TY - JOUR

T1 - Diffusion and Brownian motion in Lagrangian coordinates

AU - Fyrillas, Marios M.

AU - Nomura, Keiko K.

N1 - Funding Information:
This work was initially supported by the Rheology Group of the University of Twente, by the European Research Consortium in Informatics and Mathematics (ERCIM fellowship Grant No. 2002–06), and by the Swiss National Foundation (Research Project No. PAER2–101107). The work was partially funded by the Frederick Research Center.

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.

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U2 - 10.1063/1.2717185

DO - 10.1063/1.2717185

M3 - Article

AN - SCOPUS:34247589394

VL - 126

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 16

M1 - 164510

ER -