Droplet dynamics on a solid substrate is significantly influenced by surfactants. It remains a challenging task to model and simulate the moving contact line dynamics with soluble surfactants. In this work, we present a derivation of the phase-field moving contact line model with soluble surfactants through the first law of thermodynamics, associated thermodynamic relations and the Onsager variational principle. The derived thermodynamically consistent model consists of two Cahn–Hilliard type of equations governing the evolution of interface and surfactant concentration, the incompressible Navier–Stokes equations and the generalized Navier boundary condition for the moving contact line. With chemical potentials derived from the free energy functional, we analytically obtain certain equilibrium properties of surfactant adsorption, including equilibrium profiles for phase-field variables, the Langmuir isotherm and the equilibrium equation of state. A classical droplet spread case is used to numerically validate the moving contact line model and equilibrium properties of surfactant adsorption. The influence of surfactants on the contact line dynamics observed in our simulations is consistent with the results obtained using sharp interface models. Using the proposed model, we investigate the droplet dynamics with soluble surfactants on a chemically patterned surface. It is observed that droplets will form three typical flow states as a result of different surfactant bulk concentrations and defect strengths, specifically the coalescence mode, the non-coalescence mode and the detachment mode. In addition, a phase diagram for the three flow states is presented. Finally, we study the unbalanced Young stress acting on triple-phase contact points. The unbalanced Young stress could be a driving or resistance force, which is determined by the critical defect strength.

For solving reactive transport problems in porous media, we analyze three primal discontinuous Galerkin (DG) methods with penalty, namely, symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), and incomplete interior penalty Galerkin (IIPG). A cut-off operator is introduced in DG to treat general kinetic chemistry. Error estimates in L ² (H ¹ ) are established, which are optimal in h and nearly optimal in p. We develop a parabolic lift technique for SIPG, which leads to h-optimal and nearly p-optimal error estimates in the L ² (L ² ) and negative norms. Numerical results validate these estimates. We also discuss implementation issues including penalty parameters and the choice of physical versus reference polynomial spaces.

This paper presents a locally conservative finite element method based on enriching the approximation space of the continuous Galerkin method with elementwise constant functions. The proposed method has a smaller number of degrees of freedom than the discontinuous Galerkin method. Numerical examples on coupled flow and transport in porous media are provided to illustrate the advantages of this method. We also present a theoretical analysis of the method and establish optimal convergence of numerical solutions.

In this article, we analyze the flow of a fluid through a coupled Stokes–Darcy domain. The fluid in each domain is non-Newtonian, modeled by the generalized nonlinear Stokes equation in the free flow region and the generalized nonlinear Darcy equation in the porous medium. A flow rate is specified along the inflow portion of the free flow boundary. We show existence and uniqueness of a variational solution to the problem. We propose and analyze an approximation algorithm and establish a priori error estimates for the approximation.