Continuum and multi-scale simulation of mixed kinematics polymeric flows with stagnation points: Closure approximation and the high Weissenberg number problem

Abstract

It is a well known fact that the upper We limit encountered in continuum level viscoelastic flow simulations with typical constitutive equations for dilute polymeric solutions that predict bounded extensional viscosities in geometries with internal stagnation points on solid surfaces is a strong function of the strain hardening nature of the fluid. Specifically, the upper We limit decreases as the level of strain hardening of the fluid is increased. To provide further insight into the high We limitations in continuum level computations of this class of flows, we have performed extensive continuum and multiscale flow simulations in two benchmark flow problems, namely sedimentation of sphere in a tube and flow past a cylinder in a channel, utilizing the FENE-P (continuum and Brownian Configuration Fields (BCF)) and Giesekus (continuum) constitutive equations as well as the FENE (BCF) dumbbell micromechanical model. Extremely large stress gradients in the axial normal stress along the plane of symmetry in the wake of the cylinder and sphere are observed in FENE-P and Giesekus predictions using both multiscale and continuum numerical techniques at We of O(1) where the numerical simulations begin to breakdown for significantly strain hardening fluids, i.e., b > 300, α < 0.005. The existence of very large localized polymeric stresses and stress gradients is shown to be the consequence of significant over prediction of macromolecular extension by the closed form constitutive equations in the extensionally dominated region of the flow. The inability of the aforementioned constitutive equations to accurately describe the flow microstructure coupling downstream of stagnation points in complex kinematics flows should motivate use of regularization techniques for the polymeric stress or utilization of more sophisticated constitutive equations in flows with strong straining components.

Introduction

Polymer dynamics in strong flows typically encountered downstream of stagnation points in complex kinematics flows play a central role in determination of many essential fluid dynamics quantities. Examples include, frictional drag on particles and particle-particle interactions [17], drop and bubble shape evolution and dynamics [37], [53], [1], film thickness in coating flows [29] and a host of internal and free surface instabilities [33], [32], [38], [46], [47], [30], [29], [7], [8]. To this end, the non-Newtonian fluid community has developed a number of benchmark problems, namely, flow past a cylinder confined in a channel [2], [3] and sedimentation of sphere in a tube [58], [1] to examine flow micro-structure coupling downstream of stagnation points on the flow properties including hydrodynamics drag on the objects as well as flow transitions.

Most of the aforementioned studies have had as their focus continuum level computations based on well known constitutive equations for dilute polymeric solutions such as the Oldroyd-B (OLD- B) and Upper Convected Maxwell (UCM) models. However, recently, computations of flow past a cylinder in a channel with the OLD-B constitutive equation has provided evidence of divergent normal stresses in the wake of the cylinder at Weissenberg numbers (We, defined as the product of the mean polymer relaxation time and a characteristic shear rate) of order 1 [4]. Specifically, it has been suggested that this divergence is linked to the unbounded extensional viscosity in the wake of the cylinder. These recent observations based on self-consistent continuum level simulations support earlier assertions by Rallison and Hinch [44] that Hookean dumbbell based constitutive equations such as the UCM, and OLD-B can give rise to unbounded or very large stresses in strong straining flows. Specifically, it was suggested that in presence of unbounded stresses, if the stress tensor is either divergent free or has infinite divergence that is highly spatially localized, there is a negligible effect on the flow, i.e., the flow will not adapt to inhibit infinite stresses or stress gradients.

The influence of finite extensibility of macromolecules on the stress singularity in internal stagnation flows that exhibit divergent free stresses has also been subject of a number of studies in the past decade. Renardy [45] utilizing an idealized planar extensional Newtonian flow kinematics has shown that for the Giesekus model, which limits the growth of extensional stresses, although the stress profiles remain finite, the stress gradients can become infinite if α, the Giesekus mobility parameter, is less than $(2-\sqrt{3})/4$. Similar behavior has also been observed in idealized uniaxial extensional Newtonian flow kinematics [5], [6]. Specifically, it has been shown that although the FENE-P model predicts finite stresses at any We, for sufficiently large bs, i.e., square of the maximum molecular extension, there exist a range of We for which the stress gradients become infinite.

Although to date, observation of infinite stress gradients in strong straining flows with constitutive models that bound extensional stresses has been based on idealized Newtonian kinematics in flows that exhibit divergent free stresses (e.g., flow generated near internal stagnation points in cross-slot and four-roll mill flows), the existence of infinite stress gradients in strong straining flows in a specific parameter range that correspond to highly strain hardening fluids is very intriguing. In fact, one might question the relevance of this interesting finding to the strong dependence of the upper We limit on the finite extensibility parameter in careful flow computations of dilute polymeric solutions with constitutive equations that bound extensional stresses (e.g., Chilcott–Rallison [12], FENE-P [58], [1], the multimode version of the Verhoef et al. model [54], [1] in geometries with internal stagnation points on solid surfaces (e.g., flow past a cylinder in a channel or sedimentation of a sphere in a tube).

Currently, the cause of the upper We limit in computation of flows with internal stagnation points on solid surfaces is not known. Clearly, if there is a connection between the aforementioned infinite extensional stresses and the computed stresses in the wake of the cylinder or sphere then one can correlate the existence of an upper bound in We with appearance of infinite stress gradients in this region. On the other hand the breakdown of the numerical solution at high We could be solely due to deficiency of numerical techniques [4]. To this end, Fattal et al. [41], [42] have suggested that the high We limit could be the result of a numerical instability that can be overcome by use of a variable transformation, namely, a logarithmic transformation of the stresses. However, careful numerical computations of flow past a cylinder in a channel by Hulsen et al. [22] produced converged results with the same upper bound in We with or without use of this transformation. Finally, it should be noted that the multiscale Brownian Configuration Fields method (BCF) does not suffer from the aforementioned numerical instability. However, one can not correlate the robustness of this simulation technique and its ability to produce converged numerical results at high We [23], [28], [27] solely to this fact since in the BCF formulation a vector field, namely, the connectivity vector of the micromechanical model is convected by the fluid motion (for example in the FENE model) as oppose to a tensor field, namely, the conformation of the macromolecule in the corresponding closed form constitutive equation, i.e., the FENE-P model [36].

To provide further insight into the high We limitations in continuum level computations of flows with internal stagnation points on solid surfaces, we have performed extensive continuum and multiscale flow simulations in two benchmark flow problems, namely sedimentation of sphere in a tube and flow past a cylinder in a channel, utilizing the FENE-P (continuum and BCF) and Giesekus (continuum) constitutive equations as well as the FENE (BCF) dumbbell micromechanical model. Extremely large stress gradients in the axial normal stress along the plane of symmetry in the wake of the cylinder and sphere are observed in both FENE-P and Giesekus predictions using both multiscale and continuum numerical techniques at We of O(1) where the numerical simulations begin to breakdown for significantly strain hardening fluids, i.e., b > 300, α < 0.005. These very large stress gradients (nearly singular) are a direct consequence of significant over prediction of macromolecular extension by the closed form constitutive equations in the extensionally dominated region of the flow. The inability of the aforementioned constitutive equations to accurately describe the flow microstructure coupling downstream of stagnation points in complex kinematics flows should motivate use of regularization techniques for the stresses [9], [57], [56] or utilization of more sophisticated constitutive equations [24], [35], [15] in flows with strong straining components.