Mathematical model of the flotation complex particle–bubble within the framework of Lagrangian formalism

Abstract

A model of the interaction of a spherical gas bubble and a rigid particle is derived as a coupled system of second-order differential equations using Lagrangian mechanics. The model takes into account oscillations of the bubble surface and the attached to it solid cylindrical particle in infinite volume of ideal incompressible liquid. The capillary force holding the particle on the bubble is due to the shape of the meniscus surface, which determines the wetting edge angle. The series with respect Legendre polynomials is used to present small axisymmetric oscillations of the particle-bubble system. Potential and kinetic energies are expressed through coefficients of this series. Particle adhesion condition to bubble surface is implemented through Lagrange multipliers. The dependence of the particle size and its density is demonstrated as a result of the numerical integration of the resulting dynamic system of differential equations.