Abstract

We investigated an incompressible viscous liquid film flow over a rotating vertical cylinder of radius R and of infinite length rotating with a uniform angular velocity \(\varvec{\Omega }\) about its axis. The surface of the vertical cylinder is non-uniformly heated where the temperature varies linearly in the downstream direction. The flow is assumed to be axisymmetric, and the component of the velocity along the azimuthal direction is assumed to be constant. The surface tension of the liquid is assumed to vary linearly with temperature such that as the temperature increases, the surface tension decreases. This gives rise to Marangoni stress over the free surface of the thin film. Using the long-wave approximation method, we derived a free surface evolution equation. For linear stability analysis, we used a normal mode approach and found that the Marangoni number plays a double role. There exists a critical Marangoni number \(\left( \mathrm{Mn}^{*}\right) \) such that for \(\mathrm{Mn}<\mathrm{Mn}^{*}\), it plays a stabilizing role and for \(\mathrm{Mn}>\mathrm{Mn}^{*}\) it plays a destabilizing role. We also found that as the rotation number \(\mathrm{Ro}\) increases, the destabilizing zone increases but it decreases with the increment of the radius R of the cylinder. We further performed a weakly nonlinear analysis of the flow using the method of multiple scales. The study reveals that the Marangoni number Mn, the radius R and the rotation number \(\mathrm{Ro}\) have substantial effects on different stability zones. The study also reveals that in the supercritical stable (subcritical unstable) zone, the threshold amplitude of the nonlinear disturbance increases (decreases) with the increment of Mn and \(\mathrm{Ro}\) but decreases (increases) with the increment of R. The nonlinear wave speed in the supercritical stable zone decreases with the increment of Mn and \(\mathrm{Ro}\), whereas it increases with the increment of R. We also examined the effect of thermocapillarity and rotation on the profile of the steady travelling wave solutions of the leading order part of the evolution equation.

中文翻译：

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薄膜在旋转不均匀加热的垂直圆柱体的外表面向下流动的稳定性

摘要

我们研究了在半径R和无限长的旋转垂直圆柱上以不可比拟的角速度\（\ varvec {\ Omega} \）旋转的不可压缩粘性液体薄膜流绕其轴。垂直圆筒的表面受到不均匀加热，温度在下游方向线性变化。假定流动是轴对称的，并且沿方位角方向的速度分量假定为恒定。假定液体的表面张力随温度线性变化，使得随着温度升高，表面张力降低。这会在薄膜的自由表面上产生Marangoni应力。使用长波近似方法，我们导出了自由表面演化方程。对于线性稳定性分析，我们使用了正常模式方法，发现Marangoni数起着双重作用。存在一个临界Marangoni数\（\ left（\ mathrm {Mn} ^ {*} \ right）\）这样\（\ mathrm {Mn} <\ mathrm {Mn} ^ {*} \），它起到稳定作用，而对于\（\ mathrm {Mn}> \ mathrm {Mn} ^ {*} \），它起稳定作用角色。我们还发现，随着转数\（\ mathrm {Ro} \）的增加，不稳定区域增加，但随着圆柱半径R的增加而减少。我们还使用多尺度方法对流动进行了弱非线性分析。研究表明，马兰戈尼数Mn，半径R和转数\（\ mathrm {Ro} \）对不同的稳定区有重大影响。研究还表明，在超临界稳定（亚临界不稳定）区域，非线性扰动的阈值幅度随Mn和\（\ mathrm {Ro} \）的增加而增加（减小），而随着Mn和\（\ mathrm {Ro} \）的增加而减小（增加）。[R 。超临界稳定区内的非线性波速随Mn和\（\ mathrm {Ro} \）的增加而减小，而随R的增加而增加。我们还研究了热毛细作用和旋转对演化方程前导部分的稳定行波解轮廓的影响。