Streaming-potential phenomena in the thin-Debye-layer limit. Part 1. General theory

Abstract

AbstractElectrokinetic streaming-potential phenomena are driven by imposed relative motion between liquid electrolytes and charged solids. Owing to non-uniform convective ‘surface’ current within the Debye layer Ohmic currents from the electro-neutral bulk are required to ensure charge conservation thereby inducing a bulk electric field. This, in turn, results in electro-viscous drag enhancement. The appropriate modelling of these phenomena in the limit of thin Debye layers $\delta \ensuremath{\rightarrow} 0$ ($\delta $ denoting the dimensionless Debye thickness) has been a matter of ongoing controversy apparently settled by Cox’s seminal analysis (J. Fluid Mech., vol. 338, 1997, p. 1). This analysis predicts electro-viscous forces that scale as ${\delta }^{4} $ resulting from the perturbation of the original Stokes flow with the Maxwell-stress contribution only appearing at higher orders. Using scaling analysis we clarify the distinction between the normalizations pertinent to field- and motion-driven electrokinetic phenomena, respectively. In the latter class we demonstrate that the product of the Hartmann & Péclet numbers is $O({\delta }^{\ensuremath{-} 2} )$ contrary to Cox (1997) where both parameters are assumed $O(1)$. We focus on the case where motion-induced fields are comparable to the thermal scale and accordingly present a singular-perturbation analysis for the limit where the Hartmann number is $O(1)$ and the Péclet number is $O({\delta }^{\ensuremath{-} 2} )$. Electric-current matching between the Debye layer and the electro-neutral bulk provides an inhomogeneous Neumann condition governing the electric field in the latter. This field, in turn, results in a velocity perturbation generated by a Smoluchowski-type slip condition. Owing to the dominant convection, the present analysis yields an asymptotic structure considerably simpler than that of Cox (1997): the electro-viscous effect now already appears at $O({\delta }^{2} )$ and is contributed by both Maxwell and viscous stresses. The present paradigm is illustrated for the prototypic problem of a sphere sedimenting in an unbounded fluid domain with the resulting drag correction differing from that calculated by Cox (1997). Independently of current matching, salt-flux matching between the Debye layer and the bulk domain needs also to be satisfied. This subtle point has apparently gone unnoticed in the literature, perhaps because it is trivially satisfied in field-driven problems. In the present limit this requirement seems incompatible with the uniform salt distribution in the convection-dominated bulk domain. This paradox is resolved by identifying the dual singularity associated with the limit $\delta \ensuremath{\rightarrow} 0$ in motion-driven problems resulting in a diffusive layer of $O({\delta }^{2/ 3} )$ thickness beyond the familiar $O(\delta )$-wide Debye layer.