Steady Flow of a Navier-Stokes Fluid Around a Rotating Obstacle

Let ℬ be a body immersed in a Navier-Stokes liquid ℒ that fills the whole space. Assume that ℬ rotates with prescribed constant angular velocity ω. We show that if the magnitude of ω is not “too large”, there exists one and only one corresponding steady motion of ℒ such that the velocity field v(x) and its gradient grad v(x) decay like |x|−1 and |x|−2, respectively. Moreover, the pressure field p(x) and its gradient grad p(x) decay like |x|−2 and |x|−3, respectively. These solutions are “physically reasonable” in the sense of Finn. In particular, they are unique and satisfy the energy equation. This result is relevant to several applications, including sedimentation of heavy particles in a viscous liquid.