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The new theorem of the steady, inviscid, compressible, guided and irrotational flows through internally foliated stream surfaces which are adapted to the geometry of Laval-nozzle casings, demonstrated on a 3D flow.

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Title: The new theorem of the steady, inviscid, compressible, guided and irrotational flows through internally foliated stream surfaces which are adapted to the geometry of Laval-nozzle casings, demonstrated on a 3D flow.
Authors: Dimitrakopoulos, Panagiotis1 (AUTHOR) p.dimitrakopoulos@borsig.de
Source: Zeitschrift für Angewandte Mathematik und Physik (ZAMP). Aug2025, Vol. 76 Issue 4, p1-20. 20p.
Subjects: Geometric congruences, Curvilinear coordinates, Compressible flow, Fluid flow, Orthogonal systems, Inviscid flow
Abstract: The main idea of this new research was to use orthogonal curvilinear coordinate systems in order to find analytic solutions related to compressible flows through Laval nozzles, by assuming steady, inviscid and compressible flows under some further assumptions: as the integrability condition should finally be met pertaining to each velocity-vector field, the focus was set on irrotational flows. Moreover, the orthogonality of the coordinate lines consequently reduced the complexity of the flow-model equations, especially after generally assuming that the guided streamlines are identical with one set of curves for each of the coordinate systems. The research was limited to those common two-dimensional and three-dimensional orthogonal curvilinear systems, which could provide curvilinear coordinate lines as streamlines of non-constant curvature. The main principle pertaining to the solution procedure was the same for all of the studied coordinate systems and is in this work solely demonstrated on a degenerate form of the oblate spheroidal coordinates for a 3D flow through a Laval nozzle, where the 3D nozzle is represented by a surface of rotation of an hyperbola (hyperboloid). The interpretation of all results finally led to the formulation of a new theorem: Assuming that the velocity of the fluid was in general tangent to curvilinear streamlines of non-constant curvature and based on all common 2D and 3D orthogonal curvilinear coordinate systems, this research paper revealed that there does not exist any unique, common and real-valued isentropic relation for all of the steady, inviscid, irrotational and compressible-fluid flows, which initially were expected to be fully congruent to the geometry of the Laval nozzle casings. Remarkably, considering the steady, inviscid, irrotational and incompressible-fluid flows, the results confirmed the general statements of another, alternative theory. [ABSTRACT FROM AUTHOR]
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Abstract:The main idea of this new research was to use orthogonal curvilinear coordinate systems in order to find analytic solutions related to compressible flows through Laval nozzles, by assuming steady, inviscid and compressible flows under some further assumptions: as the integrability condition should finally be met pertaining to each velocity-vector field, the focus was set on irrotational flows. Moreover, the orthogonality of the coordinate lines consequently reduced the complexity of the flow-model equations, especially after generally assuming that the guided streamlines are identical with one set of curves for each of the coordinate systems. The research was limited to those common two-dimensional and three-dimensional orthogonal curvilinear systems, which could provide curvilinear coordinate lines as streamlines of non-constant curvature. The main principle pertaining to the solution procedure was the same for all of the studied coordinate systems and is in this work solely demonstrated on a degenerate form of the oblate spheroidal coordinates for a 3D flow through a Laval nozzle, where the 3D nozzle is represented by a surface of rotation of an hyperbola (hyperboloid). The interpretation of all results finally led to the formulation of a new theorem: Assuming that the velocity of the fluid was in general tangent to curvilinear streamlines of non-constant curvature and based on all common 2D and 3D orthogonal curvilinear coordinate systems, this research paper revealed that there does not exist any unique, common and real-valued isentropic relation for all of the steady, inviscid, irrotational and compressible-fluid flows, which initially were expected to be fully congruent to the geometry of the Laval nozzle casings. Remarkably, considering the steady, inviscid, irrotational and incompressible-fluid flows, the results confirmed the general statements of another, alternative theory. [ABSTRACT FROM AUTHOR]
ISSN:00442275
DOI:10.1007/s00033-025-02511-z