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, $\mathscr B$, 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 $\mathscr B$, $\omega(x,t)$ behaves like the vorticity field of the corresponding steady-state problem.