Inertial Stability as a Result of the Relation between Transport and Relative Fluid Rotations

Павло Володимирович Лук’янов


The aim of the paper is fluid inertial stability nature determination through the representation of potential (non-rotational) motion as the compensation of two rotations, transport one and relative one. Theoretical methods are used. It is based on well-known description of fluid motion as a sum of three types (Cauchy–Helmholz theorem), but uses theoretical mechanics approach. The motion is considered as a sum of transport and relative ones. Thransport angular  velocity corresponds to macroscopic motion, while relative one is caused by fluid parcel deformation. From the position, fluid potential motion is a particular case when the sum of transport and and relative angular velocities are equal to zero. As a result, using Rayleigh circulation theorem (inertial stabilty criterion), it has been pointed out inertial stability physical mechanism. It caused by relative rotation prevailing over transport one when angular velocities have different signs. A hypothetical was made in attempt to extend the assertion for general case. The proposed approach has been tested through agreement with known Kloosterziel–van Heijst inertial stability criterion for f-plane. The criterion derived in the paper is simpler than one because it is based on analysis of only one value – angular velocity.


Inertial stability; Potential (non-rotational) motion


Линь Цзя-цзяо. Теория гидродинамической устойчивости. – М.: Изд-во ин. лит. 1958. –195 с.

R.C. Kloosterziel and G.J.F. Heijst, “An experimental study of unstable barotropic vortices in a rotating fluid”, J. Fluid Mech., vol. 223, pp. 1–24, 1991.

R.C. Kloosterziel et al., “Inertial instability and stratified fluids: barotropic vortices”, Ibid, vol. 583, pp. 379–412, 2007.

R.C. Kloosterziel, “Viscous symmetric stability of circular flows”, Ibid, vol. 652, pp. 171–193, 2010.

G.F. Carnevale et al., “Predicting the aftermath of vortex breakup in rotating flow”, Ibid, vol. 669, pp. 90–119, 2011.

G.F. Carnevale et al., “Inertial and barotropic instabilities of free current in three-dimensional rotating flow”, Ibid, vol. 725, pp. 117–151, 2013.

Фабрикант Н.Я. Аэродинамика. Общий курс. – М.: Наука, 1964. – 814 с.

Кочин Н.Е., Кибель И.А., Розе Н.В. Теоретическая гидромеханика. – М.: Гос. изд. физ.-мат. лит., 1963. – Т. 1. – 840 с.

Лойцянский Л.Г. Механика жидкости и газа. – М.: Наука, 1987. – 840 с.

Алексеенко С.В., Куйбин П.А., Окулов ВЛ. Введение в теорию концентрированных вихрей. – Новосибирск: Ин-т теплофизики СО РАН, 2003. – 504 с.

H.Z. Baumert, “Universal equations and constants of turbulent motion” Physica Scripta, T155, 014001 (12 p), 2013.

D. Coles, “Transition in circular Couette flow”, J. Fluid Mech., vol. 21, p. 385, 1965.

Джозеф Д. Устойчивость движений жидкости. – М.: Мир, 1981. – 639 с.

Лурье А.И. Аналитическая механика. – М.: ГИФМЛ, 1961. – 675 с.

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