Introduction to diffusion¶

In the Navier-Stokes equation, \(D_t {\vec v} = - \frac{1}{\rho} {\vec\nabla} p + \nu \nabla^2 {\vec v}\), the vorticity term appears somewhat as a “diffusion term”. To understand what is meant by this, let me introduce a couple of comments about diffusion (as I don’t know how many of you have encountered it yet).

Einstein showed that the probability distrubution of a set of random walkers followed a diffusion equation

\[ \partial_t P({\vec x},t) = D \nabla^2 P \]

where \(P(x,t)\) is the probability density of the particle: that is, \(P dx\) is the probability to find the particle in an interval \([x- \frac{1}{2} dx, x + \frac{1}{2} dx]\).

This can be thought of as the equation for the density of some “tracer particles” in a static fluid, which move by random Brownian motion (every time \(\delta t\), the particle moves some fixed \(\delta x\) with uniform probability in any direction), scattering off of the fluid particles. In one dimension, the derivation is straightfoward:

\[ P(x, t + \delta t) = \frac{1}{2} (P(x + \delta x, t) + P(x - \delta x, t)) \]

Expanding the left hand side to leading nontrivial order in \(\delta t\),

\[ P(x, t + \delta t) = P(x, t) + \delta t {\dot P}(x, t) \]

When expanding the right hand side, the terms linear in \(\delta x\) cancel. The terms \(P(x,t)\) cancel in the equation so we are left with

\[ \delta T \partial_t P = \frac{1}{2} \delta x^2 \partial_x^2 P + ... \]

we have left out higher orders in \(\delta t\), \(\delta x\). This is he diffusion equation with \(D = \frac{\delta x^2}{2\delta t}\). Nete that this has the same dimension as the kinematic viscosity, as it must, since it converts the Laplacian into something with dimensions of a time derivative. For multiple diffusing particles, we can replace \(P\) with the density (thinking of the density as the probability of a single particle being at a given position, times the number of particles).

It is worth noting that if we set \(t - = i \tau\), we get something with the mathematical structure of the (free) Schrödinger equation. This is sometimes a useful trick (for example if you know solutions to one, you can do this rotation to get solutions to the other. More generally just as the wave function has an interpretation in terms of a Feynman path integral, such diffusive systems have a path integral representation.)

Solutions of this equation spread out or diffuse in time. A classic example is a solution such that

\[ P(x,0) = \delta(x) \]

corresponding to a particle (or collection of particles) starting at a fixed location. The corresponding solution to the diffusion equation is

\[ P(x,t) = \frac{1}{\sqrt{2 Dt}} e^{-\frac{x^2}{4Dt}} \]

Note in this equation that the width of the distribution scales as \(\Delta x \sim \sqrt{2 D t}\). This is characteristic of diffusion processes and can be guessed from dimensional analysis. If we assume that \(P\) varies by order \({\cal O}(1)\) over time scale \(T\) and distance scale \(X\), \(P = P(t/T, x/X)\) and we assume derivatives of the arguments are order \({\cal O}(1)\). Setting \(t = T \tau\), \(x = L y\),

\[ \partial_{\tau} P = \frac{T D}{L^2} \partial_y^2 P \]

Since by supposition the deivatives are order 1, \(L^2 \sim T D\).

These equations can be solved by Fourier transform. We know that we can write

\[ P(x,t) = \int_{-\infty}^{\infty} \frac{dk}{\sqrt{2pi}} {\tilde P}(k,t) e^{i k x} \]

In terms of this integral, the diffusion equation becomes

\[ \int_{-\infty}^{\infty} \frac{dk}{\sqrt{2\pi}} \left(\dot{\tilde P} + D k^2 {\tilde P}\right) e^{i k x} = 0 \]

Integrating the above over \(\int_{-\infty}^{\infty} \frac{dx}{\sqrt{2\pi}} e^{-i q x}\) and using

\[ \int_{-\infty}^{\infty} \frac{dx}{2\pi} e^{i x (k - q)} = \delta(k - q) \]

we find \(\dot{\tilde P} + D k^2 {\tilde P} = 0\), solved by \(P(k,t) = P(k,0) e^{- D k^2 t}\). For delta function initial conditions above, \(P(k,0) = \frac{1}{\sqrt{2\pi}}\). Thus,

(24)¶\[\begin{align} P(x,t) & = & \int_{-\infty}^{\infty} \frac{dk}{\sqrt{2pi}} {\tilde P}(k,t) e^{i k x}\\ & = & \int_{-\infty}^{\infty} \frac{dk}{2\pi} e^{i k x - D k^2 t} \end{align}\]

This is just a Gaussian integral. Complete the square in the exponential,

\[ D k^2 t - i k t = Dt (k^2 - \frac{ix}{Dt} k - \frac{x^2}{4 D^2t^2}) + \frac{x^2}{4 Dt} \]

and using

\[ \int_{-\infty}^{\infty} dx e^{-\alpha x^2} = \sqrt{\frac{\pi}{a}} \]

we get the desired answer.

The diffusion equation is central in fluid mechanics problems both for understanding viscosity and when the fluid is transporting additional quantities. This could be thermodynamic quantities like temperature, or the concentration of some chemical like salt or carbon. For example, if \(p = p(\rho, T)\) is the equation of state, the Navier-Stokes equation plus mass conservation \(D_t \rho = 0\) no longer close (there are four equations for five unknowns \({\vec v}\), \(\rho\), \(T\)); we need an equation for \(T\) which is typically

\[ D_t T = D_{temp} \nabla^2 T \]

where \(D_{temp}\) is the thermal diffusivity (do not confuse it with a derivative operator such as \(D_t\). Unfortunately this notation is canonical in the literature). Actually in the ocean and atmosphere a slightly differnt quantity known as potential temperature is used, but that is a longer story.

These equations are known as advection-diffusion equations. Advection refers to the \({\vec v}\cdot{\vec \nabla} T\) term in \(D_t\), describing the time derivative of the quanity along a fluid trajectory; without diffusion, blobs of a quantity move with the fluid. If \(D = 0\) and the velocity is constant in the \(x\) direction for example,

\[ T(x,t) = T(x - vt) \]

is a solution.

Physics 111a -- Fluid Mechanics

Introduction to diffusion

Introduction to diffusion¶