Lagrange crisis and generalized variational principle for 3D unsteady flow

• Published in: International journal of numerical methods for heat & fluid flow Vol. 30; no. 3; pp. 1189 - 1196
• Main Author: He, Ji-Huan
• Format: Journal Article
• Language: English
• Published: Bradford Emerald Publishing Limited 02.03.2020; Emerald Group Publishing Limited
• Subjects: Calculus; Crises; Design techniques; Flow control; Fractals; Fractional calculus; Inverse method; Lagrange multiplier; Mathematical analysis; Methods; Numerical analysis; Potential flow; Three dimensional flow; Unsteady flow; Variational principles; Variational theory; Plasma; Semi-inverse method; Burger equation
• ISSN: 0961-5539; 0961-5539; 1758-6585
• DOI: 10.1108/HFF-07-2019-0577

Purpose A three-dimensional (3D) unsteady potential flow might admit a variational principle. The purpose of this paper is to adopt a semi-inverse method to search for the variational formulation from the governing equations. Design/methodology/approach A suitable trial functional with a possible unknown function is constructed, and the identification of the unknown function is given in detail. The Lagrange multiplier method is used to establish a generalized variational principle, but in vain. Findings Some new variational principles are obtained, and the semi-inverse method can easily overcome the Lagrange crisis. Practical implications The semi-inverse method sheds a promising light on variational theory, and it can replace the Lagrange multiplier method for the establishment of a generalized variational principle. It can be used for the establishment of a variational principle for fractal and fractional calculus. Originality/value This paper establishes some new variational principles for the 3D unsteady flow and suggests an effective method to eliminate the Lagrange crisis.