We study the contact line dynamics of a viscous droplet deposited at the centre of a substrate subject to an axial thermal gradient. The temperature of the substrate decreases with distance from the centre, so the Marangoni stress induced at the liquid–air interface displaces the liquid radially outward. The flow experiences two stages. In the first stage, the droplet evolves towards an axially symmetric ring whose radius increases with time as $t^{1/3}$ . Using the lubrication approximation, we perform numerical simulations that confirm this law for the motion of the front and show that the maximum thickness of the profile decreases as $t^{-0.374}$ . We explain the evolution law of the contact line by balancing Marangoni and viscous stresses. In the second stage, the contact line becomes unstable and develops smooth corrugations whose amplitude increases with time and that eventually become long fingers. The temporal evolution of the Fourier spectra of the contour shows a shift of the most unstable mode from smaller to larger azimuthal wavenumbers.

中文翻译：

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轴向热毛细管向外流动中的接触线运动

我们研究了沉积在受轴向热梯度影响的基板中心的粘性液滴的接触线动力学。衬底的温度随着距中心的距离而降低，因此在液-气界面处引起的 Marangoni 应力使液体径向向外移动。流程经历两个阶段。在第一阶段，液滴向轴对称环演化，其半径随时间增加为 $t^{1/3}$ 。使用润滑近似，我们进行了数值模拟，证实了前部运动的这一规律，并表明轮廓的最大厚度随着 $t^{-0.374}$ 减小。我们通过平衡 Marangoni 和粘性应力来解释接触线的演变规律。在第二阶段，接触线变得不稳定并形成平滑的波纹，其幅度随着时间的推移而增加，最终变成长手指。轮廓的傅立叶光谱的时间演变显示了最不稳定模式从较小的方位角波数到较大的方位角波数的转变。