conditions of the atomistic domain, with the aid of a particle insertion algorithm named USHER [8] and specularly reflecting walls. An effective boundary force based on the radial distribution function was considered to compensate for the lack of surrounding structure of the fluid. The simulation results of the liquid argon flow around a carbon nanotube (CNT) from the hybrid method were in good agreement with the fully MD reference solutions. Based on the work by Werder et al. [7], Dupuis et al. [9] introduced a novel coupling scheme to solve the same problem. In this method, the lattice Boltzmann (LB) method was coupled with the MD simulation to take advantage of the inherent mesoscopic characteristics of the LB method. The LB method was used for the entire domain, while the local details in the overlap region were provided by the MD simulation. The velocity boundary condition around the CNT was presumed nonslipping to solve the Navier-Stokes (N-S) equations. A local forcing term g was added to the lattice Boltzmann equation in order to enforce Dirichlet boundary conditions. During the information exchange procedure between the continuum and the atomistic region, the velocities and the velocity gradients were both considered to improve the results. The results showed that the LB-MD hybrid model indeed had better performance than the previous coupled macro method-md model proposed by Werder et al.

5 372 W. J. ZHOU ET AL. [7] when simulating the liquid argon flow around a CNT. All the above-mentioned methods belong to the state variable exchange expressed by Eq. (1a). A primary hybrid method based on direct flux exchange was proposed by Flekkoy et al. [10]. In their method, the coupling was directly realized by the exchange of mass and momentum fluxes across the particle continuum interface between the continuum and atomistic regions. Thus the conservation laws were naturally satisfied. Steady isothermal Couette and Poiseuille flows were simulated to validate their coupling scheme. Then Delgado-Buscalioni and Conveney [11] developed a similar scheme based on the one proposed by Flekkoy et al. [10]. In their work, unsteady fluid flows were investigated and the energy flux across the particle continuum interface was also considered. The hybrid method based on direct flux exchange was further developed by Wagner and Flekkoy [12], Delgado-Buscalioni et al. [13], De Fabritiis et al. [14], and Kalweit and Drikakis [15]. In this article, the hybrid scheme coupling the MD and the LBM is adopted. The LB method is adopted for resolving the continuum formulation because of its inherent mesoscopic characteristics and its geometric flexibility [16]. Moreover, the underlying kinetic nature of the LB equation is valuable for the simulation of microfluidics [17]. The Maxwell Demon method [6] is used to impose the boundary conditions of the atomistic region from the continuum region. What are different from reference [6] includes following two aspects. First, the boundary conditions of the continuum region are obtained by reconstructing the single-particle distribution function of the LB method from the results of the MD simulation through the reconstruction operator proposed in [1, 2]. With the reconstruction operator approach, the selection of the continuum part out of the whole computational domain is more flexible because of the good geometric adaptive feature of the LB method. Second, a mesh refinement of the LB domain is needed when one wants to acquire better results. Correspondingly, the size of sampling cells of the MD domain is reduced. However, traditional weighting function used in MD to average the atomic properties cannot guarantee stability, due to the decrease of the size of sampling cells in the MD simulation. A novel weighting function is proposed and is proven to be efficient. The outline of the article is as follows. In Section 2, the basic principles of the MD simulation and the LB method and their coupling scheme are presented, and then a new weighting function used to average the atomic properties is introduced. In Section 3, the hybrid scheme is demonstrated through simulations of Couette flow and flow of liquid argon around a CNT. Finally, some concluding remarks are given. 2. HYBRID SCHEME 2.1. Molecular Dynamics Simulation In the atomistic region, the MD simulation is applied [18, 19]. The shifted Lennard-Jones (L-J) potential is used to describe the molecular interactions, " r 12 r 6 r 12 / ðþ¼4e r þ r # 6 r r r c r c ð2þ

6 MULTISCALE SCALING FOR DENSE FLUID FLOWS 373 where e ¼ J and r ¼ nm are the energy and length characteristic parameters, respectively. r c is a cutoff length beyond which the molecular interactions are not considered. The equation of motion used to update the acceleration of each molecular is d 2 r i m i dt 2 ¼ X q/ ji qr j6¼i i ð3þ These Newtonian equations of motion are integrated using the leapfrop algorithm with a time step dt of 0.005s (s ¼ m 1=2 re 1=2, m is the molecular mass). The other parameters are different for different problems and will be given in detail individually Lattice Boltzmann Method In the continuum region, the two-dimensional (2-D) LB method is used due to the absence of variation along the z direction. The main fluid flow in the bulk region away from the fluid solid interface is described by the incompressible N-S equations as follows: ru ¼ 0 ð4þ qu qt þ ðu rþu ¼ 1 rp þ n Du; ð5þ q where u is the velocity vector, p is the pressure, and q is the density. To solve Eqs. (4) and (5), the two-dimensional nine-velocity square lattice (D2Q9) model [20] is used. The nine velocities c i are c 0 ¼ 0 c i ¼ cfcos½ði 1Þp= 2Š; sin½ði 1Þp= 2Šg for i ¼ 1; 2; 3; 4 c i ¼ cfcos½ð2i 1Þp= 4Š; sin½ð2i 1Þp= 4Šg for i ¼ 5; 6; 7; 8 ð6þ where c ¼ Dx=Dt, Dx is the spatial separation of the lattice, and Dt is the time step. The lattice Boltzmann equation with the BGK model [21, 22] is given by f i ðx þ c i Dt; t þ Dt Þ ¼ f i ðx; tþ 1 h i s f i f ðeqþ i ð7þ where f ðeqþ i is the ith equilibrium distribution function and s is the relaxation time. The equilibrium distribution function is expressed as [20] f ðeqþ i ¼ t i q 1 þ c iau a c 2 s þ u au b 2c 4 s c ia c ib c 2 s d ab p where a and b are the Cartesian coordinates, c s ¼ c= ffiffiffi 3 is the speed of sound, ti is a weighting factor with t 0 ¼ 4=9, t 1 ¼ t 2 ¼ t 3 ¼ t 4 ¼ 1=9, and t 5 ¼ t 6 ¼ t 7 ¼ t 8 ¼ 1=36. ð8þ

7 374 W. J. ZHOU ET AL. The density q and the fluid velocity u are given as q ¼ X8 i¼0 f i qu ¼ X8 i¼0 c i f i ð9þ 2.3. The Hybrid Scheme The domain decomposition method based on the alternating Schwarz method [23] is represented in Figure 1. The LB method is used to describe the continuum region away from the fluid solid interface, while the MD simulation is adopted to describe the region near the interface, and both methods are assumed valid and applied in the overlap region between C 1 and C 4. The alternating Schwarz method is described as follows: In every iteration, given the boundary conditions on C 4 by the atomistic solution of the previous iteration and an outer boundary condition by the specific problem, the N-S equations for the continuum region is solved by the LB method. Then the continuum solution in turn offers a boundary condition for the MD simulation in the atomistic region between C 2 and C 3. This procedure is repeated until convergence toward a steady-state solution is achieved. Obviously, the main difficulty in the atomistic continuum coupling schemes is the appropriate imposition of the boundary conditions between the continuum and the atomistic region. The region between C 1 and C 2 is the reservoir region, which will be treated in different ways for different problems. In the C! P region between C 2 and C 3, the imposition of boundary condition from the LB to the MD method is accomplished by the Maxwell Demon approach [6]. It is implemented as follows. At every time step, the velocities of molecules in the C! P region are reset through a Maxwellian distribution with mean and variance consistent with the local continuum fluid velocity and temperature resulted from LB solution. The boundary condition for the continuum region from the atomistic region is obtained by the reconstruction operator method [1, 2]. In Figure 1, C 4 is the computational boundary of the Figure 1. Domain decomposition of the hybrid scheme.

8 MULTISCALE SCALING FOR DENSE FLUID FLOWS 375 continuum region in the atomistic region. The single-particle distribution function of the grid point D can be given by following reconstruction operator: f i ¼ f eq i 1 sdtu ib c 2 s U ia q xa u b þ nq 2 xa u b þ nq 1 S ab q xa q ð10þ where U ia ¼ c ia u a, S ab ¼ q xb u a þ q xa u b, and n is the kinematic viscosity. The gradients are calculated for the macroscopic quantities of the grid point E on the P! C boundary, which can be extracted through spatial and temporal averaging from the molecule configurations. In addition, the distance between the P! C boundary and the computational boundary of the continuum region C 4 is always the same as the grid size of the continuum region in the related direction. Usually, thermal fluctuation should be considered in the LB equation when dealing with nanoscopic flows [24, 25], especially for fluid flow problems where Brownian motion is important. However, in this article the LB method is used to describe the continuum region away from the fluid solid interface where the fluid is still in the hydrodynamic region, and the thermal fluctuation can be ignored according to the derivation process shown in [1]. Therefore the coupling between MD and LB involves only mean flow fields, as done in [7, 9]. It is worth noting that in our hybrid scheme the LB method does not cover the entire domain as was done in [9]. Therefore, the boundary conditions needed to solve the N-S equations need not be assumed in advance but can be provided by the MD simulation at each iteration The Weighting Function The methods used to extract macroscopic quantities from discrete atomic properties are crucial. In a 2-D system, the macroscopic quantity Q ij of a grid X ij, where i and j represent the locations in the x and y directions, respectively, is generally obtained from the neighboring N atoms by Q ij ¼ XN k¼1 q k W ij;k where q k is the property q of atom k. The weighting function W ij,k measures the contribution of atom k to quantity Q ij of grid node X ij and is given by W ij;k ¼ w k ðx i Þw k y j ð12þ ð11þ The simplest type of weighting functions is called nearest-grid-point (NGP). In this scheme, the properties of an atom are totally assigned to the nearest grid point, i.e., W (NGP)ij, k ¼ 1 if the distances between the atom and the grid point in the x and y directions are less than Dx and Dy, respectively, otherwise W (NGP)ij, k ¼ 0. In this study a more accurate weighting function named cloud-in-cell (CIC) [26, 27] is also adopted, which assigns the properties of an atom to the nearest four grid points as depicted in Figure 2a. It can be expressed as w k ðx i Þ¼ 1 þ x k x i Dx 1 x k x i Dx ðx i Dx < x k < x i Þ ðx i < x k < x i þ DxÞ ð13þ

9 376 W. J. ZHOU ET AL. Figure 2. Distribution schemes of atom properties to the surrounding grid points. where x k and x i are the atom and the grid point positions in the x direction, respectively, and w k (y j ) has a similar expression as w k (x i ). However, in our numerical practice it is found that when the mesh becomes so fine that there is very few atoms in each block sized by Dx Dy, the fluctuations increase when averaging the atom properties using the NGP or CIC scheme, which is due to the fact that the number of atoms assigned to each grid point is reduced. To solve this problem, a novel type of weighting function named modified cloud-in-cell (MCIC) is proposed. As shown in Figure 2b, the dots represent atoms and the triangles denote the grid points inside the circle of radius r 0. The properties of an atom are distributed to the grid points located inside the circle whose center coincides with the atom position. Here we choose the radius of the circle as r 0 ¼ 1.0r. The MCIC weighting function can be defined as 8 < W ij;k ¼ ðd ij;kþ 1 = PN ðd ij;k Þ 1 d ij;k < r 0 ð14þ : m¼1 0 otherwise where d ij,k is the absolute distance between the atom k and the grid point X ij, and N is the number of grid points inside the circle. Use of the MCIC scheme can guarantee that the number of atoms assigned to each grid point remains unchanged when the mesh of the LB method changes. 3. RESULTS AND DISCUSSION In this section, the hybrid scheme is first validated through simulations of steady Couette flows and is then used to investigate the flow of liquid argon around a CNT. Finally, the novel weighting function is used to adapt the refinement of the mesh Couette Flows As a good benchmark test, Couette flows are first simulated to validate our hybrid scheme. In our simulation, the fluid argon is confined between two parallel

10 MULTISCALE SCALING FOR DENSE FLUID FLOWS 377 walls at y ¼ 0 and 68.12r. The top wall is moving at a velocity U ¼ r=s while the bottom wall is kept still. The LB method is implemented in the upper subregion of 14.60r y 68.12r (i.e., the continuum region), the MD simulation is used in the lower subregion of 0 y 34.06r (i.e., the atomistic region), and the overlap region lies in the domain of 14.60r y 34.06r, as depicted in Figure 3. The flow in the continuum region is assumed to be fully developed, hence is one-dimensional (1-D) in the sense that it changes only in the y coordinate, while the particle motion in the atomistic region is three-dimensional (3-D). The parameters of the atomistic region are described as follows. The density of liquid argon is q ¼ 0.81mr 3, the temperature of the fluid during the whole simulation is kept constant at T ¼ 1.1ek B 1, where k B is Boltzmann s constant. Control of the temperature is achieved through a Langevin thermostat [28] with the damping rate s 1, which is applied only in the z direction, normal to the bulk flow. Thus the viscosity of the liquid argon at given density and temperature is m ¼ 2.14esr 3 [5]. The lower wall consists of two layers of atoms in face-centered cubic (FCC) (1 1 1) structure. The interactions between fluid and solid atoms are also calculated through Eq. (2), two groups of parameters (e wf =e, r wf =r, q w =q) ¼ (0.6, 1.0, 1) and (0.6, 0.75, 4), where e wf, r wf, and q w are the fluid solid energy scale, the fluid solid length scale, and the wall density, respectively. The former group of parameters yields a nonslip fluid wall boundary condition, and the latter yields a slip one [29]. All the interactions are truncated at r c ¼ 2.2r. As the sampling cell size in the atomistic region of this problem is relatively large, the simplest weighting function NGP is adopted when averaging the atomic properties. The Reynolds number based on the channel height is Re ¼ In the atomistic region, periodic boundary conditions are imposed in the x and z directions, and the simulation sizes along these two directions are l x ¼ l z ¼ 13.62r. The continuum region is discretized by uniform grids with Dx ¼ Dy ¼ 4.87r, and the periodic boundary condition is applied for the inlet and outlet boundaries. At the P! C boundary, the macroscopic quantities such as velocities are presumed to be uniform along the x direction and are taken from the atomistic region by averaging the properties of all atoms within a volume of dimensions l x Dy l z whose center is located at the grid point concerned. In the overlap region, the thicknesses Figure 3. Schematic diagram of Couette flow.

11 378 W. J. ZHOU ET AL. of the reservoir layer and the C! P region are Dy and 0.2r, respectively. To ensure that the atoms always stay in the simulation domain, an external depress force is added to the atoms in the reservoir region [30]: y y 2 f y ¼ p 0 r 1 ðy y 2 Þ=ðy 1 y 2 Þ ð15þ where p 0 represents the average pressure of the atomistic region, and y 1 and y 2 denote the coordinate values in the y direction of the upper and lower boundaries of the reservoir region, respectively. In each iteration of the hybrid scheme, the LB method runs for 20,000 steps (totally 3.5 ns), and the MD simulation runs for 60,000 steps (totally 0.65 ns), of which the last 40,000 steps are used to provide the next boundary condition for the continuum region. Figure 4 shows the steady solutions of the Couette flows from the hybrid method and the MD reference simulation, where H represents that height of the channel. The solid and the dash lines are the pure MD reference solutions of the two parameter groups and the symbols represent the solutions of the hybrid scheme. It can be seen that the hybrid results are in good agreement with the pure MD reference solutions, under slip and no-slip fluid wall boundary conditions Flow Past a CNT In this section, the flow of liquid argon around a CNT is investigated using the hybrid scheme. All the configuration parameters are the same as [7, 9]. The CNT, of chirality (16, 0) and radius r ¼ 1.836r, is located at the center of the whole computational domain of 88.11r 88.11r 13.19r (30 nm 30 nm 4.49 nm in real units). The interactions between argon and carbon atoms are also calculated through Eq. (2), with e ArC ¼ 0.572e, r ArC ¼ 1.0r. All the interactions are truncated at Figure 4. Comparisons of the steady velocity profiles from the hybrid method and the pure MD simulation results under slip and no-slip boundary conditions.

12 MULTISCALE SCALING FOR DENSE FLUID FLOWS 379 r c ¼ 2.94r. In the hybrid simulation, the size of the MD region is 29.37r 29.37r 13.19r. As shown in Figure 5, the MD region is restricted by an outer boundary C 1, while the LB region is bounded by an inner boundary C 4. Uniform grids are used for the LB region, with grid size Dx ¼ Dy ¼ 1.47r. A flow velocity of u x ¼ 0.634r=s and u y ¼ 0 is specified at the inlet (x ¼ 0) and outlet (x ¼ 88.11r) boundaries. The density is chosen to be q ¼ 0.6mr 3 ; the temperature is maintained at T ¼ 1.8ek 1 B by using the Langevin thermostat [28] in the z direction with the damping rate s 1. Thus the viscosity of the fluid argon needed by the LB method is n ¼ 1.5 esm 1 [31], and the Reynolds number based on the CNT diameter is Re ¼ 1.5. In the overlap region, the C! P region is the domain between C 2 and C 3 and has a thickness of 2.94r. The reservoir region has a thickness of 3.67r, and no restriction is imposed to atoms in this region. Due to the existence of the reservoir region, periodic boundary conditions can be imposed in all three directions x, y, andz, and there is no need to consider the insertion of atoms near the boundary as treated in [7]. For determining the reconstruction operator containing velocity gradient (Eq. (10)) on the boundary C 4, the macro quantities at the lattice nodes along two adjacent boundaries, the boundary C 4 and C 5, are needed. The information on C 4 can be obtained from the LBM solutions on that line, while the macro quantities at the lattice nodes along the boundary C 5 (i.e., the P! C boundary) are obtained by averaging the properties of all atoms within a volume of certain size whose center is coincident with the lattice node. The numbers of steps of LB and MD methods in each iteration of the hybrid scheme are the same as in the Couette flows: In each iteration, the LB method runs for 20,000 steps, and the MD simulation runs for 20,000 steps for equilibration and a further 40,000 steps to collect the next boundary condition for the continuum region. The hybrid simulation starts from the LB region with an initial condition u x ¼ 0.634r=s and u y ¼ 0 at the inner boundary C 4. To evaluate the deviation of the hybrid results from the pure MD simulation results, an error e i at iteration i is expressed as e i ¼ 1 X u i m u m;pure ð16þ N m2x u 1 Figure 5. Schematic diagram of argon flow around a CNT (color figure available online).

13 380 W. J. ZHOU ET AL. where N is the number of cells in the whole computational domain X, u 1 represents the free-stream velocity, u i m is the hybrid velocity of cell m at iteration i, andu m;pure is the corresponding pure MD result. Meanwhile, the change rate of the velocity field d i is used to describe the stability of the hybrid scheme, and it is given by d i ¼ 1 X u i m ui 1 m ð17þ N m2x To study the influence of different mesh resolutions to the hybrid results, three cases are considered as follows. u 1 Case 1. The grid size of the LB domain is Dx ¼ Dy ¼ 1.47r, the corresponding sampling cells of the MD domain are Case 2. The mesh of the LB domain is doubly refined on the basis of case 1, thus the grid size is Dx ¼ Dy ¼ 0.735r, and the corresponding sampling cells of the MD domain are Case 3. The mesh of the LB domain is doubly refined on the basis of case 2, thus the grid size is Dx ¼ Dy ¼ r, and the corresponding sampling cells of the MD domain are The above three cases are simulated using both CIC and MCIC schemes. In order to compare with the results of [7, 9], real units are used hereafter. The time evolution of errors e i of three cases using the CIC scheme is depicted in Figure 6. As can be seen, the error decreases when the mesh is refined from case 1 to case 3. The average errors between iterations 50 and 100 of case 1 to case 3 are 1.54%, 1.25%, and 1.00%, respectively. The corresponding time evolution of the rate of velocity change d i using the CIC scheme is shown in Figure 7. Unlike the trend of change Figure 6. Time evolution of error e i between the hybrid results using the CIC scheme and the pure MD reference solutions for three different cases.

14 MULTISCALE SCALING FOR DENSE FLUID FLOWS 381 Figure 7. Time evolution of the rate of velocity change of the hybrid results using the CIC scheme for three different cases. of error e i, the rate d i increases with the mesh refinement, and the average rates of velocity change between iterations 50 and 100 for case 1 to case 3 are 0.27%, 0.33%, and 0.40%, respectively. Figure 6 and 7 imply that although hybrid results get better when the mesh is refined, fluctuations between successive iterations increase at the same time. The reason is that when the mesh gets finer, the number of atoms assigned to each grid point decreases when averaging the atomic properties as the size of sampling cells becomes small. Fortunately, this problem can be avoided when using the MCIC scheme. In the MCIC scheme, the number of atoms assigned to each grid point stays fixed when the mesh is refined, as long as the radius r 0 remains unchanged. Figure 8 shows the time evolution of errors e i of three cases using the MCIC scheme. The error decreases when the mesh is refined, which is similar to the result of the CIC scheme. The average errors between iterations 50 and 100 of cases 1 to 3 when using the MCIC scheme are 1.59%, 1.18%, and 1.03%, respectively. The trend of rate of velocity change with the MCIC method, however, is different from the one with the CIC method, which can be seen from Figure 9. The average rates between iterations 50 and 100 of case 1 to case 3 when using the MCIC scheme are 0.31%, 0.28%, and 0.31%, respectively, which maintain almost at the same level. Obviously, it can be seen that the results from the MCIC scheme are much more stable than those from the CIC scheme. In Figure 10, the converged hybrid simulation results of case 3 using the MCIC method are compared with the pure MD reference solutions. Figure 10c indicates that the u x velocity contour of the hybrid results is highly consistent with the pure MD solutions, especially in the wake region of the CNT. Figure 11 shows the time evolution of the velocity u x along the horizontal centerline y ¼ 15 nm and the vertical centerline x ¼ 15 nm. After about 20 iterations, the velocity profiles of the hybrid results agree well with the MD reference solutions.

15 382 W. J. ZHOU ET AL. Figure 8. Time evolution of error e i between the hybrid results using the MCIC scheme and the pure MD reference solutions for three different cases. Moreover, the pure LB method is also adopted to see whether the fluid flow around the CNT can be appropriately captured. In the pure LB method, the CNT is treated as a cylinder, and zero velocities are imposed at its surface. The finally adopted lattice spacing Dx ¼ 0.25r and a further decrease in the lattice spacing does Figure 9. Time evolution of the rate of velocity change of the hybrid results using the MCIC scheme for three different cases.

16 MULTISCALE SCALING FOR DENSE FLUID FLOWS 383 Figure 10. Comparisons of the velocity contours between the converged hybrid solutions and the pure MD reference results. (a) Pure MD results. (b) Hybrid solutions. (c) Comparison of two results (dashed lines, hybrid solutions; solid lines, MD reference results). not affect the results. The u x distributions along x and y are presented in Figure 11. From the figure it can be seen that even though as a whole the results from the pure LB method agree with the MD reference solutions quite well, deviations much larger than 1% are observable. This comparison further proves the necessity of the hybrid scheme. It is also interesting to note that in [32] the LBM was used to simulate the whole flow domain around a nano-size obstacle, and its was indicated that in order to capture the recirculation just behind the obstacle the lattice spacing of the LB method, Dx ¼ 0.25r should be adopted. However, in this article, the flow features near the CNT is obtained by MD simulation, and the other computational domain is resolved by LB. Our results show that in our hybrid scheme, when the LB grid spacing Dx ¼ r, the deviation between results of the hybrid scheme and the pure MD method is only 1.00%.

17 384 W. J. ZHOU ET AL. Figure 11. Time evolution of the u x velocity component: (a) along the horizontal centerline y ¼ 15 nm; (b) along the vertical centerline x ¼ 15 nm. As far as the computational time is concerned, the time consumption of the hybrid scheme is only about one-fifth of that of the pure MD simulation. 4. CONCLUSIONS In this article, a hybrid scheme has been adopted for dense fluid flow. MD simulations are used in the region near the fluid solid interface, while LB methods are used in the bulk region away from the fluid solid interface, where the continuum assumption is still valid. During the coupling procedure, the Maxwell Demon [6] approach is used to impose the boundary conditions for the atomistic region from the continuum region, and the boundary conditions of the continuum region is obtained by the reconstruction operator [1, 2]. The steady Couette flow is simulated to validate the hybrid scheme, and the results agree well with the pure MD reference solutions. Then the flow of fluid argon around a CNT is investigated. The mesh refinement of the LB method accompanied by the reduction in size of sampling cells of the atomistic region lead to better results but severe fluctuations when using the CIC weighting function. A novel weighting function named MCIC is proposed. The results show that the MCIC method performs better than the CIC method when the mesh is refined. Moreover, it is not necessary to use the LB method for the entire computational domain as was done in [9] when using our hybrid scheme, because the distribution function at the boundary of the LB method can be easily reconstructed due to the reconstruction operator method. Thus the characteristics of the fluid solid interface, such as slip or no-slip boundary conditions, do not need to be known in advance, which facilities the use of our hybrid scheme for complex geometries and boundary conditions. REFERENCES 1. H. Xu, H. B. Luan, Y. L. He, and W. Q. Tao, A Lifting Relation from Macroscopic Variables to Mesoscopic Variables in Lattice Boltzmann Method: Derivation, Numerical

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