On a differential equation for a gas bubbles collapse mathematical model

In this paper we present a mathematical model for estimate the collapse time of a gas bubble in a vane of a oil gerotor pump. This amount of time cannot be greater of the total time spent by the pump for filling and then emptying out a vane in a single revolution, otherwise there is a loss of lubrication between internal and external gears. We assume that oil is incompressible and viscous, the bubble has a spherical shape and it is not translating into the external fluid. The analytical treatment of the model shows that the Navier-Stokes equations for the velocity field of the oil can be reduced to a single non linear ordinary differential equation for the variation in time of the bubble radius. The collapse time estimated by a numerical resolution of this equation and the collapse time calculated from an analytical resolution of the linearized equation are substantially equal. keywords: cavitation, Navier-Stokes equations, spherical coordinates, numerical and analytical resolution. 1 The problem We consider an oil gerotor-type pump for industrial applications, with an internal maximum pressure of about 15 bar. Experimental results show that for a rotational velocity of about 2000 rpm some phenomena of cavitation can occur. In this work we don’t consider a description of the arise of cavitation (see e.g. [1]), but we want to estimate the time, which we call collapse time tc, spent by a bubble of gas to reduce to zero its radius. In this way, we can deduce a maximum rotational velocity to avoid loss of lubrication due to the presence of bubbles in the geometrical zone where profiles of pignon and crown are coincident (see Figure 1).