Abstract
We consider the motion of compressible viscous fluids around a rotating rigid obstacle when the velocity at infinity is non zero and parallel to the axis of rotation. We prove the existence of weak solution.
Similar content being viewed by others
Notes
We shall give more details on this procedure when treating the limit \(R\rightarrow \infty \) in the next section.
References
Bresch, D., Desjardins, B., Gérard-Varet, D.: Rotating fluids in a cylinder. DCDS A 11, 47–82 (2004)
Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier. E.: Mathematical Geophysics, vol. 32 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, Oxford (2006)
Deuring, P., Kračmar, S., Nečasová, Š.: On pointwise decay of linearized stationary incompressible viscous flow around rotating and translating bodies. SIAM J. Math. Anal. 43, 705–738 (2011)
Farwig, R.: An \(L^{q}\)-analysis of viscous fluid flow past a rotating obstacle. Tôhoku Math. J. 920(58), no. 1, 129–147 (2006)
Farwig, R., Hishida, T., Müller, D.: \(L^q\)-theory of a singular “winding” integral operator arising from fluid dynamics. Pacific J. Math. 215, 297–312 (2004)
Farwig, R., Krbec, M., Nečasová, Š.: A weighted \(L^q\) approach to Oseen flow around a rotating body. Math. Methods Appl. Sci. 31(5), 551–574 (2008)
Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004)
Feireisl, E.: Low Mach number limits of compressible rotating fluids. J. Math. Fluid Mech. 14(1), 618 (2012)
Feireisl, E., Gallagher, I., Novotný, A.: A singular limit for compressible rotating fluids. SIAM J. Math. Anal. 44(1), 192–205 (2012)
Feireisl, E., Gallagher, I., Gerard-Varet, D., Novotný, A.: Multi-scale analysis of compressible viscous and rotating fluids. Commun. Math. Phys. 314, 641–670 (2012)
Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Birkhäuser, Basel (2009)
Feireisl, E., Kreml, O., Nečasová, Š., Neustupa, J., Stebel. J.: Incompressible limits of fluids excited by moving boundaries. SIAM J. Math. Anal. 46, 1456–1471 (2014)
Feireisl, E., Kreml, O., Nečasová, Š., Neustupa, J., Stebel, J.: Weak solutions to the barotropic Navier–Stokes system with slip boundary conditions in time dependent domains. J. Differ. Equ. 254(2013), 125–140 (2013)
Galdi, G.P.: On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications. In: Friedlander, S., Serre, D. (eds) Handbook of Mathematical Fluid Dynamics, Vol. 1, pp. 653–791. Elsevier, New York (2002)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems. Springer, New York, Berlin, Heidelberg (2012)
Galdi, G.P., Silvestre, A.L.: Further results on steady-state flow of a Navier–Stokes liquid around a rigid body. Existence of the wake. RIMS Kôkyûroku Bessatsu B1, 108–127 (2007)
Galdi, G.P., Kyed, M.: Steady-state Navier–Stokes flows past a rotating body: Leray solutions are physically reasonable. Arch. Rational Mech. Anal. 200, 21–58 (2011)
Geissert, M., Heck, H., Hieber, M.: \(L^{p}\) theory of the Navier–Stokes flow in the exterior of a moving or rotating obstacle. J. Reine Angew. Math. 596, 45–62 (2006)
Hishida, T.: An existence theorem for the Navier–Stokes flow in the exterior of a rotating obstacle. Arch. Rational Mech. Anal. 150, 307–348 (1999)
Jesslé, D., Jin, B.J., Novotný, A.: Navier–Stokes–Fourier system, weak solutions, relative entropy, weak-strong uniqueness. SIAM J. Math. Anal. 45(3), 1907–1951 (2013)
Kračmar, S., Nečasová, Š., Penel, P.: Anisotropic \(L^{2}\)-estimates of weak solutions to the stationary Oseen-type equations in 3D-exterior domain for a rotating body. J. Math. Soc. Jpn. 62(1), 239–268 (2010)
Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)
Lions, P.-L.: Mathematical Topics in Fluid Dynamics. Compressible Models, vol. 2. Oxford Science Publication, Oxford (1998)
Novo, S.: Compressible Navier–Stokes model with inflow-outflow boundary conditions. J. Math. Fluid Mech. 7, 485–514 (2005)
Novotný, A., Straškraba, I.: Introduction to the Mathematical Theory of Compressible Flow. Oxford University Press, Oxford (2004)
Sýkora, P.: Motion of a compressible fluid in a time dependent domain. Master thesis, Charles University of Prague (2012) (in Czech)
Author information
Authors and Affiliations
Corresponding author
Additional information
S. Kracmar acknowledges the support of the Czech Science Foundation (GAČR) project P201/11/1304. S. Nečasová acknowledges the support of the Czech Science Foundation (GAČR) project P201/11/1304 in the general framework of RVO 67985840. The work was supported by the MODTERCOM project within the APEX programme of the region Provence-Alpes-Côte d’Azur, France.
Rights and permissions
About this article
Cite this article
Kračmar, S., Nečasová, Š. & Novotný, A. The motion of a compressible viscous fluid around rotating body. Ann Univ Ferrara 60, 189–208 (2014). https://doi.org/10.1007/s11565-014-0212-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11565-014-0212-5