Stationary weak solutions for a class of non-Newtonian fluids with energy transfer

Consiglieri, Luisa

doi:10.1016/s0020-7462(96)00087-x

Cited by 22 publications

(16 citation statements)

References 2 publications

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“…The proof of 1 = τ : Du can be found in[9]. Arguing as in Theorem 4 we recognise that the weak limit e is the SOLA solution to (54).…”

mentioning

confidence: 86%

“…In [9], the problem is studied under the Fourier linear heat flux and a Fourier-type boundary condition for the scalar energy. In [10], the particular case of Navier-Stokes-Fourier problem includes the radiation behavior at the two-dimensional space.…”

Section: Definitionmentioning

confidence: 99%

“…The proof of u m u in W 1,p 0 (Ω; Γ 0 ) is classical on elliptic inequalities [21] and the convergence τ m · ∇u m τ · ∇u in L 1 (Ω) follows by similar arguments already used in [9]. Finally, the convergence, (τ m · n)u m (τ · n)u in L 1 (Γ ), is a consequence of the strong convergence u m → u in L p (Γ ) and τ m τ in L p (Γ ).…”

Section: Proof Of Theoremmentioning

confidence: 99%

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

Consiglieri

2008

Journal of Mathematical Analysis and Applications

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We state an abstract variational formulation to a coupled system constituted by an inequality and an equality motivated by the motion and energy equations, and the constitutive laws for the stress tensor and the heat flux, respectively, when non-Newtonian fluids are taken care of. Here the existence of a weak solution is proven via a fixed point argument to multivalued mappings. The nonstandard boundary conditions correspond to friction wall laws and energy transfer condition considered on a part of the boundary, whereas there exists the presence of the frictional work due to the friction of the fluid motion. We conclude by formulating the corresponding stationary heat conducting viscous incompressible flow problem and we establish an existence result.

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“…The proof of 1 = τ : Du can be found in[9]. Arguing as in Theorem 4 we recognise that the weak limit e is the SOLA solution to (54).…”

mentioning

confidence: 86%

Section: Definitionmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

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

Consiglieri

2008

Journal of Mathematical Analysis and Applications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Boundary Regularity of Shear Thickening Flows

Veiga

Kaplický

Růžička

2010

J. Math. Fluid Mech.

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This article is concerned with the global regularity of weak solutions to systems describing the flow of shear thickening fluids under the homogeneous Dirichlet boundary condition. The extra stress tensor is given by a power law ansatz with shear exponent p ≥ 2. We show that, if the data of the problem are smooth enough, the solution u of the steady generalized Stokes problem belongs to W 1,(np+2−p)/(n−2) (Ω). We use the method of tangential translations and reconstruct the regularity in the normal direction from the system, together with anisotropic embedding theorem. Corresponding results for the steady and unsteady generalized Navier-Stokes problem are also formulated. (2000). 35Q35, 35J65, 76D03. Mathematics Subject Classification

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“…In references [1,24,[50][51][52][53] the authors consider electrorheological fluids. Fluids with energy transfer, thermal viscous dependence, and related topics are treated in [14][15][16][17][18][19]. For particularly interesting results in two dimensions see [31][32][33]38] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On the global regularity of shear thinning flows in smooth domains

Veiga

2009

Journal of Mathematical Analysis and Applications

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In some recent papers we have been pursuing regularity results up to the boundary, in W 2,l (Ω) spaces for the velocity, and in W 1,l (Ω) spaces for the pressure, for fluid flows with shear dependent viscosity. To fix ideas, we assume the classical non-slip boundary condition. From the mathematical point of view it is appropriate to distinguish between the shear thickening case, p > 2, and the shear thinning case, p < 2, and between flat-boundaries and smooth, arbitrary, boundaries. The p < 2 non-flat boundary case is still open. The aim of this work is to extend to smooth boundaries the results proved in reference [H. Beirão da Veiga, On non-Newtonian p-fluids. The pseudo-plastic case, J. Math. Anal. Appl. 344 (1) (2008) 175-185]. This is done here by appealing to a quite general method, introduced in reference [H. Beirão da Veiga, On the LadyzhenskayaSmagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem, J. Eur. Math. Soc., in press], suitable for considering non-flat boundaries.

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Stationary weak solutions for a class of non-Newtonian fluids with energy transfer

Cited by 22 publications

References 2 publications

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

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

Boundary Regularity of Shear Thickening Flows

On the global regularity of shear thinning flows in smooth domains

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