Tensor-nonlinear shear flows: Material functions and the diffusion-vortex solutions

Dimitri V. Georgievskii

Russian journal of nonlinear dynamics, Tome 7 (2011) no. 3, pp. 451-463.

Résumé

This work deals with tensor-nonlinear constitutive relations connecting the deviators of stress tensor and strain rate tensor in incompressible isotropic media which are called in continuum mechanics as Reiner–Rivlin fluids. The connections of quadratic and cubic invariants of two tensors, where two material functions involve, are presented. The main attention is given to one-dimensional shear flows in various curvilinear coordinate systems. The scheme of obtaining of the material functions for shear on the basis of the steady Poiseuille flow in a plane layer is described. The self-similar solutions corresponding to the generalized diffusion of vortex layer both in plane and axially symmetric cases are derived.

Détail

Keywords: tensor nonlinearity, material function, Reiner–Rivlin fluid, shear, vortex layer.
Mots-clés : invariant, constitutive relation, diffusion of vortex

Bibliographie

[1] Reiner M., “Rheology”, Handbuch der Physik: Bd. 6: Elastizität und Plastizität, ed. Hrsg. von S. Flügge, Springer, Berlin–Göttingen–Heidelberg, 1958, 434–550 | MR

[2] Malinin N. I., “K voprosu ob effekte Vaissenberga”, Kolloidnyi zhurnal, 22:2 (1960), 201

[3] Rivlin R. S., Ericksen J. L., “Stress-deformation relations for isotropic materials”, J. Rational Mech. Anal., 4:2 (1955), 323–425 | MR | Zbl

[4] Rivlin R. S., “The formulation of constitutive equations in continuum physics: Part 1”, Arch. Ration. Mech. Anal., 4:2 (1960), 129–144 | MR

[5] Nikolaevskii V.N., “Mekhanika geomaterialov. Uslozhnennye modeli”, Itogi nauki i tekhniki. Ser. Mekhanika deformiruemogo tverdogo tela, 19, VINITI, M., 1987, 148–176

[6] Wilkinson W. L., Non-Newtonian fluids: Fluid mechanics, mixing and heat transfer, Pergamon Press, London, 1960, 138 pp. | MR

[7] Astarita G., Marrucci G., Principles of non-Newtonian fluid mechanics, McGraw Hill, London, 1974, 289 pp.

[8] Bibik E. E., Reologiya dispersnykh sistem, LGU, L., 1981, 172 pp.

[9] Litvinov V. G., Dvizhenie nelineino-vyazkoi zhidkosti, Nauka, M., 1982, 374 pp.

[10] Brutyan M. A., Krapivskii P. L., “Gidrodinamika nenyutonovskikh zhidkostei”, Itogi nauki i tekhniki. Ser. Kompleksnye i spetsialnye razdely mekhaniki, 4, VINITI, M., 1991, 3–98

[11] Andreev V. K., Bublik V. V., Bytev V. O., Simmetrii neklassicheskikh modelei gidrodinamiki, Nauka, Novosibirsk, 2003, 352 pp.

[12] Skulskii O. I., Aristov S. N., Mekhanika anomalno vyazkikh zhidkostei, RKhD, M.–Izhevsk, 2004, 156 pp.

[13] Miroshnichenko D. S., Vliyanie nenyutonovskikh svoistv dispersionnykh zhidkostei na reologicheskoe povedenie razbavlennykh suspenzii, Diss. $\dots$ kand. fiz.-mat. nauk. 01.02.05, KNU im. T. G. Shevchenko, Kiev, 2000, 129 pp.

[14] Vorotnikov D. A., “O dvizhenii nelineino-vyazkoi zhidkosti v $R^n$”, Vestn. Voronezhskogo GU. Ser. Fizika, matematika, 2002, no. 1, 102–120

[15] Eskin L. D., “Uravneniya, opisyvayuschie dinamiku nenyutonovskoi zhidkosti s reologicheskim zakonom Reinera–Rivlina: 1. Gruppovoi analiz”, Izv. vuzov. Matematika, 2004, no. 3, 64–72

[16] Khazem Ali Attia, “Chislennoe issledovanie techeniya i teplootdachi k poristomu disku, vraschayuschemusya v zhidkosti Reinera–Rivlina”, PMTF, 46:1 (2005), 85–95 | Zbl

[17] Nigmatulin R. I., Nikolaevskii V. N., “Diffuziya vikhrya i sokhranenie momenta kolichestva dvizheniya v dinamike nepolyarnykh zhidkostei”, PMM, 34:2 (1970), 318–323 | Zbl

[18] Georgievskii D. V., “Avtomodelnye resheniya v zadache ob obobschennoi diffuzii vikhrya”, Izv. RAN. MZhG, 2007, no. 2, 3–12

[19] Georgievskii D. V., “Obobschennaya diffuziya vikhrya: avtomodelnost i zadacha Stefana”, Sovrem. matemat. i eë prilozh., 62 (2009), 28–46

[20] Lokhin V. V., Sedov L. I., “Nelineinye tenzornye funktsii ot neskolkikh tenzornykh argumentov”, PMM, 27:3 (1963), 393–417 | MR | Zbl

[21] Spencer A. J. M., “Theory of invariants”, Continuum physics, v. 1, ed. A. C. Eringen, Academic Press, New York–London, 1971, 239–353 | MR

[22] Pobedrya B. E., Lektsii po tenzornomu analizu, MGU, M., 1986, 263 pp.

[23] Georgievskii D. V., “Tenzorno-nelineinye effekty pri izotermicheskom deformirovanii sploshnykh sred”, Uspekhi mekhaniki, 1:2 (2002), 150–176

[24] Nikabadze M. U., Nekotorye voprosy tenzornogo ischisleniya, V 2-kh chastyakh, TsPI MGU, M., 2007, 86, 94 pp.

[25] Dimitrienko Yu. I., Mekhanika sploshnoi sredy: T. 1: Tenzornyi analiz, MGTU, M., 2011, 464 pp.

[26] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady. Elementarnye funktsii, Nauka, M., 1981, 800 pp.

[27] Barenblatt G. I., Podobie, avtomodelnost, promezhutochnaya asimptotika, Gidrometeoizdat, L., 1982, 256 pp.

[28] Green A. E., Rivlin R. S., “Steady flow of non-Newtonian fluids through tubes”, Quart. Appl. Math., 14:3 (1956), 299–308 | MR | Zbl