Сферическое течение идеальной жидкости в пространственно-неоднородном силовом поле

Abstract

Построены точные частные решения уравнений Эйлера, определяющие стационарное сферическое движение несжимаемой невязкой жидкости. Даны примеры влияния структуры пространственной неоднородности силового поля на гидродинамические параметры течения: задача о протекании жидкости сквозь ядро сферического слоя; широтные и меридианные течения; поведение изобар и линий равных скоростей в потенциальном, соленоидальном и лапласовом силовых полях.

Exact particular solutions to Euler equations are obtained for a steady spherical flow of an incompressible inviscid fluid. The effect of the structure of the force field spatial nonuniformity on hydrodynamic parameters of the flow is studied. An exact solution to a flux problem is obtained. In this problem, the fluid flows in a spherical layer of finite thickness whose external boundary is impermeable. In the northern part of the layer, the fluid flows out of the core; in the southern part, into the core. There is no flowing at the equator. The peculiarities of the pressure gradient on the layer boundaries are discussed in detail. The intensity of mass force sources is calculated. Both exponential and power-law dependences of the flow velocity on the core surface temperature are proposed. The zonal and meridional flows occurring in potential, solenoidal, and Laplace force fields are considered. Examples of the conditions under which the velocity contours are or are not isobars are given. The behavior of these lines is shown to be mainly affected by a meridional component of the mass force. Physical models corresponding to the given solution sare presented. An example of the zonal flow inside an impermeable sphere is indicated. A zonal flow is considered in the external space of two impermeable cones. Arrangement of the cones has a sandglass-like shape. They have a common axis, a common vertex, and opposite bases. In a partial case, the impermeable boundaries are represented as a cone and an equatorial plane. The same arrangement of the cones is used for a hydrodynamic interpretation of the meridional flow, where the vertices of the cones are located in the center of the internal sphere, and the fluid flows out of the upper cone into the lower one through their permeable walls. The flow region is radially confined by external and internal impermeable spheres. In a specific case, the lower cone degenerates into a plane, and the fluid outflows from the spherical layer through a round ring located in the equatorial plane.