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.

arXiv comment29 pages, 14 figures

References in corpus (6)

• Elastic turbulence in a polymer solution flow
• Elastic turbulence in curvilinear flows of polymer solutions
• A nonlinear elastic instability in channel flows at low Reynolds numbers
• Nonmodal amplification of stochastic disturbances in strongly elastic channel flows
• Geometric decomposition of the conformation tensor in viscoelastic turbulence
• Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids

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• Subcritical and supercritical bifurcations in axisymmetric viscoelastic pipe flows
• The mean conformation tensor in viscoelastic turbulence
• The dynamics of dense particles in vertical channel flows: gravity, lift and particle clusters
• On the large-Weissenberg-number scaling laws in viscoelastic pipe flows
• Experimental observation of the origin and structure of elasto-inertial turbulence
• Prediction of flow and elastic stresses in a viscoelastic turbulent channel flow using convolutional neural networks
• Perturbation symmetries in shear-thinning viscoelastic pipe flows and the Petrov-Galerkin implementation