Based on the lubrication theory and the coupling between gas phase diffusion and droplet evaporation, a mathematical model of droplet evaporation on a solid substrate with uniform wall temperature is established, and the evolution equation of droplet thickness based on the gas phase diffusion model is derived. The quasi-static gas phase field is solved with the droplet contact line dynamics, and the influence of Ma and Pe_k on the droplet evaporation under the effect of gas phase is discussed through numerical simulation. The results show that under the same parameters, the droplet evaporation process of the gas phase diffusion model is slower than that of the one-sided model, and the contact radius and the evaporating rate is decreased, and the results are more consistent with the experimental results. Under the gas phase diffusion model, the droplet evaporating rate is increased and the droplet evaporation process is shortened by reducing Ma, thereby promoting droplet evaporation. By increasing Pe_k, the gas density near the droplet is densified, and the droplet evaporation is enhanced.

The three-dimensional multi-relaxation (MRT) Shan-Chen Lattice Boltzmann method (LBM) is used to simulate the dynamic process of a droplet impacting small solid spheres with different wettability (The ratio of droplet diameter to small sphere diameter is greater than 1.) On the basis of verifying the accuracy of the calculation model by various methods, the influence of the Weber number of the droplet and the surface wettability of the small sphere on the dynamic process and the relationship of the geometry of the lamella formed with wettability are explored. The results show that the lamella formed after impacting the hydrophilic sphere is bell-shaped, When the surface wetting degree is constant, the maximum bottom diameter of the lamella increases with the increase of Weber number. When impacting the hydrophilic small sphere, the lamella completely closes after the formation of the lamella; the larger the Weber number, the earlier the lamella is completely closed, and the lamella is prone to break during the closing process. When impacting the hydrophobic small sphere, a larger Weber number leads to an earlier breakage of the lamella, and the broken lamellas fall down in a ring strip during the opening of the lamella. The geometric characteristics of the lamella are affected by the Weber number and wettability of the small sphere.

A mathematical model of droplet spreading on surface of an immiscible liquid based on lubrication approximation was used. Influence of viscosity ratio on evolution and equilibrium state of droplet at high viscosity ratios was investigated. Crucial parameters including droplet thickness and spreading radius were examined. It shows that deformation of the liquid-liquid interface near the contact line is affected by the viscosity ratio and the surface tension ratio; Increase of viscosity ratio reduces the spreading rate and the time constant, thereby, prolongs the spreading evolution. It does not affect the final stable shape of the droplet; Relation between spreading radius and time satisfies x_max= 1 - 0.2 exp(- βt). Inertial oscillation does not appear at the final stage of droplet spreading in the case of high viscosity ratios.

To analyze instability on surface of liquid film in a wire-frame drainage experiment, we establish a three-dimensional mathematical model for drainage process of a wire-frame containing insoluble surfactants, and simulate instability at bottom of the liquid film. Influence of factors including Marangoni effect, dilational viscosity and disturbance wave number are analyzed. It shows that bottom perturbation is severe at beginning of drainage, then quickly weakens, and gradually increases in the late drainage. Perturbation at the beginning is caused by initial perturbation, and instability at the late of the drainage is related to the distribution of surfactant. A weaker Marangoni effect enhances the surface disturbance, while a stronger Marangoni effect inhibits the bottom perturbation, making the liquid film rigid and causing surface countercurrent. Higher dilational viscosity slows down the drainage process and reduces the surface velocity. It suppresses the countercurrent phenomenon caused by the Marangoni effect. A greater disturbance wave number makes the perturbation stronger in the early stage of drainage, while it does not affect the stability of late stage of the drainage.