Scalar mixing by stagnation points on solid surfaces

Abstract

Recent advances have linked mixing in porous media to the deformation of infinitesimal fluid elements. However, these Lagrangian frameworks generally neglect or simplify interactions with solid surfaces. We consider transport of thin solute blobs (strips or sheets) near single generic obstacles, in particular along separation lines (separatrices) which connect to stagnation points at solid boundaries. In this case, we demonstrate that the coupling between flow and diffusion is analytically tractable. We provide a complete theory for scalar mixing near upstream stagnation points, which we validate by numerical simulations. We identify a mixing time scale $t_B \sim Pe^{1/3}$, where $Pe$ is the Péclet number, which controls whether scalar decay is driven by fluid stretching or by interaction with the solid boundary. Asymptotically, for times $t < t_B$, we find that the maximum concentration $c_{\rm max}$ decays as $t^{-5/2}$ and that the width $σ$ grows like $t^{1/2}$. For $t > t_B$, we find exponential decay of $c_{\rm max}$ and constant width $σ\sim s_B \sim Pe^{-1/3}$, where $s_B$ is a mixing length scale distinct from that which appears in shear flows. The asymptotic scalings of $c_{\rm max}(t)$ and $σ(t)$, and their dependence on $Pe$, are universal to no-slip stagnation points and qualitatively different from stagnation points in open flows.