An efficient finite volume method for one-dimensional problems with application to the dynamics of capillary jets

Abstract

Abstract We propose a numerical method to study both the nonlinear dynamics and the stability of equilibrium solutions of one-dimensional problems modeled by systems of differential-algebraic equations (DAE) written in conservation form. The method relies on a quadrature rule (finite volume approach) for the spatial discretization, while an implicit scheme is used in time to compute the transient dynamics. The stability of equilibrium solutions is analyzed by solving the generalized eigenvalue problem obtained after a standard linearisation. The Jacobian matrices needed for the computations - which are the same independently of whether one is interested in computing the equilibrium shapes, the nonlinear transient evolution or the linear stability analysis–have been systematically and efficiently computed with the help of a symbolic toolbox. To illustrate the method, we have applied it to several problems in the one-dimensional dynamics of capillary jets. For this purpose, and apparently for the first time, we have rewritten all the equations governing the jet kinematics and dynamics in conservation form and have shown the numerical advantages of this formulation, together with the numerical method proposed, when applied to the global stability of jets stretched by gravity, to the jetting to dripping transition and to the recoil of the filament remaining after the jet breakup. For the latter problem, we show that the results obtained with the one dimensional equations are in reasonable good agreement with the those previously reported in the literature and obtained with complicated potential three dimensional codes.