Scaling approach in solving problems of complex viscoelastic flows with multi-mode constitutive equations of differential type

Abstract

For multi-mode constitutive equations (CEs), modal independence immediately results in a scaling theorem, which in cases with known flow histories, reduces the solution of any 3D unsteady and nonisothermal problem to the computations for a single Maxwell mode. Although it is easy to extend the scaling theorem for the compressible case, for simplicity, only the incompressible version is considered in the chapter. The general unified formulation for multi-mode approaches is discussed. Some similarity assumptions are made which make it possible to prove the scaling theorem. There are two examples –– start-up simple shear flows and scaling in computations of complex flows––that illustrate the use of the scaling approach for reducing computations for complex viscoelastic flows. The chapter proves the scaling theorem for very broad formulations of multimode CEs of Maxwell type. However, the very existence and stability of solutions of the CEs will highly depend on such fundamental properties as the thermodynamic consistency and stability of CEs.