Analytic and Numerical Study of N‐Waves Governed by the Nonplanar Burgers Equation

Abstract

In this article, evolution of N-waves under the nonplanar Burgers equation, which takes into account geometrical expansion or contraction, is treated analytically. An exact asymptotic solution, generalizing that for the planar Burgers equation, is given for the case of expansion. An approximate treatement, using a balancing argument, gives asymptotic analytic results for both expansion and contraction. The analysis is fortified by an accurate numerical solution of the problem. This study is brought in close conjunction with the earlier work of Crighton and Scott [13] and Sachdev, Joseph and Nair [3].

Asymptotic N‐wave Solutions of the Nonplanar Burgers Equation

“…More details are given in Section 3. The large time asymptotic solution of the nonplanar Burgers equation (1) constructed here by us would be an improvement over the N-wave solution of the linearized form of (1) for all j > 0 without any restriction on j, unlike Sachdev et al [6]. The construction of asymptotic N-wave solution of (1) here follows two steps.…”

“…Hence these equations have to be dealt with directly. Sachdev et al [6] constructed "composite N-wave" solutions of (1), taking into account the behavior of solutions of (1) as t → 0 and as t → ∞, when the parameter j takes the form m/n; m, n are integers and (m, n) = 1. Their construction of solutions of (1) follows closely the form of the exact solution of plane Burgers equation (Equation (1) with j = 0).…”

“…Their construction of solutions of (1) follows closely the form of the exact solution of plane Burgers equation (Equation (1) with j = 0). The evaluation of the solution constructed by Sachdev et al [6] is difficult, because every term of the series given in the denominator (see (53)-(54)) is of the order O(t (2+ j)/2 ) as t → ∞. In other words, every term is of the same order and hence no term can be neglected.…”

“…In other words, every term is of the same order and hence no term can be neglected. Sachdev et al [6] evaluated Reynolds number, defined by…”

“…They observed numerically that except for some initial time, the Reynolds numbers (2) evaluated from the numerical solution and the solution of (1) constructed by them agreed very well. Sachdev et al [6] considered the cases 0 < j < 2 and j = 2 separately. The reason for this is that they wanted to take into account the behavior of the inviscid solution (of (1) with δ = 0)…”