Share This Article:

Nonlinear Interaction of N Conservative Waves in Two Dimensions

Abstract Full-Text HTML XML Download Download as PDF (Size:795KB) PP. 127-142
DOI: 10.4236/ajcm.2014.43012    4,900 Downloads   5,627 Views   Citations

ABSTRACT

Kinematic Fourier (KF) structures, exponential kinematic Fourier (KEF) structures, dynamic exponential (DEF) Fourier structures, and KEF-DEF structures with constant and space-dependent structural coefficients are developed in the current paper to treat kinematic and dynamic problems for nonlinear interaction of N conservative waves in the two-dimensional theory of the Newtonian flows with harmonic velocity. The computational method of solving partial differential equations (PDEs) by decomposition in invariant structures, which continues the analytical methods of undetermined coefficients and separation of variables, is extended by using an experimental and theoretical computation in Maple?. For internal waves vanishing at infinity, the Dirichlet problem is formulated for kinematic and dynamics systems of the vorticity, continuity, Helmholtz, Lamb-Helmholtz, and Bernoulli equations in the upper and lower domains. Exact solutions for upper and lower cumulative flows are discovered by the experimental computing, proved by the theoretical computing, and verified by the system of Navier-Stokes PDEs. The KEF and KEF-DEF structures of the cumulative flows are visualized by instantaneous surface plots with isocurves. Modeling of a deterministic wave chaos by aperiodic flows in the KEF, DEF, and KEF-DEF structures with 5N parameters is considered.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Miroshnikov, V. (2014) Nonlinear Interaction of N Conservative Waves in Two Dimensions. American Journal of Computational Mathematics, 4, 127-142. doi: 10.4236/ajcm.2014.43012.

References

[1] Lamb, S.H. (1945) Hydrodynamics.6th Edition, Dover Publications, New York.
[2] Pozrikidis, C. (2011) Introduction to Theoretical and Computational Fluid Dynamics. 2nd Edition, Oxford University Press, Oxford.
[3] Miroshnikov, V.A. (2005) The Boussinesq-Rayleigh Series for Two-Dimensional Flows Away from Boundaries. Applied Mathematics Research Express, 2005, 183-227. http://dx.doi.org/10.1155/amrx.2005.183
[4] Korn, G.A. and Korn, T.A. (2000) Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. 2nd Revised Edition, Dover Publications, New York.
[5] Kochin, N.E., Kibel, I.A. and Roze, N.V. (1964) Theoretical Hydromechanics. John Wiley & Sons Ltd., Chichester.
[6] Miroshnikov, V.A. (2009) Spatiotemporal Cascades of Exposed and Hidden Perturbations of the Couette Flow. Advances and Applications in Fluid Dynamics, 6, 141-165.
[7] Miroshnikov, V.A. (2012) Dual Perturbations of the Poiseuille-Hagen Flow in Invariant Elliptic Structures. Advances and Applications in Fluid Dynamics, 11, 1-58.

  
comments powered by Disqus

Copyright © 2019 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.