Abstract
For viscous (barotropic or incompressible) fluids it is shown that, if the vorticity and the viscous force are orthogonal, vortex lines are convected by a vector field which fits with the velocity field when viscosity vanishes (extension of Helmholtz theorem); it is also found that energy remains constant along the field lines of this vector field (extension of Bernoulli theorem).
If, moreover, vorticity and velocity are orthogonal too, the magnitude of the vorticity then behaves as the density of a fluid which flows along streamsheets according to this very same vector field. These properties are mainly encountered for plane parallel flows, axially symmetrical flows, spherical flows, but also for some other miscellaneous flow geometries such as unidirectional or radial flows. The set of the former three flows can even be characterized by these properties; that enhances this set of important flow geometries, avails a general view on vorticity behavior, and explains the great simplicity of vorticity equations in these cases. Numerous examples and comments are given for illustrating.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lamb, H.,Hydrodynamics. Cambridge: University Press; New York: Dover (1945).
Prager, W.,Introduction to Mechanics of Continua. Boston: Ginn and Company (1961).
Sedov, L.,Foundations of the Non-Linear Mechanics of Continua. Oxford: Pergamon Press (1966).
Appel, P.,Traité de Mécanique Rationnelle, Tome 3, Equilibre et Mouvement des Milieux Continus. Paris: Gauthiers Villars (1903).
Ericksen, J.L., Tensor fields. In: S. Flugge (ed.),Principles of classical mechanics and field theory, Encyclopedia of Physics/Hanbuch der Physik, Vol. 3.1. Berlin: Springer Verlag (1960) pp. 794–858.
Crocco, L. and Pascucci, L., Diagrammi termodinamici dei gas di combustione; indicazioni per l'applicatione dei diagrammi al calcolo del ciclo del motore a scoppio (1947) Monograf. Scient.di aeronautica, 2, pl, 23 (Roma-Stabilimento Tipografico Fausto Failli). See also: Serrin, J., Mathematical principles of classical fluid mechanics. In: S. Flugge (ed.),Fluid Dynamics I, Encyclopedia of Physics/Handbuch der Physik, Vol. 8.1 (1959) pp. 125–263 § 38–39.
Germain, P.,Mécanique des Milieux Continus. Paris: Masson (1962).
Ionescu-Bujor, Etude intrinsèque des écoulements permanents et rotationnels d'un fluide parfait (1961). Paris: Thèse Faculté des Sciences de Paris.
Kaplun, S., The role of coordinate systems in boundary layer theory.J. Applied Math. and Phys. (ZAMP) 5 (1954) 111–135.
Legner, H., On Kaplun's optimal coordinates.J. Fluid Mech. 115 (1982) 379–393.
Schlichting, H.,Boundary Layer Theory, 7th edn. New York: McGraw-Hill (1979).
Landau, L. and Lifschitz, E.,Fluid Mechanics, Course of Theoretical Physics, Vol. 6 2nd edn. Oxford: Pergamon Press (1987).
Truesdell, C. and Toupin, R.,The classical field theories. In: S. Flugge (ed.),Encyclopedia of Physiks/Hanbuch der Physik, Vol. 3.1,Principles of Classical Mechanics and Field Theory. Berlin: Springer Verlag (1960) pp. 226–793.
Oroveanu, T., On a class of viscous fluid motions.Revue Roumaine des Sciences Techniques-Serie de Mécanique Appliquée, Vol. 33. (1988) 135–144.
Casal, P. and Gouin, H., Invariance properties of inviscid fluids of gradeN. Lecture Notes in Physics Vol. 344. Springer-Verlag (1989) pp. 85–98.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bouthier, M. Vorticity in plane parallel, axially symmetrical or spherical flows of viscous fluid. Appl. Sci. Res. 50, 1–27 (1993). https://doi.org/10.1007/BF01086450
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01086450