Bubbles and Mathematical Modelling
In this talk we are concerned about singular boundary value problems arising in hydrodynamics and cosmology. In the case of spherical simmetry, the orginal partial differential equation may be reduced to a second order nonlinear ordinary differential equation (ODE). This is the case, for example, of the formation of spherical bubbles or droplets in a mixture gas-liquid. We are interested on solutions of the resulting ODE which are strictly increasing on the positive semi-axis and have finite limits at 0 and infinity (bubble-type solutions). Necessary and sufficient conditions for the existence of such solutions are obtained in the form of a restriction on the equation coefficients. The asymptotic behavior of certain solutions of this equation is analysed near the two singularities (when r tends to 0 and r tends to infinity), where the considered boundary conditions define one-parameter families of solutions. Based on the analytical study, an efficient numerical method is proposed to compute approximately the needed solutions of the above problem. Some results of the numerical experiments are displayed and their physical interpretation is discussed.
