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Exact N-Wave Solutions for the Non-Planar Burgers Equation

P. L. Sachdev, K. T. Joseph and K. R. C. Nair
Proceedings: Mathematical and Physical Sciences
Vol. 445, No. 1925 (Jun. 8, 1994), pp. 501-517
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/52515
Page Count: 17
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Exact N-Wave Solutions for the Non-Planar Burgers Equation
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Abstract

An exact representation of N-wave solutions for the non-planar Burgers equation ut + uux + 1/2ju/t = 1/2δ uxx, j = m/n, m < 2n, where m and n are positive integers with no common factors, is given. This solution is asymptotic to the inviscid solution for |x| < √ (2Q0 t), where Q0 is a function of the initial lobe area, as lobe Reynolds number tends to infinity, and is also asymptotic to the old age linear solution, as t tends to infinity; the formulae for the lobe Reynolds numbers are shown to have the correct behaviour in these limits. The general results apply to all j = m/n, m < 2n, and are rather involved; explicit results are written out for j = 0, 1, 1/2, 1/3 and 1/4. The case of spherical symmetry j = 2 is found to be `singular' and the general approach set forth here does not work; an alternative approach for this case gives the large time behaviour in two different time regimes. The results of this study are compared with those of Crighton & Scott (1979).

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