### 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)¿¿¿=¿¿y2¿+iV(y)¿-¿'¿, where ¿ 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 ¿, 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 Veff(y)=V(y)/w(y), corresponding to the effective potential in the imaginary potential formulation. If Veff(y) is smooth, then ||¿||L2(¿)=exp(-¿'¿-b¿3), where b>0 is a constant. Conversely, if the effective potential is singular, then ||¿||L2(¿)=1-a¿¿ with a>0. The exponent ¿ 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.

Original language | English |
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Article number | 063011 |

Pages (from-to) | 063011-1/15 |

Number of pages | 15 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 87 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2013 |

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## Cite this

Giona, M., Anderson, P. D., & Garofalo, F. (2013). Short-time behavior of advecting-diffusing scalar fields in Stokes flows.

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,*87*(6), 063011-1/15. [063011]. https://doi.org/10.1103/PhysRevE.87.063011