Exact analytic solutions of nonlinear boundary value problems in fluid mechanics (Blasius equations)

Abstract

In this work it is shown that, by a series of admissible functional transformations, the generalized Blasius equation in fluids can be exactly reduced to a three-term generalized Emden–Fowler equation. Furthermore, the restricted in axisymmetric flows and simplified forms of this equation can be reduced to (i) two-term generalized Emden–Fowler equations; (ii) generalized Emden–Fowler equations; (iii) Emden–Fowler equations of the normal form; and (iv) Abel equations of the second kind. By means of a recently developed mathematical solution methodology (Panayotounakos, Fifth Greek Congress on Mechanics, Xania, Crete, 22–25 June 2004, Hellas, Greece), we provide exact analytic solutions for the simplified as well as for the restricted forms of the above-mentioned Blasius equations. Thus, it is proved that important, unsolvable in exact form problems in nonlinear fluid dynamics now can be analytically solved.

Exact analytic solutions of the porous media and the gas pressure diffusion ODEs in non-linear mechanics

“…The reduction procedure introduced in the paper and the constructed solutions are very general, and can be applied to a large number of nonlinear ODEs in mathematical physics and nonlinear mechanics including the Van der Pol nonlinear oscillator, the Blasius equation in fluids [7], the Langmuir equation in current flow, etc.…”

“…Our goal is the development of the construction of exact analytic solutions of the above equations based on a mathematical technique leading to the derivation of exact analytic solutions of the Abel equation of the second kind of the normal form (see Refs. [6], [7]). …”

“…[6,7], we will provide the exact analytic solution of the Kidder equation (2.1). Exactly the same mathematical technique follows the gas pressure diffusion equation (2.14).…”

Closed-Form Solutions of the Thomas-Fermi in Heavy Atoms and the Langmuir-Blodgett in Current Flow ODEs in Mathematical Physics

“…In what follows, based on a mathematical construction recently developed in [14,15] concerning the closed-form analytic solution of an Abel equation of the second kind of the normal form − = ( ), we provide the closed-form solutions of the reduced Abel equation 18, that is to say, the construction of the intermediate integral of the original TF equation 3in the phase plane, as well as the final solution in the physical plane in accordance with the given boundary conditions.…”

“…In this paper using series of admissible functional transformations, at first both LB and TF nonlinear ODEs are reduced to equivalent Abel's equations of the second kind of the normal form. According to a mathematical methodology recently developed [14][15][16], we extract closed-form solution of the above Abel equations in both phase and physical planes under initial data in accordance with the physical problems.…”