Exact n-wave solutions for the non-planar Burgers equation

Abstract

An exact representation of N-wave solutions for the non-planar Burgers equation u_t + uu_x + ½ju/t = ½δ u_xx, 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| < surd (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, ½ , 1/3 and ¼. 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).