Invariance group properties and exact solutions of equations describing time-dependent free surface flows under gravity
Authors:
P. L. Sachdev and Varughese Philip
Journal:
Quart. Appl. Math. 43 (1986), 463-480
MSC:
Primary 76B15; Secondary 35C05
DOI:
https://doi.org/10.1090/qam/846158
MathSciNet review:
846158
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Abstract: Using the method of infinitesimal transformations, a 6-parameter family of exact solutions describing nonlinear sheared flows with a free surface are found. These solutions are a hybrid between the earlier self-propagating simple wave solutions of Freeman, and decaying solutions of Sachdev. Simple wave solutions are also derived via the method of infinitesimal transformations. Incomplete beta functions seem to characterize these (nonlinear) sheared flows in the absence of critical levels.
N. C. Freeman, Simple waves on shear flows: Similarity solutions, J. Fluid Mech. 56, 257–263 (1972)
- P. L. Sachdev, Exact, self-similar, time-dependent free surface flows under gravity, J. Fluid Mech. 96 (1980), no. 4, 797–802. MR 566646, DOI https://doi.org/10.1017/S0022112080002364
- P. L. Sachdev and A. Venkataswamy Reddy, Some exact solutions describing unsteady plane gas flows with shocks, Quart. Appl. Math. 40 (1982/83), no. 3, 249–272. MR 678197, DOI https://doi.org/10.1090/S0033-569X-1982-0678197-8
- P. L. Sachdev and M. C. Singh, Exact time-dependent flows for nonshallow water equations, Z. Angew. Math. Phys. 35 (1984), no. 4, 585–591 (English, with German summary). MR 759159, DOI https://doi.org/10.1007/BF00945076
P. A. Blythe, Y. Kazakia and E. Varley, The interaction of large amplitude shallow-water waves with an ambient shear flow: Noncritical flows, J. Fluid Mech. 56, 241–255 (1972)
- L. V. Ovsjannikov, Gruppovye svoĭ stva differentsial′nykh uravneniĭ ., Izdat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk], 1962 (Russian). MR 0142695
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Y. L. Luke, Special functions and their approximations, vol. 1,2 Academic Press, New York, 1969
G. I. Barenblatt and Ya. B. Zeldovich, Self-similar solutions as intermediate asymptotics, Annual Review of Fluid Mechanics 4, 285–312 (1972)
- Michael S. Longuet-Higgins, Self-similar, time-dependent flows with a free surface, J. Fluid Mech. 73 (1976), no. 4, 603–620. MR 416223, DOI https://doi.org/10.1017/S0022112076001523
- E. Varley, J. Y. Kazakia, and P. A. Blythe, The interaction of large amplitude barotropic waves with an ambient shear flow: critical flows, Philos. Trans. Roy. Soc. London Ser. A 287 (1977), no. 1342, 189–236. MR 462157, DOI https://doi.org/10.1098/rsta.1977.0144
N. C. Freeman, Simple waves on shear flows: Similarity solutions, J. Fluid Mech. 56, 257–263 (1972)
P. L. Sachdev, Exact self-similar time-dependent free surface flows under gravity, J. Fluid Mech. 96, 797–802 (1980)
P. L. Sachdev and A. V. Reddy, Some exact solutions describing unsteady plane gas flows with shocks, Quart. Appl. Math. XL, 249–272 (1982)
P. L. Sachdev and M. C. Singh, Exact time-dependent flows for non-shallow water equations, J. Appl. Math. and Phys. (ZAMP) 35, 585–591 (1984)
P. A. Blythe, Y. Kazakia and E. Varley, The interaction of large amplitude shallow-water waves with an ambient shear flow: Noncritical flows, J. Fluid Mech. 56, 241–255 (1972)
L. V. Ovsjannikov, Group properties of differential equations, translation by G. W. Bluman of Gruppovye Svoysta Differentsialny Uraveni Novosibirsk, U.S.S.R., 1962
G. W. Bluman and J. D. Cole, Similarity methods for differential equations, Springer-Verlag, New York, 1974
J. D. Logan and J. D. J. Perez, Similarity solutions for reactive shock hydrodynamics, SIAM J. Appl. Math. 39, 512–527 (1980)
Y. L. Luke, Special functions and their approximations, vol. 1,2 Academic Press, New York, 1969
G. I. Barenblatt and Ya. B. Zeldovich, Self-similar solutions as intermediate asymptotics, Annual Review of Fluid Mechanics 4, 285–312 (1972)
M. S. Longuet-Higgins, Self-similar, time-dependent flows with a free surface, J. Fluid Mech. 73, 603–620 (1976)
E. Varley, J. Y. Kazakia and P. A. Blythe, The interaction of large amplitude barotropic waves with an ambient shear flow: Critical flows, Phil. Trans. Roy. Soc. 287, 189–236 (1977)
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Article copyright:
© Copyright 1986
American Mathematical Society
