On the stationary Navier–Stokes flows around a rotating body

Abstract

Consider the stationary motion of an incompressible Navier-Stokes fluid around a rotating body K = R 3 \ which is also moving in the direction of the axis of rotation. We assume that the translational and angular velocities U, ω are constant and the external force is given by f = div F. Then the motion is described by a variant of the stationary Navier-Stokes equations on the exterior domain for the unknown velocity u and pressure p, with U, ω, F being the data. We first prove the existence of at least one solution (u, p) satisfying ∇u, p ∈ L 3/2,∞ ( ) and u ∈ L 3,∞ ( ) under the smallness condition on |U | + |ω| + F L 3/2,∞ ( ) . Then the uniqueness is shown for solutions (u, p) satisfying ∇u, p ∈ L 3/2,∞ ( ) ∩ L q,r ( ) and u ∈ L 3,∞ ( ) ∩ L q * ,r ( ) provided that 3/2 < q < 3 and F ∈ L 3/2,∞ ( ) ∩ L q,r ( ). Here L q,r ( ) denotes the well-known Lorentz space and q * = 3q/(3 − q) is the Sobolev exponent to q. IntroductionLet be an exterior domain in R 3 with smooth boundary ∂ . Consider the motion of an incompressible Navier-Stokes fluid around the rigid body K = R 3 \ which is rotating about an axis with constant angular velocity ω = ce 3 = (0, 0, c) T . We also assume that the body K is moving in the direction of the axis of rotation with constant velocity U = ke 3 . Then with respect to a coordinate system attached to the body, the velocity u = (u 1 , u 2 , u 3 ) T and pressure p of the fluid is governed by the following initial boundary value problem for a variant of the Navier-Stokes equations in (see [7,12,18] for a detailed derivation):

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

“…In this subsection, we consider the following equations in the whole space : The following results summarize the known results (see previous studies).…”

“…By using the so‐called localization technique, Kračmar et al obtained the existence and uniqueness of weak solutions in D 1, r (Ω) for 3/2< r <3. More recently, this result was improved by Heck et al in the framework of Lorentz spaces. In this paper, we extend the previous results because of Galdi and Kračmar et al to the full range 4/3< r <4, by using Theorem and interior regularity of very weak solutions.…”

“…The second aim of this paper is to obtain existence and uniqueness results of weak and strong solutions for the rotating Oseen equations (O) satisfying appropriate D 1, r and D 2, s ‐estimates. During the last years, the rotating problem was studied by several authors and we will refer some of them; see previous studies for instance. As we mentioned previously, the rotating Oseen equations can by no means be considered as a perturbation of the classical Oseen equations, even for small | ω |.…”

Oseen System With Coriolis Term

“…Applying again Theorem 3.2 we get that there is a solution ( , ) v � � � in the space W 1,q (V; (11), we have e u � � W � 1,q (�\V; � 3 ) by Proposition 2.3.…”

A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation

“…A case where also translation is included see [16]. Concerning nonlinear L q setting we refer to the work of Farwig, Hishida [19], Heck et al [50]. A case with nondecaying initial data was studied by Giga et al [43].…”