Unsteady asymptotic solutions of the two-dimensional Euler equations
Author:
Radhakrishnan Srinivasan
Journal:
Quart. Appl. Math. 54 (1996), 211-223
MSC:
Primary 35Q35; Secondary 76C05
DOI:
https://doi.org/10.1090/qam/1388013
MathSciNet review:
MR1388013
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Abstract: A technique is described for deducing a class of unsteady asymptotic solutions of the two-dimensional Euler equations. In contrast to previously known analytical results, the vorticity function $\left [ {\omega \left ( {x, y, t} \right )} \right ]$ for these solutions has a complicated dependence on the spatial coordinates $\left ( {x, y} \right )$ and time $\left ( t \right )$. The results obtained are in implicit form and are valid in those regions of space and time where $t\omega \to {o^ + }$ or $t\omega \to + \infty$. These asymptotic solutions may be split into an unsteady, two-dimensional and irrotational basic flow and a disturbance that is strongly nonlinear at appropriate locations within the domain of validity. The generality and complexity of these solutions make them theoretically interesting and possibly useful in applications.
G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967, Chapter 7
R. Berker, Intégration des équations du mouvement d’un fluide visqueux incompressible, in Handbuch der Physik, ed. S. Flügge, vol. VIII/2, Springer-Verlag, Berlin, 1963, pp. 1–384
C. Y. Wang, Exact solutions of the unsteady Navier-Stokes equations, Appl. Mech. Rev. 42, S269–282 (1989)
K. B. Ranger, A complex variable integration technique for the two-dimensional Navier-Stokes equations, Quart. Appl. Math. 49, 555–562 (1991)
K. B. Ranger, Solutions of the Navier-Stokes equations in implicit form, Quart. Appl. Math. 50, 793–800 (1992)
G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967, Chapter 7
R. Berker, Intégration des équations du mouvement d’un fluide visqueux incompressible, in Handbuch der Physik, ed. S. Flügge, vol. VIII/2, Springer-Verlag, Berlin, 1963, pp. 1–384
C. Y. Wang, Exact solutions of the unsteady Navier-Stokes equations, Appl. Mech. Rev. 42, S269–282 (1989)
K. B. Ranger, A complex variable integration technique for the two-dimensional Navier-Stokes equations, Quart. Appl. Math. 49, 555–562 (1991)
K. B. Ranger, Solutions of the Navier-Stokes equations in implicit form, Quart. Appl. Math. 50, 793–800 (1992)
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Article copyright:
© Copyright 1996
American Mathematical Society