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Updated: Jul 27, 2026
08:01
The Diffusion of Passive Tracers in Laminar Shear Flow
Published on: May 1, 2018
Mathematical analysis of thermal diffusion shock waves
Vitalyi Gusev1, Walter Craig, Roberto LiVoti
1Université du Maine, av. Messiaen, 72085 LeMans, Cedex 09 France.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 31, 2005
Summary
Thermal diffusion, or the Ludwig-Soret effect, separates mixtures using temperature gradients. This study provides an exact solution for a sinusoidal temperature field, revealing shock wave formation and halting, with diffusion
Area of Science:
- Physics
- Physical Chemistry
Background:
- Thermal diffusion, the Ludwig-Soret effect, describes mixture separation under a temperature gradient.
- Binary mixtures' concentration changes are modeled by nonlinear partial differential equations.
Purpose of the Study:
- To derive an exact solution for the Ludwig-Soret equation without mass diffusion.
- To analyze shock wave formation and halting in a sinusoidal temperature field.
- To investigate the influence of diffusion on concentration profiles via numerical methods.
Main Methods:
- Exact analytical solution of the nonlinear partial differential equation for the Ludwig-Soret effect.
- Analysis of shock wave dynamics and shock time expressions for limiting density fractions.
- Numerical integration to study the effects of mass diffusion.
Main Results:
- An exact solution reveals the formation of counterpropagating shock waves.
- These shock waves decelerate and eventually cease motion.
- Expressions for shock time were derived for specific initial density fractions.
- Numerical simulations show diffusion's impact on concentration profile evolution.
Conclusions:
- The study provides a precise mathematical description of thermal diffusion phenomena.
- Shock wave dynamics are a key feature in the absence of diffusion.
- Mass diffusion plays a significant role in modifying concentration profiles over time and space.
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