Theory of electrohydrodynamic instabilities in electrolytic cells

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Abstract
The classical Nernst theory for diffusion‐limited electrical current transport in electrolytic cells predicts limiting currents of order microamps for unstirred cells. For unstirred cells, one encounters much higher current densities for large enough voltages. In this paper we develop the theory of the hydrodynamic stability of an electrolytic cell as a function of the imposed electric current. A new electrohydrodynamic instability is encountered when the current is forced to exceed the Nernst limit. The convection is driven by the volume force exerted by the electrical field on space charges in the electrolyte. This intrinsic instability is found to be easily masked by extrinsic convection sources such as gravity or stirring. We perform a linear stability analysis and derive a dimensionless number Le whose value determines the convection pattern. The quiescent cell becomes hydrodynamically unstable if (i) the current density exceeds the Nernst limit and if (ii) Le≫1. We expect full hydrodynamic convection if Le/Pr≫1 with Pr the Prandtl number.

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