Time Periodic Solutions of the Navier–Stokes Equations with Nonzero Constant Boundary Conditions at Infinity

Abstract

Abstract. We construct solutions for the Navier-Stokes equations in three dimensions with a time periodic force which is of compact support in a frame that moves at constant speed. These solutions are related to solutions of the problem of a body which moves within an incompressible fluid at constant speed and rotates around an axis which is aligned with the motion. In contrast to other authors who analyze stationary solutions in a frame of reference attached to the body, the analysis for the present problem is done in a frame which is moving at constant speed but is not rotating. This avoids the unpleasant unbounded linear terms which are present in a description with respect to a rotating frame. 1. Introduction. The classic paper of Weinberger [9] concerning the steady fall of a body in a Navier-Stokes liquid starts with the following definition: "We say a body undergoes a steady falling motion in an infinite viscous fluid if the motion of the fluid as seen by an observer attached to the body is independent of time."One of the interesting possible cases is a body that is falling steadily, and is rotating around an axis that is parallel to the direction in which the body is falling.A first proof of the existence of such solutions for this case has been given only recently in the three papers by Galdi and Silvestre [4,3,2]. Their method for solving the problem is to consider the equations, as proposed by Weinberger, in a frame attached to the body, where the flow is stationary. In this frame the Navier-Stokes equations have an additional linear term with unbounded coefficients, which is due to the transformation into the rotating frame. This complicates the problem considerably when compared to the situation without rotation.It is important to note that even without the rotation the problem is difficult because of the slow decay of the vorticity in the downstream region. This leads to a very strong asymmetry in the behavior at infinity, and the main difficulty is to encode this behavior when choosing function spaces.Once the existence of a solution is established one is interested in giving detailed information concerning its behavior at infinity. As in related problems, this behavior is expected to be independent of the details of the body. It turns out to be possible to use this fact in order to simplify the analysis of the asymptotic behavior, by considering first the problem in the whole space, and to mimic the body by a smooth force of compact support (see the end of this section for details). The case with a body, i.e., the case of an exterior domain, can then be treated in a second step, once the behavior at infinity is understood. For a related problem in two dimensions, this strategy has

Leading Order Asymptotics of Stationary Navier–stokes Flows in the Presence of a Wall

“…Therefore,û 1;1;1 andû 1;1;2 are the leading order terms ofû 1 in Fourier space. We …rst prove (5), which is an immediate consequence of: Proposition 7 Letû 1 (k; t) be as above. Then we have…”

Viscous flow around a rigid body performing a time-periodic motion

“…In a short note, Serrin [46] proposed the examination of the corresponding time-periodic configuration, and Prodi [44], Yudovich [54] and Prouse [45] initiated the study of time-periodic Navier-Stokes flow in bounded domains. Through the years, this investigation has been continued and extended to other types of domains and fluid-flow problems by several authors, see for example [27,49,41,40,51,37,38,39,28,53,11,21,22,50,52,47,14,29,30,32,42,23,10,5,18]. We also refer to [19] for a more detailed overview.…”

Viscous Flow Around a Rigid Body Performing a Time-periodic Motion

“…The study of time-periodic Navier-Stokes flows was proposed in a short note by Serrin [47], which induced Prodi [45], Yudovich [56] and Prouse [46] to initiate the examination in bounded domains. Through the years, this investigation has been continued and extended to other types of domains and fluid-flow configurations by several authors; see for example [5,10,11,14,18,[21][22][23][28][29][30][31]33,[38][39][40][41][42][43]49,[51][52][53][54][55]. We refer to [19] for a more detailed overview.…”