Asymptotic properties of the steady fall of a body in viscous fluids

Integr. equ. oper. theory

Neustupa

2008

We present the description of the spectrum of a linear perturbed Oseen-type operator which arises from equations of motion of a viscous incompressible fluid in the exterior of a rotating compact body. Considering the operator in the function space L 2 σ (Ω) we prove that the essential spectrum consists of an infinite set of overlapping parabolic regions in the left half-plane of the complex plane. Our approach is based on a reduction to invariant closed subspaces of L 2 σ (Ω) and on a Fourier series expansion with respect to an angular variable in a cylindrical coordinate system attached to the axis of rotation. Mathematics Subject Classification (2000). Primary 35Q35; Secondary 35P99, 76D07.

Section: Motivation and Introductionmentioning

confidence: 99%

On the Spectrum of an Oseen-Type Operator Arising from Flow past a Rotating Body

Integr. equ. oper. theory

Neustupa

2008

“…We refer to [3], [9], [10], [12], [13], [14], [15], [16], [18], [19], [20], [21], [23], [24], [25], [26], [27], [28], [33], [34] and [36]. Nevertheless, our mathematical understanding is still far from complete.…”

Section: Introductionmentioning

confidence: 99%

Stationary Navier-Stokes Flow Around a Rotating Obstacle

Hishida

2007

Abstract. Consider a viscous incompressible fluid filling the whole 3-dimensional space exterior to a rotating body with constant angular velocity o. By using a coordinate system attached to the body, the problem is reduced to an equivalent one in a fixed exterior domain. The reduced equation involves the crucial drift operator ðo5xÞ Á ', which is not subordinate to the usual Stokes operator. This paper addresses stationary flows to the reduced problem with an external force f ¼ div F , that is, time-periodic flows to the original one. Generalizing previous results of G. P. Galdi [19] we show the existence of a unique solution ð'u; pÞ in the class L 3=2; y when both F A L 3=2; y and o are small enough; here L 3=2; y is the weak-L 3=2 space.

“…The spaceŴ k,q (Ω) consists of equivalence classes of L 1 loc -functions being unique only up to elements from Π k−1 and is equipped with the norm |α|=k ∂ α u q , where · q denotes the L q -norm. However, sometimes being less careful, we will consider v ∈Ŵ k,q (Ω) as a function (representative) rather than an equivalence class of functions, i.e., v ∈ L 1 loc (Ω) such that ∂ α v ∈ L q (Ω) for every multi-index α with |α| = k. For further results on similar problems we refer to [4], [8], [9], [10], [11], [12] and [13].…”

mentioning

confidence: 99%

Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle

Banach Center Publications

2005

Abstract. Consider the problem of time-periodic strong solutions of the Stokes system modelling viscous incompressible fluid flow past a rotating obstacle in the whole space R 3 . Introducing a rotating coordinate system attached to the body yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In a recent paper [2] the author proved L q -estimates of second order derivatives uniformly in the angular and translational velocities, ω and k, of the obstacle, whereas the transport terms fails to have L q -estimates independent of ω. In this paper we clarify this unexpected behavior and prove weighted L q -estimates of first order terms independent of ω.1. Introduction. Consider the Navier-Stokes equations modelling viscous flow past a rotating body K ⊂⊂ R 3 with axis of rotation ω =ωe 3 =ω(0, 0, 1) T ,ω = |ω| > 0, and with velocity u ∞ = ke 3 , k > 0, at infinity. Then the velocity v and the presure p satisfy the systemin Ω(t), t > 0, v(y, t) = ω ∧ y on ∂Ω(t), t > 0, v(y, t) → u ∞ = 0 as |y| → ∞,