Dispersion effects in the miscible displacement of two fluids in a duct of large aspect ratio

Abstract

We study miscible displacements in long ducts in the dispersive limit of small $\varepsilon \hbox{\it Pe}$, where $\varepsilon \,{\ll}\, 1$ is the inverse aspect ratio and $\hbox{\it Pe}$ the Péclet number. We consider the class of generalized Newtonian fluids, with specified closure laws for the fluid properties of the concentration-dependent mixture. Regardless of viscosity ratio and the constitutive laws of the pure fluids, for sufficiently small $\varepsilon \hbox{\it Pe}$ these displacements are characterized by rapid cross-stream diffusion and slow streamwise dispersion, i.e. the concentration appears to be near-uniform across the duct and spreads slowly as it translates. Using the multiple-scales method we derive the leading-order asymptotic approximation to the average fluid concentration $\bar{c}_0$. We show that $\bar{c}_0$ evolves on the slow timescale $t \sim (\varepsilon \hbox{\it Pe})^{-1}$, and satisfies a nonlinear diffusion equation in a frame of reference moving with the mean speed of the flow. In the case that the two fluids have identical rheologies and the concentration represents a passive tracer, the diffusion equation is linear. For Newtonian fluids we recover the classical results of Taylor (l953), Aris (1956), and for power-law fluids those of Vartuli et al. (1995). In the case that the fluids differ and/or that mixing is non-passive, $\bar{c}_0$ satisfies a nonlinear diffusion equation in the moving frame of reference. Given a specific mixing/closure law for the rheological properties, we are able to compute the dispersive diffusivity $D_T(\bar{c}_0)$ and predict spreading along the channel. We show that $D_T(\bar{c}_0)$ can vary significantly with choice of mixing law and discuss why. This also opens the door to possibilities of controlling streamwise spreading by the rheological design of reactive mixtures, i.e. including chemical additives such that the rheology of the mixture behaves very differently to the rheology of either pure fluid. Computed examples illustrate the potential effects that might be achieved.