Sliding motion and bifurcation in saline oscillator's model and liquid's density measurement using saline oscillator

The saline oscillator has been modeled by several Filippov systems in the literature. In this paper, we examine the effects of the novel feature of the recently introduced model, onto its output. We then investigate the bifurcation process of the saline oscillator's model in the framework of Filippov theory. We find that the recently introduced modification substantially affects the period of the output of the model in case the ratio a/b of radii of the orifice and the inner container is considerable. The theoretical analysis predicts that the model goes through a non-conventional Hopf bifurcation which could appear as result of a sliding motion on the switching boundary. In case a/b is insignificant, a relationship between the midpoint of oscillation, the system geometrical parameters and the liquids densities is analytically derived. To validate this formula, we compute the dependencies on system parameters of the midpoint of oscillation by numerically solving the model. Then, we compare these dependencies of the midpoint of oscillation with those directly obtained from the analytical formula. We then show, from this relation, that another practical application of the saline oscillator is the measurement of the density of a liquid. Finally, we describe the method of measurement and propose a sketch for a density meter, based on saline oscillator, which is suitable for samples of small volume and does not need calibration procedure.