A novel 3D model for non‐Newtonian fluid flows in a pipe network

Abstract

In this paper, we propose a novel mathematical model that describes steady‐state 3D flows of a non‐Newtonian fluid with shear‐dependent viscosity in a pipe network. Our approach is based on the rejection of averaging of the velocity field and the application of conjugation conditions that provide the mass balance for interior joints of the network. Using boundary conditions involving the pressure, we formulate the corresponding boundary value problem and introduce the concept of weak solutions by integral indentities. The main result of the work is an existence theorem in the class of weak solutions for large data. The proof of this result is performed with the help of the Galerkin procedure, methods of the topological degree theory, the technique of monotone operators as well as compactness arguments.