Consistent evaporation formulation for the phase-field lattice Boltzmann method

A consistent evaporation model is developed for the conservative Allen-Cahn-based phase-field lattice Boltzmann method that uses an appropriate source term to recover the advection-diffusion equation for the specific humidity. To evaluate the accuracy of the proposed scheme, simulations are conducted of a steady-state one-dimensional Stefan flow for a flat interface and a three-dimensional evaporating sessile droplet on a flat substrate for a curved interface. It is confirmed that the results for the evaporative mass flux of the Stefan flow agree well with those obtained from the analytical solution within a specific humidity range of 0.8 or less at the liquid-gas interface. For the sessile droplet case, the results for the dependence of the contact angle on the evaporative mass flux and its profile show good agreement with those obtained from the model of Hu and Larson [J. Phys. Chem. B 106, 1334 (2002)].

Figure 1

Computational domain used for the simulations of the Stefan flows.

Figure 2

Nondimensional evaporative mass flux in the simulations of the Stefan flows.

Figure 3

Schematic of the computational domain used for the simulations of the evaporating droplet on the flat substrate.

Figure 4

Specific humidity field around the droplet obtained in the simulations of a droplet on a flat substrate at different contact angles: (a) $\theta ={30}^{\circ }$, (b) $\theta ={60}^{\circ }$, and (c) $\theta ={90}^{\circ }$.

Figure 5

Velocity field inside the droplet obtained in the simulation of a droplet on a flat substrate at $\theta ={90}^{\circ }$.

Figure 6

Nondimensional evaporative mass flux profiles obtained in the simulations of a droplet on a flat substrate at different contact angles: (a) $\theta ={30}^{\circ }$, (b) $\theta ={60}^{\circ }$, and (c) $\theta ={90}^{\circ }$.

Figure 7

Relative error of the evaporative mass flux at the top of the droplet from the model proposed by Hu and Larson [11].

Figure 8

Normal velocity profile from the droplet apex along the $z$-direction, where $d$ denotes the distance from the droplet apex.