When trying to understand the relative dominance of diffusion vs. advection in a hydrodynamic system, we usually compare the characteristic length scales associated with each process and define their ratio as a dimensionless group often referred to as a Peclét number. This involves the following steps:
1. Determine what is the system you want to analyze. Are you interested in the motion of a single particle or of a chemical species in a fluid? Are you interested in motion happening in one dimension or in three? Are you looking at the displacements occurring over a fixed interval of time or the displacements that happened within a fixed length/volume?
2. Estimate the diffusive/advective displacements. 432
Diffusion is almost always a result of Brownian motion, which means that the root-mean-square displacement occurring through diffusion is proportional to the square root of the elapsed time $t$ multiplied by a diffusive constant $D$ (with some numerical factor depending on the dimension of the problem). Advective displacement is almost always the product of the bulk fluid velocity $u$ and the elapsed time $t$; this needs to be altered if the speed is not uniform or changing with time.
3. Take the ratio and simplify. Using the expressions above, divide the advective displacement by the diffusive displacement and simplify the expression⁠—chances are some terms will simplify or cancel out. This ratio (the <PERSON> number) now indicates to you which parameters in your physical system influence the balance of advective/diffusive displacement and which don't.
As an example, consider a molecule that is part of a fluid flowing unidirectionally and steadily in 3-D⁠—such a molecule undergoes both bulk advective motion and random Brownian motion. If we wanted to compare the total diffusive and advective displacement taken by that molecule over some specific time interval, we would find the RMS diffusive displacement in 3D to be $d_{diff} = \sqrt{6Dt}$, where $D$ is the diffusivity constant associated with the molecule. The advective displacement would simply be $d_{adv} = ut$ where $u$ is the bulk fluid velocity.
Hence, when taking the ratio of each, we find
$$Pe = \frac{d_{adv}}{d_{diff}}=\frac{ut}{\sqrt{6Dt}}=u\sqrt{\frac{t}{6D}}$$
The most common definition of the <PERSON> number is for 1-D motion over a fixed length interval, in which case the $6$ above becomes a $2$ and the $t$ is usually replaced with $\frac{L}{u}$, where $L$ is the length of the interval. 453