Abstract

Assuming that the streamlines are given by keeping constant one of the two orthogonal curvilinear coordinates in the Euclidean two-dimensional space, while considering a steady, two-dimensional, isentropic flow of an ideal gas through a convergent-divergent nozzle, and thus parallel to the curvilinear upper and lower walls of the nozzle, the theory of differential geometry together with the balance equations of physics was used to study the existence (or non-existence !) of real-valued, differentiable and integrable mass-density solutions to the problem, by means of analytical solutions, and corresponding to an expansion or compression of the gas along the streamlines. Thus, by initially assuming that the partial mass-density derivatives with respect to both curvilinear coordinates satisfy the integrability condition of Schwarz, the resulting system of four scalar partial differential equations led to an analytically derived quadratic equation for the determination of the ideal-gas mass density, based on generalised orthogonal curvilinear coordinates: Finally, the four orthogonal curvilinear coordinate systems, defined by the Killing two-tensors for the Euclidean two-dimensional space, were used, in order to examine whether these coordinate systems could satisfy the already mentioned generalised curvilinear-geometry equation as a quadratic equation, and the related requirements with regard to the partial mass-density derivatives, or not. Only real and nonzero, positive values for the mass density were considered, based on curvilinear streamlines of nonzero curvature.

中文翻译：

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通过考虑在欧几里得正交曲线坐标系中通过拉瓦尔喷嘴的稳定、等熵、二维流，关于不存在对应于理想气体沿流线的膨胀或压缩的实值、解析质量密度解二维空间

摘要

假设流线是通过保持欧几里得二维空间中两个正交曲线坐标之一恒定而给出的，同时考虑理想气体通过收敛-发散喷嘴的稳定、二维、等熵流，因此平行于喷嘴的曲线上壁和下壁，微分几何理论以及物理平衡方程被用来研究问题的实值、可微分和可积质量密度解的存在（或不存在！） ，通过解析解，对应于气体沿流线的膨胀或压缩。因此，通过最初假设关于两个曲线坐标的偏质量密度导数满足 Schwarz 可积条件，所得到的四个标量偏微分方程组导致了用于确定理想气体质量的解析导出的二次方程基于广义正交曲线坐标的密度：最后，使用由欧几里得二维空间的Killing二张量定义的四个正交曲线坐标系，以检验这些坐标系是否能够满足已经提到的广义正交曲线坐标系。曲线几何方程作为二次方程，以及关于部分质量密度导数的相关要求，或不是。基于非零曲率的曲线流线，仅考虑质量密度的实数和非零正值。