Abstract

A new exact solution of the Oberbeck–Boussinesq equations has been presented which describes the steady-state Marangoni convection of a vertical swirling liquid in the shear flow of a viscous incompressible liquid. The velocity field of the nonuniform vortical liquid flow is studied in detail. It is shown that there are stagnation points in the liquid flow, the number of which can be as large as five. The existence of stagnation points leads to the formation of counterflows with very complex topology resembling cellular flow patterns. The tangential stresses in the liquid flow are not constant. They are also stratified from the zone of tensile and compressive stresses. Because of the structure of the exact solution, the vorticity components coincide to within a dissipative coefficient with the expressions for the tangential stresses, which means that there are several counterrotating vortices in the liquid. Similar studies of polynomial exact solutions for temperature and pressure determined the number of zones of stratification of the corresponding fields.

中文翻译：

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垂直涡旋液体的热毛细管对流

摘要

提出了一种新的Oberbeck-Boussinesq方程精确解，它描述了粘性不可压缩液体的剪切流中垂直回旋液体的稳态Marangoni对流。详细研究了非均匀涡流的速度场。结果表明，在液流中存在停滞点，其数量可以多达五个。停滞点的存在导致具有非常复杂的拓扑结构（类似于细胞流动模式）的逆流形成。液体流中的切向应力不是恒定的。它们也从拉伸应力和压缩应力区域分层。由于精确解的结构，涡度分量与切向应力的表达式在一个耗散系数内重合，这意味着液体中有几个反向旋转的涡流。对温度和压力的多项式精确解的类似研究确定了相应区域的分层区域数。