On Seliger and Whitham’s variational principle for hydrodynamic systems from the point of view of ‘fictitious particles’

Abstract

According to Miles [9, p. 103], variational principles in fluid dynamics have been associated with important names as, e.g., Finlayson [2], Luke [7], Miloh [10], Seliger and Whitham [12], Serrin [13], Truesdell and Toupin [14]. In the paper by Seliger and Whitham [12], a comprehensive treatment for problems within fluid dynamics, plasma dynamics and elasticity is presented, where general variational principles are stated for a given system of equations. In the context of fluid dynamics, difficulties arisen from the usage of the Eulerian description are overcome by representing the Eulerian velocity through a representation for isentropic flow, first suggested by Clebsh in 1859, and by introducing Lagrange multipliers as variational constraints that enable one to recover the missed identity of the fluid particles. In other words, Seliger and Whitham [12] emphasize that the traditional form of Hamilton’s principle, convenient within the Lagrangian description, must be equipped with side conditions, within the Eulerian approach, if non-trivial flows are studied. As suggested by Lin [6], the identity is preserved through the invariance of the initial coordinates of each particle along its path. Seliger and Whitham [12] showed more: just one initial coordinate is enough. Moreover, such an addition of side conditions enables one to elegantly write the Lagrangian density solely as the pressure field. This is a quite interesting and general result, being valid for rotational and compressible flows. The point to be reinforced is that the paper by Seliger and Whitham bridges the Lagrangian and the Eulerian descriptions through Hamilton’s principle. Whereas the usual form of Hamiltons’s principle is written within a Lagrangian description (see [12], p. 4, Eq. (8)), i.e., considering a “sum over particles” or “volume of the material”, Seliger and Whitham’s version of Hamilton’s principle is written within an Eulerian description, for an arbitrary region of the space. Indeed, in hydrodynamics, many problems are preferably treated by taking open (control) volumes. In the context of marine hydrodynamics, for example, the water impact problem and the oscillating floating body problem are important examples to be mentioned. In the first problem, see, e.g., [5] the splashing jets are cut out and only the ‘bulk’ of the liquid is considered for the sake of formulating the problem mathematically. Thus, the so-called jet root plays the role of a moving control surface, so enabling kinetic energy to flow through it. In the second mentioned problem, when the radiated waves are taken into account, the liquid domain is usually supposed to be bounded by a distant vertical cylindrical surface, a fixed control surface, through which