Geodesic
The exact solutions of the problem of a viscous fluid flow in a cylindrical domain with varying radius
Russian journal of nonlinear dynamics, Tome 11 (2015) no. 1, pp. 89-97.

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In the frameworks of a class of exact solutions of the Navier–Stokes equations with linear dependence of part the speed components on one spatial variable the axisymmetrical nonselfsimilar flows of viscous fluid in the cylindrical area which radius changes over the time under some law calculated during the solution are considered. The problem is reduced to two-parametrical dynamic system. The qualitative and numerical analysis of the system allowed to allocate three areas on the phase plane corresponding to various limit sizes of a pipe radius: radius of a pipe and stream velocity tend to infinity for finite time, the area of a cross section of the cylinder tend to zero during a finite time span, radius of the tube infinitely long time approaches to a constant value, and the flow tend to the state of rest. For a case of ideal fluid flow the solution of the problem is obtained in the closed form and satisfying the slip condition.
Keywords: Navier–Stokes equations, pipe flow.
Mots-clés : exact solutions 
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Denis V. Knyazev; Ilia Yu. Kolpakov. The exact solutions of the problem of a viscous fluid flow in a cylindrical domain with varying radius. Russian journal of nonlinear dynamics, Tome 11 (2015) no. 1, pp. 89-97. http://geodesic.mathdoc.fr/item/ND_2015_11_1_a3/

[1] Uchida Sh., Aoki H., “Unsteady flows in a semi-infinite contracting or expanding pipe”, J. Fluid Mech., 82:2 (1977), 371–387 | DOI

[2] Skalak F. M., Wang C. Y., “On the unsteady squeezing of a viscous fluid from a tube”, J. Austral. Math. Soc. Ser. B. Appl. Math., 21:1 (1979), 65–74 | DOI

[3] Wang C. Y., “Arbitrary squeezing of fluid from a tube at low squeeze numbers”, Z. Angew. Math. Phys., 31:5 (1980), 620–627 | DOI

[4] Blyth M. G., Hall Ph., Papageorgiou D. T., “Chaotic flows in pulsating cylindrical tubes: A class of exact Navier–Stokes solutions”, J. Fluid Mech., 481 (2003), 187–213 | DOI

[5] Bellamy-Knights P. G., “An unsteady two-cell vortex solution of the Navier–Stokes equations”, J. Fluid Mech., 41:3 (1970), 673–687 | DOI

[6] Goldshtik M. A., Shtern B. H., Yavorskii H. I., Techeniya vyazkoi zhidkosti s paradoksalnymi svoistvami, Nauka, Novosibirsk, 1989, 336 pp.

[7] Andronov A. A., Leontovich E. A., Gordon I. I., Maier A. G., Kachestvennaya teoriya dinamicheskikh sistem vtorogo poryadka, Nauka, M., 1966, 568 pp.