Abstract
The equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.
Highlights
We investigate the fluid flow past a rigid body B that moves through an infinite three-dimensional liquid reservoir with prescribed velocity
We only consider the case where the angular velocity η is constant, but the translation velocity ξ may depend on time
We investigate a configuration where the rigid body B translates periodically with some prescribed time period T > 0
Summary
T ∈ R and x ∈ R3 denote time and spatial variable, respectively, ξ. B with respect to its center of mass. We only consider the case where the angular velocity η is constant, but the translation velocity ξ may depend on time. In a frame attached to the body, with origin at its center of mass xC, the motion of an incompressible Navier–Stokes fluid around B that adheres to B at the boundary is described by the equations. The constants ρ > 0 and μ > 0 denote density and viscosity, respectively. We investigate a configuration where the rigid body B translates periodically with some prescribed time period T > 0.
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