¶q/¶t =
r 1 - s ¶/¶r (r s -
1q m
¶q/¶r) in
0 £ r £ ¥ with q
³ 0, s > 0. (A)
Here s is not necessarily integral; m is
initially unrestricted. Material-conserving instantaneous source solutions of A
are reviewed as an entrée to material-losing solutions. Simple physical
arguments show that solutions for a finite slug losing material at infinity at a
finite nonzero rate can exist only for the following m-ranges: 0 <
s < 2, -2s -1 < m
£ -1; s > 2, -1 < m
< -2s -1. The result for s = 1 was
known previously. The case s = 2, m = -1, needs further
investigation. Three different similarity schemes all lead to the same ordinary
differential equation. For 0 < s < 2, parameter g (0 < g < ¥) in that equation
discriminates between the three classes of solution: class 1 gives the
concentration scale decreasing as a negative power of (1 + t / T);
2 gives exponential decrease; and 3 gives decrease as a positive power of (1 -
t / T), the solution vanishing at t = T < ¥. Solutions for s = 1, m = -3/2 are presented
graphically. The variation of concentration and flux profiles with increasing
g is physically explicable in terms of increasing flux
at infinity. An indefinitely large number of exact solutions are found for
s = 1, g = 1. These demonstrate the systematic
variation of solution properties as m decreases from -1 toward -2 at
fixed g.
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Author
J. R. Philip
CSIRO Centre for Environmental Mechanics, Canberra, Australia,
2061.
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