Natural convective diffusion between two nonconcentric spheres

A study is made of the natural convective diffusion in a nonconcentric spherical layer formed by a ball of radius R₁ in a sphere of radius R₂. The centers of the ball and sphere are separated by a distance d R₁. The spherical layer is filled with a binary electrolyte, and the outer surface of the ball and the inner surface of the sphere serve as the anode and cathode, respectively, of an oxidation—reduction reaction. It is assumed that the reaction proceeds in accordance with diffusion kinetics, i.e., the current in the circuit limits the rate at which the reacting substances reach the electrodes 1]. If a current passes in the system, a concentration gradient develops in the reacting substances, and in a gravitational field a convective motion of the fluid is generated, which changes the rate at which the reacting substances arrive at the electrodes. For a binary eletrolyte in the leading approximation in the small parameter r_D(R₂ – r₁)^–1, where r_D is the Debye radius, the migration current can be eliminated, and one need consider only the diffusion and convective fluxes of the ions 2]. If the centers of the ball and the sphere coincide, the integrated diffusion flux at small Grashof numbers is not changed, and there is merely a local redistribution.3–5]. At small Grashof numbers, a strong dependence of the integrated diffusion flux on the eccentricity of the spherical layer must be expected. In the present paper, the hydrodynamic velocity field of the fluid, and also the change in the integrated diffusion flux due to the convective transport of the ions are found in the linear approximation in the small parameter = d/R₁.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 151–154, January–February, 1984.