On the necessity of surface growth terms for the consistency of jump relations at a singular surface

Abstract

When a singular surface is present in the region under consideration, the local forms of the equations of balance must be accompanied by jump relations in order to connect the different values of the discontinuous entities at the two sides of the singular surface. The classical general form of these jump relations is provided by the Kotchine relation. In the present contribution, we exemplary deal with balance of mass, momentum and kinetic energy, as well as with balance of total energy, denoted as the first law of thermodynamics, and with balance of internal energy. We first derive an extension of the Kotchine relation with respect to surface growth terms. By a surface growth term we mean an entity which equivalently describes the balance of some quantity associated with the material that is instantaneously located at or in the vicinity of the singular surface. In a more detailed modelling, a singular surface is often described by a thin shell-type region or layer of transition. Surface growth terms may represent a non-vanishing rate of change of the respective quantity, or they may characterise some sources of this quantity. As a main result of the present paper, we show that surface growth terms are needed in order to ensure consistency between the jump relations for balance of mass, momentum and kinetic energy, and between the jump relations for balance of total energy and balance of internal energy. Even when surface growth terms for mass, balance of momentum and total energy are absent, one generally must take into account surface growth terms for balance of kinetic energy and balance of internal energy. The presented results refer to both, a singular surface, as well as to a thin region of transition. The jump relations in the latter case are referred to an equivalent singular surface, replacing the region of transition. The uni-axial flow of a fluid in a diffusor is used to demonstrate that the present results may be used even in cases in which the region of transition is not thin.