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Electrophoresis of bubbles

Ory Schnitzer (Technion), Itzchak Frankel (), Ehud Yariv ()

Electrohydrodynamics and electrokinetics of fluid systems

Mon 9:00 - 10:30

Barus-Holley 161

Smoluchowski’s formula is clearly inadequate for drops and bubbles, as the no-slip condition underlying its derivation does not apply at a free surface. We address the simplest model problem, namely a spherical gas bubble of uniform surface-charge density. The thin-double-layer limit is analyzed using inner-outer asymptotic expansions, the inner region being a Debye diffuse layer in quasi-equilibrium and the outer region constituting an electro-neutral bulk with uniform salt concentration at leading order. As in the classical problem of a solid particle, asymptotic matching with the bulk results in an effective electro-osmotic slip. However, the bubble motion is not determined by the slip mechanism, but rather by the shear-stress balance at the interface. The interfacial shear is driven by the salt perturbation in the bulk, itself engendered by the electric field and modified by Debye-scale advection. The resulting macroscale model consists of a set of differential equations in the bulk, governing the leading-order flow and the excess-salt perturbation, together with effective “boundary” conditions representing asymptotic matching with the diffuse-layer fields. The excess salt is thus governed by an advection–diffusion equation together with a Neumann-type condition which accommodates an effective surface-convection term, rendering a coupling to the Stokes flow. This flow, in turn, is animated by an effective shear-stress condition, wherein surface gradients of the salt perturbation result in a Marangoni-like stress jump. Because of the advection term, the problem is inherently nonlinear and thus does not admit a closed-form solution. A relevant limit for typical bubble size is that of strong fields, where the intense advection results in a vanishing excess salt. This trivial solution cannot however satisfy the effective boundary condition; a diffusive boundary layer therefore develops. The bubble speed scale as the 2/3-power of the applied field.