Non Conservative Products in Fluid Dynamics

TL;DR

This paper develops a mathematical framework for fluid flow in pipes with complex geometries, addressing the challenges of non-conservative products and establishing existence of solutions for such systems.

Contribution

It introduces a new definition of solutions for BV geometries with non-conservative sources and proves their existence, including the role of curvature in smooth pipes.

Findings

Existence of solutions for BV geometries with non-conservative products

Justification of curvature effects in the momentum equation for smooth pipes

A general approach to balance laws with non-conservative source terms

Abstract

Fluid flow in pipes with discontinuous cross section or with kinks is described through balance laws with a non conservative product in the source. At jump discontinuities in the pipes' geometry, the physics of the problem suggests how to single out a solution. On this basis, we present a definition of solution for a general BV geometry and prove an existence result, consistent with a limiting procedure from piecewise constant geometries. In the case of a smoothly curved pipe we thus justify the appearance of the curvature in the source term of the linear momentum equation. These results are obtained as consequences of a general existence result devoted to abstract balance laws with non conservative source terms.