Non-homogeneous generalized Newtonian fluids (original) (raw)
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Strong Solutions for Generalized Newtonian Fluids
Journal of Mathematical Fluid Mechanics, 2005
We consider the motion of a generalized Newtonian fluid, where the extra stress tensor is induced by a potential with p-structure (p = 2 corresponds to the Newtonian case). We focus on the three dimensional case with periodic boundary conditions and extend the existence result for strong solutions for small times from p > 5 3 (see ) to p > 7 5 . Moreover, for 7 5 < p ≤ 2 we improve the regularity of the velocity field and show that u ∈ C([0, T ], W
Nonlinear Analysis: Real World Applications 45(2):704-720, 2019
We investigate the steady-state equations of motion of the generalized Newtonian fluid in a bounded domain Ω ⊂ R N , when N = 2 or N = 3. Applying the tools of nonlinear analysis (Smale's theorem, theory of Fredholm operators, etc.), we show that if the dynamic stress tensor has the 2-structure then the solution set is finite and the solutions are C 1-functions of the external volume force f for generic f. We also derive a series of properties of related operators in the case of a more general p-structure, show that the solution set is compact if p > 3N/(N + 2) and explain why the same approach as in the case p = 2 cannot be applied if p is different than 2.
St Petersburg Mathematical Journal
The authors consider the following extension of the stationary Navier-Stokes equations -div(T(·,ε(u)))+∇π+[∇u]u=g, posed in a bounded and Lipschitz domain Ω of ℝ n , n=2 or 3. Here ε(u) is the symmetric gradient of u and T is the gradient with respect to its second argument of a potential f(x,ε) which satisfies some growth conditions on its second order partial derivatives D ε 2 and D x D ε . The volumic forces g are supposed to belong to L ∞ (Ω,ℝ n ). Homogeneous Dirichlet boundary conditions are imposed on the boundary ∂Ω. The main result proves the existence of a solution (v,π)∈W o p 1 ∩W t,loc 2 (Ω,ℝ n )×W s,loc 1 (Ω), where t and s, greater than 1, are linked to the above growth conditions, which involve p and some q ¯. This result is proved assuming that p is large enough and q ¯ is small enough. For the proof, the authors use their previous results which consider various situations for f. The key argument is the proof of some estimates on the weak solution (in a variational f...
Generalized Newtonian fluids in moving domains
Journal of Differential Equations, 2018
In this paper we prove the existence of weak solutions for the equations describing the unsteady motion of an incompressible, viscous and homogeneous generalized Newtonian fluid in a non-cylindrical domain t∈I {t} × (t).
Well-posedness for a class of non-Newtonian fluids with general growth conditions
Nonlocal and Abstract Parabolic Equations and their Applications, 2009
The paper concerns uniqueness of weak solutions to non-Newtonian fluids with nonstandard growth conditions for the Cauchy stress tensor. We recall the results on existence of weak solutions and additionally provide the proof of existence of measure-valued solutions. Motivated by the fluids of strongly inhomogeneous behaviour and having the property of rapid shear thickening we observe that the described situation cannot be captured by power-lawtype rheology. We describe the growth conditions with the help of general x-dependent convex functions. This formulation yields the existence of solutions in generalized Orlicz spaces. These considerations are motivated by e.g. electrorheological fluids, magnetorheological fluids, and shear thickening fluids.
Existence of Strong Solutions for Incompressible Fluids with Shear Dependent Viscosities
2010
Certain rheological behavior of non-Newtonian fluids in engineering sciences is often modeled by a power law ansatz with p ∈ (1, 2]. In the present paper the local in time existence of strong solutions is studied. The main result includes also the degenerate case (δ = 0) of the extra stress tensor and thus improves previous results of [L. Diening and M. Růžička,
Existence Results for Steady Flows of Quasi-Newtonian Fluids Using Weak Monotonicity
Journal of Mathematical Fluid Mechanics, 2005
This work is concerned with the study of steady flows of an incompressible quasi-Newtonian fluid, with viscosity depending on the second invariant of the rate of deformation tensor, in bounded domains. We first establish a result of existence of measure-valued solutions. Next, we prove that under some natural monotonicity conditions, these solutions are weak.
Computers & Mathematics with Applications, 2007
We consider the strong solution of an initial boundary value problem for a system of evolution equations describing the flow of a generalized Newtonian fluid of power law type. For a rather large scale of growth rates we prove local initial regularity results such as higher integrability of the pressure function or the existence of the second spatial derivatives of the velocity field. Acknowledgement: The second author's research was supported by the Humboldt foundation.
On non-Newtonian Fluids with Energy Transfer
2009
The model combining incompressible Navier-Stokes' equation in a non-Newtonian p-power-law modification and the nonlinear heat equation is considered. Existence of its (very) weak solutions is proved for p > 11/5 under mild assumptions of the temperature-dependent stress tensor by careful successive limit passage in a Galerkin approximation. . 35Q35, 76A05, 80A20.