## 2 Transport Processes

### 2.1 Diffusion Without Shear

#### (i) Instantaneous plane source in 3-dimensions

Suppose that the initial distribution of is dependent only on and and that , and are constant. Then Eq. (14) reduces to

We now transform to a frame of reference moving with the flow speed :

Then Eq. (15) becomes

Suppose that is constant. Then the solution to Eq. (16) is

It can be shown that

The solution (17) can be interpreted as an instantaneous release of an amount kgm of pollutant at in the plane . Eq. (17) is the Gaussian puff'' solution because it describes the release of a small puff'' of pollutant.

An important quantity is the mean square distance which the pollutant has spread from . This is given by

Therefore

The pollutant therefore spreads at a rate proportional to .

#### (ii) Instantaneous line source in 3-dimensions

In this case we assume that so that the equation for is

As in the previous case we make the transformation and also let . Then

The solution is

It can be shown that and . Hence

#### (iii) Instantaneous point source in 3-dimensions

This is the most general and practical case where . The equation for is

We could put in line with the cases of plane and line sources but it is difficult to think of real cases where there is a mean vertical velocity so we keep . The transsformed equation for is then

The solution is

In this case , and so that

Fig. 2. Concentration distributions at various times after pollutant release for the diffusion of an instantaneous plane source (Eq. (20)).

### 2.2 Advection Without Diffusion in a Shear Flow

Suppose that we have a uni-directional shear flow:

and suppose that we are interested in the flow between and .

Fig. 3. Shear flow .

If we neglect diffusion, then the equation for is

If we again put Eq. (26) becomes

and so , i.e. the concentration is independent of time moving with the local flow . Consider as in the previous section the case of an instantaneous plane source at , . Then at

This must be the solution for all time, because . Therefore

Fig. 4. A line (or plane in 3-D) of pollutant initially lying on is stretched and rotated by a shear flow . At time the extent of the line of pollutant is denoted by .

Consider the horizontal extent of the pollutant. At time it is clearly

We can see that the pollutant spreads out at a rate proportional to . This contrasts with the rate proportional to for diffusion, so it would appear that advection is always more important than diffusion, except perhaps for a short time after release. However, we shall see in the next section that this is not the case.

### 2.3 Advection and Diffusion in a Shear Flow

When advection and diffusion occur simultaneously, the stretching of regions of high concentration, as occurs for pure advection, is less effective because transverse diffusion of the high concentration region is enhanced by the stretching caused by the shear. This is shown schematically in Fig. 5.

Fig. 5. An initial slab (a) of pollutant is rotated and stretched by shear (b), leading to enhanced diffusion in the transverse direction (c). The spreading of the line of maximum concentration is clearly reduced by the diffusion.

The processes described here are known as Taylor's mechanism.

back to syllabus