Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-18T21:53:46.905Z Has data issue: false hasContentIssue false
  • English
  • Français

Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves

Published online by Cambridge University Press:  26 April 2006

N. Sugimoto
N. Sugimoto
Affiliation:
Department of Mechanical Engineering, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper deals with initial-value problems for the Burgers equation with the inclusion of a hereditary integral known as the fractional derivative of order ½. Emphasis is placed on the difference between the local and global dissipation due to the second-order and the half-order derivatives, respectively. Exploiting the smallness of the coefficient of the second-order derivative, an asymptotic analysis is first developed. When a discontinuity appears, the matched-asymptotic expansion method is employed to derive a uniformly valid solution. If the coefficient of the half-order derivative is also small, as is usually the case, the evolution comprises three stages, namely a lossless near field, an intermediate Burgers region, and a hereditary far field. In view of these results, the equation is then solved numerically, under various initial conditions, by finite-difference and spectral methods. It is revealed that the effect of the fractional derivative accumulates slowly to give rise to a significant dissipation and distortion of the waveform globally, which is to be contrasted with the effect of the second-order derivative, significant only locally, in a thin 'shock layer’.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 225 , April 1991 , pp. 631 - 653
Copyright
© 1991 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Basdevant, C., Deville, M., Haldenwang, P., Lacroix, J. M., Ouazzani, J., Peyret, R., Orlandi, P. & Patera, A. T., 1986 Spectral and finite difference solutions of the Burgers equation. Computers Fluids 14, 23–41.Google Scholar
Chester, W.: 1964 Resonant oscillations in closed tubes. J. Fluid Mech. 18, 44–64.Google Scholar
Cole, J. D.: 1968 Perturbation Methods in Applied Mathematics. Blaisdell.
Crighton, D. G. & Scott, J. F., 1979 Asymptotic solutions of model equations in nonlinear acoustics. Phil. Trans. R. Soc. Lond. A 292, 101–134.Google Scholar
Gazdag, J.: 1973 Numerical convective schemes based on accurate computation of space derivatives. J. Comput. Phys. 13, 100–113.Google Scholar
Gel'fand, I. M. & Shilov, G. E. 1964 Generalized Functions, Vol. 1. Academic.
Gittler, Ph. & Kluwick, A., 1989 Dispersive Wandreibungseffekte bei hochfrequenten Wellen in gasgefüllten Rohren. Z. Angew. Math. Mech. 69, 578–579.Google Scholar
Kakutani, T. & Matsuuchi, K., 1975 Effect of viscosity on long gravity waves. J. Phys. Soc. Japan 39, 237–246.Google Scholar
Keller, J. J.: 1981 Propagation of simple non-linear waves in gas-filled tubes with friction. Z. Angew. Math. Phys. 32, 170–181.Google Scholar
Lee-Bapty, I. P. & Crighton, D. G. 1987 Nonlinear wave motion governed by the modified Burgers equation. Phil. Trans. E. Soc. Lond. A 323, 173–209.Google Scholar
Lighthill, M. J.: 1956 Viscosity effects in sound waves of finite amplitude. In Surveys in Mechanics (ed. G. K. Batchelor & R. M. Davies), pp. 250–351. Cambridge University Press.
Miksis, M. J. & Ting, L., 1990 Effective equations for multiphase flows – Waves in bubbly liquid. Adv. Appl. Mech. (to be published).Google Scholar
Mitchell, A. R. & Griffiths, D. F., 1980 The Finite Difference Method in Partial Differential Equations. John Wiley & Sons.
Sachdev, P. L.: 1987 Nonlinear Diffusive Waves. Cambridge University Press.
Sugimoto, N.: 1989 'Generalized' Burgers equations and fractional calculus. In Nonlinear Wave Motion (ed. A. Jeffrey), pp. 162–179. Longman Scientific & Technical.
Sugimoto, N.: 1990 Evolution of nonlinear acoustic waves in a gas-filled pipe. In Frontiers of Nonlinear Acoustics, 12th Intl Symp. on Nonlinear Acoustics (ed. M. F. Hamilton & D. T. Blackstock), pp. 345–350. Elsevier Applied Sciences.
Whitham, G. B.: 1974 Linear and Nonlinear Waves. Wiley-Interscience.