The General Equations of Diffusion and Flow in a Straight Tube

THE GENERAL EQUATIONS OF DIFFUSION AND FLOW IN A STRAIGHT TUBE 

Consider an infinite tube whose axis is parallel to the axis Ox of a rectangular co-ordinate system Oxyz- Let S denote the domain occupied by the interior of the tube in the plane Oyz, and let s be its area and the curve T its perimeter. In steady uniform flow the velocity u is everywhere in the direction Ox and is a function of y and z given by 

Let C x, y, z, t) denote the concentration of solute at the point x, y, z and time t, and let D p(y, z) be its diffusion coefficient. The function defines the variation of the diffusion coefficient and D is its mean value over the cross-section of the tube. The equation governing C is thus 

It is convenient to take an origin moving with the mean speed of the stream and to reduce the variables to dimensionless form. Let a be a dimension characteristic of the cross-section S and let 

If equation (9) is averaged over the cross-section, the use of Green s theorem and the condition (11) reduces it to 

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