Abstract The one-dimensional diffusion equation can be written as $$\begin{aligned} \frac{\partial \phi }{\partial t}=\alpha \frac{\partial ^{2}\phi }{\partial x^{2}}\,. \end{aligned}$$ The dependent variable $$\phi $$ (such as temperature, species, momentum) diffuses in an infinite medium in both directions (to the left and right, $$x^{+}$$ and $$x^{-}$$ ) without any preference due to molecular activity. On the macroscopic scale, the rate of diffusion depends on the parameter $$\alpha $$ , where $$\alpha $$ stands for the thermal diffusion coefficient, mass diffusion coefficient, or kinematics viscosity, for energy, species, and momentum diffusion, respectively. The diffusion process becomes faster as the parameter $$\alpha $$ increases. An order of magnitude analysis of the above equation yields $$\begin{aligned} \frac{1}{\tau }\approx \alpha \frac{1}{\ell ^{2}}\,, \end{aligned}$$ where $$\tau $$ and l are time and length scales, respectively.
The Diffusion Equation
Lattice Boltzmann Method ; 53-80
2nd ed. 2019
2019-01-01
28 pages
Article/Chapter (Book)
Electronic Resource
English
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