The dynamics of spreading and recoiling of a liquid droplet after colliding with a partially wet solid surface is studied analytically. Using a mass-spring-damper analogy and based on the variational principle, the energy balance equation is developed, and a nonlinear ordinary differential equation is obtained. Predicted results are validated against the experimental data of others, and it is shown that the model is able to closely predict the effects of inertia, viscosity, and surface tension by reproducing many of the features of droplet dynamics. In addition to the maximum spreading diameter, the conditions under which a droplet recoils or bounces are obtained. Assuming a negligible contact angle hysteresis and no formation of rim or splash, results are discussed for partially wet surfaces in a wide range of viscosity and impact velocities. Compared to other models, which are mostly linear, the proposed model has the advantage of providing a simple and accurate description of droplet dynamics.

中文翻译：

---

液滴撞击固体表面的非线性模型

分析性地研究了与部分润湿的固体表面碰撞后液滴散布和回卷的动力学。利用质量-弹簧-阻尼器的类比，并基于变分原理，建立了能量平衡方程，得到了非线性常微分方程。预测的结果与其他实验数据进行了验证，结果表明该模型通过重现液滴动力学的许多特征，能够紧密预测惯性，粘度和表面张力的影响。除了最大铺展直径之外，还可以获得液滴回弹或反弹的条件。假设接触角滞后可以忽略不计并且没有形成轮缘或飞溅，讨论了在粘度和冲击速度范围很广的部分潮湿表面的结果。