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

Giona, Massimiliano; Anderson, P.D. and Garofalo, Fabio LU (2013) In Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)2001-01-01+01:002016-01-01+01:00 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)
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author
publishing date
type
Contribution to journal
publication status
published
subject
in
Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)2001-01-01+01:002016-01-01+01:00
volume
87
issue
6
publisher
American Physical Society
external identifiers
  • scopus:84879669284
ISSN
1539-3755
DOI
10.1103/PhysRevE.87.063011
language
English
LU publication?
no
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a48ddabd-cc44-4df7-a2cc-509cb825f4bb
date added to LUP
2016-06-27 10:30:39
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2017-01-01 08:29:09
@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.},
  articleno    = {063011},
  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)2001-01-01+01:002016-01-01+01:00},
  title        = {Short-time behavior of advecting-diffusing scalar fields in Stokes flows},
  url          = {http://dx.doi.org/10.1103/PhysRevE.87.063011},
  volume       = {87},
  year         = {2013},
}