Dynamic analysis of the flotation of small spheres*

Small spheres experience a dynamic process to float or sink after they contact the liquid surface. Thus, their final status should be judged by dynamic analysis instead of static equilibrium. The motion equation of small spheres after contacting the liquid surface with zero velocity is established in consideration of the gravity, surface tension, and buoyancy, combining a quasi-static assumption of liquid surface shape, and the motion process of spheres is numerically simulated. The results show that a finally floating sphere experiences a sinking process of first accelerating and then decelerating, during which the static equilibrium position is crossed due to sphere inertia. There exists a limit position for the sphere, beyond which the surrounding liquid surface will collapse. The sphere will float if its velocity decreases to zero before reaching the limit position, else it will sink ultimately. Based on such dynamic analysis, the critical contact angle and limit density for small spheres to float are determined, and agree well with the published experimental results, while the limit density predicted by static equilibrium has a large deviation.