# A linear magnetic Bénard problem with tensorial electrical conductivity

A. Georgescu; L. Palese; A. Redaelli

Bollettino dell'Unione Matematica Italiana (2006)

- Volume: 9-B, Issue: 1, page 197-214
- ISSN: 0392-4033

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topGeorgescu, A., Palese, L., and Redaelli, A.. "A linear magnetic Bénard problem with tensorial electrical conductivity." Bollettino dell'Unione Matematica Italiana 9-B.1 (2006): 197-214. <http://eudml.org/doc/289623>.

@article{Georgescu2006,

abstract = { For normal mode perturbations, in the hypothesis that the principle of exchange of stabilities holds, the eigenvalue problem defining the neutral curves of the linear stability for a magnetic electroanisotropic Benard problem is solved by Budiansky-DiPrima method. The unknown functions are taken as Fourier series on some total sets of separable Hilbert spaces and the expansion functions satisfied only part of the boundary conditions of the problem. This introduces some constraints to be satisfied by the Fourier coefficients. In order to keep the number of these constraintsas low as possible we are lead to use total sets for the even velocity and temperature fields different from the case when velocity and temperature are odd.The splitting ofthe unknown functions into even and odd parts leads to two problems of the sameorder as the given one each of which containing even as well as odd order parts of these functions. The secular equations involve series which are truncated to one and two terms, the last situation corresponding to best results. A closed form of the neutral curve is obtained. The presence of the Hall currents is proved to be destabilizing.},

author = {Georgescu, A., Palese, L., Redaelli, A.},

journal = {Bollettino dell'Unione Matematica Italiana},

language = {eng},

month = {2},

number = {1},

pages = {197-214},

publisher = {Unione Matematica Italiana},

title = {A linear magnetic Bénard problem with tensorial electrical conductivity},

url = {http://eudml.org/doc/289623},

volume = {9-B},

year = {2006},

}

TY - JOUR

AU - Georgescu, A.

AU - Palese, L.

AU - Redaelli, A.

TI - A linear magnetic Bénard problem with tensorial electrical conductivity

JO - Bollettino dell'Unione Matematica Italiana

DA - 2006/2//

PB - Unione Matematica Italiana

VL - 9-B

IS - 1

SP - 197

EP - 214

AB - For normal mode perturbations, in the hypothesis that the principle of exchange of stabilities holds, the eigenvalue problem defining the neutral curves of the linear stability for a magnetic electroanisotropic Benard problem is solved by Budiansky-DiPrima method. The unknown functions are taken as Fourier series on some total sets of separable Hilbert spaces and the expansion functions satisfied only part of the boundary conditions of the problem. This introduces some constraints to be satisfied by the Fourier coefficients. In order to keep the number of these constraintsas low as possible we are lead to use total sets for the even velocity and temperature fields different from the case when velocity and temperature are odd.The splitting ofthe unknown functions into even and odd parts leads to two problems of the sameorder as the given one each of which containing even as well as odd order parts of these functions. The secular equations involve series which are truncated to one and two terms, the last situation corresponding to best results. A closed form of the neutral curve is obtained. The presence of the Hall currents is proved to be destabilizing.

LA - eng

UR - http://eudml.org/doc/289623

ER -

## References

top- ABANI, KAMLA - SRIVASTAVA, K.M., Rayleigh-Taylor instability of a viscous plasma inthe Presence of Hall current, Il Nuovo Cimento, 26, 2 (1975), 419-432.
- CHANDRASEKHAR, S., Hydrodynamic and hydromagnetic stability, Clarendon, Oxford, 1968. MR128226
- DI PRIMA, R. C., Some variational principles for problems in hydrodynamic and hydromagnetic stability, Quart. Appl. Math., 18, 4 (1961), 375-385. MR116767DOI10.1090/qam/116767
- EBEL, D. - SHEN, M.C., On the linear stability of a toroidal plasma with resistivity, viscosity and Hall current, J. Math.Anal. Appl., 125 (1987), 81-103. Zbl0649.76023MR891351DOI10.1016/0022-247X(87)90166-1
- EBEL, D. - SHEN, M. M., Linearization principle for a toroidal Hall current plasma with viscosity and resistivity, Annali Mat. Pura Appl., 150 (1988), 39-65. Zbl0666.76153MR946029DOI10.1007/BF01761463
- GEORGESCU, A., Hydrodynamic stability theory, Kluwer, Dordrecht, 1985. Zbl0608.76035MR850008DOI10.1007/978-94-017-1814-1
- GEORGESCU, A., Variational formulation of some nonselfadjoint problems occurring in Benard instability theory, I, Series in Mathematics35/1977INCREST, Bucharest.
- GEORGESCU, A. - PALESE, L. - PASCA, D. - BUICAN, M., Critical hydromagnetic stability of a thermodiffusive state, Rev. Roumaine Math. Pures et Appl., 38, 10 (1993), 831840. MR1264602
- GEORGESCU, A. - PALESE, L., Neutral stability hypersurfaces for an anisotropic M.H.D. thermodiffusive mixture. III. Detection of false secular manifolds among the bifurcation characteristic manifolds, Rev. Roumaine Math. Pures et Appl., 41, 12 (1996), 35-49. Zbl0857.76032MR1404641
- GRADSHTEYN, J. S. - RYZHIK, I. M., Table of integrals, series, and products, Academic, New York, 1980. Zbl0521.33001MR669666
- JOSEPH, D. D., Stability of fluid motions, vols. I, II, Springer, Berlin, 1976. MR627612
- MAIELLARO, M. - PALESE, L., Sui moti M.H.D. stazionari di una miscela binaria in uno strato obliquo poroso in presenza di effetto Hall e sulla loro stabilita, Rend. Accad. Sc. Mat. Fis., Napoli, IV, XLVI (1979), 471-481. Zbl0441.76041
- MAIELLARO, M. - PALESE, L., Electrical anisotropic effects on thermal instability. Int. J. Engng. Sc., 22, 4 (1984), 411-418. Zbl0534.76045
- MAIELLARO, M. - PALESE, L. - LABIANCA, A., Instabilizing-stabilizing effects of M.H.D. anisotropic currents, Int. J. Engng. Sc., 27, 11 (1989), 1353-1359. Zbl0693.76059MR998286DOI10.1016/0020-7225(89)90124-9
- MAIELLARO, M. - LABIANCA, A., On the non linear stability in anisotropic MHD with applications to Couette Poiseuille flows. Int. J. Engng. Sc., 40, (2002), 1053-1068. Zbl1211.76153
- MIKHLIN, S. G., Mathematical physics, an advanced course, North Holland, Amsterdam, 1970. Zbl0202.36901MR286325
- MULONE, G. - RIONERO, S., On the non linear stability of the rotating Benard problemvia the Lyapunov direct method, J. Math.Anal. Appl., 144 (1989), 109-127. Zbl0682.76037MR1022564DOI10.1016/0022-247X(89)90362-4
- MULONE, G. - RIONERO, S., On the stability of the rotating Benard problem, Bull. Tech.Univ. Istanbul, 47 (1994), 181-202. Zbl0864.76030MR1321950
- MULONE, G. - SALEMI, F., Some continuous dependence theorems in M.H.D. with Hall and ion-slip currents in unbounded domains, Rend. Accad. Sci. Fis. Mat. Napoli, IV, 55 (1988), 139-152. Zbl1145.76473MR1136744
- S PAI, H. I., Magnetohydrodynamics and plasma dynamics, Springer, Berlin, 1962.
- PALESE, L., Sull'instabilita gravitazionale e sulla propagazione ondosa per un fluido elettroconduttore anisotropo inquinato, Atti Sem. Mat. Fis. Univ. Modena,XLII (1994), 1-17.
- PALESE, L., Electroanisotropic effects on the thermal instability of an anisotropic binary fluid mixture, J. of Magnetohydrodynamics and Plasma Research, 7, 2/3 (1997), 101-120.
- PALESE, L. - GEORGESCU, A. - PASCA, D., Stability of a binary mixture in a porous medium with Hall ion-slip effect and Soret Dufour currents, Analele Univ. Oradea, 3 (1993), 92-96.
- PALESE, L. - Georgescu, A., A linear magnetic Benard problem with Hall effect. Application of Budiansky-DiPrima method, Rapporti Int. Dip. Mat. Bari, 15 (2003).
- PALESE, L. - GEORGESCU, A. - PASCA, D. - BONEA, D., Thermosolutal instability of a compressible Soret-Dufour mixture with Hall and ion-slip currents through a porous medium, Rev. Roumaine Mec. Appl., 42, 3-4, (1997) 279 -296. MR2165211
- RIONERO, S. - MULONE, G., A non linear stability analysis of the magnetic Benard problem through the Lyapunov direct method, Arch. Rational Mech. Anal., 103 (1988), 347-368. Zbl0666.76068MR955532DOI10.1007/BF00251445
- SHARMA, R. C. - SHARMA, K. C., Thermal instability of compressible fluids with Hallcurrents through porous medium, Instanbul Univ. Fen. Fak. Mec. A, 43 (1978), 89-98. MR948330
- SHARMA, R. C. - RANI, NEELA, Hall effects on thermosolutal instability of a plasma, Indian J. of Pure Appl. Mat., 19, 2 (1988), 202-207. Zbl0637.76045
- SHARMA, R. C. - CHAND, TRILOK, Thermosolutal instability of compressible Hall plasma in porous medium, Astrophysics and Space Science, 155 (1989), 301-310. Zbl0671.76072
- SOLONNIKOV, V. A. - MULONE, G., On the solvability of some initial boundary value problems in magnetofluidmechanics with Hall and ion-slip effects, Rend. Mat. Acc. Lincei, 9, 6 (1995), 117-132. Zbl0834.76094MR1354225
- SUTTON, G. W. - SHERMAN, A., Engineering magnetohydrodynamics, Mc Graw Hill, New York, 1965.

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