Bubble capture by a propeller

Abstract

A small air bubble (radius $a$) is injected in water (kinematic viscosity $\nu$) in the vicinity (distance $r_0$) of a propeller (radius $r_p$, angular frequency $\omega$). We study experimentally and theoretically the conditions under which the bubble can be ‘captured’, i.e. deviated from its vertical trajectory (imposed by gravity $g$) and moved toward the centre of the propeller ($r\,{=}\,0$). We show that the capture frequency $\omega_{\hbox{\scriptsize\it capt}}$ follows the relationship \[\omega_{\hbox{\scriptsize\it capt}}=\left(\frac{2ga^2}{9\beta\nu r_p f(\hbox{\it Re}_b)}\right)\left(\frac{r_0}{r_p}\right)^2(1+\cos\varphi_0),\] where $\beta$ is a dimensionless parameter characterizing the propeller, $f(\hbox{\it Re}_b)$ is an empirical correction to Stokes' drag law which accounts for finite-Reynolds-number effects and $\pi/2-\varphi_0$ is the angle between the axis of the propeller and the line between the centre of the propeller and the point where the bubble is injected. This law is found to be valid as long as the distance $d$ between the propeller and the water surface is larger than $3r_0$. For smaller distances, the capture frequency increases; using an image technique, we show how the above expression is modified by the presence of the surface.