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Steady flow of a Navier-Stokes fluid around a rotating obstacle. (English) Zbl 1156.76367
Summary: Let \(\mathcal B\) be a body immersed in a Navier-Stokes liquid \(\mathcal L\) that fills the whole space. Assume that \(\mathcal B\) rotates with prescribed constant angular velocity \(\omega\). We show that if the magnitude of \(\omega\) is not “too large”, there exists one and only one corresponding steady motion of \(\mathcal L\) 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.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
76E07 Rotation in hydrodynamic stability
76T20 Suspensions
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