A Variational Model for Two-Phase Immiscible Electroosmotic Flow at Solid Surfaces (Communications in Computational Physics, 2012)
We develop a continuum hydrodynamic model for two-phase immiscibleflows that involve electroosmotic effect in an electrolyte and moving contact line atsolid surfaces. The model is derived through a variational approach based on the Onsager principle of minimum energy dissipation. This approach was first presented inthe derivation of a continuum hydrodynamic model for moving contact line in neutral two-phase immiscible flows (Qian, Wang, and Sheng, J. Fluid Mech. 564, 333–360(2006)). Physically, the electroosmotic effect can be formulated by the Onsager principle as well in the linear response regime. Therefore, the same variational approachis applied here to the derivation of the continuum hydrodynamic model for chargedtwo-phase immiscible flows where one fluid component is an electrolyte exhibitingelectroosmotic effect on a charged surface. A phase field is employed to model thediffuse interface between two immiscible fluid components, one being the electrolyteand the other a nonconductive fluid, both allowed to slip at solid surfaces. Our modelconsists of the incompressible Navier-Stokes equation for momentum transport, theNernst-Planck equation for ion transport, the Cahn-Hilliard phase-field equation forinterface motion, and the Poisson equation for electric potential, along with all thenecessary boundary conditions. In particular, all the dynamic boundary conditions atsolid surfaces, including the generalized Navier boundary condition for slip, are derived together with the equations of motion in the bulk region. Numerical examplesin two-dimensional space, which involve overlapped electric double layer fields, havebeen presented to demonstrate the validity and applicability of the model, and a fewsalient features of the two-phase immiscible electroosmotic flows at solid surface. Thewall slip in the vicinity of moving contact line and the Smoluchowski slip in the electricdouble layer are both investigated.