Computing the yield limit in three-dimensional flows of a yield stress fluid about a settling particle

https://doi.org/10.1016/j.jnnfm.2020.104374Get rights and content

Highlights

  • We directly compute the yield limit of viscoplastic flows around a settling particle.

  • No iteration in the rheological parameters is needed to approach the stopping limit.

  • The method is based on primal–dual optimization and a consistent discretization.

  • Its results are compared against accurate axisymmetric flow computations.

  • Sample computations for 3D particle shapes without axial symmetry are presented.

Abstract

Calculating the yield limit Yc (the critical ratio of the yield stress to the driving stress), of a viscoplastic fluid flow is a challenging problem, often needing iteration in the rheological parameters to approach this limit, as well as accurate computations that account properly for the yield stress and potentially adaptive meshing. For particle settling flows, in recent years calculating Yc has been accomplished analytically for many antiplane shear flow configurations and also computationally for many geometries, under either two dimensional (2D) or axisymmetric flow restrictions. Here we approach the problem of 3D particle settling and how to compute the yield limit directly, i.e. without iteratively changing the rheology to approach the yield limit. The presented approach develops tools from optimization theory, taking advantage of the fact that Yc is defined via a minimization problem. We recast this minimization in terms of primal and dual variational problems, develop the necessary theory and finally implement a basic but workable algorithm. We benchmark results against accurate axisymmetric flow computations for cylinders and ellipsoids, computed using adaptive meshing. We also make comparisons of accuracy in calculating Yc on comparable fixed meshes. This demonstrates the feasibility and benefits of directly computing Yc in multiple dimensions. Lastly, we present some sample computations for complex 3D particle shapes.

Introduction

The ability to resist a shear stress at rest is the key qualitative feature of a yield stress fluid. This feature arises in the earliest classical flows studied with simple yield stress fluid models: for sufficiently large yield stresses: dense particles are statically suspended in fluid, a layer of yield stress fluid on a vertical surface does not flow, a stress-controlled vane viscometer cannot turn, etc. This competition is captured physically as the dimensionless balance Y, between the yield stress and whatever is the driving force (stress) of the flow, e.g. pressure drop in flow along a pipe, buoyancy in settling of a particle or in bubble rise. Many 1D flows allow analysis and explicit determination of limiting values of the yield number Y, such that there is no flow for YYc. General determination of Yc remains elusive although Yc has been given a formal mathematical definition for various classes of flow e.g. [1], [2], the roots of which go back to the 1965 analysis of [3] for anti-plane shear flows.

In this paper our concern is computation of Yc for 3D settling particles and more specifically direct computation. To explain direct, we first discuss more general particle computations for yield stress fluids. The best known work here by Beris et al. [4] considered flow around a sphere and was the first to convincingly frame the theoretical question of a limiting Y and to calculate it. There have since been extensive calculations of flows around isolated axisymmetric and 2D particles e.g. [1], [5], [6], [7], [8], [9], [10], [11], [12], [13]. For Stokes flows we must distinguish 2 different formulations for these problems: a resistance problem and a mobility problem. Simply put, a resistance problem imposes motion on a particle and computes the force & moment; a mobility problem imposes the forces/moments and computes velocity. Either formulation can be used to estimate Yc. For the mobility formulation, for fixed imposed forces/moments one increases the yield stress incrementally until a zero velocity is computed. For the resistance formulation an asymptotic approach is needed, via rescaling of variables, again with successive computations; see e.g. [12].

The need to compute a flow at successively large values of a dimensionless parameter makes these methods iterative or indirect, in terms of finding Yc. The limiting flows are generally characterized by diminishing relevance of the viscous dissipation with respect to the plastic dissipation. This leads naturally to the question of whether we can simply neglect the viscous aspect of the flow a priori, whether such a computation is viable and whether the solution allows us to compute the limit Yc directly. The motivation for this is partly intellectual and partly pragmatic. Calculation of Yc for 2D and axisymmetric flows using indirect methods has been most convincing when an augmented Lagrangian method has been used and when some form of mesh adaptivity is implemented, so that the mesh can deform to relevant geometric features of the limiting flow. This approach was originally developed by Saramito and co-workers [14] and has been used extensively over the past decade for particle settling, e.g. [1], [12]. Although highly effective, the mesh refinement requires successive computation and each flow computation using augmented Lagrangian approaches is itself slow to converge iteratively. While viable in 2D/axisymmetric situations, moving to 3D has not been explored. Indeed there are relatively few 3D particle computations for viscoplastic fluids at all and none that we know with adaptive meshing and augmented Lagrangian methods. Coupling this to an iterative parametric increase to attain Yc is likely to be prohibitive computationally. Secondly, numerical implementation of the yielding problem via Mobility ([M]) formulations to calculate Yc is not always trivial which leads to the alternative usage of Resistance ([R]) formulations to study this limit (the definitions of [R] and [M] formulations are in Section 2). Investigating the yield limit using the [R] formulations has its own drawbacks as well. For instance, calculating the yield limit needs some computations at moderate/finite yield stresses and then extrapolating to an infinitely large yield stress which is: (i) time-consuming; although some new accelerating augmented Lagrangian approaches have been proposed recently such as FISTA [15] and PAL method [16] and (ii) sensitive to extrapolation procedure as well (for instance, see Fig. 23 of [10]). Thus, direct methods become attractive.

The idea of a direct calculation is not new. For 1D shear flows, we often first compute the shear stress and then the velocity. If the shear stress does not exceed the yield stress there is no flow and no need to compute the velocity: a direct calculation of Yc, e.g. for flow in a channel/pipe we require the wall shear stress to exceed the yield stress. The analysis of Mosolov & Miasnikov [3] turned the limiting variational problem (velocity minimization) into a geometric problem, solved directly for Yc: see the almost algorithmic application of this in [17]. In [18] we have extended the scope of [3] to anti-plane particle settling flows, and used a set theoretic approach which again directly calculates Yc without recourse to the viscous solutions. More general anti-plane flows, including non-convex and multiple particle arrays were considered in [19], again for anti-plane shear flows.

In moving to 2D flows there are physically based direct approaches such as the theory of perfect plasticity, which involves constructing a network of sliplines around a particle. As explored in [12] the slipline method does not always produce velocity and stress fields that are representative on the limiting flow, although sometimes the estimates of Yc are exact. The latter often happens when the flow has a specific geometric structure that allows one to estimate terms in the optimization problem directly. Thus, for example in [20] flow onset in a convectively driven flow was observed as yielding along the boundary of a cylindrical domain; an observation that allowed direct calculation of Yc, albeit heuristically.

In the broader context, there are other reasons for calculating Yc. First, as well as being a limiting value for the static steady problem, for many flows YYc also means that the associated unsteady flows converge to the static steady state, i.e. stability; see [2], [21]. This type of result, originally established by [22] for 1D channel flow, has much wider application, as discussed in [23]. A long term goal of work in this area is to be able to characterize statically stable limits for viscoplastic suspensions i.e. particles in a simple yield stress fluid. Industrially used suspensions are often buoyant and it is of some value to understand idealized behaviors. There are a number of computational studies that deal with multiple particles, e.g. [24], [25], [26], [27], [28], [29] in both steady and transient situations. Ostensibly these studies are focused at the micro-scale hydrodynamics, and they do uncover details of yielding, bridging between particles and contact. However, it is not clear what the next step is for such studies: unlike Newtonian fluids, the viscoplastic Stokes equations are not linear in the velocity, so superposition principles fail and methods such as Stokesian dynamics are not valid. A pure computational approach has been followed in [21], [30] in which a fictitious domain approach is used. This approach models the full fluid–solid domain and is implemented within the augmented lagrangian framework. Although suited to suspensions the computational algorithms are relatively slow to converge at each timestep and transient calculations have thus been limited to small numbers of particles in two dimensions. Recent work [31] has looked at steady Stokes flow of 2D particles arranged in random configurations, to represent a given solids fraction. These calculations are interesting in that, by repeating with successively large yield stress, we begin to estimate the critical yield limit Yc of a suspension, i.e. at which the entire suspension is static. Also the relative economy of the Stokes flow calculations means that a good level of mesh refinement is feasible. We are beginning to uncover some of the true complexity of the stress field in suspensions.

An outline of the paper is as follows. Below (Section 2) we introduce the physical problem and give the formal definition of Yc. We show that the primal problem has a solution in a subspace of the space of vector fields of bounded deformation BD (Theorems 1 & 2) and then explore two dual problem formulations. Section 3 explains the computational algorithms for both the direct method (Section 3.1) and for axisymmetric particle computations carried out using a more conventional method (Section 3.2). In Section 4 we present a range of results for flows around 3D particles (axisymmetric or not, convex or not). The paper finishes with a discussion and conclusions.

Section snippets

Problem statement

In this paper we focus on the motion of an isolated particle in a large bath of yield-stress fluid. We are specifically interested in computing the static stability of particles or the yield limit, i.e. when the force on the particle is just enough to move it. Hence we only consider inertia-less flows. The particle is denoted by X, X is the boundary of the particle, Ω represents the entire domain (fluid+particle) and Ω is its outer boundary.

For any fixed finite yield stress and body force on

Computational methods

This paper is focused at proposing a general method for computing Yc directly, based on the primal–dual formulation of the previous section. We develop and describe this method below in Section 3.1. In order to benchmark this method we test against computations carried out on axisymmetric particles, which are computed using the full viscoplastic problem. We outline this method in Section 3.2.

Results

We first present comparative results between the full viscoplastic flow computations and those of the direct method. These comparisons are confined to axisymmetric particles. Having established that the direct method is effective, we present new results on non-axisymmetric particles, to act as benchmarks for future computation.

Conclusions

Although the yield limit in many classical yield-stress fluid mechanics problem is well-documented as the maximization/minimization problems [55], practical methods for capturing this important limit are mostly based on the full computations of the Stokes equations. This involves computing the flow in a yield stress fluid and iteratively approaching the yield limit by increasing Bingham/yield numbers in [R]/[M] formulations. To ensure an accurate computation, augmented Lagrangian methods and

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work has been supported by the Austrian Science Fund (FWF) within the national research network ‘Geometry+Simulation’, project S11704. Part of the research has been carried out at the University of British Columbia, supported by Natural Sciences and Engineering Research Council of Canada via their Discovery Grants programme (Grant No. RGPIN-2015-06398).

References (62)

  • TreskatisT. et al.

    Practical guidelines for fast, efficient and robust simulations of yield-stress flows without regularisation: a study of accelerated proximal gradient and augmented Lagrangian methods

    J. Non-Newton. Fluid Mech.

    (2018)
  • DimakopoulosY. et al.

    The PAL (Penalized Augmented Lagrangian) method for computing viscoplastic flows: A new fast converging scheme

    J. Non-Newton. Fluid Mech.

    (2018)
  • HuilgolR.R.

    A systematic procedure to determine the minimum pressure gradient required for the flow of viscoplastic fluids in pipes of symmetric cross-section

    J. Non-Newton. Fluid Mech.

    (2006)
  • WachsA. et al.

    Particle settling in yield stress fluids: limiting time

    J. Non-Newton. Fluid Mech.

    (2016)
  • LiuB.T. et al.

    Interactions of two rigid spheres translating collinearly in creeping flow in a Bingham material

    J. Non-Newton. Fluid Mech.

    (2003)
  • TokpaviD.L. et al.

    Interaction between two circular cylinders in slow flow of Bingham viscoplastic fluid

    J. Non-Newton. Fluid Mech.

    (2009)
  • FahsH. et al.

    Pair-particle trajectories in a shear flow of a Bingham fluid

    J. Non-Newton. Fluid Mech.

    (2018)
  • YuZ. et al.

    A fictitious domain method for dynamic simulation of particle sedimentation in Bingham fluids

    J. Non-Newton. Fluid Mech.

    (2007)
  • AttouchH. et al.

    Duality for the sum of convex functions in general Banach spaces

  • TreskatisT. et al.

    An accelerated dual proximal gradient method for applications in viscoplasticity

    J. Non-Newton. Fluid Mech.

    (2016)
  • ChaparianE. et al.

    Cloaking: particles in a yield-stress fluid

    J. Non-Newton. Fluid Mech.

    (2017)
  • ChaparianE. et al.

    An adaptive finite element method for elastoviscoplastic fluid flows

    J. Non-Newton. Fluid Mech.

    (2019)
  • DubashN. et al.

    What is the final shape of a viscoplastic slump?

    J. Non-Newton. Fluid Mech.

    (2009)
  • BerisA. et al.

    Creeping motion of a sphere through a Bingham plastic

    J. Fluid Mech.

    (1985)
  • TZ. et al.

    Viscoplastic flow around a cylinder kept between parallel plates

    J. Non-Newton. Fluid Mech.

    (2002)
  • Deglo de BessesB. et al.

    Sphere drag in a viscoplastic fluid

    AIChE J.

    (2004)
  • ChaparianE. et al.

    Yield limit analysis of particle motion in a yield-stress fluid

    J. Fluid Mech.

    (2017)
  • FrigaardI. et al.

    Critical yield numbers of rigid particles settling in Bingham fluids and cheeger sets

    SIAM J. Appl. Math

    (2017)
  • IglesiasJ.A. et al.

    Critical yield numbers and limiting yield surfaces of particle arrays settling in a Bingham fluid

    Appl. Math. Optim.

    (2020)
  • KarimfazliI. et al.

    A novel heat transfer switch using the yield stress

    J. Fluid Mech.

    (2015)
  • MB.

    Application de la méthode des éléments finis à la résolution numérique d’inéquations variationnelles de type Bingham

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