The stationary Navier–Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet-type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suitable power law depending on $p\in (1,2)$ (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier–Stokes and the Stokes cases. Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments.

非牛顿不可压缩流体的稳态纳维-斯托克斯方程与稳态热方程耦合并服从狄利克雷型边界条件。粘度应该取决于温度，应力取决于应变，通过适当的幂律取决于$p\epsilon （1,2）$（剪切稀化情况）。对于这个问题，我们建立了弱解的存在，并证明了纳维-斯托克斯和斯托克斯情况的一些规律性结果。然后，通过有限元方案对卡罗幂律的后一种情况进行近似，并获得一些误差估计。然后通过一些二维数值实验验证这些估计。