Contact angle hysteresis is an important physical phenomenon omnipresent in nature and various industrial processes, but its effects are not considered in many existing multiphase flow simulations due to modeling complexity. In this work, a multiphase lattice Boltzmann method (LBM) is developed to simulate the contact-line dynamics with consideration of the contact angle hysteresis for a broad range of kinematic viscosity ratios. In this method, the immiscible two-phase flow is described by a color-fluid model, in which the multiple-relaxation-time collision operator is adopted to increase numerical stability and suppress unphysical spurious currents at the contact line. The contact angle hysteresis is introduced using the strategy proposed by Ding and Spelt [Ding and Spelt, J. Fluid Mech. 599, 341 (2008)JFLSA70022-112010.1017/S0022112008000190], and the geometrical wetting boundary condition is enforced to obtain the desired contact angle. This method is first validated by simulations of static contact angle and dynamic capillary intrusion process on ideal (smooth) surfaces. It is then used to simulate the dynamic behavior of a droplet on a nonideal (inhomogeneous) surface subject to a simple shear flow. When the droplet remains pinned on the surface due to hysteresis, the steady interface shapes of the droplet quantitatively agree well with the previous numerical results. Four typical motion modes of contact points, as observed in a recent study, are qualitatively reproduced with varying advancing and receding contact angles. The viscosity ratio is found to have a notable impact on the droplet deformation, breakup, and hysteresis behavior. Finally, this method is extended to simulate the droplet breakup in a microfluidic T junction, with one half of the wall surface ideal and the other half nonideal. Due to the contact angle hysteresis, the droplet asymmetrically breaks up into two daughter droplets with the smaller one in the nonideal branch channel, and the behavior of daughter droplets is significantly different in both branch channels. Also, it is found that the contact angle hysteresis is strengthened with decreasing the viscosity ratio, leading to an earlier droplet breakup and a decrease in the maximum length that the droplet can reach before the breakup. These simulation results manifest that the present multiphase LBM can be a useful substitute to Ba et al. [Phys. Rev. E 88, 043306 (2013)PLEEE81539-375510.1103/PhysRevE.88.043306] for modeling the contact angle hysteresis, and it can be easily implemented with higher computational efficiency.

Droplet dynamics in microfluidic applications is significantly influenced by surfactants. It remains a research challenge to model and simulate droplet behaviour including deformation, breakup and coalescence, especially in the confined microfluidic environment. Here, we propose a hybrid method to simulate interfacial flows with insoluble surfactants. The immiscible two-phase flow is solved by an improved lattice Boltzmann colour-gradient model which incorporates a Marangoni stress resulting from non-uniform interfacial tension, while the convection–diffusion equation which describes the evolution of surfactant concentration in the entire fluid domain is solved by a finite difference method. The lattice Boltzmann and finite difference simulations are coupled through an equation of state, which describes how surfactant concentration influences interfacial tension. Our method is first validated for the surfactant-laden droplet deformation in a three-dimensional (3D) extensional flow and a 2D shear flow, and then applied to investigate the effect of surfactants on droplet dynamics in a 3D shear flow. Numerical results show that, at low capillary numbers, surfactants increase droplet deformation, due to reduced interfacial tension by the average surfactant concentration, and non-uniform effects from non-uniform capillary pressure and Marangoni stresses. The role of surfactants on the critical capillary number ($Ca_{cr}$) of droplet breakup is investigated for various confinements (defined as the ratio of droplet diameter to wall separation) and Reynolds numbers. For clean droplets, $Ca_{cr}$ first decreases and then increases with confinement, and the minimum value of $Ca_{cr}$ is reached at a confinement of 0.5; for surfactant-laden droplets, $Ca_{cr}$ exhibits the same variation in trend for confinements lower than 0.7, but, for higher confinements, $Ca_{cr}$ is almost a constant. The presence of surfactants decreases $Ca_{cr}$ for each confinement, and the decrease is also attributed to the reduction in average interfacial tension and non-uniform effects, which are found to prevent droplet breakup at low confinements but promote breakup at high confinements. In either clean or surfactant-laden cases, $Ca_{cr}$ first remains almost unchanged and then decreases with increasing Reynolds number, and a higher confinement or Reynolds number favours ternary breakup. Finally, we study the collision of two equal-sized droplets in a shear flow in both surfactant-free and surfactant-contaminated systems with the same effective capillary numbers. It is identified that the non-uniform effects in the near-contact interfacial region immobilize the interfaces when two droplets are approaching each other and thus inhibit their coalescence.

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In this work, we develop a two-phase lattice Boltzmann method (LBM) to simulate axisymmetric thermocapillary flows. This method simulates the immiscible axisymmetric two-phase flow by an improved color-gradient model, in which the single-phase collision, perturbation and recoloring operators are all presented with the axisymmetric effect taken into account in a simple and computational consistent manner. An additional lattice Boltzmann equation is introduced to describe the evolution of the axisymmetric temperature field, which is coupled to the hydrodynamic equations through an equation of state. This method is first validated by simulations of Rayleigh-Bénard convection in a vertical cylinder and thermocapillary migration of a deformable droplet at various Marangoni numbers. It is then used to simulate the thermocapillary migration of two spherical droplets in a constant applied temperature gradient along their line of centers, and the influence of the Marangoni number (Ca), initial distance between droplets (S 0), and the radius ratio of the leading to trailing droplets (Λ) on the migration process is systematically studied. As M a increases, the thermal wake behind the leading droplet strengthens, resulting in the transition of the droplet migration from coalescence to non-coalescence; and also, the final distance between droplets increases with M a for the non-coalescence cases. The variation of S 0 does not change the final state of the droplets although it has a direct impact on the migration process. In contrast, Λ can significantly influence the migration process of both droplets and their final state: at low M a, decreasing Λ favors the coalescence of both droplets; at high M a, the two droplets do not coalesce eventually but migrate with the same velocity for the small values of Λ, and decreasing Λ leads to a shorter equilibrium time and a faster migration velocity.