A derivation of the Green-Naghdi equations for irrotational flows

A new derivation of the Green-Naghdi (GN) equations for 'sheet-like' flows is made by use of the principle of virtual work. Divergence-free virtual di splacements are used to formulate the momentum equations weakly. This resul ts in the elimination of the internal pressure from the GN equations. As is well-known in particle dynamics, the principle of virtual work can be inte grated to obtain Hamilton's principle. These integrations can be performed in a straightforward manner when the Lagrangian description of fluid motion is adopted. When Hamilton's principle is written in an Eulerian reference frame, terms must be added to the Lagrangian to impose the Lin constraint t o account for the difference between the Lagrangian and Eulerian variables (Lin). If, however, the Lin constraint is omitted, the scope of Hamilton's principle is confined to irrotational flows (Bretherton). This restricted H amilton's principle is used to derive the new GN equations for irrotational flows with the same kinematic approximation as in the original derivation of the GN equations. The resulting new hierarchy of governing equations for irrotational flows (referred to herein as the IGN equations) has a conside rably simpler structure than the corresponding hierarchy of the original GN governing equations that were not limited to irrotational flows. Finally, it will be shown that the conservation of both the in-sheet and cross-sheet circulation is satisfied more strongly by the IGN equations than by the or iginal GN equations.