On the applicability of the Brinkman equation in soft surface electrokinetics

Abstract

The Stokes equation is commonly used within the field of electrokinetics of hard impermeable surfaces while the Brinkman equation is adopted for tackling hydrodynamics in the framework of soft (permeable) surface electrokinetics (SSE). The latter was initially proposed for modeling the hydrodynamics in so-called hybrid systems that consist of a porous medium and an adjacent fluid phase basically because the conventional Darcy law or Debye and Bueche model initially proposed for that purpose failed to provide the required velocity and shear stress-continuity conditions at the porous media–fluid interface. However, even though the physical background of the Brinkman equation and its boundary conditions have been discussed when applied to the hydrodynamics of hybrid systems, controversy still remains with respect to their applicability in the field of SSE. Indeed, recent experiments pointed out better agreement between shear flow into a regular array of rods oriented across the flow and the solution of the Brinkman equation for hybrid systems providing a stress-jump boundary condition is taken into account (M.F. Tachie et al., J. Fluid. Mech. 493 (2003) 319). As there is identity in the Brinkman model for hybrid systems and for SSE, the question arises whether the above discontinuity of viscous stress must be incorporated or not into SSE modeling. Recent determination of hydrodynamic penetration length λ o - 1 of swollen and collapsed thermo-responsive films (J.F.L. Duval, R. Zimmermann, A.L. Cordeiro, N. Rein, C. Werner, Langmuir 25 (2009) 10691) suggests that there is no need for a cardinal revision of the Brinkman model, although further experimental investigations are required to support such a conclusion. With regard to these experiments, almost complete agreement between independent determination of λ o - 1 by swelling experiments and its derivation according to Brinkman model was obtained.