A two-dimensional model for slow convection at infinite Marangoni number
A two-dimensional model for slow convection at infinite Marangoni number
复制标题
无限马兰戈尼数下慢对流的二维模型
DOI：
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发表时间：
1997
影响因子：
3.7
通讯作者：
B. Jüttner
中科院分区：
文献类型：
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作者：
A. Thess;Daniel Spirn;B. Jüttner
如果粘性流体的表面张力分布不均匀，则其自由表面是对流流动(Marangoni对流)的来源。这种不均匀是由于表面张力依赖于标量，无论是表面活性剂浓度还是温度。表面张力诱导的速度使标量重新分布，形成闭合环路相互作用。结果表明，在(I)小雷诺数和(Ii)扩散系数为零的假设下，这一非线性过程可用自由表面标量场的单个自洽二维演化方程来描述，该方程可从三维基本方程出发而不经近似得到。对于一个特定的系统，这个方程的表述只需要闭合定律的知识，闭合定律将表面速度表示为自由表面上活动标量的线性泛函。对于平面不偏转且水平范围无限的各种系统，包括无限深的流体、有限深度的流体、旋转的流体和在磁场作用下的导电流体，我们都显式地推导出了这些闭合定律。对于无限深层的正则问题，我们证明了奇异(点状)表面活性剂或温度分布的动力学可以进一步归结为一个常微分方程组，等价于二维理想流体中的点-涡动力学。通过数值模拟，我们进一步证明了初始光滑标量场的动力学演化一般会导致有限时间的奇异性。该理论为高Prandtl数流体或高Schmidt数体系中强非线性Marangoni对流的简化模拟提供了一个合理的框架。
The free surface of a viscous fluid is a source of convective flow (Marangoni convection) if its surface tension is distributed non-uniformly. Such non-uniformity arises from the dependence of the surface tension on a scalar quantity, either surfactant concentration or temperature. The surface-tension-induced velocity redistributes the scalar forming a closed-loop interaction. It is shown that under the assumptions of (i) small Reynolds number and (ii) vanishing diffusivity this nonlinear process is described by a single self-consistent two-dimensional evolution equation for the scalar field at the free surface that can be derived from the three-dimensional basic equations without approximation. The formulation of this equation for a particular system requires only the knowledge of the closure law, which expresses the surface velocity as a linear functional of the active scalar at the free surface. We explicitly derive these closure laws for various systems with a planar non-deflecting surface and infinite horizontal extent, including an infinitely deep fluid, a fluid with finite depth, a rotating fluid, and an electrically conducting fluid under the influence of a magnetic field. For the canonical problem of an infinitely deep layer we demonstrate that the dynamics of singular (point-like) surfactant or temperature distributions can be further reduced to a system of ordinary differential equations, equivalent to point-vortex dynamics in two-dimensional perfect fluids. We further show, using numerical simulations, that the dynamical evolution of initially smooth scalar fields leads in general to a finite-time singularity. The present theory provides a rational framework for a simplified modelling of strongly nonlinear Marangoni convection in high-Prandtl-number fluids or systems with high Schmidt number.