Spatial decay of the vorticity field of time-periodic viscous flow past a body

Abstract

We study the asymptotic spatial behavior of the vorticity field, $\omega (x,t)$, associated to a time-periodic Navier-Stokes flow past a body, $\mathcal{B}$, in the class of weak solutions satisfying a Serrin-like condition. We show that, outside the wake region, $\mathcal{R}$, $\omega$ decays pointwise at an exponential rate, uniformly in time. Moreover, denoting by $\bar{\omega }$ its time-average over a period and by ${\omega }_P:=\omega -\bar{\omega }$ its purely periodic component, we prove that inside $\mathcal{R}$, $\bar{\omega }$ has the same algebraic decay as that known for the associated steady-state problem, whereas ${\omega }_P$ decays even faster, uniformly in time. This implies, in particular, that "sufficiently far" from $\mathcal{B}$, $\omega (x,t)$ behaves like the vorticity field of the corresponding steady-state problem.

References in corpus (1)

• Existence, Uniqueness and Asymptotic Behavior of Regular Time-Periodic Viscous Flow around a Moving Body: Rotational Case