The effects of curvature and buoyancy on the laminar dispersion of solute in a horizontal tube

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Abstract
When a solute is injected into a straight circular tube through which a solvent is in steady laminar flow it spreads out longitudinally under the combined effect of molecular diffusion and advection with the flow. Taylor (1953) showed that, provided there is no density difference between the solute and the solvent, the distribution of mean concentration satisfies a diffusion equation with a certain longitudinal diffusivity for large times and with respect to axes moving with the discharge velocity. In the present paper it is shown by a general argument that this statement remains true if the pipe is uniformly curved. An expression is given for the diffusivity, valid when the radius of curvature is sufficiently large. For all common liquids and most gases the diffusivity is reduced by the curvature. The rest of the paper deals with the effects of buoyancy forces caused by a density difference between the solute and the solvent, when the tube is horizontal. It is shown that in general a longitudinal diffusivity does not exist and an equation is derived that replaces the diffusion equation if the effects of buoyancy are small. A prediction from this equation is that buoyancy should not have a noticeable effect on the longitudinal dispersion for Péclet numbers near a certain value at which the two opposing influences of horizontal spreading due to gravity and lateral mixing due to secondary flow are in balance. This prediction is consistent with some observations made by Reejhsinghani, Gill & Barduhn (1966).

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