Affiliations: [a] Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ-115 67 Praha 1, Czech Republic. E-mail: feireisl@math.cas.cz | [b] Department of Mathematics, Nanjing University, 22 Hankou Road, Gulou District, 210093 Nanjing, China. E-mails: luyong@nju.edu.cn, sunyz@nju.edu.cn
Abstract: We consider a non–homogeneous incompressible and heat conducting fluid confined to a 3D domain perforated by tiny holes. The ratio of the diameter of the holes and their mutual distance is critical, the former being equal to ε3, the latter proportional to ε, where ε is a small parameter. We identify the asymptotic limit for ε→0, in which the momentum equation contains a friction term of Brinkman type determined uniquely by the viscosity and geometric properties of the perforation. Besides the inhomogeneity of the fluid, we allow the viscosity and the heat conductivity coefficient to depend on the temperature, where the latter is determined via the Fourier law with homogenized (oscillatory) heat conductivity coefficient that is different for the fluid and the solid holes. To the best of our knowledge, this is the first result in the critical case for the inhomogenous heat–conducting fluid.
