Abstract
Several problems of motion of a viscous incompressible fluid with a point source in the flow region are considered. The corresponding initial-boundary-value problems for the Navier-Stokes equations have no solutions in the standard class of functions because the flow velocity field contains an infinite Dirichlet integral. Problem regularization allows one to prove its solvability under certain constraints on the initial data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Russo and A. Tartaglione, “On the Singular Solutions of the Stationary Navier–Stokes Problem,” Lithuan. Math. J. 53 (4), 423–437 (2013).
V. V. Pukhnachev, “Singular Solutions of Navier–Stokes Equations,” “Advances in Mathematical Analysis of Partial Differential Equations. Dedicated to the Memory of O. A. Ladyzhenskaya,” AMS Transl. Ser. 2, 232, 193–218 (2014).
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow (Nauka, Moscow, 1970; Gordon and Breach, New York, 1969).
G. B. Jeffery, “The Two-Dimensional Steady Motion of a Viscous Fluid,” Philos. Mag. Ser. 6, 29 (172), 455–465 (1915).
G. Hamel, “Spiralformige Bewegungen zahen Flussigkeiten,” Jahresber. Deutsch. Math.-Verein. 25, 34–60 (1917).
L. D. Akulenko, D. V. Georgievskii, S. A. Kumakshev, and S. V. Nestereov, “State of the Art of the Problem of the Viscous Fluid Flow in Converging Channels,” in Advanced Problems of Mechanics, Mechanics of the Fluid, Gas, and Plasma (Nauka, Moscow, 2008), pp. 144–169 [in Russian].
J. Serrin, “On the Mathematical Basis for Prandtl’s Boundary Layer Theory: an Example,” Arch. Rational Mech. Anal. 28, 217–225 (1968).
V. V. Pukhnachev, “Invariant Solutions of the Navier–Stokes Equations that Describe Motions with Free Boundaries,” Dokl. Akad. Nauk SSSR, 202 (2), 302–305 (1972).
M. A. Goldshtik, V. N. Shtern, and N. I. Yavorskii, Viscous Flows with Paradox Properties (Nauka, Novosibirsk, 1989) [in Russian].
L. Rivkind and V. A. Solonnikov, “Jeffery–Hamel Asymptotics for Steady State Navier–Stokes Flow in Domains with Sector-Like Outlet to Infinity,” J. Math. Fluid Mech. 2 (4), 324–352 (2000).
V. Shverak, “On the Landau Solutions of the Navier–Stokes Equations,” Probl. Mat. Anal., No. 61, 175–191 (2011).
A. V. Shapeev, “Unsteady Self-Similar Flow of a Viscous Incompressible Fluid in a Plane Diffuser,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 1, 41–46 (2004).
H. Bateman and A. Erdelyi, Higher Transcendental Functions. 1. The Gamma Function. The Hypergeometric Functions. Legendre Functions (McGraw-Hill, 1953).
J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Noordhoff, Leyden, Netherlands, 1965).
A. V. Shapeev, “Viscous Incompressible Flows in Sectors and Cones,” Candidate’s Dissertation (Novosibirsk, 2009) [in Russian].
V. V. Pukhnachev, “Point Vortex in a Viscous Incompressible Fluid,” Prikl. Mekh. Tekh. Fiz. 55 (2), 180–187 (2014) [J. Appl. Mech. Tech. Phys. 55 (2), 345–351 (2014)].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.V. Pukhnachev.
Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, 2019, Vol. 60, No. 2, pp. 19–30, March–April, 2019.
Rights and permissions
About this article
Cite this article
Pukhnachev, V.V. Problem of a Point Source. J Appl Mech Tech Phy 60, 200–210 (2019). https://doi.org/10.1134/S0021894419020020
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0021894419020020