Stationary Navier-Stokes Flow Around a Rotating Obstacle

Farwig, Reinhard; Hishida, Toshiaki

doi:10.1619/fesi.50.371

Cited by 50 publications

(58 citation statements)

References 31 publications

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“…In particular, Kozono and Yamazaki showed in [24] that if U = ω = 0 and F L 3/2,∞ ( ) is small, then the problem (0.2) has at least one solution (u, p) satisfying ∇u, p ∈ L 3/2,∞ ( ) and u ∈ L 3,∞ ( ). This result was extended by Farwig and Hishida [8] to the case of non-zero angular velocity ω. However, it remains still open to prove the uniqueness of solutions (u, p) of (0.2) satisfying ∇u, p ∈ L 3/2,∞ ( ) and u ∈ L 3,∞ ( ); see e.g.…”

Section: Introductionmentioning

confidence: 53%

On the stationary Navier–Stokes flows around a rotating body

2011

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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):

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Section: Introductionmentioning

confidence: 53%

On the stationary Navier–Stokes flows around a rotating body

2011

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“…We will essentially use this property to obtain the resolvent estimates in Section 3.3. As a result, the constant C in (7) and (8) obtained so far is highly depending on α ̸ = 0 and tends to ∞ as α → 0, although (7) and (8) are known to hold for the limit case α = 0.…”

Section: Theorem 13 There Exists a Positive Constant δ Such That If mentioning

confidence: 97%

“…The spectrum of the linear operator arising in this problem is studied by Farwig and Neustupa [10]; see also the linear analysis by Hishida [21]. The existence of steady-state solutions to the associated system is proved in [2], Silvestre [42], Galdi [13], and Farwig and Hishida [7]. In particular, in [13] the steady flows with the decay order O(|x| −1 ) are obtained.…”

Section: Theorem 13 There Exists a Positive Constant δ Such That If mentioning

confidence: 99%

On Stability of Steady Circular Flows in a Two-Dimensional Exterior Disk

Maekawa

2017

Arch Rational Mech Anal

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We study the stability of some exact stationary solutions to the two-dimensional Navier-Stokes equations in an exterior domain to the unit disk. These stationary solutions are known as a simple model of the flow around a rotating obstacle, while their stability has been open due to the difficulty arising from their spatial decay in a scale-critical order. In this paper we affirmatively settle this problem for small solutions. That is, we will show that if these exact solutions are small enough then they are asymptotically stable with respect to small L 2 perturbations.Keywords Navier-Stokes equations · two-dimensional exterior flows · scale-criticality · stability of stationary flows · flow around a rotating obstacle

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“…Also motivated by the important role played in the theory of liquid-solid interactions, for example [27], over the recent years there has been a substantial effort devoted to the study of similar questions when the body is also allowed to rotate with a constant angular velocity, in both steady and unsteady cases [1,[7][8][9][10][11][12][15][16][17][18][19][20][21][22][23][24][25][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][51][52][53][54][55][58][59][60]62], [31,Chapter XI].…”

Section: Introductionmentioning

confidence: 99%

Exponential Decay of the Vorticity in the Steady-State Flow of a Viscous Liquid Past a Rotating Body

Deuring

Galdi

2016

Arch Rational Mech Anal

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Consider the flow of a Navier-Stokes liquid past a body rotating with a prescribed constant angular velocity, ω, and assume that the motion is steady with respect to a body-fixed frame. In this paper we show that the vorticity field associated to every "weak" solution corresponding to data of arbitrary "size" (Leray Solution) must decay exponentially fast outside the wake region at sufficiently large distances from the body. Our result improves and generalizes in a nontrivial way famous results by Clark (Indiana Univ Math J 20:633-654, 1971) and Babenko and Vasil'ev (J Appl Math Mech 37:651-665, 1973) obtained in the case ω = 0.

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Stationary Navier-Stokes Flow Around a Rotating Obstacle

Cited by 50 publications

References 31 publications

On the stationary Navier–Stokes flows around a rotating body

On the stationary Navier–Stokes flows around a rotating body

On Stability of Steady Circular Flows in a Two-Dimensional Exterior Disk

Exponential Decay of the Vorticity in the Steady-State Flow of a Viscous Liquid Past a Rotating Body

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