Analysis of Micropolar Fluids: Existence of Potential Microflow Solutions, Nearby Global Well-Posedness, and Asymptotic
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Analysis of Micropolar Fluids: Existence of Potential Microflow Solutions, Nearby Global Well-Posedness, and Asymptotic Stability Noah Stevenson1 · Ian Tice1
Received: 23 January 2020 / Accepted: 24 September 2020 / Published online: 30 September 2020 © Springer Nature B.V. 2020
Abstract In this paper we concern ourselves with an incompressible, viscous, isotropic, and periodic micropolar fluid. We find that in the absence of forcing and microtorquing there exists an infinite family of well-behaved solutions, which we call potential microflows, in which the fluid velocity vanishes identically, but the angular velocity of the microstructure is conservative and obeys a linear parabolic system. We then prove that nearby each potential microflow, the nonlinear equations of motion are well-posed globally-in-time, and solutions are stable. Finally, we prove that in the absence of force and microtorque, solutions decay exponentially, and in the presence of force and microtorque obeying certain conditions, solutions have quantifiable decay rates. Mathematics Subject Classification (2010) Primary 76A05 · 35B35 · 76D03 · Secondary 35B04 · 74A60 · 35K40 Keywords Micropolar fluids · Potential microflows · Stability
1 Introduction 1.1 Overview The theory of micropolar fluids, first introduced by Eringen [6] to describe the mechanics of a microcontinuum, is an extension of the classical theory of fluid mechanics. Among the novelties of the former theory are the effects of microstructure on the fluid. In essence, the angular velocity and rotational inertia of the microstructure are accounted for at each I. Tice was supported by an NSF CAREER Grant (DMS #1653161). N. Stevenson was supported by the summer research support provided by this grant.
B I. Tice
[email protected] N. Stevenson [email protected]
1
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA
904
N. Stevenson, I. Tice
point in the fluid, and the dynamics of these quantities couples to the bulk dynamics. In the case of a viscous and incompressible fluid, the system is governed by a variant of the Navier-Stokes equations, coupled to dissipative evolution equations for the microangular momentum. Micropolar fluids are common, and examples include: blood [3, 19, 22], certain lubricants [1, 4, 20, 24], liquid crystals [6, 11, 14], and ferrofluids [21]. In this paper we shall concern ourselves with the viscous and incompressible micropolar model. For the sake of simplicity, our fluids are taken to be spatially periodic, and our microstructure is assumed to be isotropic and homogeneous. Thus, our micropolar fluid occupies the three dimensional flat torus, T3 = R3 /Z3 , and the microstructure has no preferred direction of rotation nor spatial dependence modulo proper rotation. The state of our micropolar fluid is described by three variables related via a system of nonlinear partial differential equations. The velocity and microangular velocity are a pair of evolving vector fields u, ω : R+ × T3 → R3 . The pressure, on
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