Abstract—

A new class of exact solutions of the Oberbeck–Boussinesq equations for incompressible media is constructed taking into account body forces, heat sources (sinks), and Joule dissipation. The expressions for the velocities are quadratic forms with respect to two coordinates, generalizing the class of Lin–Sidorov–Aristov solutions. Temperature, pressure, and the field of body forces are described by forms of the fourth degree. The possibility of using this class for convective flows in the Stokes and Oseen approximation is considered, and the possibilities of a new class for describing rotating liquid masses are demonstrated. A simple example illustrates the complex structure of the velocity field for a creeping convective Couette-type fluid flow in a layer with a permeable boundary moving inhomogeneously.

中文翻译：

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描述不可压缩流体的 Oberbeck-Boussinesq 方程组的一类新精确解

摘要-

考虑了体力、热源（汇）和焦耳耗散，构造了一类新的不可压缩介质的 Oberbeck-Boussinesq 方程的精确解。速度的表达式是关于两个坐标的二次形式，概括了 Lin-Sidorov-Aristov 解的类别。温度、压力和体力场用四度的形式来描述。考虑了在 Stokes 和 Oseen 近似中将此类用于对流流动的可能性，并证明了描述旋转液体质量的新类的可能性。一个简单的例子说明了蠕变对流库埃特型流体在具有不均匀运动的可渗透边界层中的速度场的复杂结构。