A NONLINEAR ELLIPTIC PROBLEM RELATED TO FLOWINGGRANULAR MATERIALS

PIERRE A. GREMAUD∗

Abstract. Similarity solutions for the flow of granular materials are constructed. Unlike pre-vious work, the present approach can be applied to non-axisymmetric containers. The steady stateequations are reduced to a nonlinear Helmholtz on a subdomain of the sphere. Existence and localuniqueness of the solutions are established. A spectral numerical method using Fourier/Chebyshev-Gauss-Radau collocation for discretization and Newton-GMRES as solver is proposed and imple-mented. Corresponding numerical experiments are discussed.

Key words. elliptic, granular, similarity, spectral

AMS subject classifications. 65N35, 35J65, 76T25

1. Introduction. This paper is about the determination of the flow of granularmaterial under gravity in hoppers of simple geometry. This problem is common tomany industrial processes. In spite of its apparent simplicity, it presents a formidablearray of difficulties. In practice, the withdrawal of stored materials from hoppers andbins is well known to be problematic. No flow, segregation, flooding (uncontrolledflow) and structural failures are often encountered [11]. Improved design criteria aresought through a better understanding of such flows. All the previous contributionswe are aware of [4], [5], [7]–[9], [13]–[17], [20], to cite but a few, deal only withaxisymmetric containers. This significant restriction is removed here.

Apart from the geometry of the hopper, two important factors are the internalfriction of the material and the friction between wall and material. Those parametersare further discussed in Section 2. The flows themselves can reach from establishedmass flows where the material moves and deforms everywhere in a “smooth way” tohighly time-dependent funnel flows where motion only takes place in the central partof the silo [14]. In the first case, steady state models are often appropriate. Indeed,current design criteria are based on mass flow calculations through the resolution ofsteady state models, a view point that is also taken in this paper. Throughout, thematerial is assumed to be an incompressible, perfectly plastic, cohesionless Coulombpowder with a yield surface of von Mises type. Further, the eigenvectors of the strainrate and stress tensors are assumed to be parallel. Those assumptions are standard inthis field, although the alignment condition is somewhat controversial. Those issuesare commented on in Section 2. Apart from the above constitutive relations, theequations derive from the basic principles of Continuum Mechanics: conservationof mass and momentum. Conservation of energy does not enter the problem as heatlosses through friction influence the temperature which is not related in any significantway to the variables under study.

A study of some of the mathematical properties of the model corresponding to theabove assumptions can be found in [17]. The steady state equations can be elliptic,hyperbolic or have no definite type depending in the material parameters.

Much remains to be done about the analysis of the above problems and as aresult, the design of mass flow hoppers still greatly relies on an observation due to

∗Department of Mathematics and Center for Research in Scientific Computation, North CarolinaState University, Raleigh, NC 27695-8205, USA (gremaud@math.ncsu.edu). Partially supported bythe Army Research Office (ARO) through grant DAAD19-99-1-0188 and by the National ScienceFoundation (NSF) through grant DMS-9818900.

1

2 P.A. GREMAUD

Jenike [8], [9] in the late 1950’s. It was noticed that for three dimensional conicalhoppers as well as for two-dimensional wedge shaped hoppers the above equationsadmit similarity solutions. Those radial solutions correspond to particle paths thatare radial lines. The similarity is reflected in scalings of the stresses and velocitieswith respect to the radial distance r from the vertex of the hopper. In those simplegeometries, the equations for the radial fields reduce to systems of ODEs that can beeasily numerically integrated. The stability of the solutions to perturbations, changeof the geometry (opening angle) and/or physical properties of the hoppers were studiedin [4], [15] and [16] through numerical resolutions of the full PDE system (in [16], thematerial is assumed to be compressible) and in [5] and [15] through linear stabilitystudies. Experimental evidences [14] confirm the important role played by the radialfields in practice.

The purpose of this paper is to show that Jenike’s construction of similarity so-lutions can be generalized to general mathematical cones, i.e., pyramidal domains ofarbitrary cross section. This is significant as previous works in this area deal exclu-sively with axisymmetric containers, even though those are the exception rather thanthe rule in practice (see e.g. [18] for remarks on the influence of the hopper geometryon the flowing properties). The loss of axisymmetry considerably complicates thestructure of the stress tensor and the ensuing equations. As is the case for Jenike’sradial solutions, the radial structure leads to significant simplifications. However, in-stead of ODEs, the radial stress field must here be obtained through the resolution ofa nonlinear Helmholtz equation in a subdomain of the sphere.

The paper is organized as follows. The model, geometry and physical assumptionsare discussed in Section 2. An existence and local uniqueness result is stated andproved in Section 3. In Section 4, a pseudospectral numerical method is proposed.It uses Fourier collocation in longitude and Chebyshev-Gauss-Radau collocation inlatitude, to account for the boundary conditions at hand. Section 5 is devoted thedescription of several numerical experiments. Conclusions are offered in Section 6.

2. The model. The physical quantities and corresponding equilibrium equationsare expressed in spherical polar coordinates, with the origin corresponding to thevertex of the hopper. For non-axisymmetric domains, coordinate systems that arebetter suited to the geometry at hand can usually be constructed. However, suchsystems are typically not orthogonal, complicating greatly the structure of the basicequations of Continuum Mechanics [1]. To simplify the numerics, such alternatecoordinate systems are introduced in Section 4 through a change of variables, but theindividual components of the stress tensor, velocity, etc..., are still measured in termsin the original spherical coordinates.

The strain rate tensor V = − 12 (∇v +∇vT ), v being the velocity, and the stress

tensor T take respectively the form (see e.g.[19], p.184)1

V =

−∂rvr − 1

2

(1r ∂θvr − vθ

r + ∂rvθ

) − 12

(1

r sin θ∂φvr − vφ

r + ∂rvφ

)· − 1

r (vr + ∂θvθ) − 12r

(∂θvφ − cot θ vφ + 1

sin θ∂φvθ

)· · − 1

r

(vr + cot θ vθ + 1

sin θ∂φvφ

)

,

T =

Trr Trθ Trφ

· Tθθ Tθφ

· · Tφφ

.

1We omit to write the lower triangular part of symmetric tensors.

FLOWING GRANULAR MATERIALS 3

The equations of equilibrium are

∇ · T = ρg,(2.1)

where g is the acceleration vector due gravity, or equivalently

∂rTrr +1

r sin θ∂φTrφ +

1r∂θTrθ +

1r(2Trr − Tφφ − Tθθ + Trθ cot θ) = −ρg cos θ,

∂rTrφ +1

r sin θ∂φTφφ +

1r∂θTφθ +

1r(3Trφ + 2Tφθ cot θ) = 0,

∂rTrθ +1

r sin θ∂φTθφ +

1r∂θTθθ +

1r

(3Trθ + (Tθθ − Tφφ) cot θ) = ρg sin θ,

where g = |g | and ρ is the density.For the plasticity model, the von Mises yield condition is assumed to hold. Ex-

pressed in terms of the principal stresses, σi, i=1,2,3, i.e., the eigenvalues of T , thiscondition reads

(σ1 − p)2 + (σ2 − p)2 + (σ3 − p)2 = 2 p2s2,(2.2)

where

p =13tr T =

13(Trr + Tθθ + Tφφ) =

13(σ1 + σ2 + σ3)

is the average stress and s = sin δ, δ being the angle of internal friction.A flow rule completes the model. The eigenvectors of the strain rate tensor V

and the deviatoric part of the stress tensor T are assumed to be parallel. This isthe Levy flow rule, which can be equivalently expressed as the existence of a positivescalar function λ > 0 such that

V = λ(T − pI).(2.3)

The alignment condition of the eigenvectors of T and V in effect neglects therotation of a material element during deformation, a controversial assumption. Thereis experimental evidence that misalignment may occur under some circumstances.Alternative models which allow for the above eigenvectors to be somewhat out ofalignment have been proposed, see e.g. [20]. However, to the best of our knowledge,there does not seem to be enough experimental data to favor one type of models overthe other. We refer to [7] for a lucid, if somewhat dated, account of the situation.Further, and more importantly, the fact that we look here exclusively at radial flows,renders the distinction between the two types of model less of an issue. Indeed, simpleexplicit calculations show that in the case of radial flows in three dimensional conicalhoppers of circular cross section, for instance, the misalignment predicted by Spencer’sdouble shearing model [20] is less than 10 in all cases of physical interest.

The unknowns characterizing the flow are the velocity v and the stress tensorT , assuming constant density. They are determined by (2.1) which corresponds toconservation of momentum, the yield condition (2.2) and the above flow rule (2.3).Note that (2.3) also implies incompressibility, and thus conservation of mass, as

∇ · v = −trV = λ tr(T − pI) = 0.

The type of the system (2.1–2.3), which has to be supplemented with side conditionsdiscussed below, can be determined. In [17], it was shown that, depending among

4 P.A. GREMAUD

other things on the internal friction, the system for fully three-dimensional flows canbe mixed hyperbolic-elliptic, elliptic or have no definite type.

Following Jenike [8], [9], we seek similarity solutions with radial symmetry. Moreprecisely, the velocity field is assumed to be purely radial, i.e.

[vr, vθ, vφ] = [vr, 0, 0],

while the stress components are of the form

Trr = r τrr(θ, φ),

and similarly for the other stress components. The strain rate tensor simplifies to

V =

−∂rvr − 1

2r ∂θvr − 12r sin θ ∂φvr

· − 1r vr 0

· · − 1r vr

.

The flow rule (2.3) has several fundamental consequences. First, since again ∇·v = 0,we obtain

∂rvr +2rvr = 0.

The proper scaling for the velocity is consequently

vr(r, θ, φ) = − 1r2

υ(θ, φ),

where we further have

∂θυ =2 Trθ

Tθθ − pυ.(2.4)

Finally, (2.3) implies

τθθ = τφφ and τθφ = 0.

The yield condition (2.2) can be expressed in terms of the four stress unknowns τrr,τrθ, τrφ and τθθ by using the invariants of T , see (2.8) below. This results in thefollowing system of three partial differential equations and one algebraic constraint inthe four remaining stress unknowns τrr, τrθ, τrφ and τθθ

1sin θ

∂φτrφ + ∂θτrθ + 3τrr − 2τθθ + τrθ cot θ = −ρg cos θ,(2.5)

1sin θ

∂φτθθ + 4 τrφ = 0,(2.6)

∂θτθθ + 4 τrθ = ρg sin θ,(2.7)

(τrr − τθθ)2 + 3 τ2rθ + 3 τ2

rφ =13(τrr + 2 τθθ)2 s2.(2.8)

The yield condition (2.8) deserves further comments. It can be written

T T AT = −3τ2rφ, with T =

τrr

τrθ

τθθ

, A =

1− s2

3 0 −1− 2s2

30 3 0

−1− 2s2

3 0 1− 4s2

3

.

FLOWING GRANULAR MATERIALS 5

00

0

|τrφ| = 0

τrr

|τrφ| > 0

τrθ

τ θθ

Fig. 2.1. Two representative level surfaces corresponding to the yield condition. In each case,only the physically relevant nappe is showed.

The symmetric matrix A can be diagonalized

A = XΛXT , with Λ =

λ1

λ2

λ3

, X =

x1 x2 x3

,

where

λ1 = 3, λ2,3 = 1− 5s2

6±

√1 +

43s2 + (

5s2

6)2

,

x1 = [0, 1, 0]T , x2,3 =[− 1

3 + 2s2

(3s2

2± 1

2

√36 + 48s2 + 25s4

), 0, 1

]T

.

As is easily seen from those expressions, λ1 and λ2 are positive for any δ > 0, whileλ3 is negative. Let Ξ =

[ξ η ζ

]T = XTT ; the yield condition takes the form

ΞT ΛΞ ≡ 3ξ2 + λ2η2 + λ3ζ

2 = −3τ2rφ,(2.9)

which corresponds to a family of hyperboloids of two sheets parametrized by |τrφ|,see Figure 2.1. The case τrφ = 0 is a cone. Only one of the two sheets is physicallyrelevant. Indeed, granular materials can only support compressive stresses, i.e., σi >0, i = 1, 2, 3. This condition is satisfied on only one of the sheets, provided δ < 60.The corresponding result given below can be found in [17] where it is stated andproved using exclusively the properties of the principal stresses (i.e., (2.2) instead of(2.8)).

Lemma 2.1. For any value of τrφ, the hyperboloid of two sheets (2.8) has onesheet corresponding to compressive stresses, σi > 0, i = 1, 2, 3, if and only if theinternal angle of friction satisfies 0 < δ < 60.

6 P.A. GREMAUD

Proof. As a direct consequence of the structure of the scaled stress tensor T =

τrr τrθ τrφ

τrθ τθθ 0τrφ 0 τθθ

, it can be seen that the principal stresses are

σ1 = τθθ,(2.10)

σ2,3 =τθθ + τrr

2±

√(τθθ − τrr)2 + 4τ2

rθ + 4τ2rφ,(2.11)

without assuming any ordering of the σi’s. The component τθθ has thus to be positive.Further, we have

det T = τθθ(τrrτθθ − τ2rθ − τ2

rφ) = σ1σ2σ3,

and thus

τrrτθθ − τ2rθ − τ2

rφ = σ2σ3,(2.12)

Therefore, σ2 and σ3 only have the same sign if τrr > 0. Solving for τrr directlyfrom the yield condition (2.8) shows that τrr > 0 for positive values of τθθ provided2s2 − 3

√3s + 3 > 0, see (2.13) below. Since s = sin δ, it is equivalent to δ < 60.

Combining (2.8) with (2.11), we observe that the remaining σ2 and σ3 are positiveif

(s2

3− 1)τ2

rr + (4s2

3− 1)τ2

θθ + (4s2

3+ 2)τrrτθθ < 3τrrτθθ,

which is clearly satisfied if again δ < 60.At this point, the system (2.5–2.8) can be rewritten in several ways. One could for

instance introduce a la Sokolovskii variables [14] which essentially would correspondto solving (2.9) by replacing ξ and η by one new variable ψ

ξ = −√−13

(λ3ζ2 + 3 τ2rφ) sin 2ψ, η =

√−1λ2

(λ3ζ2 + 3 τ2rφ) cos 2ψ.

Note that the above expressions make sense because of the yield condition (2.9) andthe fact that λ2 > 0. The above approach would lead to an elliptic system of threefully nonlinear first order equations in the variables ζ, ψ and τrφ. We choose ratherto work with the original scaled variables after having solved (2.8) for τrr

τrr = f±(τrθ, τrφ, τθθ)

≡ 13− s2

((3 + 2s2)τθθ ± 3

√3s2τ2

θθ − (3− s2)τ2rθ − (3− s2)τ2

rφ

).(2.13)

As can obviously be seen from both the above expression and Figure 2.1, for any triple(τrθ, τrφ, τθθ), one can have zero, one or two corresponding values of τrr. The zerosolution case, i.e.

τ2θθ <

3− s2

3s2(τ2

rθ + τ2rφ),

corresponds, by assumption here, to stress states incompatible with the physical prop-erties of the material under consideration.

FLOWING GRANULAR MATERIALS 7

The two solutions in (2.13) are related to the so-called active (+ sign) and passive(– sign) states of the flowing material, see e.g. [14]. As the passive state is the onethat tends to be observed experimentally upon discharge of the hopper, we adopt, tofix the ideas, τrr = f−(τrθ, τrφ, τθθ). However, the analysis presented below appliesequally to both cases.

The unknowns τrφ and τrθ can be eliminated from the system by using (2.6) and(2.7). Relation (2.5) becomes

− 1sin2 θ

∂φφτθθ − 1sin θ

∂θ(sin θ∂θτθθ) + 12τrr − 8τθθ = −6ρg cos θ.

One can recognize the above differential operator − 1sin2 θ

∂φφ ·− 1sin θ∂θ(sin θ∂θ·) = −∆

as Laplace’s operator on the sphere. The system is closed by expressing τrr as afunction of τθθ and its first derivatives through (2.6), (2.7) and (2.13), leading to

−∆τθθ + χτθθ = −6ρg cos θ + F (θ, τθθ, ∂θτθθ, ∂φτθθ),(2.14)

where χ = 4 3+8s2

3−s2 and

F (θ, τθθ, ∂θτθθ, ∂φτθθ) =9

3− s2

√48s2τ2

θθ − (3− s2)(

(ρg sin θ − ∂θτθθ)2 + (∂φτθθ

sin θ)2

).

The above nonlinear Helmholtz equation (2.14) has to be solved in a subdomainΩ of the sphere corresponding to a “triangle”, with two edges corresponding to arcsof great circles and a third one corresponding to the outer boundary, see Figure 2.2 inthe case of a domain with rectangular cross section. The boundary ∂Ω of Ω consistsof the four parts

∂Ω = Γ0 ∪ Γ1 ∪ Γ2 ∪ Γ3,

where Γ0 is the trivial boundary (θ, φ); θ = 0 and Γ1,3 = (θ, φ); 0 < θ < θw, φ =∓φw, with θw > 0 and φw > 0. The boundary Γ2 is represented as the graph of aneven function C of class C1, θw = C(±φw)

Γ2 = (θ, φ); θ = C(φ),−φw < φ < φw.

The domain Ω in which the problem is to be solved is thus

Ω = (θ, φ);−φw < φ < φw, 0 < θ < C(φ),

see Figure 2.2.The boundary conditions are derived from physical considerations. On Γ1 and

Γ3, we impose

τθθ(·,−φw) = τθθ(·, φw), ∂φτθθ(·,−φw) = ∂φτθθ(·, φw) = 0.(2.15)

Indeed, by symmetry, it is clear that one should have

τθθ(·,−φw) = τθθ(·, φw), ∂φτθθ(·,−φw) = −∂φτθθ(·, φw).

Further, by a formal regularity argument, one obtains that ∂φτθθ vanishes on Γ1

and Γ3. Indeed, consider the problem in the full domain Ωfull = (θ, φ); 0 < θ <

8 P.A. GREMAUD

Γ

Γ

Γ1

2

3

Ω w

θw

φ

Γ Γ

Γ

Γ0

3

2

1

θ

φ

θ

w w

φ

φ φ(0,− ) )(0,

=

( )

Fig. 2.2. Geometry of the computational domain.

Cper(φ), φ ∈ [0, 2π], where Cper(φ) is constructed from C(φ) by using the periodicityof the entire hopper section. Freezing the coefficients in the nonlinear part F of (2.14),classic regularity results, see [6] §3.2 and Remark 3.2.4.6 yield H2-regularity of thesolutions. For such a solution u, one has on each φ = cst line Γφ

γ ∂φu+ = γ ∂φu−,

where u± correspond to the values from each side of Γφ and the mapping γ : H2(Ωfull) →H1/2(Γφ) which takes u to γ∂φu is the normal trace operator, see [6], Lemma 1.5.1.8.

The condition ∂φτθθ(±φw) = 0 has also been verified numerically by varying thedomain of resolution (for instance, by rotating it by π/4 in case of a square domain).

On Γ2, the law of sliding friction applies, i.e.

| TT | = µwTN , on Γ2,

where TT and TN are the scaled tangential and normal stresses on the hopper wall,and µw is the coefficient of wall friction. Since the outer unit normal to the wall is

N =[0, sin θ,−C′(φ)]√

sin2 θ + C′(φ)2,

it follows that

TN = τθθ and TT = [Nθτrθ + Nφτrφ, 0, 0]T .

The boundary condition on Γ2 then reads by (2.6), (2.7)

ρg sin2 C(φ)− sin C(φ)∂θτθθ +C′(φ)

sin C(φ)∂φ∂φτθθ =

−4 µw

√sin2 C(φ) + C′(φ)2τθθ,(2.16)

which is essentially a Robin boundary condition.

FLOWING GRANULAR MATERIALS 9

3. Mathematical analysis. The above problem presents several nonstandardfeatures and difficulties (Laplace-Beltrami operator, Robin boundary condition, nonLipschitz coefficients, restricted range of admissible values). In this section, an exis-tence and local uniqueness result is established.

We equip L2(Ω) with the inner product (u, v) =∫ φw

−φw

∫ C(φ)

0uv sin θ dθdφ. Let

V = v ∈ L2(Ω);∇v ≡ [∂θv,1

sin θ∂φv] ∈ L2(Ω)2,

u (·,−φw) = u(·, φw) in the sense of the trace.

The space V is equipped with inner product and norm

(u, v)V = (u, v) + (∇u,∇v), ‖u ‖V =√

(u, u)V .

Formally, if u stands for a classical solution to problem (2.14) and if v ∈ V , weobtain after integration by parts and use of the boundary conditions (2.15) and (2.16)

−∫∫

Ω

∆u v dω =∫ φw

−φw

∫ C(φ)

0

∇u · ∇v sin θ dθdφ

−∫ φw

−φw

(4 µw

√sin2 C(φ) + C′(φ)2 u(C(φ), φ) + ρg sin2 C(φ)

)v(C(φ), φ) dφ.

Let a : V × V → R be the bilinear form defined by

a(u, v) =∫ φw

−φw

∫ C(φ)

0

∇u · ∇v + χuv sin θ dθdφ

− 4µw

∫ φw

−φw

√sin2 C(φ) + C′(φ)2 u(C(φ), φ)v(C(φ), φ) dφ.

For Γ2, i.e., C of class C1, it is well known that the trace operator γ : V → H1/2(Γ2)is linear and continuous [6], Th. 1.5.1.3, p.38. Consequently, the form a : V ×V → Ris continuous. Further, under a condition that links wall friction, internal friction andthe geometry of the domain (a precise condition is given as part of the proof of thenext result), the bilinear form a(·, ·) is V -elliptic.

Proposition 3.1. The bilinear form a(·, ·) is V -elliptic, provided Γ2 is a curveof class C1 and µw is sufficiently small.

Proof. Let eθ = [1, 0] and h(φ) =√

sin2 C(φ) + C′(φ)2; we have

∫∫

Ω

h(φ)∇(u2) · eθ dω = 2∫∫

Ω

h(φ) u ∂θu dω.

On the other hand, the divergence theorem leads to∫∫

Ω

h(φ)∇(u2) · eθ dω = −∫∫

Ω

u2∇ · (h(φ) eθ) dω +∫

∂Ω

h(φ) u2eθ ·N dσ

=∫

Γ2

h(φ) u2eθ ·N dσ,

10 P.A. GREMAUD

where N stands again for the unit outer normal and where we have used (2.15). OnΓ2, we observe that

h(φ) eθ ·N =√

sin2 C(φ) + C′(φ)2 eθ ·N = sin C(φ), φ ∈ [−φw, φw],

and thus∫

Γ2

h(φ) u2eθ ·N dσ ≥ sin θmin√sin2 θmax + C′∞2

∫

Γ2

h(φ)u2 dσ,

where θmin(max) = minφ∈[−φw,φw](max)C(φ) and C′∞ = maxφ∈[−φw,φw] | C′(φ) |. Com-bining the last three expressions and using dσ = sin θ dφ yields, for any η > 0

φw∫

−φw

h(φ) u2(C(φ), φ) dφ ≤ 2

√sin2 θmax + C′∞2

sin2 θmin

∫∫

Ω

|h(φ)| |u| |∂θu| dω

≤ sin2 θmax + C′∞2

sin2 θmin

(1η‖u‖2L2(Ω) + η‖∂θu‖2L2(Ω)

).

From the definition of a(·, ·), sufficient conditions of V -ellipticity are then found to be

χ− 4µw

η

sin2 θmax + C′∞2

sin2 θmin

> 0 1− 4µwηsin2 θmax + C′∞2

sin2 θmin

> 0,

for some η > 0. A sufficient condition is then for instance

χ− 32µ2w

(sin2 θmax + C′∞2

sin2 θmin

)2

> 0.

We can now state a weak formulation of our problem: Find a function u ∈ V suchthat

a(u, v) =∫∫

Ω

F (u)v dω − 6 ρg

∫∫

Ω

cos θv dω + ρg

∫ φw

φw

sin2 C(φ)v(C(φ), φ) dφ,

for any v ∈ V and where, with a slight abuse of notation, F still denotes the nonlinearfunction (2.14). We define A ∈ L(V, V ?) and Φ ∈ V ? by respectively

< Au, v > = a(u, v), ∀u, v ∈ V,

< Φ, v > = −6 ρg

∫∫

Ω

cos θv dω + ρg

∫ φw

−φw

sin2 C(φ)v(C(φ), φ) dφ, ∀v ∈ V,

where < ·, · > denotes the V, V ?-duality product. Identifying in the obvious way F (u)with an element of V ?, the problem becomes: Find u ∈ V such that

Au = F (u) + Φ.(3.1)

Let us denote by G : V → V ? the mapping

G(u) = Au− F (u),

FLOWING GRANULAR MATERIALS 11

and let u0 = −ρg cos θ ∈ V . Note that G(u0) ∈ V ? is well defined as F (u0) is itselfwell defined by construction, provided θmax < π/2. Direct calculations show

L(V, V ?) 3 DG(u) = A−DF (u),

where

< DF (u)h, v >=9

3− s2

∫∫

Ω

48s2 uh− (3− s2)[−(ρg sin θ − ∂θu)∂θh + ∂φu∂θhsin2 θ

]√48s2u2 − (3− s2)[(ρg sin θ − ∂θu)2 +

(∂φusin θ

)2

]

v dω,

for any h, v ∈ V .Proposition 3.2. Under the assumptions of Proposition 3.1, there are neighbor-

hoods U ⊂ V and U? ⊂ V ? of u0 and G(u0) respectively such that for each Φ ∈ U?,the functional equation (3.1) has one and only one solution in U .

Proof. From the previous expression and the Inverse Mapping Theorem, G is aC1-diffeomorphism between a neighborhood of u0 in V and one of G(u0) in V ?. Theproposition follows.

Arguments following the lines of the formal remarks at the end of §3 could beused to obtain H2(Ω)-regularity of the solutions. We do not pursue this issue furtherhere.

4. Numerical analysis. In order to simplify the numerics, the problem is nowmapped onto a simple rectangular computational domain. From now on, the functionC which describes Γ2 is assumed to be of class C2. We define the new coordinates

Θ = θwθ

C(φ)and Φ = φ.

Note that r, Θ, Φ is not an orthogonal coordinate system. Keeping in mind thatθ = ΘC(φ)/θw, the problem for the transformed unknown U(Θ, Φ) = u(θ, φ) takesthe form

−∂ΦΦU + 2ΘC′(Φ)C(Φ)

∂ΘΦU − 1C(Φ)2

(Θ2C′(Φ)2 + θ2

w sin2 θ)∂ΘΘU

+(

ΘC2(Φ)

[C′′(Φ)C(Φ)− 2 C′(Φ)2]− 12

θw sin 2θ

C(Φ)

)∂ΘU + χ sin2 θ U

= −6ρg cos θ sin2 θ + F(Θ, Φ, U, ∂ΘU, ∂ΦU), (Θ, Φ) ∈ (0, θw)× (−φw, φw),(4.1)

where

F =9 sin θ

3− s2

s48s2 sin2 θ U2 − (3− s2)

[ρg sin2 θ − θw sin θ

C(Φ)∂ΘU ]2 + [∂ΦU − ΘC′(Φ)

C(Φ)∂ΘU ]2

.

The boundary condition on Γ2 becomes

ρg sin2 C(Φ)− sin C(Φ)θw

C(Φ)∂ΘU +

C′(Φ)sin C(Φ)

(−θw

C′(Φ)C(Φ)

∂Θ + ∂Φ

)U =

−4 µw

√sin2 C(Φ) + C′(Φ)2U, Φ ∈ (−φw, φw).(4.2)

The above problem is discretized by collocation; Chebyshev collocation at theChebyshev-Gauss-Radau points in used in Θ, while Fourier-cosine collocation at the

12 P.A. GREMAUD

Fourier collocation points is used in Φ. More precisely, we set

UNM (Θ,Φ) =N−1∑n=0

M/2−1∑

m=−M/2

Unmψn(Θ)eim(4Φ+π),(4.3)

where ψnN−1n=0 are the Lagrange interpolation polynomials at the Chebyshev-Gauss-

Radau nodes on [0, θw], i.e.

Θj =θw

2

(1 + cos(

2πj

2N − 1))

, j = 0, . . . , N − 1.(4.4)

This choice, as opposed to the more standard Chebyshev-Gauss-Lobatto collocation,see e.g. [2], §2.4, results from the nature of the boundary condition along Θ = θw.For completeness, we derive the expression of the collocation derivative below (whichwe have not been able to find in the literature).

Lemma 4.1. The Lagrange interpolation polynomials on the Chebyshev-Gauss-Radau nodes (4.4) are given by

ψj(Θ) =1cj

θw −ΘΘ−Θj

(1N

T ′N

(2Θθw

− 1)

+1

N − 1T ′N−1

(2Θθw

− 1))

, j = 0, . . . , N−1,

where

c0 = 1− 2N,

cj = − θw

2Θj

(N cos

2πNj

2N − 1+ (N − 1) cos

2π(N − 1)j2N − 1

), j = 1, . . . , N − 1,

where TN (x) = cos(N arccos x), |x| ≤ 1, is the Chebyshev polynomial of degree N .

The above result can easily be verified through the use of l’Hospital’s rule andelementary properties of the Chebyshev polynomials. Interpolation at the nodes (4.4)of a function u of Θ defined in [0, θw] simplify takes the form

INu(Θ) =N−1∑

j=0

u(Θj)ψj(Θ).

By definition, the Chebyshev collocation derivative of u at those nodes is then

(INu)′(Θl) =N−1∑

j=0

u(Θj)ψ′j(Θl) =N−1∑

j=0

Dlju(Θj),

with Dlj = ψ′j(Θl). The collocation derivative at the nodes can then be obtainedthrough matrix multiplication. Elementary albeit tedious calculations lead to the

FLOWING GRANULAR MATERIALS 13

following expressions

Dlj =

23

1θw

N(N − 1), if l = j = 0,

1c0θw

2sin2 2πl

2N−1

(N cos 2Nπl

2N−1 + (N − 1) cos 2(N−1)πl2N−1

)if j = 0,

l = 1, . . . , N − 1,

cl

cj

1Θl−Θj

if j = 1, . . . , N − 1,

j 6= l, l = 0, . . . , N − 1,

− 14Θj

3θw−2Θj

θw−Θj− θw

41

cjΘj

1√Θj(θw−Θj)

(N2 sin(N arccos( 2Θj

θw− 1))+

(N − 1)2 sin((N − 1) arccos( 2Θj

θw− 1))

)if j = l = 1, . . . , N − 1.

In the Φ direction, the collocation points are taken as the usual Fourier collocationnodes, i.e.,

Φl = φw(2l

M− 1), l = 0, . . . , M − 1.(4.5)

Let U be the N ×M matrix of coefficients Unm, n = 0, . . . , N − 1, m = 0, . . . , M/2−1,−M/2, . . . ,−1 and let W be the M ×M Fourier matrix

W =

1 1 1 . . . 11 ωM ω2

m . . . ωM−1M

1 ω2M ω4

m . . . ω2(M−1)M

. . . . . . . . . . . . . . .

1 ωM−1M ω

2(M−1)m . . . ω

(M−1)2

M

,

where ωM = ei2π/M is the primitive M -th root of unity. Further, if L is the M ×Mdiagonal matrix with diagonal [0, . . . , M/2− 1,−M/2, . . . ,−1], then for any j, l, j =0, . . . , N − 1, l = 0, . . . ,M − 1, the nodal values of UNM and its derivatives can beexpressed as follows

UNM (Θj , Φl) = (UW)jl,

∂ΦUNM (Θj , Φl) = 4i (ULW)jl,

∂ΦΦUNM (Θj ,Φl) = −16 (UL2W)jl,

∂ΘUNM (Θj , Φl) = (DUW)jl,

∂ΘΦUNM (Θj ,Φl) = 4i (DULW)jl,

∂ΘΘUNM (Θj , Φl) = (D2UW)jl.

The N ×M matrix of unknown coefficients U clearly satisfies a matrix equationof the type

A1UB1 + . . . ApUBp = F (U),

where the Ai’s are N × N matrices while the Bi’s are M × M . The above systemis obtained by enforcing the conditions that first, the discrete solution UNM from(4.3) satisfies the PDE (4.1) at the collocation points (Θj ,Φl), j = 1, . . . , N − 1,

14 P.A. GREMAUD

l = 0, . . . , M−1 and second, that the boundary condition (4.2) is verified at the nodesΘ0,Φl), l = 0, . . . , M − 1. Note that no side conditions of any form are imposed inthe neighborhood at Θ = 0. We denote by A⊗B the NM×NM matrix correspondingto the Kronecker product of an N ×N matrix A by an M ×M matrix B. Further, forany matrix A, vec(A) denotes the vector formed by stacking the columns of A. Onecan check [12], p.410

vec(AUB) = (BT ⊗A) vec(U).

A direct consequence of this elementary relation is that the above matrix equationcan be rewritten

H vec(U) = vec(F (U)),(4.6)

where H =∑p

j=1(BTj ⊗Aj) is a non sparse NM ×NM -matrix.

The above nonlinear system is numerically solved as follows. First, the matrixH is factorized into H = LV, where V is an upper triangular matrix and L is a“psychologically” lower triangular matrix (LU factorization). Then, the nonlinearequation

vec(U) = V−1L−1vec(F (U)),

is solved by a Newton-GMRES solver [10].

5. Numerical results. The numerical approach is tested by comparing with anexact solution obtained for a simplified linear problem in a domain corresponding toa circular cone. More precisely, we consider the following problem

− 1sin θ

(sin θu′(θ))′ + χu = F for 0 < θ < θw,(5.1)

u′(0) = 0 u′(θw) = ρg sin θw + 4µwu(θw),(5.2)

where F is taken as constant. The above equation and boundary conditions of thismodel problem corresponds exactly to (4.1, 4.2) whith a very simplified right-handside.

The change of variable v(z) = u(θ) with z = cos θ leads to

((1− z2)v′(z))′ + ν(ν + 1)v(z) = −F,

where ν(ν+1) = −χ. A fundamental system of solution to the homogeneous equation

((1− z2)v′(z))′ + ν(ν + 1)v(z) = 0,

is provided by the Legendre functions Pν(z) and Qν(z) of the first and second kindrespectively, see e.g. [3], §8.82, 8.83. The coefficient ν defined by ν(ν + 1) = −χsatisfies

ν = −12±

√14− χ = −1

2± iλ,

where λ =√

χ− 14 is a real parameter as, obviously, χ = 4 3+8s2

3−s2 ≥ 4 for any material.We thus get the solution in terms of P− 1

2+iλ(z) and Q− 12+iλ(z) which are conical

functions, see [3], §8.84. The general solution to (5.1) is then

u(θ) = AP− 12+iλ(cos θ) + BQ− 1

2+iλ(cos θ) + F/χ,(5.3)

FLOWING GRANULAR MATERIALS 15

N = M 8 10 12 14error 7.061 (-8) 1.723 (-10) 3.567 (-13) 5.995 (-15)

Table 5.1Maximum norm of the error for the model problem (5.1), (5.2).

where the last term is a particular solution to the considered problem. Since for realθ, Q− 1

2+iλ(θ) is always imaginary while P− 12+iλ is always real, we get B = 0. After

noticing that u′(0) = 0, the value of A can be found from (5.2)

A =ρg sin θw + 4µwF/χ

p′(θw)− 4µwp(θw), where p(θ) = P− 1

2+iλ(cos θ).

Our numerical solver is now applied to the present model problem (5.1), (5.2).The corresponding numerical solutions are compared to the exact solution (5.3). Theresults are summarized in Table 5.1, where the maximum norm of the error is givenas a function of the number of nodes, taken here as N = M . This illustrates twofacts. First, as expected, the method is found to be spectrally accurate. Second, forthis simple problem, a mesh as small as 14 × 14 leads to errors of the order of theround-off errors.

The standard fully axisymmetric case of a circular cone for the full problemprovides an additional way of testing the approach, this time in the nonlinear regime.Again, axisymmetry considerably simplifies the problem. Since no variable dependson φ in that case, a direct consequence of the flow rule (2.3) is that the (scaled) stresstensor T reduces to

T =

τrr τrθ 0τrθ τθθ 00 0 τθθ

.

Further, the yield condition (2.8) reduces, in (τrr, τrθ, τθθ)-space, to the cone illus-trated in Figure 2.1 (Trφ = 0), as opposed to the family of hyperboloids discussed in§2. This cone can be parametrized by the Sokolovskii variables (σ, ψ) where σ is theaverage stress and ψ is a new variable, as follows [9], [14], [17]

τrr = σ(1− 2√3s cos 2ψ), τrθ = −σs sin 2ψ, τθθ = σ(1 +

1√3s cos 2ψ).

The above parametrization “solves” the yield condition. The remaining unknowns(σ, ψ) are determined by solving the equations of equilibrium (2.1), which now onlyyield two ordinary differential equations. The corresponding boundary value problemcan easily be solved numerically. We omit the details, see, e.g., [5]. A comparison interms of the average stress σ between results from the ODE code using the Sokolovskiivariables on the one hand, and results from the present approach on the other hand,is illustrated in Figure 5.1. The agreement is excellent (relative error < .02%).

Having gained some confidence in the approach, we now apply it to the generalfull problem. Results are presented for the following kind of hoppers. For 0 ≤ λ ≤ 1,we consider the following family of domains

C(φ) = (1− λ)θw + λ arctan(tan θw cos φw

cos φ),

16 P.A. GREMAUD

0 5 10 15 20 25 300.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

angle θ

aver

age

stre

ss σ

Fig. 5.1. Comparison between ODE solver (solid curve) and present spectral PDE solver ()for a circular (axisymmetric) cone; θw = 30, δ = 30, µw = tan(15), N = M = 20.

corresponding to• conical hopper with circular cross-section λ = 0,• pyramidal hopper with transitional cross-section 0 < λ < 1,• pyramidal hopper with square cross-section λ = 1,

In each case treated below, φw = π/4 and θw is a free parameter, both angles beingdefined in Figure 2.2.

The results presented below are meant to illustrate the qualitative effects of thegeometry of the hopper on the flows. All results are represented on horizontal cross-sections taken at the same vertical height. This requires appropriate scaling accordingto the principle outlined in §2 (stress = O(r), velocity = O(r−2)). Unlike the stress,the velocity field, which is computed from (2.4), is only defined up to a scalar multiple.This indeterminacy can be solved by imposing an outflow rate for instance. Inciden-tally, in practice, the flow through outlets at the bottom of the hopper is almostalways imposed, through some kind of feeder device for instance. Here, the velocityfields are normalized to have maximum value 1. In Figure 5.2, a comparison betweena circular conical hopper, a transitional hopper λ = 1/2 and a square pyramidal onewith “same” opening angle is offered. By same angle, we mean that at a given com-mon height, the geometric domains corresponding to horizontal cross-sections admitinscribed circles of equal diameter. The material constants are chosen as δ = 40 andµw = tan(20).

All calculations were done on a quarter domain (the solid diagonal line appearingin all figures is an artifact of the plotting software). Several observations can be drawnfrom the above experiment. First, flows in pyramidal hoppers seem to exhibit largerstresses than flows in purely conical containers. To derive more precise practical con-clusions, arguments related to the volume of material effectively contained would haveto be considered. Second, and somewhat surprisingly, the presence of corners seemsto have relatively little effect on the velocity flow. Comparisons with both labora-tory experiments and different plasticity models would be extremely interesting. Inthe usual axisymmetric case, the latter kind of comparison can be easily performed.Models based on a Tresca yield condition for instance, rather than the present von

FLOWING GRANULAR MATERIALS 17

0.3

0.4

0.5

0.6

0.7

0.8

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

average stress

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

velocity

0.3

0.4

0.5

0.6

0.7

0.8

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

average stress

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

velocity

0.3

0.4

0.5

0.6

0.7

0.8

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

average stress

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

velocity

Fig. 5.2. Horizontal cross-sections of flows in similar hoppers (see text) of various shapes;

θconew = 30.68, θ

1/2w = 35.01, θpyramid

w = 40, δ = 40, µw = tan(20), N = M = 20.

Mises condition, tend to predict flows that are more sensitive to wall friction, i.e. thematerial does not flow as well near the wall, [5], [13], [14].2 Unfortunately, how togeneralize Tresca’s yield condition to non-axisymmetric flows is not entirely straight-forward, as one looses the type of symmetry properties that are precisely used to close

2It is also interesting to note that in that case, the steady state equations are always hyperbolic.In depth numerical calculations under a Tresca yield condition can be found in [4], [13].

18 P.A. GREMAUD

the system in the usual case; see [17], p.29, for further comments.

6. Conclusions. We have shown how to generalize to non-axisymmetric con-tainers the notion of similarity solutions introduced by Jenike [8] in the case of purelyconical (or wedge) containers. This generalization comes at the price of having to solvea nonlinear Helmholtz equation on a part of the sphere, as opposed to a boundaryproblem for a simple system of ODEs in the previous cases.

The present approach applies to “conical” domains, in the mathematical sense,i.e., any domain invariant under the transformation (r, θ, φ) 7→ (cr, θ, φ), where c >0. It is acknowledged that, clearly, not all industrial hoppers satisfy this property.However, failing this, no similarity solutions are expected to exist, and the full three-dimensional equilibrium equations (2.1) would have to be solved.

Acknowledgments. The author thanks Tim Kelley, Matt Matthews, Tony Royal,David Schaeffer and Michael Shearer for many helpful discussions.

REFERENCES

[1] L. Brillouin, Les lois de l’elasticite en coordonnees quelconques, Congres International deMathematique, Toronto, 1924, Annales de Physique, 3 (1925), pp. 251–298.

[2] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dy-namics, Springer-Verlag, 1988.

[3] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press,1965.

[4] P.A. Gremaud and J.V. Matthews, On the Computation of Steady Hopper Flows: I, StressDetermination for Coulomb Materials, NCSU-CRSC Tech Report CRSC-TR99-35, to bepublished in J. Comput. Phys.

[5] P.A. Gremaud, J.V. Matthews and M. Shearer, Similarity solutions for granular materialsin hoppers, in: Nonlinear PDE’s, dynamics, and continuum physics, J. Bona, K. Saxton andR. Saxton Eds., pp. 79–95, Contemporary Mathematics, #255, AMS, 2000.

[6] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, 1985.[7] R. Jackson, Some mathematical and physical aspects of continuum models for the motion of

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[8] A.W. Jenike, Gravity flow of bluk solids, Bulletin No. 108, Utah Eng. Expt. Station, Universityof Utah, Salt Lake City (1961).

[9] A.W. Jenike, A theory of flow of particulate solids in converging and diverging channels basedon a conical yield function, Powder Tech., 50 (1987), pp. 229–236.

[10] C.T. Kelley, Iterative methods for linear and nonlinear equations, SIAM, Frontiers in AppliedMathematics #16, 1995.

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[12] P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd Ed., Academic Press, 1985.[13] J.V. Matthews, An analytical and numerical study of granular flows in hoppers, PhD Thesis,

Dept. of Mathematics, North Carolina State Univ., 2000.[14] R.M. Nedderman, Static and kinematic of granular materials, Cambridge University Press,

1992.[15] E.B. Pitman, The stability of granular flow in converging hoppers, SIAM J. Appl. Math., 48

(1988), pp. 1033–1052.[16] J.R. Prakash and K.K. Rao, Steady compressible flow of cohesionless granular materials

through a wedge-shaped bunker, J. Fluid Mech., 225 (1991), pp. 21–80.[17] D.G. Schaeffer, Instability in the evolution equations describing incompressible granular flow,

J. Diff. Eq., 66 (1987), pp. 19–50.[18] D. Schulze, Storage, Feeding, Proportioning, in: Powder Technology and Pharmaceutical Pro-

cesses, Handbook of Powder Technology Vol. 9, D. Chulia, M. Deleuil and Y. Pourcelot Eds,pp. 285–317, Elsevier, 1994.

[19] I.S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, 1956.

FLOWING GRANULAR MATERIALS 19

[20] A.J.M. Spencer, Remarks on coaxiality in fully developed gravity flows of dry granular mate-rials, in: IUTAM Symposium on Mechanics of Granular and Porous Materials, N.A. Fleckand A.C.F. Cocks Eds, pp. 227–238, Kluwer, 1997.