Physics Letters A 371 (2007) 39–40 www.elsevier.com/locate/pla Variational principle for two-dimensional incompressible inviscid flow Ji-Huan He Institute of Physics of Fibrous Soft Matter, Modern Textile Institute, Donghua University, 1882 Yan’an Xilu road, Shanghai 200051, China Received 26 January 2007; accepted 21 March 2007 Available online 24 March 2007 Communicated by A.R. Bishop Abstract A new special function is introduced, and a variational formulation is established for two-dimensional incompressible inviscid flow, which might find potential applications in numerical simulation and various inverse problems.  2007 Elsevier B.V. All rights reserved. Keywords: Navier–Stokes equation; Variational theory; Semi-inverse method 1. Introduction Recently variational principle for fluid mechanics has been caught much attention [1–8]. Variational-based finite element method has been, and continues to be, a popular numerical tool, however it is very difficult to establish a variational formulation for fluid mechanics [3,4]. For example Zuckerwar and Ash es- tablished a variational principle of the dynamics of a fluid with volume viscosity by introducing an additional field variable [9], however, Scholle found the established theory shows inconsis- tencies [10]. 2. Variational formulation This Letter considers a two-dimensional incompressible in- viscid flow, its governing equations are as follows [3] (1) u ∂u ∂x + v ∂u ∂y =- 1 ρ ∂p ∂x , (2) u ∂v ∂x + v ∂v ∂y =- 1 ρ ∂p ∂y , (3) ∂u ∂x + ∂v ∂y = 0, where u and v are the velocity components, ρ is a constant density, and p is pressure. E-mail address: jhhe@dhu.edu.cn. In order to apply the semi-inverse method [11,12] to es- tablish a variational formulation of the discussed problem, we re-write Eqs. (1) and (2) in conservative forms, which read (4) ∂ ∂x ( u 2 + P ) + ∂ ∂y (uv) = 0, (5) ∂ ∂x (uv) + ∂ ∂y ( v 2 + P ) = 0, where P = p/ρ . From Eqs. (4) and (5), we can obtain the following equation (6) ∂ 2 ∂x 2 ( u 2 + P ) - ∂ 2 ∂y 2 ( v 2 + P ) = 0. According to Eqs. (4) and (5), or Eq. (6) we can introduce a special function ϕ defined as (7) u 2 + P = ∂ 2 ϕ ∂y 2 , (8) v 2 + P = ∂ 2 ϕ ∂x 2 , (9) uv =- ∂ 2 ϕ ∂x∂y . Our aim is to establish a variational formulation whose station- ary conditions satisfy Eqs. (6)–(8) and (3). By the semi-inverse 0375-9601/$ – see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.03.044