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Modelling particle migration in rod suspensions: from
micro-mechanics to a macroscopic field description
Marta Perez, Emmanuelle Abisset-Chavanne, Francisco Chinesta, Roland
Keunings
To cite this version:
Marta Perez, Emmanuelle Abisset-Chavanne, Francisco Chinesta, Roland Keunings. Modelling
par-ticle migration in rod suspensions: from micro-mechanics to a macroscopic field description. The
19th International ESAFORM Conference on Material Forming ESAFORM 2016, Apr 2016, Nantes,
France. �hal-01591169�
Modelling particle migration in rod suspensions: from
micro-mechanics to a macroscopic field description
Marta Perez
1,a),b), Emmanuelle Abisset-Chavanne
1,c), Francisco Chinesta
1,d)and
Roland Keunings
2,e)1ESI GROUP Chair& High Performance Computing Institute, Ecole Centrale Nantes
1 Rue de la Noe, BP 92101, F-44321 Nantes cedex 3, France
2ICTEAM, Universit´e Catholique de Louvain. Av. Georges Lemaitre 4, B-1348 Louvain-la-Neuve, Belgium a)Corresponding author: Marta.Perez-Miguel@ec-nantes.fr
b)Marta.Perez-Miguel@ec-nantes.fr c)Emmanuelle.Abisset-Chavanne@ec-nantes.fr
d)Francisco.Chinesta@ec-nantes.fr e)Roland.Keunings@uclouvain.be
Abstract. This work proposes a micro-mechanical model describing the interactions of a test particle immersed in a semi-concentrated suspension of rigid rods. For that purpose, after revisiting the main elements related to suspensions exhibiting second-order velocity gradient effects, a micro-mechanical model is proposed that allows us to define a sort of interaction field giving the force acting on a test particle due to the rod kinematics at any point of the flow domain. Finally, a macroscopic description is derived that only involves the second-order orientation tensor and its spatial gradient. The model presented in this work shows that migration in fibre suspensions depends on the fibre concentration and orientation distribution and not only on the flow shear rate as is the case in suspensions of spheres.
PRELIMINARIES
In this work, we consider a suspension of rigid rods of length 2L immersed in a Newtonian fluid of viscosity η. Due to the flow-induced fibre rotation, any point of the flow domain, at which a test particle is located, will be subjected to a number of rod-particle hydrodynamic interactions, whose intensity depends on the concentration of fibres about that point and on the fibre rotary velocity that itself depends on the local fluid velocity gradient. In fact, the interaction intensity depends on the relative velocity between the test particle and the fibre at the contact point. By considering the resultant of all possible interactions between the fibres and the imaginary test particle, we can derive a resultant force. Now, such a force could be considered as a sort of mean field. Applying those forces at each point along an imaginary test fibre, we could obtain the resultant force acting on a fibre and then evaluate its possible migration.
For the sake of simplicity, we consider the 2D planar case depicted in Fig. 1. In the discussion that follows, we consider a generic rod whose orientation is defined by the unit vector p, having its centre of gravity located at point G. The flow kinematics are defined by the velocity field v(x), assumed unperturbed by the rod presence and orientation. We consider a small virtual spherical particle located at position S, close enough to the rod to assume that an hydrodynamic interaction occurs, and having a velocity VS. For an inertialess spherical particles, and in absence
of other forces, its velocity coincides with that of the fluid at its location.
For addressing the rod kinematics, we consider a second-gradient model because the velocity gradient is assumed to vary along the rod length. The rod can be regarded as a rigid system composed of two beads joined by a rigid connector. The forces are assumed to act on the dumbbell beads. In absence of rod interactions (they will be introduced later), only an hydrodynamical force FHacts on each bead, located with respect to the rod centre of gravity at positions
FH(pL) = ξ (v(G) +∇v · pL + H : (p ⊗ p)L2− VG− ˙pL), (1)
where ξ is the friction coefficient, v(G) is the unperturbed fluid velocity at the rod center of gravity, VGthe rod center
of gravity velocity and H the third-order tensor involving the second derivatives of the fluid velocity, Hi jk = 12∂x∂vi
j∂xk. In this expression, the first and second velocity gradients are evaluated at the rod center of gravity. Later, this fact will be expressed explicitly by employing the notation∇v|Gand H|G.
FIGURE 1. Rod - virtual test particle interaction.
The resulting force acting on the opposite bead at−Lp reads
F(−pL) = ξ (v(G) − ∇v · pL + H : (p ⊗ p)L2− VG+ ˙pL). (2)
By enforcing the force balance and neglecting inertia effects, we obtain
v(G)− VG=−H : (p ⊗ p)L2, (3)
which means that the rod centre of gravity has a relative velocity with respect to that of the fluid at this position. However, as proved in [1], the torque balance results in the standard Jeffery equation
˙p =∇v · p − (∇v : (p ⊗ p)) p. (4) The associated mesoscopic description uses the pdf – probability distribution function – ψ(x, t) whose evolution is governed by the so-called Fokker-Planck equation
Dψ
Dt +∇p· (˙pψ) = 0, (5) where where ˙p is given by Eq. (4), DtD denotes the material derivative, and∇p is the del operator with respect to the
conformational coordinates p.
When addressing semi-concentrated suspensions, it is usual to introduce in the previous equation a diffusion term [3]
Dψ
Dt +∇p· (˙pψ) = Dr∇
2
pψ. (6)
The last equation can be rewritten as:
Dψ
with the effective velocity ˙˜p expressed as
˙˜p= ˙p − Dr
∇pψ
ψ , (8)
where ˙p is given by Eq. (4).
A coarser description can be defined from the pdf moments [2]. The second and fourth-order moments, a and A respectively, are defined as
a= Z C p ⊗ pψ dp, (9) and A= Z C p ⊗ p ⊗ p ⊗ pψ dp, (10) where C denotes the surface of the unit sphere (unit circle in the 2D case). For notational simplicity, in the previous expression, we do not indicate the dependence of a on the space and time coordinates (x, t) and that of the pdf ψ on (x, t, p). In what follows, the space dependence will sometimes be explicitly indicated; the time dependence, however, will not. All the developments apply at each time t.
By taking the time derivative of Eq. (9), using Eq. (5), integrating by parts and using Eqs. (9) and (10), we obtain ˙a= ∇v · a + a · (∇v)T − A : ∇v, (11) with ˙a being the material derivative that includes advective effects.
Using the same rationale, but now considering Eq. (7) instead of Eq. (5), results in ˙a= ∇v · a + a · (∇v)T− A : ∇v − 6D r a − I 3 ! . (12)
MICRO-MECHANICAL MODELLING
From Fig. 1, it can be understood that all fibres having their center of gravity inside the sphere centered at S and having a radius L can interact with the test particle S. In that figure, we show a rod having its center of gravity at position G= S − ρp.
The number of rodsΓ(G, p) having their centre of gravity at position G = S − ρp and being oriented in direction p is given by, using the Bayes rule:
Γ(G, p) = φ(G) ψ(p; G), (13) where
φ(G) = φ(S) − ρ ∇φ|S· p, (14)
and similarly
ψ(p; G) = ψ(p; S) − ρ ∇ψ|S· p. (15)
The interaction force depends on the rod velocity Vr(S) at the interaction point S= G + ρp, that reads
Vr(S)= VG+ ρ˙p = VG+ ρ (∇v|G· p − (∇v|G: (p ⊗ p))p) . (16)
With the velocity of the virtual particle denoted by VS, the resulting interaction force between the test particle
located at S and a rod having its centre of gravity at position G, fG,SI , scales with the relative velocity, i.e.
fG,SI ∝(Vr(S) − VS)= (VG+ ρ (∇v|G· p − (∇v|G: (p ⊗ p))p) − VS), (17)
with according to Eq. (3), VG= v(G) + H|G: (p ⊗ p)L2and
v(G)= v(S) − ρ∇v|S· p+ ρ2H|S: (p ⊗ p) ∇v|G= ∇v|S−ρH|S· p H|G≈ H|S , (18)
from which the rod-particle interaction force can be rewritten as involving only quantities defined at position S: fG,SI ∝ v(S) − ρ∇v|S· p+ ρ2H|S: (p ⊗ p)+ L2H|S: (p ⊗ p)+
ρ∇v|S· p −ρ(∇v|S: (p ⊗ p))p − ρ2H|S: (p ⊗ p)+ ρ2(H|S∵ (p ⊗ p ⊗ p ⊗ p) − VS =
(v(S) − VS) −ρ(∇v|S: (p ⊗ p))p+ L2(H|S: (p ⊗ p) − H|S∵ (p ⊗ p ⊗ p ⊗ p)) . (19)
This force must be multiplied by the number of rodsΓ(G, p) = φ(G)ψ(p; G) having their centre of gravity at G and being oriented in direction p, to obtain by using Eqs. (14) and (15), FI
G,S:
FIG,S∝ φ(S) − ρ ∇φ|S· p · ψ(p; S) − ρ ∇ψ|S· p · n
(v(S) − VS) −ρ(∇v|S: (p ⊗ p))p+ L2(H|S: (p ⊗ p) − H|S∵ (p ⊗ p ⊗ p ⊗ p))o . (20)
In the resulting force expression (20), two contributions can be identified, the first one related to the test particle FI,tpG,S, and the other involving the rod conformation FI,conG,S , defined respectively from:
FG,SI,tp= φ(S) − ρ ∇φ|S· p · ψ(p; S) − ρ ∇ψ|S· p · (v(S) − VS), (21)
and
FI,conG,S = φ(S) − ρ ∇φ|S· p · ψ(p; S) − ρ ∇ψ|S· p ·
n
−ρ(∇v|S: (p ⊗ p))p+ L2(H|S: (p ⊗ p) − H|S∵ (p ⊗ p ⊗ p ⊗ p))o . (22)
As previously indicated, inertialess particles move with the fluid velocity, i.e. VS = v(S), that implies at it turn
FI,tpG,S = 0.
With v(S), φ(S) and ψ(p; S) known, as well as their spatial gradients ∇v|S, H|S, ∇φ|Sand ∇ψ|Sat position S, the
terms FI,tpG,Sand FI,conG,S can be integrated in the cercle B(S, L)= [0, L] × C of radius L centered at position S, illustrated in Fig. 1: FI,tp(S)= Z L 0 Z S FG,SI,tpdρ dp, (23) and FI,con(S)= Z L 0 Z S FI,conG,S dρ dp. (24) The resultant interaction force at position S is thus
QI(S)= QI,tp(S)+ QI,con(S) ∝ (FI,tp(S)+ FI,con(S)). (25) It is easy to check that QI,tp = 0 for a test particle moving with the fluid, and that as soon as the second-order velocity gradient vanishes, i.e. H|S= 0, the conformational component also vanishes, QI,con= 0, if Γ(G, p) is uniform
in space. We illustrate this behaviour in the next section.
Now, one can assimilate an imaginary (test) rigid rod as an assembly of spherical particles distributed all along the fibre axis parameterized by the coordinate x. Thus, it is easy to obtain the forces FI,tpG,S(x)and FI,conG,S(x)at each position S(x) following the rationale described before. Then, the integral along the fiber length results in the contributions
FI,tp= Z x Z L 0 Z S FI,tpG,S(x)dx dρ dp, (26) and FI,con= Z x Z L 0 Z S FI,conG,S(x)dx dρ dp. (27)
NUMERICAL TEST
In this section, we address a simple test to check the previous developments. We consider the planar fully established Poiseuille flow defined inΩ = [0, W] × [−H, H], with W = 2 and H = 1, whose kinematics is given by
v= u(x, y) v(x, y) ! = ˙γ(H20− y2) ! , (28)
and consider ˙γ = 1, S = (1, 0.5)T, and the rod length L= 0.1. That flow implies a second-order velocity gradient H
that is constant in the whole flow domain, with only one non-zero component H122= −2˙γ.
In this numerical example, we consider a uniform rod distribution, that results in φ = cte. The orientation distribution is the solution of the steady-state Fokker-Planck equation (7), where the diffusion coefficient was set to Dr= 0.1 and the distribution normalization condition was enforced by using a Lagrange multiplier.
0.1 0.08 0.06 0.04 ρ 0.02 0 0 100 θ 200 300 1 0 0.5 1.5 Γ
FIGURE 2. Joint probability distribution function.
Figure 2 illustrates the joint probability function πLρ2Γ(ρ, p) =
ρ
πL2Γ(S − ρp, p), whereas Fig. 3 represents its distribution at ρ= L. In the latter figure, one notices a slight asymmetry related to the slight evolution of the velocity gradient across the channel width. Moreover, due to the competition between the induced flow orientation and the diffusion randomizing effects, the steady-state orientation distribution is slightly deviated from the flow direction. When considering first-order kinematics with ∇|G = ∇v|S, ∀G ∈ B(S, L) (ball of radius L centered at position S),
the orientation distribution still deviates from the flow direction, but now, as it can be noticed in Fig. 4, the symmetry θ ↔ θ+π is retained. In order to analyze migration effects, expected to take place along the y-direction, we compute the y-component of the force for different configuration (θ, ρ). Figure 5 depicts the y-component of FG(ρ,p),SI,con (FG,SI,tp = 0). Even though it is sometimes positive and sometimes negative, the net value in the circumstances analyzed here is positive, thus inducing in the present case a net force on the test particle towards the regions of higher shear rate. The situation is richer than for a suspension of rigid spheres, for which at constant concentration the net force points towards the region of lower shear rate. When considering first-order kinematics, with constant velocity gradient, the resultant force vanishes.
We consider now a rigid rod as an assembly of 5 spherical particles. The coordinates of the spherical particles placed at both extremities are S1= (1, 0.5)Tand S5= (1.071, 0.571)T, with the total length L= 0.1. The y-component
θ 0 50 100 150 200 250 300 350 ψ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
FIGURE 3. Probability distribution for ρ= L for H , 0.
θ 0 50 100 150 200 250 300 350 ψ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
FIGURE 4. Probability distribution for ρ= L for H = 0.
FI,conremains positive, as in the case of only one test particle, showing a migration towards the regions with higher shear rate.
0.1 0.05 ρ 0 0 100 θ 200 300 0.02 0.03 0 -0.02 -0.03 -0.01 0.01 F I ( y ρ , θ )
FIGURE 5. y-component of the interaction force parameterized by ρ and p.
FIGURE 6. y-component of the interaction force for each spherical particle composing the rod.
MACROSCOPIC MODELLING
In view of the dependence of the interaction force on the orientation distribution function, the latter depending implic-itly on the considered angular direction p, i.e. ψ(p; G), the expressions derived above do not admit simple macroscopic expressions involving the usual orientation tensors.
The simplest closure consists in approximating the number of rods aligned in direction p at point S by
ψ(p; S) ≈ β (a(S) : (p ⊗ p)) , (29) where β ensuring the normalization condition. This allows us to express the interaction contributions (21) and (22) as FI,tpG,S= φ(S) − ρ ∇φ|S· p · a(S) : (p ⊗ p) − ρ ∇a|S∵ (p ⊗ p ⊗ p) · (v(S) − VS), (30)
and
FI,conG,S = φ(S) − ρ ∇φ|S· p · a(S) : (p ⊗ p) − ρ ∇a|S∵ (p ⊗ p ⊗ p) ·
n
−ρ(∇v|S: (p ⊗ p))p+ L2(H|S: (p ⊗ p) − H|S∵ (p ⊗ p ⊗ p ⊗ p))o . (31)
Integration in B(S, L)= [0, L] × C, according to Eqs. (23) and (24), gives
FI,tp(S)= Z L
0
Z
S
FI,tpG,Sdρ dp = G(a(S), ∇a|S; aiso, Aiso), (32)
and FI,con(S)= Z L 0 Z S
FI,conG,S dρ dp = H(a(S), ∇a|S; Aiso, Aiso, Aiso), (33)
where G(•) and H (•) denote functional dependences on the second (aiso), fourth (Aiso), sixth (Aiso) and eighth (Aiso)
order tensors aiso= Z C p ⊗ p dp, (34) Aiso= Z C p ⊗ p ⊗ p ⊗ p dp, (35) Aiso= Z C p ⊗ p ⊗ p ⊗ p ⊗ p ⊗ p dp, (36) Aiso= Z C p ⊗ p ⊗ p ⊗ p ⊗ p ⊗ p ⊗ p ⊗ p dp. (37) All these tensors can be calculated exactly as they do not involve the distribution function ψ.
CONCLUSION
In this work, an interaction field acting on any test particle has been derived. It is expected to quantify flow-induced migration effects. Our numerical experiments reveal that depending on the concentration distribution and the flow conditions inducing the orientation distribution, the interaction field can evolve significantly. Thus, in some conditions, a net force is predicted that pushes the test particle towards the regions of higher rate of strain. In order to validate the approach and its predictions, direct numerical simulation must be performed. This constitutes work in progress.
REFERENCES
[1] E. Abisset-Chavanne, J. Ferec, G. Ausias, E. Cueto, F. Chinesta, R. Keunings. A second-gradient theory of dilute suspensions of flexible rods in a Newtonian fluid.Archives of Computational Methods in Engineering, 22, 511-527, 2015.
[2] S. Advani, Ch. Tucker. The use of tensors to describe and predict fiber orientation in short fiber composites.J.
Rheol., 31, 751-784, 1987.
[3] F. Folgar, Ch. Tucker. Orientation behavior of fibers in concentrated suspensions.J. Reinf. Plast. Comp., 3, 98-119, 1984.
