Abstract
To address the scientific inquiry regarding the dynamic mechanism of deformation of droplets on the jet surface, this study establishes a stochastic differential equation describing deformation of a droplet on a uniform jet surface, considering the affine deformation of a semi-ellipsoidal droplet driven by the pulsation velocity of the jet. The results indicate that the equilibrium point set of the model conforms to an inverse proportional function concerning the major and minor axes of the droplet. Additionally, the assumption of the physical model is satisfied only when the droplet position parameter \(\theta \in \left( {\left. {\frac{{{\pi }}}{2},{{\pi }}} \right]} \right.\) . The solution process, initialized with the equilibrium point set, tends to reach the state of mean square instability within a very short duration. For dimensionless initial semi-minor axes of the droplet below 0.1 and within the constraint of the equilibrium point set, the probability of the droplet being in a stretched state during the later stages of deformation ranges between 60 and 65%. In this model, during the late stage of droplet deformation, the dimensionless semi-minor axis of the droplet approaches zero. Thus, when the droplet is in the stretched state, it is close to breaking. During retraction, the length of semi-major axis of the droplet fluctuates and gradually decreases. However, in the stretched state, the length of semi-major axis of the droplet increases rapidly, with its growth rate accelerating with time.
