The Tube of Circular Cross-Section

Before proceeding with the discussion of the general case, it is interesting to consider the case of a tube of circular cross-section and to note how the present treatment agrees with Taylor s. In this case we take Ox to be the axis and a to be the radius of the tube and transform to polar co-ordinates 

In particular, dm0/dr = 0 so that m0 is a constant which we may take as unity. This of course merely states that the total quantity of solute is constant. 

In this integral the constant term vanishes since the mean value of (1 - 2p ) is zero and the terms with m 0 vanish by integration round the circle. Hence 

Thus as t— a), drai/dr— 0 and the centre of gravity ultimately moves with the mean speed of the stream. Choosing the origin in the original plane of the centre of gravity, mio = 0 and 

Thus the centre of gravity ultimately moves to a position 

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