A mathematical model is established to investigate a vertical gravity-driven drainage flow containing a soluble surfactant when considering the effect of wall slip. The lubrication theory is employed to obtain the evolution equations describing film thickness, surface velocity, surfactant concentrations at the air–liquid, solid–liquid interface, and in the bulk. The influence of constant slip length b_c and variable slip length b_s varying with surfactant concentration on the drainage dynamics is investigated compared with the case of no-slip b_o, and the mechanism of the film thinning and the backflow caused by wall slip is examined. Simulated results show that the wall slip has a significant impact on the dynamics of the film drainage compared with the no-slip case. For the case of constant slip length, the wall slip accelerates the film thinning in the early stage. At the middle stage, the wall slip enhances the Marangoni effect and surface velocity rapidly decreases, causing a surface backflow phenomenon at the film bottom; the higher the slip length, the more obvious surface backflow. In the late stage, surface backflow weakens, and the film thickness is less than that of b_o. For the case of variable slip length, in the early stage, the film thickness and surface velocity are between those of b_o and b_c; at the middle stage, a weak surface backflow is evolved at the film bottom; in the late stage, the film thickness is close to that of b_c, and the surfactant concentration is lower than those of b_o and b_c.

中文翻译：

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可溶性表面活性剂存在下墙滑对垂直膜排水的影响

建立了数学模型，以研究考虑壁滑效应的垂直重力驱动的排水流，其中包含可溶表面活性剂。润滑理论被用来获得描述薄膜厚度，表面速度，气液界面，固液界面以及整体中表面活性剂浓度的演化方程。与无滑移b _o的情况相比，研究了随表面活性剂浓度变化的恒定滑移长度b _c和可变滑移长度b _s对排水动力学的影响_。，并研究了由壁滑引起的薄膜变薄和回流的机理。模拟结果表明，与无滑移情况相比，壁滑对薄膜排水的动力学有显着影响。对于滑动长度恒定的情况，壁滑在早期加速了薄膜的变薄。在中间阶段，壁滑增强了马兰戈尼效应，表面速度迅速降低，在薄膜底部引起表面回流现象。滑移长度越高，表面回流越明显。在后期，表面回流减弱，膜厚小于b _o。对于滑动长度可变的情况，在早期阶段，膜厚和表面速度在b _{o的范围内。}和b _c ; 在中间阶段，在薄膜底部产生了弱的表面回流。在后期，膜厚接近于b _c，表面活性剂浓度低于b _o和b _c。