Short-time behavior of advecting-diffusing scalar fields in Stokes flows

Abstract

This article addresses the short-term decay of advecting-diffusing scalar fields in Stokes flows. The analysis is developed in two main subparts. In the first part, we present an analytic approach for a class of simple flow systems expressed mathematically by the one-dimensional advection-diffusion equation w(y)∂(ξ)φ=ε∂(y)(2)φ+iV(y)φ-ε'φ, where ξ is either time or axial coordinate and iV(y) an imaginary potential. This class of systems encompasses both open- and closed-flow models and corresponds to the dynamics of a single Fourier mode in parallel flows. We derive an analytic expression for the short-time (short-length) decay of φ, and show that this decay is characterized by a universal behavior that depends solely on the singularity of the ratio of the transverse-to-axial velocity components V(eff)(y)=V(y)/w(y), corresponding to the effective potential in the imaginary potential formulation. If V(eff)(y) is smooth, then ||φ||(L(2))(ξ)=exp(-ε'ξ-bξ(3)), where b>0 is a constant. Conversely, if the effective potential is singular, then ||φ||(L(2))(ξ)=1-aξ(ν) with a>0. The exponent ν attains the value 5/3 at the very early stages of the process, while for intermediate stages its value is 3/5. By summing over all of the Fourier modes, a stretched exponential decay is obtained in the presence of nonimpulsive initial conditions, while impulsive conditions give rise to an early-stage power-law behavior. In the second part, we consider generic, chaotic, and nonchaotic autonomous Stokes flows, providing a kinematic interpretation of the results found in the first part. The kinematic approach grounded on the warped-time transformation complements the analytical theory developed in the first part.