The kernel-conformation constitutive laws

Abstract

This work presents a generic kernel-conformation tensor transformation for a large class of differential constitutive models, in which several matrix kernel-transformation families can be applied to the conformation tensor evolution equation. The discretization of this generic analytical framework generates numerical results consistent with the standard conformation tensor discretization at low Weissenberg numbers (Wi). The numerical efficiency of High Weissenberg Number Problems is ultimately related with both the specific kernel function used in the matrix transformation, but is also related to the existence of mathematical singularities introduced either by the physical description of the flow or by the characteristics of the adopted constitutive equation. We also show that the log-conformation [R. Fattal, R. Kupferman, J. Non-Newtonian Fluid Mech., 123(2-3) (2004) 281-285] and the square-root-conformation [N. Balci, B. Thomases, M. Renardy, C.R. Doering, J. Non-Newtonian Fluid Mech., 166(11) (2011) 546-553] approaches are two relevant particular cases of the kernel-conformation tensor transformation. Simulations for the benchmark confined cylinder flow of an Oldroyd-B fluid provide a preliminary assessment of the merits of some tested transformation functions.