# Critical Yield Numbers and Limiting Yield Surfaces of Particle Arrays Settling in a Bingham Fluid

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## Abstract

We consider the flow of multiple particles in a Bingham fluid in an anti-plane shear flow configuration. The limiting situation in which the internal and applied forces balance and the fluid and particles stop flowing, that is, when the flow settles, is formulated as finding the optimal ratio between the total variation functional and a linear functional. The minimal value for this quotient is referred to as the critical yield number or, in analogy to Rayleigh quotients, generalized eigenvalue. This minimum value can in general only be attained by discontinuous, hence not physical, velocities. However, we prove that these generalized eigenfunctions, whose jumps we refer to as limiting yield surfaces, appear as rescaled limits of the physical velocities. Then, we show the existence of geometrically simple minimizers. Furthermore, a numerical method for the minimization is then considered. It is based on a nonlinear finite difference discretization, whose consistency is proven, and a standard primal-dual descent scheme. Finally, numerical examples show a variety of geometric solutions exhibiting the properties discussed in the theoretical sections.

## Keywords

Bingham fluid Exchange flow Settling Critical yield number Total variation Piecewise constant solutions## Mathematics Subject Classification

49Q20 76A05 76T20 49M25## 1 Introduction

An important property of Bingham fluid flows is the occurrence of plugs, which are regions where the fluid moves like a rigid body. Such rigid movements occur at positions where the stress does not exceed the yield stress.

In this paper we consider anti-plane shear flow in an infinite cylinder, where an ensemble of inclusions move under their own weight inside a Bingham fluid of lower density, and in which the gravity and viscous forces are in equilibrium [cf (6)], therefore inducing a flow which is *steady* or *stationary*, that is, in which the velocity does not depend on time. For such a configuration, we are interested in determining the ratio between applied forces and the yield stress such that the Bingham fluid stops flowing completely. This ratio is called *critical yield number*.

*Related work* To our knowledge, the first mathematical studies of critical yield numbers were conducted by Mosolov and Miasnikov [27, 28], who also considered the anti-plane situation for flows inside a pipe. In particular, they discovered the geometrical nature of the problem and related the critical yield number to what in modern terminology is known as the Cheeger constant of the cross-section of the region containing the fluid. Very similar situations appear in the modelling of the onset of landslides [18, 19, 22], where non-homogeneous coefficients and different boundary conditions arise. Two-fluid anti-plane shear flows that arise in oilfield cementing are studied in [15, 16]. Settling of particles under gravity, not necessarily in anti-plane configurations is also considered in [23, 30]. Finally, the previous work [17] also focuses in the anti-plane settling problem. There, the analysis is limited to the case in which all particles move with the same velocity and where the main interest is to extract the critical yield numbers from geometric quantities. In the current work we lift this restriction and focus on the calculations of the limiting velocities, also from a numerical point of view. Various applications of the critical yield stress of suspensions are pointed out in [4, Sect. 4.3]. On the numerical aspects, there are several methods available in the literature for the computation of limit loads [8] and Cheeger sets [6, 7, 9], and both of these problems are closely related to ours, as we shall see below.

*Structure of the paper* We begin in Sect. 2 by recalling the mathematical models describing the stationary Bingham fluid flow in an anti-plane configuration, and an optimization formulation for determining the critical yield number.

Next, in Sect. 3 we consider a relaxed formulation of this optimization problem, which is naturally set in spaces of functions of bounded variation, and show that the limiting velocity profile as the flow stops is a minimizer of this relaxed problem.

In Sect. 4, as in the case of a single particle [17], we prove that there exists a minimizer that attains only two non zero velocity values.

Finally, in Sect. 5 we present a numerical approach to compute minimizers. This approach is based on the non-smooth convex optimization scheme of Chambolle–Pock [10] and an upwind finite difference discretization [11]. We prove the convergence of the discrete minimizers to continuous ones as the grid size decreases to zero. We then use this scheme to illustrate the theoretical results of Sect. 4.

## 2 The Model

*v*is its velocity (for which incompressibility implies \({\text {div}}v = 0\)), and \(\mathcal E v = (\nabla v + \nabla v^\top )/2\) is the linearized strain, \(\nabla v \in \mathbb R^{3 \times 3}\) being the Jacobian matrix of the vector

*v*. We denote by \(\sigma _D\) the deviatoric part of the Cauchy stress tensor \(\sigma (x,y,z) \in \mathbb R^{3 \times 3}_{\text {sym}}\), that is

*p*is the pressure and \(\mathrm {tr}\,\sigma _D = 0\). These equations state that as long as a certain stress is not reached, there is no response of the fluid (see Fig. 1).

*exchange flow problem*, meaning that we require that the total flux across the horizontal slice is zero,

### 2.1 Eigenvalue Problems

*m*is a scalar multiplier for the exchange flow condition (3). Writing the Euler–Lagrange equations in the \(\hat{\omega }\) argument at an optimal pair for the saddle point problem, we obtain a solution of our constitutive and balance Eqs. (4) and (6), with

*Y*and a velocity scale \(\hat{\omega }_0\) by

*h*is \(\int _{\Omega _f} \nabla \omega _Y \cdot \nabla h\), differentiating in the direction \(v-\omega _Y\), as done in [13, Sect. I.3.5.4] shows that for every \(v \in H_\diamond \),

## 3 Relaxed Problem and Physical Meaning

*E*might not attain a minimizer in \(H_\diamond \), we consider a relaxed formulation on a subset of functions of bounded variation.

### 3.1 Functions of Bounded Variations and Their Properties

We recall the definition of the space of functions of bounded variation and some properties of such functions that we will use below. Proofs and further results can be found in [1], for example.

### Definition 1

Let \(A \subset \mathbb R^2\) be open. A function \(v \in L^1(A)\) is said to be of bounded variation if its distributional gradient \(\nabla v\) is a Radon measure with finite mass, which we denote by \(\mathrm {TV}(v)\). In particular, if \(\nabla v \in L^1(A)\), then \(\mathrm {TV}(v)=\int _A \left|\nabla v \right|\). Similarly, for a set *B* with finite Lebesgue measure \(|B| < +\infty \) we define its perimeter to be the total variation of its characteristic function \(1_B\), that is, \({\text {Per}}(B)=\mathrm {TV}(1_{B})\).

### Theorem 1

*A*, denoted \(\mathrm {BV}(A)\), is a Banach space when associated with the norm

The space of functions of bounded variation satisfies the following compactness property [1, Theorem 3.44]:

### Theorem 2

*w*in \(L^1\),

We frequently use the *coarea* and *layer cake formulas*:

### Lemma 1

An important role in characterizing constrained minimizers of the \(\mathrm {TV}\) functional is played by Cheeger sets, which we now define.

### Definition 2

A set is called Cheeger set of \(A \subseteq \mathbb R^2\) if it minimizes the ratio \({\text {Per}}(\cdot ) / |\cdot |\) among the subsets of *A*.

The following result is well known and has been stated for instance in [25, Proposition 3.5, iii] and [29, Proposition 3.1]:

### Theorem 3

For every non-empty measurable set \(A \subseteq \mathbb R^2\) open, there exists at least one Cheeger set, and its characteristic function minimizes the quotient \(u \mapsto \mathrm {TV}(u)/\Vert u\Vert _{L^1(A)}\) in \(L^1(A)\setminus \{0\}\). Moreover, almost every level set of every minimizer of this quotient is a Cheeger set.

### Remark 1

Some sets may have more than one Cheeger set, which introduces nonuniqueness in the minimizers of the quotient \(\mathrm {TV}(\cdot )/\Vert \cdot \Vert _{L^1(A)}\). One example is the set \(\Omega \) of Fig. 6 below.

### 3.2 Generalized Minimizers of *E*

*E*is invariant with respect to scalar multiplication, and we can therefore add the constraintto \(\mathcal {B}\) without changing the minimal value of the functional

*E*. Thus, the problem of minimizing

*E*over \(\mathcal {B}\) is equivalent to the following problem:

### Problem 1

By using standard compactness and lower semicontinuity results in \(\mathrm {BV}(\mathbb R^2)\), it is easy to see [17] that there is at least one solution to Problem 1. In particular, we emphasize that all the constraints above are closed with respect to the \(L^1\) topology.

### 3.3 The Critical Yield Limit

We investigate the limit of \(\omega _Y\) [the minimizer of \(G_Y^\diamond \), defined in (11)] when \(Y \rightarrow Y_c\). For this purpose we first prove

### Proposition 1

The quantity \(\int _{\Omega _f} |\nabla \omega _Y|\) is nonincreasing with respect to \(0 \leqslant Y \leqslant Y_C\). In particular, it is bounded.

### Proof

We are now ready to investigate the convergence of \(\omega _Y\) and its rate.

### Theorem 4

### Proof

*Y*, and since \(\omega _{Y_c}=0\), the above implies

From the above result, we see that a minimizer of the quotient \(\frac{\int _\Omega |\nabla v|}{\int _{\Omega _s} v}\) can be obtained as a limit of rescaled physical velocities, and therefore carries information about their geometry. For this reason, we will focus on these minimizers in the following.

## 4 Piecewise Constant Minimizers

We prove the existence of solutions of Problem 1 with particular properties. In our previous work [17] this problem was considered under the assumption that the velocity is constant in the whole \(\Omega _s\). In the situation considered here, the physical velocity \(\omega \) is constant only on every connected component of \(\Omega _s\), and the velocity of *each* solid particle is an unknown. Therefore, the candidates of limiting profiles *v* over which we optimize (belonging to \(\mathrm {BV}_\diamond \)) also satisfy \(\nabla v = 0\) on \(\Omega _s\).

### 4.1 A Minimizer with Three Values

### Theorem 5

There is a solution of Problem 1 that attains only two non-zero values.

- 1.
We prove the existence of a minimizer for Problem 1 which attains only finitely many values. This is accomplished by convexity arguments reminiscent of slicing by the coarea (16) and layer cake (17) formulas, but more involved.

- 2.
When considered over functions with finitely many values, the minimization of the total variation with integral constraints is a simple finite-dimensional optimization problem, and standard linear programming arguments provide the result.

### Lemma 2

In turn our proof of Lemma 2 is based on the following minimizing property of level sets, which we believe could be of interest in itself.

### Lemma 3

Let \(\Omega _0, \Omega _1, \nu \) and \(\mathcal {A}_\nu (\Omega _0, \Omega _1)\) be as in Lemma 2, and *u* a minimizer of \(\mathrm {TV}\) in \(\mathcal {A}_\nu (\Omega _0, \Omega _1)\). Assume further that *u* has values only in [0, 1], and denote \(E_s := \{u > s\}\). Let \(s_0\) be a Lebesgue point of \( s \mapsto {\text {Per}}(E_s)\) and \(s \mapsto |E_s|\) (these two functions are measurable, so almost every \(s \in [0,1]\) is a Lebesgue point for them). Then \(1_{E_{s_0}}\) minimizes \(\mathrm {TV}\) in \(\mathcal {A}_{|E_{s_0}|}(\Omega _0, \Omega _1)\).

The proofs of these two lemmas are located after the proof of Theorem 5.

### Proof of Theorem 5

**Step 1. A minimizer with finite range**

*u*of the total variation in \(\mathrm {BV}_\diamond \), that is, a solution of Problem 1. We represent \(\Omega _s\) by its connected components \(\Omega ^i_s\), \(i=1,\ldots ,N\),

*u*belongs to \(\mathrm {BV}_\diamond \),

*u*is constant on every \(\Omega ^i_s\), and we introduce the constants \(\gamma _i\) such that

As a result, the function \(v_i := \frac{u_i - \gamma _i}{\gamma _{i+1} - \gamma _i} \) minimizes the total variation with constraints \(\left. v_i \right| _{\mathbb R^2 \setminus \{u > \gamma _i\}} \equiv 0\), \(\left. v_i \right| _{\{u \geqslant \gamma _{i+1}\}} \equiv 1\) and prescribed integral. Lemma 2 (applied with \(\Omega _0 = \{u > \gamma _i\}\) and \(\Omega _1 = \{u \geqslant \gamma _{i+1} \}\)) shows that \(v_i\) can be replaced by a five level-set function \(\tilde{v}_i\) which has total variation smaller or equal to \(\mathrm {TV}(v_i)\). Hence \(u_i\) can be replaced by the five level-set function \(\tilde{u}_i := \gamma _i + \tilde{v}_i (\gamma _{i+1} - \gamma _i)\) without increasing the total variation.

*u*and \(\tilde{u}\) coincide on \(\Omega _s\), so the constraint Open image in new window is satisfied).

**Step 2. Construction of a three-valued minimizer**

*p*constraints are active at it. Therefore, at least \(p-2\) of these constraints should be of those defining the quadrant \(\sigma : x \geqslant 0\), meaning that at a vertex, at least \(p-2\) coefficients of

*x*are zero.

This polyhedron could be unbounded, but since \(a \geqslant 0\) and \(\sigma :x \geqslant 0\) componentwise, the minimization of \(a^T (\sigma :x)\) must have at least one solution in it. Moreover, since it is contained in a quadrant (\(\sigma : x \geqslant 0\)), it clearly does not contain any line, so it must have at least one vertex [5, Theorem 2.6]. Since the function to minimize is linear in *x*, it has a minimum at one such vertex [5, Theorem 2.7]. That proves the existence of a minimizer of (25) with at least \(p-2\) of the \((\alpha _i)\) being zero. This corresponds to a minimizer for Problem 1 which has only two level-sets with nonzero values, finishing the proof of Theorem 5.\(\square \)

#### 4.1.1 Proof of Lemma 2

### Proof of Lemma 2

*w*be an arbitrary minimizer of \(\mathrm {TV}\) in \(\mathcal {A}\). Splitting

*w*at 0 and 1 we can write

*u*a generic one. In what follows, we denote by \(E_s := \{u > s\}\) the level-sets of

*u*.

*s*, \(1_{E_{s}}\) minimizes \(\mathrm {TV}\) in \(\mathcal {A}^{(0,1)}_{|E_{s}|}\). That implies in particular that for a.e.

*s*, \(E_s\) minimizes perimeter with fixed mass. We introduce \(E_s^{(1)}\) the set of points of density 1 for \(E_s\) and \(E_s^{(0)}\) the set of points of density 0 for \(E_s\), that is

### Claim

If \(E^\pm \) is not empty, \(1_{E^\pm }\) minimizes total variation in \(\mathcal {A}^{(0,1)}_{|E^\pm |}\), with \(|E^+| \leqslant \mu \leqslant |E^-|.\)

### Proof of claim

*s*. Then, let us select a decreasing sequence \(s_n \searrow s_\mu \) such that for each

*n*, \(1_{E_{s_n}^{(1)}}\) minimizes total variation in \(\mathcal {A}^{(0,1)}_{|E_{s_n}|}\). Since \(E_{s_n}^{(1)} \rightarrow E^+\) in \(L^1\), one has \(|E^+| = \lim |E_{s_n}^{(1)}| = \lim |E_{s_n}|\) and the semicontinuity for the perimeter gives

*n*such that \( |E^+| \geqslant |E_{s_n}| \geqslant |E^+| - \delta \) and

Selecting an increasing sequence \(\tilde{s}_n \nearrow s_\mu \) and such that \(\Omega \setminus E_{s_n}^{(0)}\) minimizes \(\mathrm {TV}\) in \(\mathcal {A}^{(0,1)}_{|E_{s_n}|}\), we obtain similarly that \(1_{E^-}\) minimizes \(\mathrm {TV}\) in \(\mathcal A^{(0,1)}_{|E^-|}.\) \(\square \)

#### 4.1.2 Proof of Lemma 3

### Proof of Lemma 3

*u*.

*B*be a ball such that \({\text {Per}}(B) \leqslant \frac{\varepsilon }{4 \Vert u_0 \Vert _\infty }\). There exists \(\alpha \) such that the function \(u_0 + \alpha 1_B\) satisfies

*h*if needed, one can enforce that \(|\alpha | \leqslant 2 \Vert u_0 \Vert _\infty \).

### 4.2 Minimizers with Connected Level-Sets

In this subsection, we refine our analysis slightly, and show the existence of three-valued minimizers for Problem 1 with additional properties. We start with the following definition:

### Definition 3

A set of finite perimeter *A* is called indecomposable, if there are no two disjoint finite perimeter sets *B*, *C* such that \(|B|>0\), \(|C|>0\), \(A = B \cup C\) and \({\text {Per}}(A)={\text {Per}}(B)+{\text {Per}}(C)\).

This notion is in fact a natural measure-theoretic sense of connectedness for sets for finite perimeter, for more information about it see [2].

### Remark 3

*k*for the perimeter and mass of

*E*, and in case it can be decomposed in the sense of Definition 3, the same lower bounds also hold for each set in such a decomposition. In consequence,

*E*can only be decomposed in at most a finite number of sets. The proof of these statements relies heavily on the results of [2], and is presented in [12] for the unconstrained case, and [21] for the case with Dirichlet constraints, as used here.

Assuming these results, one can simplify the level sets of solutions further:

### Theorem 6

There exists a minimizer for Problem 1 attaining exactly three values for which all non-zero level-sets are indecomposable.

### Proof

*u*can be written as

*u*of the form

*u*is a minimizer of \(\mathrm {TV}\), it follows that

*k*,

*l*are linear in

*h*, one can replace

*h*by \(-h\) and obtain

*h*such that \(h=-\alpha \) or \(k = -\alpha \) without violating \(|l|\le \beta \), and therefore produce a minimizer whose positive part is either \(\Omega _2\) or \(\Omega _1\), respectively. We proceed similarly for the negative part and therefore obtain an indecomposable negative level-set. \(\square \)

### Remark 4

*h*as long as \(k \geqslant -\alpha - \beta - l\). The equality case in this last constraint corresponds to joining \(\Omega _2\) to \(\Omega _-\), and avoiding creating a “hole” in \(\Omega _-\) by the procedure mentioned above (which replaces \( \alpha 1_{\Omega _2}\) by zero). Clearly, this procedure can also be performed for the positive level set, and in fact the “holes” to be deleted could also be connected components of the zero level set. Therefore, a solution in which both the positive and negative level set are simply connected can be obtained.

### Remark 5

The intuition behind these last results is that, like in the proof of Theorem 5, the constraints of the problem are linear with respect to the values, and the total variation is also linear as long as the signs of the differences of values at the interfaces do not change. In particular, the points at which the topology of the level sets changes are situations in which these signs change (that is, the values of two adjacent level sets are equal).

## 5 Numerical Scheme and Results

This restriction corresponds to the case in which either \(\Omega _s\) is connected, so that there is only one solid particle, or all the particles are constrained to move with the same velocity. In Sect. 5.4 we point out the required modifications for the multi-particle case and present a variety of computed examples.

*q*, whereas the conditions \(v = 0\) on \(\partial \Omega \) and \(v = 1\) on \(\Omega _s\) are encoded as indicator functions. Our discretization of choice is finite differences on a rectangular grid \(\{1,\ldots , m\}\times \{1,\ldots n\}\), where in this whole section, for simplicity, we assume that \(n=m\) and \(\Omega \Subset (0,1)^2\). This leads to a saddle point problem of the form

*A*is denoted by \(\upchi _A\), so that \(\upchi _A(x)=0\) if \(x \in A\), and \(+\infty \) otherwise. \(\nabla \) stands for a suitable discrete gradient, whose choice we now discuss.

### 5.1 Discretization

*n*one needs to extends the functions outside the grid, but for the problem at hand any choice will do, since \(\Omega ^n\) never touches the boundary of the grid.

### 5.2 Convergence of the Discretization

We first prove the following lemma, which states that the continuous total variation may be computed with multipliers with positive components, mimicking the discrete definition.

### Lemma 4

### Proof

*p*,

*q*be admissible in the right hand side of (33). Then we notice that \(p-q\) is also admissible in (34), because

*p*,

*q*being componentwise positive implies

We can now prove Gamma-convergence of the discrete problems, implying convergence of the corresponding minimizers.

### Theorem 7

### Proof

*n*big enough. That implies \(\upchi _{C^n}(v_n) = + \infty \) and the \(\Gamma \)-liminf inequality is trivially true. If \(\upchi _{C}(v) < \infty \), then \(\upchi _{C}(v) =0\) and the inequality is also true since \(C^n \subset C\).

Then, \(\mathrm {TV}(v_\delta ) \rightarrow \mathrm {TV}(v)\) ([3, Theorem 1.3], noticing that *v* is constant around \(\partial [0,1]^2\)) and, thanks to (32), if \(\delta \leqslant \frac{1}{n},\) we have \(\upchi _{C^n}(v_\delta ) = 0.\)

*n*is large enough and therefore \(\mathrm {TV}^n(v_{\delta ,n}) \rightarrow \mathrm {TV}(v_\delta ).\) By a diagonal argument on \(\delta \) and

*n*, we conclude. \(\square \)

### 5.3 Single Particle Results

In this section, we again restrict ourselves to the case in which there is either only one particle, or the particles are constrained to move with the same velocity.

### 5.4 Several Particles

*i*-th component of the discrete domain, corresponding to \(\Omega _s^i\). The set \(C^n\) is the discrete counterpart to the set \(\mathrm {BV}_\diamond \) used in Sects. 3 and 4.

We also give an example where uniqueness of the minimizer is not expected. In Fig. 9, we consider a grid of circular particles in a square. It is easy to see analytically that any subset of the particles can be chosen as positive part of the minimizer. We present two computations at different numerical resolutions that pick two different subsets.

Since the solutions we compute correspond to limit profiles of the original flows (Theorem 7), the results presented both here and in Sect. 4.1 mean that near the stopping regime \(Y \rightarrow Y_c\) the transition between yielded and unyielded regions of the fluid typically happens closer and closer to the particle boundaries and the domain boundaries. This is consistent with the Cheeger set interpretation of the buoyancy case (which was already present in [17]) and the many previous works on non-buoyancy cases ([19, 28], for example).

### 5.5 A Random Distribution of Small Particles

We also present two examples of random distribution of square particles in a bigger square. Figure 10 shows the same number of particles distributed in two different ways and the corresponding minimizers. This example shows that the yield number depends strongly on the geometry of the problem, not only on the ratio solid/fluid. An interesting problem would be to investigate the optimal distribution to maximize/minimize this yield number.

## Notes

### Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). This research was supported by the Austrian Science Fund (FWF) through the National Research Network ‘Geometry+Simulation’ (NFN S11704). We would like to thank Ian Frigaard (UBC) for useful discussions.

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