Singularity of the potential field at the vertex of a symmetric tip with four identical perpendicular wings of wedge-shaped form
DOI:
https://doi.org/10.17721/1812-5409.2026/1.18Keywords:
potential theory, mixed boundary-value problem, solution singularity, integral transformation method, Wiener-Hopf techniqueAbstract
The study addresses a spatial problem of the potential theory outside a symmetric tip with four identical perpendicular wings of wedge-shaped form. The main objective is to determine the singularity of the solution at the tip's vertex.
By utilizing the symmetric shape of the tip, the problem is reduced to finding a harmonic function in a quarter-space that satisfies mixed boundary conditions on its sides.
The homogeneous solutions of the problem are expressed in spherical coordinates as a Mehler – Fock-type integral involving Legendre functions. Using the method of integral transforms, the problem is reduced to a Wiener – Hopf-type functional equation, which is transformed into an infinite system of linear algebraic equations by the Wiener – Hopf method. The solutions of this system allow for the derivation of analytical representations for the homogeneous solutions of the original boundary-value problem. The constructed homogeneous solutions enable the determination of the singularity of the potential field at the tip's vertex.
As a result of solving the given problem, a relationship between the singularity exponent and the deviation angle of the tip's wing edges has been established.
The constructed homogeneous solutions enable the determination of the nature of the potential field singularity at the tip's vertex and establish a quantitative relationship between the singularity exponent and the deviation angle of the tip's wing edges. The obtained results provide an assessment of the qualitative behavior of the potential field at the tip's vertex, as well as determine the quantitative characteristics describing this behavior. It makes it possible to find correct solutions of boundary-value problems in complex domains with corner points with the same geometry, especially when numerical methods are used.
Pages of the article in the issue: 136 - 142
Language of the article: English
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