We study the asymptotic behavior of solutions of the problem that describes small motions of a viscous incompressible fluid filling a domain Ω with a large number of suspended small solid interacting particles concentrated in a small neighborhood of a certain smooth surface Γ ⊂ Ω. We prove that, under certain conditions, the limit of these solutions satisfies the original equations in the domain Ω\Γ and some averaged boundary conditions (conjugation conditions) on Γ.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. A. Galitsin and V. A. Trotsenko, “On the calculation of frequencies and apparent masses of a fluid in a rectangular container with separators in the transverse plane of its symmetry,” Prykl. Hidromekh., No. 1, 20–27 (2000).
V. A. Trotsenko, “On the influence of ring separators on the efficiency of damping of wave motions of a fluid in a vessel,” Dopov. Nats. Akad. Nauk Ukr., No. 6, 50–56 (2005).
G. N. Mikishev, Experimental Methods in Dynamics of Space Vehicles [in Russian], Mashinostroenie, Moscow (1978).
D. I. Borisov, “Small motions of an ideal fluid in a vessel with perforated separators,” Visn. Kharkiv. Nats. Univ., Ser. Mat., Prikl. Mat., Mekh., No. 749, 86–95 (2006).
L. V. Berlyand and E. Ya. Khruslov, “Homogenized non-Newtonian viscoelastic rheology of a suspension of interacting particles in a viscous Newtonian fluid,” SIAM J. Appl. Math., 64, No. 3, 1002–1034 (2004).
M. A. Berezhnyi, “The asymptotic behaviour of viscous incompressible fluid small oscillations with solid interacting particles,” J. Math. Phys., Anal., Geometry, 3, No. 2, 135–156 (2007).
M. Berezhnyi, L. Berlyand, and E. Khruslov, “The homogenized model of complex fluids,” NHM, Networks Heterogeneous Media, 3, No. 4 (2008).
R. G. Larson, The Structure and the Rheology of Complex Fluids, Oxford University, Oxford–New York (1999).
J. N. Pernin and E. Jacquet, “Elasticity and viscoelasticity in highly heterogeneous composite medium: threshold phenomenon and homogenization,” Int. J. Eng. Sci., 39, 1655–1689 (2001).
W. B. Russel, D. A. Saville, and W. R. Schowalter, Colloidal Dispersions, Cambridge University, Cambridge (1989).
J. Sanchez-Hubert, “Asymptotic study of the macroscopic behaviour of solid–liquid mixture,” Math. Meth. Appl. Sci., 2, 1–11 (1980).
O. A. Oleinic, A. S. Shamaev, and G. A. Iosif’yan, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam (1982).
M. Berezhnyy and L. Berlyand, “Continuum limit for three-dimensional mass-spring networks and discrete Korn’s inequality,” J. Mech. Phys. Solids, 54, No. 3, 635–669 (2006).
A. Bendali, J. M. Dominguez, and S. Gallic, “A variational approach for the vector potential formulation of the Stokes and Navier–Stokes problems in three-dimensional domains,” J. Math. Anal. Appl., 107, 537–560 (1985).
A. I. Markushevich, Theory of Analytic Functions [in Russian], Vol. 2, Nauka, Moscow (1968).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 3, pp. 302–321, March, 2009.
Rights and permissions
About this article
Cite this article
Berezhnoi, M.A. Small oscillations of a viscous incompressible fluid with a large number of small interacting particles in the case of their surface distribution. Ukr Math J 61, 361–382 (2009). https://doi.org/10.1007/s11253-009-0219-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-009-0219-8