The velocity discretization is a critical step in deriving the lattice Boltzmann (LBE) from the continuous Boltzmann equation. This problem is considered in the present paper, following an alternative approach and giving the minimal discrete velocity sets in accordance with the order of approximation that is required for the LBE with respect to the continuous Boltzmann equation and with the lattice structure. Considering to be the order of the polynomial approximation to the Maxwell-Boltzmann equilibrium distribution, it is shown that solving the discretization problem is equivalent to finding the inner product in the discrete space induced by the inner product in the continuous space that preserves the norm and the orthogonality of the Hermite polynomial tensors in the Hilbert space generated by the functions that map the velocity space onto the real numbers space. As a consequence, it is shown that, for each order N of approximation, the even-parity velocity tensors are isotropic up to rank 2N in the discrete space. The norm and the orthogonality restrictions lead to space-filling lattices with increased dimensionality when compared with presently known lattices. This problem is discussed in relation with a discretization approach based on a finite set of orthogonal functions in the discrete space. Two-dimensional square lattices intended to be used in thermal problems and their respective equilibrium distributions are presented and discussed.

The velocity discretization is a critical step in deriving the lattice Boltzmann (LBE) from the Boltzmann equation. The velocity discretization problem was considered in a recent paper (Philippi et al., From the continuous to the lattice Boltzmann equation: the discretization problem and thermal models, Physical Review E 73: 56702, 2006) following a new approach and giving the minimal discrete velocity sets in accordance with the order of approximation that is required for the LBE with respect to the Boltzmann equation. As a consequence, two-dimensional lattices and their respective equilibrium distributions were derived and discussed, considering the order of approximation that was required for the LBE. In the present work, a Chapman-Enskog (CE) analysis is performed for deriving the macroscopic transport equations for the mass, momentum and energy for these lattices. The problem of describing the transfer of energy in fluids is discussed in relation with the order of approximation of the LBE model. Simulation of temperature, pressure and velocity steps are also presented to validate the CE analysis.

In this paper, a lattice BGK (Bhatnagar-Gross-Krook) model is proposed for immiscible fluids. Collision operator is decoupled considering mutual and cross collisions between lattice particles, with three independent parameters related to the species diffusivity and to the viscosity of each fluid. Field mediator's concept, described by Santos and Philippi [Phys. Rev. E 65, 046305 (2002)], is extended to the framework of the lattice-Boltzmann equation and interference between mediators and particles is modeled by considering that there is a deviation in particles velocity, proportional to the mediators' distribution at the site. A Chapman-Enskog analysis is performed leading to theoretical predictions of the macroscopic equations inside the transition layer and to the transition-layer thickness. Chapman-Enskog analysis is restricted to near-equilibrium states and was unable to predict the correct second-order interfacial tension dependence on the modeled long-range fields intensity. Interfacial tension was, only, correctly retrieved using a nonequilibrium solution. Theoretical predictions are compared with simulation results and the model is tested considering its ability in describing the dynamical behavior of the interface and Galilean invariance.

In this work, we present a derivation for the lattice-Boltzmann equation directly from the linearized Boltzmann equation, combining the following main features: multiple relaxation times and thermodynamic consistency in the description of non isothermal compressible flows. The method presented here is based on the discretization of increasingly order kinetic models of the Boltzmann equation. Following a Gross-Jackson procedure, the linearized collision term is developed in Hermite polynomial tensors and the resulting infinite series is diagonalized after a chosen integer N, establishing the order of approximation of the collision term. The velocity space is discretized, in accordance with a quadrature method based on prescribed abscissas (Philippi et al., Phys. Rev E 73, 056702, 2006). The problem of describing the energy transfer is discussed, in relation with the order of approximation of a two relaxation-times lattice Boltzmann model. The velocity-step, temperature-step and the shock tube problems are investigated, adopting lattices with 37, 53 and 81 velocities.

A Lattice-Boltzmann method based on field mediators is proposed to simulate the capillary rise between parallel plates by considering the effect of long-range interactions between the fluids and the solid walls. As a starting point, a liquid-vapor system was employed, which was modeled using a known model described in the literature. The simulations were compared with theoretical solutions of the Bosanquet equation. The results obtained are in good agreement with theoretical predictions, particularly when the dependence between the dynamic contact angle and the capillary number is taken into account. This dependence shows that on smooth and homogeneous solid surfaces, where pinning effects in the contact line are weak, the dynamic contact angle is linearly dependent on the capillary number, in good agreement with theoretical and experimental results available in the literature. Some discrepancies were observed in the first stages of the capillary rise when the distance between the parallel plates is large, considering the high complexity involved in predicting the initial meniscus formation. The results presented in this work appear to indicate that the inclusion of long-range forces does not change significantly the fluid flow dynamics at the mesoscopic level, at least, when ideally flat and homogeneous solid surfaces are used in the simulations.

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In fluid mechanics, multicomponent fluid systems are generally treated either as homogeneous solutions or as completely immiscible parts of a multiphasic system. In immiscible systems, the main task in numerical simulations is to find the location of the interface evolving over time, driven by normal and tangential surface forces. The lattice-Boltzmann method (LBM), on the other hand, is based on a mesoscopic description of the multicomponent fluid systems, and appears to be a promising framework that can lead to realistic predictions of segregation in non-ideal mixtures of partially miscible fluids. In fact, the driving forces in segregation are of a molecular nature: there is competition between the intermolecular forces and the random thermal motion of the molecules. Since these microscopic mechanisms are not accessible from a macroscopic standpoint, the LBM can provide a bridge linking the microscopic and macroscopic domains. To this end, the first purpose of this article is to present the kinetic equations in their continuum forms for the description of the mixing and segregation processes in mixtures. This paper is limited to isothermal segregation; non-isothermal segregation was discussed by Philippi et al. (Phil. Trans. R. Soc., vol. 369, 2011, pp. 2292–2300). Discretization of the kinetic equations leads to evolution equations, written in LBM variables, directly amenable for numerical simulations. Here the dynamics of the kinetic model equations is demonstrated with numerical simulations of a spinodal decomposition problem with dissolution. Finally, some simplified versions of the kinetic equations suitable for immiscible flows are discussed.

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