Abstract
A mathematical model for axisymmetric eddy motion of a perfect incompressible fluid in a long tube with thin elastic walls is proposed. Necessary and sufficient conditions for hyperbolicity of the system of equations of motion for flows with monotonic radial velocity profiles are formulated. The propagation velocities of the characteristics of the system under study and the characteristic shape of this system are calculated. The existence of simple waves continuously attached to a given steady shear flow is proved. The group of transformations admitted by the system is found, and submodels that determine invariant solutions are given. By integrating factor‐systems, new classes of exact solutions of equations of motion are found.
Similar content being viewed by others
REFERENCES
T. Pedly, The Fluid Mechanics of Large Blood Vessels, Cambridge Univ. Press (1980).
A. N. Volobuev, “Fluid flow in tubes with elastic walls, ” Usp. Fiz. Nauk, 165, No. 2, 177–186 (1995).
V. M. Teshukov, “Hyperbolicity of the equations of motion of a long wave, ” Dokl. Akad. Nauk SSSR, 284, No. 3, 555–562 (1985).
L. V. Ovsyannikov, Group Analysis of Differential Equations, Academic Press, New York (1982).
V. M. Teshukov, “Long-wave approximation model for a gas shear ow in a channel of varying area, ” Prikl. Mekh. Tekh. Fiz., 39, No. 1, 15–27 (1998).
B. J. Benney, “Some properties of long nonlinear waves, ” Stud. Appl. Math., 52, 45–50 (1973).
V. E. Zakharov, “Benney's equations and the quasi-classic approximation in the inverse-problem method, ” Funkts. Anal. Prilozhen., 14, No. 2, 15–24 (1980).
V. M. Teshukov, “Long waves in an eddying barotropic uid, ” Prikl. Mekh. Tekh. Fiz., 35, No. 6, 17–26 (1994).
V. M. Teshukov, “Simple waves on a free-boundary ow of an ideal incompressible uid, ” Prikl. Mekh. Tekh. Fiz., 38, No. 2, 48–57 (1997).
P. L. Sachdev and Ph. Varugheze, “Invariance group properties and exact solutions of equations describing time-dependent free surface ows under gravity, ” Quart. Appl. Math., 43, No. 2, 465–482 (1986).
E. Varley and P. A. Blythe, “Long eddies in sheared ows, ” Stud. Appl. Math., 68, 103–187 (1983).
A. A. Chesnokov, “Exact solutions of the equations of vortex shallow water, ” Prikl. Mekh. Tekh. Fiz., 38, No. 5, 44–55 (1997).
A. A. Chesnokov, “Vortex motion of uid in a narrow channel, ” Prikl. Mekh. Tekh. Fiz., 39, No. 4. 38–49 (1998).
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Banach-Space Differential Equations [in Russian], Nauka, Moscow (1970).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chesnokov, A.A. Axisymmetric Vortex Fluid Flow in a Long Elastic Tube. Journal of Applied Mechanics and Technical Physics 42, 628–637 (2001). https://doi.org/10.1023/A:1019251730222
Issue Date:
DOI: https://doi.org/10.1023/A:1019251730222