Péclet number

Remarks

With $x = L \hat x$, $t = (L / U) \hat t$ and $u = U \hat u$, the equation $c_t + u \cdot \nabla c = D \Delta c$ becomes $c_{\hat t} + \hat u \cdot \hat\nabla c = \mathrm{Pe}^{-1} \hat\Delta c$. So $\mathrm{Pe}$ is the only parameter of the dimensionless equation.

For heat, $\mathrm{Pe} = \mathrm{Re} \cdot \mathrm{Pr}$, the product of the Reynolds number $U L / \nu$ and the Prandtl number $\nu / \alpha$; for mass, $\mathrm{Pe} = \mathrm{Re} \cdot \mathrm{Sc}$ with the Schmidt number $\mathrm{Sc} = \nu / D$.

Example: water flowing at $U = 1$ cm/s in a channel of width $L = 1$ cm carrying a solute with $D = 10^{-5}$ cm$^2$/s has $\mathrm{Pe} = 10^{5}$, so advection dominates.

Non-example: $U L / \nu$, with the kinematic viscosity $\nu$ in place of $D$, is the Reynolds number, not the Péclet number; it compares advection with diffusion of momentum, not of the scalar.