Lattice Boltzmann simulation of thermal nonideal fluids G. Gonnella Dipartimento di Fisica, Università di Bari and Istituto Nazionale di Fisica Nucleare, Sezione di Bari, via Amendola 173, 70126 Bari, Italy A. Lamura Istituto Applicazioni Calcolo, CNR, via Amendola 122/D, 70126 Bari, Italy V. Sofonea Center for Fundamental and Advanced Technical Research, Romanian Academy, Bd. Mihai Viteazul 24, 300223 Timişoara, Romania Received 19 February 2007; revised manuscript received 14 May 2007; published 7 September 2007 A thermal lattice Boltzmann model for a van der Waals fluid is proposed. In the continuum, the model reproduces at second order of a Chapman-Enskog expansion, the theory recently introduced by A. Onuki Phys. Rev. Lett. 94, 054501 2005. Phase separation has been studied in a system quenched by contact with external walls. Pressure waves favor the thermalization of the system at initial times and the temperature, soon with respect to typical times of phase separation, becomes homogeneous in the bulk. Alternate layers of liquid and vapor form on the walls and disappear at late times. DOI: 10.1103/PhysRevE.76.036703 PACS numbers: 47.11.-j, 64.70.Fx, 05.70.Ln, 47.20.Hw I. INTRODUCTION In fluid systems the coupling between hydrodynamics and thermodynamics produces a variety of complex effects of fundamental and technological importance 1. An example is the piston effect consisting of the speeding up of the ther- mal equilibration of a near-critical fluid warmed by external walls 2,3. This is due to pressure waves created by thermal expansion near the walls. In phase transitions, heat transport and thermal gradients, often neglected, can be quite relevant 4,5. When a fluid is quenched from a temperature above the critical point into a coexistence region, the temperature jump is generally assumed instantaneous in all of the system, and most of the theories are based on isothermal evolution 6. However, in realistic situations, the system is quenched by contact with colder external walls. The temperature of the bulk does not set immediately at the value of the walls and this can influence the phase separation. Simulations are gen- erally useful for analyzing these phenomena, but a full ther- mohydrodynamic study has been so far not considered. An efficient approach to numerical simulations of fluids, which has been developed in the last years, is based on the so-called lattice Boltzmann method LBM7. A set of dis- tribution functions representing densities of particles moving along fixed lattice directions evolve following Boltzmann equations with collision terms linearized in a single relax- ation time approximation 8. Conservation laws and proper choice of equilibrium distribution functions in the collision terms ensure that the correct hydrodynamic equations are recovered in the continuum limit. LBM has been applied to study single fluids, dynamics and phase ordering in multi- component and complex fluids 9; the method can easily handle complex geometries and is suitable for parallel imple- mentation 7. In this paper, inspired by the Klimontovich approach to kinetic theory for nonideal gases 10, we introduce a lattice Boltzmann scheme which allows to simulate the thermohy- drodynamic equations for a multiphase fluid including inter- face free-energy contributions. This is an example of LBM able to correctly reproduce in the continuum, at second order of a Chapman-Enskog expansion, the transport equations re- cently established by Onuki 11. We will study the phase separation of a van der Waals fluid. Our results show that thermalization occurs faster due to the “piston effect.” After a very early stage of phase separation the temperature in the bulk of the system can be considered homogeneous. Alter- nate layers of liquid and vapor form close to the external walls, but do not survive at late times. LBM for thermal fluids has been so far only set up for ideal fluids 12 or do not consider all terms present in con- tinuum equations 13. We define density, velocity, and tem- perature as momenta of various orders of the same set of particle distributions—see Eqs. 1–3. In other models, where the temperature results form a separate set of distribu- tions, the energy equation reduces to a convection-diffusion equation not including all the stress contributions 14. Moreover, the Prandtl number can be varied in our model, and due to the large variability of this number in real sys- tems, this is a major goal in LBM for nonideal fluids 12. We will use a finite difference lattice Boltzmann method FDLBM15, where the relationship c = s / t among the lattice speed c and the space and time steps s and t are no longer considered. In thermal LBM, for each direction, dif- ferent distribution functions are introduced moving with dif- ferent speeds and FDLBM allows to choose the more conve- nient discrete velocity sets 16. In addition, FDLBM enables to use different discretization schemes. This is useful for im- proving numerical stability and when nonuniform grids or mixtures with different masses are considered. II. THE MODEL We generalize the two-dimensional lattice Boltzmann model introduced in Ref. 16. There are four sets of veloci- ties defined by e 0 =0, e ki = cosi -1  4 , sini -1  4 c k  with PHYSICAL REVIEW E 76, 036703 2007 1539-3755/2007/763/0367035 ©2007 The American Physical Society 036703-1