Steady-State Flows of Thermal Viscous Incompressible Fluids With Convective-Radiation Effects

Abstract

The work deals with the coupled system constituted by the equations of motion and energy with nonlinear and nonlocal boundary conditions in order to describe the thermal flow motion of a class of non-Newtonian fluids and the convective-radiation balance, respectively. For the constitutive laws in an n-dimensional space (n = 2, 3), the stress tensor and the heat flux are considered related with the (p, q) coercivity parameters for p > 2n/(n + 1) and q > np/(p(n + 1) − n), respectively. The radiation character on the boundary condition presents an additional difficulty to the problem which already includes energy-dependent viscosity and conductivity behaviors. In the framework of nonstandard Sobolev spaces we prove the existence of steady-state solutions applying a fixed point argument. We also study the existence of a solution to a coupled system motivated by the buoyancy driven flows from geophysical and astrophysical models.

Navier–Stokes Equations with Shear-Thickening Viscosity. Regularity up to the Boundary

“…In particular, the same property holds by replacing ∇ by D. The above considerations, together with the nonnegativity of the integrands that appear on the left-hand side of inequality (5.11), allow us to pass to the limit by using Fatou's lemma. This yields 12) for each k = 3. Hence,…”

“…Without any claim of completeness, in addition to the articles already quoted, we would like to mention the following articles related to the problems treated in this paper: [1], [4], [9], [10], [11], [12], [14], [15], [24], [28], [29], [30], [31], [36], [37], [38], [39], and all the relevant references therein.…”

A (p−q) coupled system in elliptic nonlinear problems with nonstandard boundary conditions

“…In [10], the particular case of Navier-Stokes-Fourier problem includes the radiation behavior at the two-dimensional space. In [12], the Joule effect is neglected and Dirichlet boundary condition for the fluid velocity is taken into account, since we deal with the existence of a weak solution capturing the radiation behavior in the three-dimensional space and the shear thinning phenomenon exhibited by a broader class of non-Newtonian fluids. That is, we provide upper bounds to the exponent of the nonlinear convective-radiative term and lower bounds to the exponent relative to the principal nonlinear elliptic operator than the classical restriction, p > 3n/(n + 2), due to the existence of the convective term.…”

Boundary Regularity of Shear Thickening Flows

“…In the article [46] the problem (1.1) is studied provided p = n = 2 by the method of straightening the boundary. In [15][16][17][18][19] very interesting, physical meaningful, problems are deeply studied. In [34] it has been shown that the unique weak solution u of (1.4) with (1.2) satisfies u ∈ L ∞ (I, W 2,2+ (Ω)) provided p ∈ [2, 4) and n = 2.…”