Stability and bifurcation analysis of a pendent drop using a novel dynamical model

• Published in: Archive of applied mechanics (1991) 2023-02, Vol.93 (2), p.487-501
• Main Authors: Ghorbanifar, Shahram, Rahni, Mohammad Taeibi, Zareh, Masoud, Nobakhti, Mohammad Hasan
• Format: Article
• Language: English
• Subjects: Bifurcations; Classical Mechanics; Dimensionless numbers; Displacement; Dynamic models; Dynamic stability; Dynamical systems; Elongation; Engineering; Equations of motion; Fluid dynamics; Fluid flow; Investigations; Mathematical analysis; Nonlinear dynamics; Nonlinear systems; Numerical prediction; Original; Polynomials; Saddle points; Stability analysis; Surface tension; Theoretical and Applied Mechanics; Viscous damping
• ISSN: 0939-1533; 1432-0681
• DOI: 10.1007/s00419-022-02278-z

Drop dynamics is often used to study liquid-fluid interfacial problems. Also, oscillatory pendent drops are a suitable and alternative tool to study non-linear dynamical systems. According to the paper by Ghorbanifar et al. (J Appl Fluid Mech 1:234, 10.47176/jafm.14.01.31313, 2021) it was predicted numerically that elastic force excreted by a pendent drop is a complete cubic polynomial function. The present work deals with the analytical analysis and proof of the force–displacement function of a pendent drop (which was realized by Ghorbanifar et al.) and using this function the bifurcation of drop was investigated. The oscillatory pendent drop equation of motion (OPDEM) presented here, can completely describe the force–displacement and damping functions of the pendent drop. In this article, the equation of motion of a pendent drop is presented, which paves the way for analytical investigations of drop dynamics. The presented novel dynamical model allows following the oscillation, growth, and detachment of a pendent drop and studying its elastic and non-linear behavior. Here, a drop was modeled as a combination of an elastic rod and a dashpot simulating the drop related surface tension and viscous damping forces, respectively. The displacement of the drop mass center was then related to the rod elongation. Then, the damped OPDEM was derived. It was found that the forcing and damping terms of OPDEM are complete cubic and quadratic polynomials, respectively. Using Lyapunov's method for stability analysis, non-linear dynamics of pendent drop was studied. It was shown the drop growth condition is that the displacement should lie between the local extrema of the forcing term of OPDEM. Exploiting the results presented by Ghorbanifar et al. effects of changing Bond and Capillary numbers on the bifurcation of the governing ODE of drop oscillations were investigated. It was realized that for the Bond numbers greater than about 0.11 saddle-point bifurcation occurs. This causes equilibrium points of OPDEM vanish and system becomes unstable. Also, increasing Capillary number to the values higher than 1.8E−5 causes transcritical bifurcation which leads to the removal of one of the balance points. For this case, by increasing Capillary number the system tends towards one equilibrium point. These results open a way to relate some dimensionless numbers in fluid flow to the dynamics of non-linear systems described by OPDEM and help in studying their stability.