Further Results on Steady-State Flow of a Navier-Stokes Liquid Around a Rigid Body. Existence of the Wake

摘要

A rigid body R is moving in a Navier-Stokes liquid ￡ that fills the whole space. We assume that all data with respect to a frame, F, attached to R, namely, the body force acting on ￡, the boundary conditions on R as well as the translational velocity, U, and the angular velocity, Ω, of R are independent of time. We assume Ω ≠ 0 (the case Ω = 0 being already known) and take, without loss of generality, Ω parallel to the base vector e_1 in F. We show that, if the magnitude of these data is not "too large", there exists at least one steady motion of ￡ in F, such that the velocity field and its gradient decay like (l + |x|)~(-1)(l+2Res(x))~(-1) and (l + |x|)~(-3/2)(1+2Re s(x))~(-3/2), respectively, where Re is the Reynolds number and s(x) := |x| + x_1 is representative of the "wake" behind the body. This motion is unique in the (larger) class of motions having velocity field decaying like |x|~(-1). Since Re is proportional to |U·-e_1|, the above formulas show that the ￡ exhibits a wake behind R if and only if U is not orthogonal to Ω.