Mechanics of variable-mass systems—Part 1: Balance of mass and linear momentum

Abstract

The equations of balance of momentum and energy usually are formulated under the assumption of conservation of mass. However, mass is not conserved when sources of mass are present or when the equations of balance are applied to a non-material volume. Mass then is said to be variable for the system under consideration. It is the scope of the present contribution to review the mechanical equations of balance for variable-mass systems. Our review remains within the framework of the classical, non-relativistic continuum mechanics of solids and fluids. We present general formulations and refer to various fields of applications, such as astronomy, machine dynamics, biomechanics, rocketry, or fluid dynamics. Also discussed are the equations for a single constituent of a multiphase mixture. The present review thus might be of interest to workers in the field of heterogeneous media as well. We first summarize the general balance law and review the Reynolds transport theorem for a non-material volume. Then the latter general formulations are used to derive and to review the equations of balance of mass and linear momentum in the presence of sources of mass in the interior of a material volume. We also discuss the appropriate modeling of such sources of mass. Subsequently, we treat the equations of balance of mass and linear momentum when mass is flowing through the surface of a non-material volume in the absence of sources of mass in the interior, and we point out some analogies to the previously presented relations. A strong emphasis is given in this article to the historical evolution of the balance equations and the physical situations to which they have been applied. The equations of balance of angular momentum and energy for variable-mass systems will be treated separately, in a second part of the review, to be published later. This review article cites 96 references.