Matrix approach to Lagrangian fluid dynamics

Abstract

A new approach to ideal-fluid hydrodynamics based on the notion of continuous deformation of infinitesimal material elements is proposed. The matrix approach adheres to the Lagrangian (material) view of fluid motion, but instead of Lagrangian particle trajectories, it treats the Jacobi matrix of their derivatives with respect to Lagrangian variables as the fundamental quantity completely describing fluid motion.A closed set of governing matrix equations equivalent to conventional Lagrangian equations is formulated in terms of this Jacobi matrix. The equation of motion is transformed into a nonlinear matrix differential equation in time only, where derivatives with respect to the Lagrangian variables do not appear. The continuity equation that requires constancy of the Jacobi determinant in time takes the form of an algebraic constraint on the Jacobi matrix. An accompanying linear consistency condition, which is responsible for the dependence on spatial variables and does not include time derivatives, ensures completeness of the system and reconstruction of the particle trajectories by the Jacobi matrix.A class of exact solutions to the matrix equations that describes rotational non-stationary three-dimensional motions having no analogues in the conventional formulations is also found and investigated. A distinctive feature of these motions is precession of vortex lines (rectilinear or curvilinear) around a fixed axis in space. Boundary problems for the derived exact solutions including matching of rotational and potential motions across the boundary of a vortex tube are addressed. In particular, for the cylindrical vortex of elliptical cross-section involved in three-dimensional precession, the outer potential flow is constructed and shown to be a non-stationary periodic straining flow at a large distance from the vortex axis.