| Abstract: |
| We consider the flow of a generalized non-Newtonian incompressible heat-conducting fluid in a bounded domain $\Omega \subset \mathbb{R}^3$, subject to Dirichlet boundary conditions for both velocity and temperature. While we assume homogeneous Dirichlet boundary conditions for the velocity, which prohibit the exchange of mass with the surroundings, we allow nonhomogeneous Dirichlet boundary conditions for the temperature, which enforces heat exchange with the surroundings.
The fluid obeys a power-law constitutive relation for the Cauchy stress, and the thermal conductivity is assumed to be a nonlinear function of the temperature, which may degenerate or become unbounded. No external forces are considered.
We study the long-time behaviour of the system. We show that the steady state of the system is nonlinearly stable within a suitably defined class of solutions. We also prove the existence of such solutions.
This is follow-up research to the work done together with A. Abbatiello and M. Bul\`\i\v cek. It is joint work with Aneta Wroblevska and Karol Hajduk. |
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