Abstract
In this paper we consider the convection-diffusion problem of a passive scalar in Lagrangian coordinates, i.e., in a coordinate system fixed on fluid particles. Both the convection-diffusion partial differential equation and the Langevin equation are expressed in Lagrangian coordinates and are shown to be equivalent for uniform, isotropic diffusion. The Lagrangian diffusivity is proportional to the square of the relative change of surface area and is related to the Eulerian diffusivity through the deformation gradient tensor. Associated with the initial value problem, we relate the Eulerian to the Lagrangian effective diffusivities (net spreading), validate the relation for the case of linear flow fields, and infer a relation for general flow fields. Associated with the boundary value problem, if the scalar transport problem possesses a time-independent solution in Lagrangian coordinates and the boundary conditions are prescribed on a material surface/interface, then the net mass transport is proportional to the diffusion coefficient. This can be also shown to be true for large Ṕclet number and time-periodic flow fields, i.e., closed pathlines. This agrees with results for heat transfer at high Ṕclet numbers across closed streamlines.
| Original language | English |
|---|---|
| Article number | 164510 |
| Journal | Journal of Chemical Physics |
| Volume | 126 |
| Issue number | 16 |
| DOIs | |
| Publication status | Published - 2007 |
| Externally published | Yes |
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ASJC Scopus subject areas
- Atomic and Molecular Physics, and Optics
Cite this
Diffusion and Brownian motion in Lagrangian coordinates. / Fyrillas, Marios M.; Nomura, Keiko K.
In: Journal of Chemical Physics, Vol. 126, No. 16, 164510, 2007.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Diffusion and Brownian motion in Lagrangian coordinates
AU - Fyrillas, Marios M.
AU - Nomura, Keiko K.
PY - 2007
Y1 - 2007
N2 - In this paper we consider the convection-diffusion problem of a passive scalar in Lagrangian coordinates, i.e., in a coordinate system fixed on fluid particles. Both the convection-diffusion partial differential equation and the Langevin equation are expressed in Lagrangian coordinates and are shown to be equivalent for uniform, isotropic diffusion. The Lagrangian diffusivity is proportional to the square of the relative change of surface area and is related to the Eulerian diffusivity through the deformation gradient tensor. Associated with the initial value problem, we relate the Eulerian to the Lagrangian effective diffusivities (net spreading), validate the relation for the case of linear flow fields, and infer a relation for general flow fields. Associated with the boundary value problem, if the scalar transport problem possesses a time-independent solution in Lagrangian coordinates and the boundary conditions are prescribed on a material surface/interface, then the net mass transport is proportional to the diffusion coefficient. This can be also shown to be true for large Ṕclet number and time-periodic flow fields, i.e., closed pathlines. This agrees with results for heat transfer at high Ṕclet numbers across closed streamlines.
AB - In this paper we consider the convection-diffusion problem of a passive scalar in Lagrangian coordinates, i.e., in a coordinate system fixed on fluid particles. Both the convection-diffusion partial differential equation and the Langevin equation are expressed in Lagrangian coordinates and are shown to be equivalent for uniform, isotropic diffusion. The Lagrangian diffusivity is proportional to the square of the relative change of surface area and is related to the Eulerian diffusivity through the deformation gradient tensor. Associated with the initial value problem, we relate the Eulerian to the Lagrangian effective diffusivities (net spreading), validate the relation for the case of linear flow fields, and infer a relation for general flow fields. Associated with the boundary value problem, if the scalar transport problem possesses a time-independent solution in Lagrangian coordinates and the boundary conditions are prescribed on a material surface/interface, then the net mass transport is proportional to the diffusion coefficient. This can be also shown to be true for large Ṕclet number and time-periodic flow fields, i.e., closed pathlines. This agrees with results for heat transfer at high Ṕclet numbers across closed streamlines.
UR - http://www.scopus.com/inward/record.url?scp=34247589394&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=34247589394&partnerID=8YFLogxK
U2 - 10.1063/1.2717185
DO - 10.1063/1.2717185
M3 - Article
VL - 126
JO - Journal of Chemical Physics
JF - Journal of Chemical Physics
SN - 0021-9606
IS - 16
M1 - 164510
ER -