The Boundary Effects and Zero Angular and Micro-rotational Viscosities Limits of the Micropolar Fluid Equations
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The Boundary Effects and Zero Angular and Micro-rotational Viscosities Limits of the Micropolar Fluid Equations Xiuli Zhu1
· Zhonghai Xu1 · Huapeng Li1
Received: 17 August 2015 / Accepted: 9 September 2016 © Springer Science+Business Media Dordrecht 2016
Abstract In this paper, we consider an initial-value problem to the two-dimensional incompressible micropolar fluid equations. Our main purpose is to study the boundary layer effects as the angular and micro-rotational viscosities go to zero. It is also shown that the boundary layer thickness is of the order O(γ β ) with (0 < β < 23 ). In contrast with Chen et al. (Z. Angew. Math. Phys. 65:687–710, 2014), the BL-thickness we got is thinner than that in Chen et al. (Z. Angew. Math. Phys. 65:687–710, 2014). In addition, the convergence rates are also improved. Keywords Incompressible micropolar fluid · Convergence rates · Boundary layer
1 Introduction The two-dimensional motion of incompressible micropolar fluids is governed by (cf. [3, 4, 10]) ut + u · ∇u + ∇π = (ν + ζ )u + 2ζ ∇ ⊥ w,
(1.1a)
wt + u · ∇w = γ w − 2ζ ∇ ⊥ · u − 4ζ w,
(1.1b)
∇ ·u=0
(1.1c)
This work was partially supported by the Science and Technology Developing Project of Jilin Province of China (Grant no. 20150101002JC), the Science and Technology Developing Project of Jilin city Jilin Province of China (Grant no. 20156405).
B X. Zhu
[email protected]
B Z. Xu
[email protected] H. Li [email protected]
1
College of Science, Northeast Dianli University, Jilin, Jilin 132013, P.R. China
X. Zhu et al.
in the half plane R2+ = {(x, y) | x > 0, −∞ < y < +∞}, with the initial-boundary conditions u(x, y, 0) = u0 (x, y), u(0, y, t) = 0,
w(x, y, 0) = w0 (x, y) (x, y) ∈ R2+ ,
w(0, y, t) = w1 (y),
(y, t) ∈ R × (0, T ),
(1.2a) (1.2b)
where u = (u1 , u2 ), w and π are the velocity vector field, micro-rotational velocity, and pressure, respectively. ν is the Newtonian kinetic viscosity. γ the angular viscosity, and ζ the dynamic micro-rotation viscosity. The modern theory of micropolar fluid dynamics began over 40 years ago in 1966, when Eringe published his pioneering works on the micropolar fluid motion equations [6]. Physically, micropolar fluid may represent fluids that consist of rigid, randomly oriented (or spherical particles) suspended in a viscous medium, where the deformation of fluid particles is ignored. It can describe many phenomena that appear in a large number of complex fluids such as the suspensions, animal blood, liquid crystals which cannot be characterized appropriately by the Navier–Stokes system, and that it is important to the scientists working with the hydrodynamic-fluid problems and phenomena. When the micro-rotation effects are neglected or w = 0, the micropolar fluid flows reduces to the incompressible Navier–Stokes equations (see, for example, [8]). There is a lot of literature devoted to the mathematical theory of the micropolar fluid system because of its importance in mathematics and physics, see, for example, [2, 4, 5, 13, 15, 16] and the references therein.
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