# Two-dimensional off-lattice Boltzmann model for van der Waals fluids with variable temperature

## TL;DR

This paper introduces a novel two-dimensional off-lattice Boltzmann model for van der Waals fluids with variable temperature, demonstrating high accuracy and stability in simulating phase separation and thermal flows.

## Contribution

It presents a new off-lattice Boltzmann approach with fourth-order accuracy for thermal liquid-vapor systems, including variable temperature effects and phase separation.

## Key findings

- Achieves at least fourth order convergence in shear and wave propagation.
- Maintains small spurious velocities (<1%) even at high temperature ratios.
- Preserves Galilean invariance up to second order.

## Abstract

We develop a two-dimensional Lattice Boltzmann model for liquid-vapour systems with variable temperature. Our model is based on a single particle distribution function expanded with respect to the full-range Hermite polynomials. In order to ensure the recovery of the hydrodynamic equations for thermal flows, we use a fourth order expansion together with a set of momentum vectors with 25 elements whose Cartesian projections are the roots of the Hermite polynomial of order Q = 5. Since these vectors are off-lattice, a fifth-order projection scheme is used to evolve the corresponding set of distribution functions. A fourth order scheme employing a 49 point stencil is used to compute the gradient operators in the force term that ensures the liquid-vapour phase separation and diffuse reflection boundary conditions are used on the walls. We demonstrate at least fourth order convergence with respect to the lattice spacing in the contexts of shear and longitudinal wave propagation through the van der Waals fluid. For the planar interface, fourth order convergence can be seen at small enough lattice spacings, while the effect of the spurious velocity on the temperature profile is found to be smaller than 1.0%, even when T w ' 0.7 T c . We further validate our scheme by considering the Laplace pressure test. Galilean invariance is shown to be preserved up to second order with respect to the background velocity. We further investigate the liquid-vapour phase separation between two parallel walls kept at a constant temperature T w smaller than the critical temperature T c and discuss the main features of this process.