Short-time behavior of advecting-diffusing scalar fields in Stokes flows
(2013) In Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) 87(6).- 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=É∂y2+iV(y)-É′, 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 , and show that this decay is characterized by a universal behavior that depends solely on the singularity of the ratio... (More)
- 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=É∂y2+iV(y)-É′, 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 , 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(ξ)=exp(-É′ξ-bξ3) , where b>0 is a constant. Conversely, if the effective potential is singular, then ||||L2(ξ)=1-aξν with a>0. The exponent ν attains the value 53 at the very early stages of the process, while for intermediate stages its value is 35. 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. © 2013 American Physical Society. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/a48ddabd-cc44-4df7-a2cc-509cb825f4bb
- author
- Giona, Massimiliano ; Anderson, P.D. and Garofalo, Fabio LU
- publishing date
- 2013-06-17
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)
- volume
- 87
- issue
- 6
- article number
- 063011
- publisher
- American Physical Society
- external identifiers
-
- scopus:84879669284
- pmid:23848776
- ISSN
- 1539-3755
- DOI
- 10.1103/PhysRevE.87.063011
- language
- English
- LU publication?
- no
- id
- a48ddabd-cc44-4df7-a2cc-509cb825f4bb
- date added to LUP
- 2016-06-27 10:30:39
- date last changed
- 2022-01-30 04:47:55
@article{a48ddabd-cc44-4df7-a2cc-509cb825f4bb,
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=É∂y2+iV(y)-É′, 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 , 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(ξ)=exp(-É′ξ-bξ3) , where b>0 is a constant. Conversely, if the effective potential is singular, then ||||L2(ξ)=1-aξν with a>0. The exponent ν attains the value 53 at the very early stages of the process, while for intermediate stages its value is 35. 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. © 2013 American Physical Society.}},
author = {{Giona, Massimiliano and Anderson, P.D. and Garofalo, Fabio}},
issn = {{1539-3755}},
language = {{eng}},
month = {{06}},
number = {{6}},
publisher = {{American Physical Society}},
series = {{Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)}},
title = {{Short-time behavior of advecting-diffusing scalar fields in Stokes flows}},
url = {{http://dx.doi.org/10.1103/PhysRevE.87.063011}},
doi = {{10.1103/PhysRevE.87.063011}},
volume = {{87}},
year = {{2013}},
}