The Lattice Boltzmann Method by unknow

Author:unknow
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

The gradient term in (9.22) penalises changes in the density. This is key to the formulation of surface tension in this model. To appreciate this statement, let us first derive the chemical potential .

The chemical potential is defined as the free energy cost (gain) for adding (removing) materials to (from) the system. Mathematically, this is given by

(9.23)

In thermodynamic equilibrium, the chemical potential is constant everywhere in space. If it is not constant, there will be a free energy gain by transferring fluid material from one part of the system to another. In other words, there will be a thermodynamic force.

When the system is in one of the bulk free energy minimum solutions, either in the liquid or in the gas phase, then ν ρ 2 −β τ w = 0 and the gradient term in (9.23) also vanishes. Therefore, we find μ = μ 0 in the liquid and gas bulk phases. Now, our statement that the chemical potential is constant everywhere in space includes the liquid-gas interface where the density varies. For simplicity, let us assume the interface is flat and is located at x = 0. The differential equation in (9.23), after setting μ = μ 0, thus reads

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