Article

ANNALI DELL'UNIVERSITA' DI FERRARA

, Volume 52, Issue 1, pp 19-36

First online:

On stationary thermo-rheological viscous flows

  • Stanislav N. AntontsevAffiliated withDepartamento de Matemática, Universidade da Beira Interior, Rua Marquês d’Ávila e Bolama, 6201-001 Covilhã, Portugal, Tel.: 00351-275 319 757, Fax: 00351-275 329 972
  • , José F. RodriguesAffiliated withCMUC and Universidade de Lisboa/CMAF, Av. Prof. Gama Pinto 2, 1649-003 Lisboa

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Abstract

We study the system of equations describing a stationary thermoconvective flow of a non-Newtonian fluid. We assume that the stress tensor S has the form

\(\displaystyle \mathbf{S}=-P\mathbf{I}+\left( \mu (\theta )+\tau (\theta ){|\mathbf{D(u)}|}^{p(\theta )-2}\right) {\mathbf{D(u)}}, \)

where u is the vector velocity, P is the pressure, θ is the temperature and μ ,p and τ are the given coefficients depending on the temperature. D and I are respectively the rate of strain tensor and the unit tensor. We prove the existence of a weak solution under general assumptions and the uniqueness under smallness conditions.

Keywords: Non-Newtonian fluids, Nonlinear thermal diffusion equations, Heat and mass transfer

Mathematics Subject Classification (2000): 76A05, 76D07, 76E30, 35G15