Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-19T03:10:46.186Z Has data issue: false hasContentIssue false
  • English
  • Français

Collapse in conical viscous flows

Published online by Cambridge University Press:  26 April 2006

M. A. Goldshtik  and
V. N. Shtern
M. A. Goldshtik
Affiliation:
Institute of Thermophysics, Novosibirsk 630090, USSR
V. N. Shtern
Affiliation:
Institute of Thermophysics, Novosibirsk 630090, USSR
Get access Rights & Permissions [Opens in a new window]

Abstract

A class of steady conically similar axisymmetrical flows of viscous incompressible fluid is studied. The motion is driven by a vortex half-line or conical vortex in the presence of a rigid conical wall or in free space. The dependence of the solutions on parameters (say, the vortex circulation) is analysed. At some finite parameter values the solutions lose existence, i.e. a flow collapse takes place. This is correlated with the appearance of a sink singularity at the symmetry axis. Analytical estimates of the critical parameter values are performed, together with numerical calculations of sounds on the solution existence region in parameter space.

Asymptotic analysis of the near-critical regimes shows that a strong axial jet develops. The jet momentum becomes infinite at the critical parameter value and a singularity occurs. These paradoxical features seem to be typical of conically similar viscous flows. Reasons for the paradox and ways of overcoming it are discussed. Solution non-uniqueness and a hysteresis phenomenon are found in the Serrin problem. Possible applications of the results to model some geophysical and asrophysical phenomena are outlined.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 218 , September 1990 , pp. 483 - 508
Copyright
© 1990 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

Arnol'd, V. I.: 1983 Catastrophe Theory. Moscow University Press, 80 pp. (English trans. R. K. Thomas, 1984, Springer.)
Boyarevich, V. V., Freiberg, Ya. Zh., Shilova, Ye. I. & Shcherbinin, A., 1985 Electrovortex Flow. Riga: Zinatne, 315 pp.
Goldshtik, M. A.: 1960 A paradoxical solution of the Navier-Stokes equations. Appl. Mat. Mech. (Sov.) 24, 610–621.Google Scholar
Goldshtik, M. A.: 1979 On swirling jets. Mech. Fluid Gas (Sov.) 1, 26–36.Google Scholar
Goldshtik, M. A.: 1981 Vortex Flows. Novosibirsk: Nauka, 366 pp.
Goldshtik, M. A. & Shtern, V. N., 1987 Induced jets and critical phenomena in viscous flows. Novosibirsk: Institute of Thermophysics, preprint N 159, 50 pp.
Goldshtik, M. A. & Shtern, V. N., 1988 Conical flows of fluid with variable viscosity. Proc. R. Soc. Lond. A 419, 91–106.Google Scholar
Goldshtik, M. A. & Shtern, V. N., 1989 On a mechanism of astrophysical jets. Sov. Phys. Dokl. 304, No 5, 1069–1072.Google Scholar
Goldshtik, M. A., Shtern, V. N. & Yavorsky, N. T., 1989 Viscous Flows with Paradoxical Features. Novosibirsk: Nauka, 340 pp.
Hamel, G.: 1916 Spiralförmige Bewegungen zäher Flüssigkeiten. Jber. D. Mat. Verein 25, 334–360.Google Scholar
Henricsen, K. H. (ed.) 1986 Proceedings of conference on jets from stars and galaxies. Can. J. Phys. 64, 351–535.
Jeffery, G. B.: 1915 Two-dimensional steady motion of a viscous fluid. Phil. Mag. 29, 455–465.Google Scholar
Königl, A.: 1986 Stellar and galactic jets: theoretical issues. Can. J. Phys. 64, 362–368.Google Scholar
Lada, C. J.: 1985 Cold outflows, energetic winds and enigmatic jets around young stellar objects. Ann. Rev. Astr. Astrophys. 23, 267–317.Google Scholar
Landau, L. D.: 1944 On an exact solution of the Navier-Stokes equations. Dokl. Akad. Nauk. SSSR 43, 299–301.Google Scholar
Paull, R. & Pillow, A. F., 1985b Conically similar viscous flows. Part 2. One-parametric swirlfree flows. J. Fluid Mech. 155, 343–358.Google Scholar
Paull, R. & Pillow, A. F., 1985b Conically similar viscous flows. Part 3. Characterization of axial causes in swirling flows and one-parameter flow generated by a uniform half-line source of kinematic swirl angular momentum. J. Fluid Mech. 155, 359–380.Google Scholar
Pillow, A. F. & Paull, R., 1985 Conically similar viscous flows. Part 1. Basic conservation principles and characterization of axial causes in swirl-free flow. J. Fluid Mech. 155, 327–342.Google Scholar
Schlichting, H.: 1965 Grenzschicht Theorie. Karlsruhe: G. Braun, 742 pp.
Schneider, W.: 1981 Flow induced by jets and plumes. J. Fluid Mech. 108, 55–65.Google Scholar
Schneider, W.: 1985 Decay of momentum flux in submerged jets. J. Fluid Mech. 154, 91–110.Google Scholar
Schneider, W., Zauner, E. & Bohm, H., 1987 The recirculatory flow induced by a laminar axisymmetric jet issuing from a wall. Trans. ASME I: J. Fluids Engng 109, 237–241.Google Scholar
Serrin, J.: 1972 The swirling vortex. Phil. Trans. R. Soc. Lond. A 271, 325–360.Google Scholar
Slezkin, N. A.: 1934 On a case of integrability of the full differential equations of viscous fluid motion. Sci. Pap. Moskow Univ. 2, 89–90.Google Scholar
Sozou, C.: 1971 On fluid motions induced by an electric current source. J. Fluid Mech. 46, 25–32.Google Scholar
Squire, B.: 1952 Some viscous fluid flow problems. 1. Jet emerging from a hole in a plane wall. Phil. Mag. 43, 942–945.Google Scholar
Tayler, R. J.: 1978 Galaxies: Structure and Evolution. London and Basingstoke: Wykeham, 223 pp.
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. Parabolic, 310 pp.
Wang, C. V.: 1971 Effect of spreading of material of the surface of a liquid. Intl J. Nonlinear Mech. 6, 255–262.Google Scholar
Yatseev, V. N.: 1950 On a class of exact solutions of the viscous fluid motion equations. Sov. Phys., J. Exp. Theor. Phys. 20, 1031–1034.Google Scholar
Yih, C.-S., Wu, F., Garg, A. K. & Leibovich, S., 1982 Conical vortices: A class of exact solutions of the Navier-Stokes equations. Phys. Fluids 25, 2147–2157.Google Scholar
Zauner, E.: 1985 Visualization of the viscous flow induced by a round jet. J. Fluid Mech. 154, 111–120.Google Scholar
Zubtsov, A. V.: 1984 On a self-similar solution for weak-swirling jet. Mech. Fluid Gas (Sov.) N4, 45–50.Google Scholar