A non-reactive solvent is released around the origin at t = 0 into a parallel flow u(y) in a
two-dimensional channel of half-width h. The solvent’s concentration c(x, y, t) is governed by the
convection-diffusion equation
ct + u(y)cx = D(cxx + cyy) with cy = 0 at y = 0 and y = h.
1. Rewrite the problem in terms of the non-dimensional variables ˜x, ˜y, ˜t and ˜u given by x = L˜x,
y = h˜y, t = L
u0
˜t, and u = u0˜u. Here u0 represents the maximum of the flow velocity and L is a
longitudinal length scale chosen so that the reference (observation) time L/u0 is large compared
with the transverse diffusion time h2/D. Hence show that the problem can be written, after
dropping tildes, as
ǫPe (ct + u(y)cx) = ǫ2cxx + cyy with cy = 0 at y = 0 and y = 1,
and determine the non-dimensional parameters ǫ and Pe in terms of L, h, u0 and D.
[5 marks]
2. Denoting transverse averages by bars, and writing
c(x, y, t) = ¯c(x, t) + c′(x, y, t) where ¯c = Z 1
0
c dy and c′ = 0,
show that ¯c is governed by
ǫPe ?¯ct + ¯u ¯cx + uc′
x = ǫ2¯cxx .
[5 marks]
3. Show that c′ is governed by
ǫPe ?c′
t + u′¯cx + uc′
x
− uc′
x = ǫ2c′
xx + c′
yy,
where u′
u(y) − ¯u is the deviation of the velocity from its mean value.
[3 marks]
4. Show that in the distinguished limit ǫ ! 0 with ǫPe ! 0, we have c′ = ǫPe¯cxF(y) to leading
order and determine the function F(y).
[8 marks]
5. Using the result of the previous question, determine uc′
x to leading order. Hence, show that ¯c
is governed by
ǫPe (¯ct + ¯u ¯cx) = ǫ2 ?1 + αPe2 ¯cxx,
and determine the number α. What is the value of α in the particular case of a Poiseuille flow
given by u = 1 − y2.
[8 marks]
6. Returning to dimensional variables, show that ¯c is governed by a convection-diffusion equation
with effective diffusion coefficient Deff which is to be determined.
[5 marks]
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