Perturbative expansions of the conformation tensor in viscoelastic flows

Abstract

We consider the problem of formulating perturbative expansions of the conformation tensor, which is a positive-definite tensor representing polymer deformation in viscoelastic flows. The classical approach does not explicitly take into account that the perturbed tensor must remain positive definite-a fact that has important physical implications, e.g. extensions and compressions are represented similarly to within a negative sign, when physically the former are unbounded and the latter are bounded from below. Mathematically, the classical approach assumes that the underlying geometry is Euclidean, and this assumption is not satisfied by the manifold of positive definite tensors. We provide an alternative formulation that retains the conveniences of classical perturbation methods used for generating linear and weakly nonlinear expansions, but also provides a clear physical interpretation and a mathematical basis for analysis. The approach is based on treating a perturbation as a sequence of successively smaller deformations of the polymer. Each deformation is modelled explicitly using geodesics on the manifold of positive-definite tensors. Using geodesics, and associated geodesic distances, is the natural way to model perturbations to positive-definite tensors because it is consistent with the manifold geometry. Approximations of the geodesics can then be used to reduce the total deformation to a series expansion in the small perturbation limit. We illustrate our approach using direct numerical simulations of the nonlinear evolution of Tollmien-Schlichting waves. †

Critical-Layer Structures and Mechanisms in Elastoinertial Turbulence

“…In both analyses, the concept of critical layers, i.e., wall-normal positions where the fluid velocity equals the wavespeed of an eigenmode or resolvent mode, is important. While some recent studies suggest the importance of critical-layer mechanisms in viscoelastic shear flows [12,[15][16][17], they do not make as direct a connection to EIT as we illustrate here. Figure 3a shows the result of linear stability analysis (the eigenvalues c) for Wi = 20, k x L x /2π = 2, k z = 0, the wavenumber corresponding to the dominant structures observed in the nonlinear simulations.…”

“…Typically, about 200 Chebyshev polynomials are sufficient for the resolvent calculations, whereas as many as 400 are required for the TS eigenmode. The norm used in the resolvent calculations is the sum of the kinetic energy and a measure of the conformation tensor perturbation magnitude that is consistent with the non-Euclidean geometry of positive-definite tensors [12]. Nonlinear simulation results: Fig.…”

Self-sustained elastoinertial Tollmien–Schlichting waves

“…where A is the conformation tensor in the laminar state. The second term provides a measure of the conformation tensor perturbation magnitude that is motivated by the non-Euclidean geometry of positive-definite tensors (Hameduddin et al 2019). For both the linear stability and resolvent analyses, the equations are discretized with a Chebyshev pseudospectral method using 401 Chebyshev polynomials.…”

Dilute suspension of neutrally buoyant particles in viscoelastic turbulent channel flow

“…The algorithm for the numerical solution of the conformation-tensor equations (2.3) was adopted in a number of high-fidelity simulations of instability waves and transition (Lee & Zaki 2017; Hameduddin, Gayme & Zaki 2019) as well as fully turbulent flows (Hameduddin et al. 2018; Hameduddin & Zaki 2019).…”