We study a nonlinear weighted least-squares finite element method for the Navier- Stokes equations governing non-Newtonian fluid flows by using the Carreau- Yasuda model. The Carreau-Yasuda model is used to describe the shear-thinning behavior of blood. We prove that the least-squares approximation converges to linearized solutions of the non-Newtonian model at the optimal rate. By using continuous piecewise linear finite element spaces for all variables and by appropriately adjusting the nonlinear weighting function, we obtain optimal L2-norm error convergence rates in all variables. Numerical results are given for a Carreau fluid in the 4-to-1 contraction problem, revealing the shear-thinning behavior. The physical parameter effects are also investigated.

There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other is the Bondi mass loss formula, which are both global in nature. The positivity of the quasi-local mass can potentially lead to a local description at null infinity. This is confirmed for the Vaidya spacetime in this note. We study the Wang-Yau quasi-local mass on surfaces of fixed size at the null infinity of the Vaidya spacetime. The optimal embedding equation is solved explicitly and the quasi-local mass is evaluated in terms of the mass aspect function of the Vaidya spacetime.

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Based on the fully compressible Navier-Stokes equations, the linear stability of thermal convection in rapidly rotating spherical shells of various radius ratios eta is studied for a wide range of Taylor number Ta, Prandtl number Pr and the number of density scale height N-rho. Besides the classical inertial mode and columnar mode, which are widely studied by the Boussinesq approximation and anelastic approximation, the quasi-geostrophic compressible mode is also identified in a wide range of N-rho and Pr for all eta considered, and this mode mainly occurs in the convection with relatively small Pr and large N.. The instability processes are classified into five categories. In general, for the specified wavenumber m, the parameter space (Pr, N-rho) of the fifth category, in which the base state loses stability via the quasi-geostrophic compressible mode and remains unstable, shrinks as eta increases. The asymptotic scaling behaviours of the critical Rayleigh numbers Ra-c and corresponding wavenumbers m(c) to Ta are found at different eta for the same instability mode. As eta increases, the flow stability is strengthened. Furthermore, the linearized perturbation equations and Reynolds-Orr equation are employed to quantitatively analyse the mechanical mechanisms and flow instability mechanisms of different modes. In the quasi-geostrophic compressible mode, the time-derivative term of disturbance density in the continuity equation and the diffusion term of disturbance temperature in the energy equation are found to be critical, while in the columnar and inertial modes, they can generally be ignored. Because the time-derivative term of the disturbance density in the continuity equation cannot be ignored, the anelastic approximation fails to capture the instability mode in the small-Pr and large-N-rho system, where convection onset is dominated by the quasi-geostrophic compressible mode. However, all the modes are primarily governed by the balance between the Coriolis force and the pressure gradient, based on the momentum equation. Physically, the most important difference between the quasi-geostrophic compressible mode and the columnar mode is the role played by the disturbance pressure. The disturbance pressure performs negative work for the former mode, which appears to stabilize the flow, while it destabilizes the flow for the latter mode. As eta increases, in the former mode the relative work performed by the disturbance pressure increases and in the latter mode decreases.