Unified elementary treatment of transport processes

The rough mean-free path development to be given in this section will be used subsequently for viscous stresses and for thermal conduction it may also be applied to diffusion, but the results in this case are not so satisfactory as are those of the preceding section. 

It will be assumed that at each collision, both colliding molecules exchange their property Q, equalizing it between them, and that their resulting Q equals the Q appropriate to the level at which they collide. Both of these assumptions are, at best, approximations that may be valid on the 

It may be expected that, on the average, molecules crossing the plane X = 0 from below will have Q = Q( — l) (that is a value of Q appropriate to the position x = —/), and those crossing from above will have Q — 

When the mean free path is small compared with the distance over which Q changes appreciably, the quantities Q( — l) and Q( + /) in equation (29) may be expanded in a Taylor series about x = 0, and terms of order higher than the first may be neglected. The result is 

The final expression for the flux of Q in terms of the gradient of its average value is obtained by substituting equation (27) into equation (30), omitting the subscript 0 (since the choice of the position x = 0 was arbitrary) 

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