On the non-existence of real-valued, analytical mass-density solutions corresponding to an expansion or compression of an ideal gas along the streamlines, by considering a steady, isentropic, 2D-flow through a Laval nozzle in orthogonal curvilinear coordinates in the Euclidean 2D-space

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.