Dispersion enhancement and damping by buoyancy driven flows in two-dimensional networks of capillaries

Abstract

The influence of a small relative density difference (Δρ∕ρ≃3×10−4) on the displacement of two miscible Newtonian liquids is studied experimentally in transparent two-dimensional square networks of microchannels held vertically; the channel width distribution is log normal with a mean value of a=0.33mm. Maps of the local relative concentration are obtained by an optical light absorption technique. Both stable displacements in which the denser fluid enters at the bottom of the cell and displaces the lighter one and unstable displacements in which the lighter fluid is injected at the bottom and displaces the denser one are realized. Except at the lowest mean flow velocity U, the average C(x,t) of the relative concentration satisfies a convection-dispersion equation. The relative magnitude of ∣U∣ and of the velocity Ug of buoyancy driven fluid motions is characterized by the gravity number Ng=Ug∕∣U∣. At low gravity numbers ∣Ng∣<0.01 (or equivalently high Péclet numbers Pe=Ua∕Dm>500), the dispersivities ld in the stable and unstable configurations are similar to ld∝Pe0.5. At low velocities such that ∣Ng∣>0.01, ld increases like 1/Pe in the unstable configuration (Ng<0), while it becomes constant and close to the length of individual channels in the stable case (Ng>0). Isoconcentration lines c(x,y,t)=0.5 are globally flat in the stable configuration, while in the unstable case, they display spikes and troughs with a rms amplitude σf parallel to the flow. For Ng>−0.2, σf increases initially with the distance and reaches a constant limit, while it keeps increasing for Ng<−0.2. A model taking into account buoyancy forces driving the instability and the transverse exchange of tracer between rising fingers and the surrounding fluid is suggested and its applicability to previous results obtained in three-dimensional media is discussed.