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Scaling‐equivalent rotating flows
Author(s) -
Magyari Eugen
Publication year - 2021
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201900229
Subject(s) - scaling , magnetohydrodynamic drive , exact solutions in general relativity , flow (mathematics) , physics , solution set , boundary value problem , rotational symmetry , magnetohydrodynamics , mathematical analysis , mathematics , range (aeronautics) , set (abstract data type) , mechanics , magnetic field , geometry , computer science , materials science , quantum mechanics , composite material , programming language
Three outstanding rotating disk flows described by an exact solution of the Navier–Stokes equations are revisited in this paper. The purpose is to find out to what extent the corresponding boundary value problems can be mapped on each other by scaling transformations. The three addressed, and seemingly basically different axisymmetric flows are (A) the flow induced by a rough rotating disk , (B) the flow induced by a simultaneously rotating and radially stretching disk , and (C) the classical von Kármán swirl . The main results of the paper can be summarized as follows. (i) The continuous set of solutions of the problem (A) corresponding to ( λ = 0 , η > 0 ) is scaling‐equivalent to the solution of von Kármán, (C), (ii) Every given solution of the problem (A) with ( λ > 0 , η ≥ 0 ) generates by a suitable scaling transformation a unique solution of (B) with c > c c r i t = 2.3848 , (iii) every given solution of the problem (B) with a c > c c r i tcan generate a continuous set of solutions of the problem (A), (iv) in the stretching dominated subcritical range c < c c r i tof (B), no scaling‐equivalent solutions of the three problems exist. All the above results are also valid for the magnetohydrodynamic (MHD) extensions of the problems (A), (B), and (C). For these cases an approximate solution could be found in a closed analytical form which furnishes for large values of the magnetic parameter M highly accurate results.