Advection and diffusion in a chemically induced compressible flow

Abstract

We study analytically the joint dispersion of Gaussian patches of salt and colloids in linear flows, and how salt gradients accelerate or delay colloid spreading by diffusiophoretic effects. Because these flows have constant gradients in space, the problem can be solved almost entirely for any set of parameters, leading to predictions of how the mixing time and the Batchelor scale are modified by diffusiophoresis. We observe that the evolution of global concentrations, defined as the inverse of the patches areas, are very similar to those obtained experimentally in chaotic advection. They are quantitatively explained by examining the area dilatation factor, in which diffusive and diffusiophoretic effects are shown to be additive and appear as the divergence of a diffusive contribution or of a drift velocity. An analysis based on compressibility is developed in the salt-attracting case, for which colloids are first compressed before dispersion, to predict the maximal colloid concentration as a function of the parameters. This maximum is found not to depend on the flow stretching rate nor on its topology (strain or shear flow), but only on the characteristics of salt and colloids (diffusion coefficients and diffusiophoretic constant) and the initial size of the patches.

Diffusiophoresis, Batchelor scale and effective Péclet numbers

“…The case of the joint evolution of Gaussian patches of salt and colloids released at the origin in a pure strain flow ( ) was solved analytically in Raynal et al. (2018) so that we only briefly recall the steps leading to the solution. The Péclet numbers are for the salt and for the colloids.…”

“…In the case of an initially round patch of radius , one still has and the equations for and read (Raynal et al. 2018) This system has a complicated analytical solution given in Raynal et al. (2018).…”

“…2016) and already used in Raynal et al. (2018), will allow for a comparison of diffusiophoretic effects in all situations without changing the shape of the profiles.…”

“…In this case, the patch of colloids is exponentially stretched in the -direction and compressed towards the modified Batchelor scale along the -direction (Raynal et al. 2018). In this first example, it is possible to define an effective Péclet number for the colloids based on the modified Batchelor scale, , which takes into account the effects of diffusiophoresis.…”

“…Using this effective Péclet number, we find that it is possible to rescale all the values of the mixing time, computed from the analytical solution of Raynal et al. (2018), on a single curve derived from the case with no diffusiophoresis.…”