On a general balance law for continua with an interface
In the literature two different types of reasoning are developed in order to obtain local evolution equations and jump conditions for a continuous system with an interface (C,S). The first one starts with an integral balance equation valid only for the whole system. The second one, on the other hand, starts assuming a general integral balance equation valid for all material volumes included in (C,S). The local form of this integral equation is then obtained owing to the arbitrariness of material volumes. Notwithstanding its more clear physical justification this second approach makes use of the concept of material volume, which is ambiguous when this volume includes a portion of a nonmaterial (i.e. not made up with always the same particles) and adsorbing interface (for instance in the case of phase transition). We formulate a general integral balance law (GIBL) valid for a continuously moving family of Eulerian volumes at any instant t included in the region \(C(t)\subset R^ 3\) occupied by the continuous system (C,S). We define (when the projection of velocity on S is continuous across S) a family \(\{c_ m(t)\}\) of Eulerian regions which can be regarded as a ``first order'' material volume and assume that for this \(\{c_ m(t)\}\) the GIBL holds. It results that it reads as the usually postulated integral balance law for material volumes. We deduce even when the projection of the velocity on S is discontinuous across S the local balance law by applying GIBL to a fixed arbitrary Eulerian volume included in (C,S). We finally observe that when applied to the region occupied at the instant t by the whole system (C,S), GIBL reduces to a global integral balance law.
scientific article; zbMATH DE number 4024963
scientific article