### Abstract

The stability of the steady flows of an ideal incompressible fluid of uniform density in a magnetic field is investigated. Only those MHD flows are considered which possess one of the types of symmetry (translational, axial, rotational or helical). The sufficient conditions for non-linear stability of the flows in question with respect to perturbations of this symmetry are obtained. These conditions are proved by the method of coupling the integrals of motion [1, 21 in the form [3-81, based on constructing functionals having absolute minima on specified steady solutions. Each of the functionals constructed is the sum of the kinetic energy, the integral of an arbitrary function of the Lagrangian coordinate and another integral, specific for the flows being investigated. The use of Lagrangian ooordinate fields leads to a whole family of new definitions of stability. According to these definitions, deviations of the perturbed flows from the unperturbed ones are measured by the integrals of the squares of the velocity-field and Lagrangiancoordinate perturbations. The stability conditions obtained are extended to existing results [5-7, 9] on new types of flows. These conditions are of an a priori nature since the corresponding theorems of existence of the solutions are not proved.

Original language | English |
---|---|

Pages (from-to) | 415-423 |

Number of pages | 9 |

Journal | Journal of Applied Mathematics and Mechanics |

Volume | 59 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1995 |

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### ASJC Scopus subject areas

- Mechanical Engineering
- Mechanics of Materials
- Applied Mathematics
- Modelling and Simulation

### Cite this

*Journal of Applied Mathematics and Mechanics*,

*59*(3), 415-423. https://doi.org/10.1016/0021-8928(95)00049-U

**The conditions for the non-linear stability of plane and helical mhd flows.** / Vladimirov, V. A.; Gubarev, Yu G.

Research output: Contribution to journal › Article

*Journal of Applied Mathematics and Mechanics*, vol. 59, no. 3, pp. 415-423. https://doi.org/10.1016/0021-8928(95)00049-U

}

TY - JOUR

T1 - The conditions for the non-linear stability of plane and helical mhd flows

AU - Vladimirov, V. A.

AU - Gubarev, Yu G.

PY - 1995

Y1 - 1995

N2 - The stability of the steady flows of an ideal incompressible fluid of uniform density in a magnetic field is investigated. Only those MHD flows are considered which possess one of the types of symmetry (translational, axial, rotational or helical). The sufficient conditions for non-linear stability of the flows in question with respect to perturbations of this symmetry are obtained. These conditions are proved by the method of coupling the integrals of motion [1, 21 in the form [3-81, based on constructing functionals having absolute minima on specified steady solutions. Each of the functionals constructed is the sum of the kinetic energy, the integral of an arbitrary function of the Lagrangian coordinate and another integral, specific for the flows being investigated. The use of Lagrangian ooordinate fields leads to a whole family of new definitions of stability. According to these definitions, deviations of the perturbed flows from the unperturbed ones are measured by the integrals of the squares of the velocity-field and Lagrangiancoordinate perturbations. The stability conditions obtained are extended to existing results [5-7, 9] on new types of flows. These conditions are of an a priori nature since the corresponding theorems of existence of the solutions are not proved.

AB - The stability of the steady flows of an ideal incompressible fluid of uniform density in a magnetic field is investigated. Only those MHD flows are considered which possess one of the types of symmetry (translational, axial, rotational or helical). The sufficient conditions for non-linear stability of the flows in question with respect to perturbations of this symmetry are obtained. These conditions are proved by the method of coupling the integrals of motion [1, 21 in the form [3-81, based on constructing functionals having absolute minima on specified steady solutions. Each of the functionals constructed is the sum of the kinetic energy, the integral of an arbitrary function of the Lagrangian coordinate and another integral, specific for the flows being investigated. The use of Lagrangian ooordinate fields leads to a whole family of new definitions of stability. According to these definitions, deviations of the perturbed flows from the unperturbed ones are measured by the integrals of the squares of the velocity-field and Lagrangiancoordinate perturbations. The stability conditions obtained are extended to existing results [5-7, 9] on new types of flows. These conditions are of an a priori nature since the corresponding theorems of existence of the solutions are not proved.

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UR - http://www.scopus.com/inward/citedby.url?scp=58149362475&partnerID=8YFLogxK

U2 - 10.1016/0021-8928(95)00049-U

DO - 10.1016/0021-8928(95)00049-U

M3 - Article

VL - 59

SP - 415

EP - 423

JO - Journal of Applied Mathematics and Mechanics

JF - Journal of Applied Mathematics and Mechanics

SN - 0021-8928

IS - 3

ER -