2.2 Partial Differential Equations

2.2.6 Convection-Diffusion

In practice, both convection and diffusion are important phenomenon governing fluid dynamics. While convection may be the dominant transport mechanism in most of a flow, accounting for diffusion near solid boundaries may be essential for accurately computing engineering quantities such as drag or heat transfer. The convection-diffusion equation is derived from a general conservation law including both convective and diffusive fluxes. In differential form, the convection-diffusion equation is

 $\dfrac {\partial U}{\partial t} + \vec{v} \cdot \nabla U - \nabla \cdot ( \mu \nabla U ) = S.$ (2.36)

The convection-diffusion equation arises in many engineering applications ranging from heat transfer to statistical mechanics (and even in finance!). Solutions of the convection-diffusion equation exhibit behavior which is the combined effect of both convection and diffusion. A key question which naturally arises is how significant are the relative effects of convection and diffusion. This is governed by the non-dimensional Peclet number (in aerodynamics applications, the Reynolds number is equivalent).

 $Pe = \frac{|\vec{v}| L}{\mu }$ (2.37)

where $$L$$ is a characteristic length scale for the problem of interest. When $$Pe \ll 1$$ diffusion effects are dominant, and convection plays a relatively small role. On the other hand, when $$Pe \gg 1$$ convection is the dominant transport mechanism. Figures 2.52.62.7 show solutions for the one-dimensional convection-diffusion problem with Peclet numbers $$10, 100$$ and $$1000$$ ($$\vec{v}=1$$ is fixed). As the Peclet number increases the solution of the convection-diffusion problem approaches that of the pure convection problem.

Figure 2.5: One-dimensional convection-diffusion problem, $$Pe=10$$
Figure 2.6: One-dimensional convection-diffusion problem, $$Pe=100$$
Figure 2.7: One-dimensional convection-diffusion problem, $$Pe=1000$$