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

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