Asymptotic Properties of the Plane Shear Thickening Fluids with Bounded Energy Integral
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Journal of Mathematical Fluid Mechanics
Asymptotic Properties of the Plane Shear Thickening Fluids with Bounded Energy Integral Shuai Li, Tao Wang and Wendong Wang Communicated by G. Seregin
Abstract. In this note we investigate the asymptotic behavior of plane shear thickening fluids around a bounded obstacle. Different from the Navier–Stokes case considered by Gilbarg–Weinberger in Gilbarg and Weinberger (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(2):381-404, 1978), where the good structure of the vorticity can be exploited and weighted energy estimates can be applied, we have to overcome the nonlinear term of high order. The decay estimates of the velocity was obtained by combining Point-wise Behavior Theorem in Galdi (An Introduction to the Mathematical Theory of the Navier–Stokes Equations Springer, New York, 2011) and Brezis–Gallouet inequality in Brezis and Gallouet (Nonlinear Anal. 4(4):677-681, 1980) together, which is independent of interest. Mathematics Subject Classification. 35Q30, 76D03, 76D07. Keywords. Asymptotic behavior, Shear thickening fluids, Generalized Navier–Stokes equations.
1. Introduction As Ladyzhenskaya suggests in her monograph in [23], it is interesting to investigate “new equations for the description of the motion of viscous incompressible fluids”, which roughly speaking means to consider viscosity coefficients, which depend on the modulus of the symmetric gradient, 1 1 ε(u) = (Du + (Du)T ) = (∂i uk + ∂k ui )1≤i,k≤2 2 2 of the velocity field u, for example, in a monotonically increasing way (shear thickening case). In this note we will consider this problem in a very special situation restricting ourselves to stationary flows through an exterior domain Ω ⊂ R2 with smooth boundary ∂Ω. More precisely, consider the solution u : Ω → R2 , π : Ω → R of the following system −div[T (ε(u))] + uk ∂k u + Dπ = 0, in Ω, (1.1) div u = 0, in Ω, where the Ω = R2 \ BR0 (0) with R0 > 0. More details on viscous incompressible flow, we refer to [9–11,13,15,16,23] and the references therein. The system (1.1) describes the stationary flow of an incompressible generalized Newtonian fluid, where u is the velocity field, π is the pressure function, uk ∂k u is the convective term, and T represents the stress deviator tensor. We assume that the stress tensor T is the gradient of a potential H : S 2×2 → R defined on the space S 2×2 of all symmetric 2 × 2 matrices of the following form H(ε) = h(|ε|), 3
where h is a nonnegative function of class C . Thus
T (ε) = DH(ε) = μ(|ε|)ε, μ(t) = 0123456789().: V,-vol
h (t) . t
(1.2)
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Note that the Navier–Stokes equations for incompressible Newtonian fluids follow from the system (1.1) if μ is a constant. If μ is not a constant, it means that the viscosity coefficient depends on ε, and system (1.1) describes the motion of continuous media of generalized Newtonian fluids. As in [12], assume that the potential h satisfies the follow conditions: h is strictly increasing and convex together with h (0) > 0 and lim h(t) t =0 ; t→0
(A1)
there
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