New SubmissionSolution of the extended van der Waals capilarity model

The dynamics of bubble formation in metastable liquids are crucial in various applications, particularly in the formation of laser-induced ultrasound waves for medical diagnostics and therapy. We describe two distinct numerical approaches to solving the Euler-Lagrange equation arising from energy minimization in the extended van der Waals capillary theory.

The first approach involves expressing the governing partial differential equation (PDE) as a system of two coupled ordinary differential equations and seeking its solution using a shooting method augmented by a Newton method to enforce the far undisturbed metastable condition. The numerical solution reveals the vapor-liquid interface profile for the metastable states that are sufficiently far from the critical condition. On the other hand, when the critical point is approached, eigenvalue analysis and numerical results demonstrate the increasingly ill-posed nature of the problem.

The second approach involves the Newton solution of the original PDE, augmented with an unsteady term controlled to have a finite or negligible magnitude compared to the other terms. Numerical solutions reveal the sensitivity of the interface profile to the initial guess of the interface profile in the Newton iterations, as well as the simple wave-like propagation of the interface.