Sistemas basados en comunicación: subastas

¯

σ=−p¯′I+ 2η(1)D¯ + 2c

η(2)−η(1)D¯(2)+cτ¯(2)

e . (4.37)

Thus, it can be seen that for given macroscopic strain rate ¯D, particle volume fraction c, and viscosities η(1) _{and η}(2)_{, the macroscopic Cauchy stress tensor ¯σ may determined by means of}

expression (4.33) for ¯D(2)_{in terms of the current values of the average of the extra stress tensor over}

the particle ¯τ(2)_e , together with the current values of the average aspect ratio and orientation of the

particles, as defined by expression (4.15). It should be emphasized that the above results reduce to the corresponding exact results of Gao et al. (2011) for dilute concentrations (c <<1) and vanishing viscosity (η(2)_{= 0) of the particles. In the next subsection, we address the characterization of these}

variables by means of appropriate evolution equations, starting from an appropriate initial state where the particles are initially spherical and unstressed.

4.3.2

Evolution equations for the microstructural variables and particle

elastic stress

So far, we have made use of Eulerian kinematics to describe the incremental behavior of the host fluid and particle phases, and accordingly generated estimates for theinstantaneous response of the suspension for a given state of the microstructure. However, when subjected to simple flows, the microstructure in the suspensions generally evolves in time as the applied deformation progresses. Therefore, in order to predict the effective time-dependent behavior of suspensions from a given instantaneous state of the microstructure, it is crucial to first characterize the evolution of relevant microstructural variables. In addition, given that the instantaneous response depends on the current values of elastic stresses acting on the particles, which are determined by incremental constitutive equations of the type (4.8), it is also necessary to develop evolution laws for the average elastic stresses in the particles.

Recalling from Appendix C.1 that our estimates for the instantaneous response of the suspension are based on the HSW variational approximation (Ponte Casta˜neda and Willis, 1995) implying that the local fields are (approximately) uniform inside the particle phase, as already anticipated in section 4.2.2, it follows that the initially spherical particles will deform through a sequence of ellipsoidal shapes throughout the deformation process, in such a way that the set of equations used to determine the instantaneous stress and strain rate fields inside particles will continue to apply at each increment

of time, except that at each step the current values of the microstructural variables and of the elastic stresses in the particles will need to be used.

First, recalling that the fluid and particle phases have been assumed to be incompressible, it follows that the volume fraction of the particles will remain constant throughout any deformation process,i.e.,

c=const. (4.38)

On the other hand, the evolution for the particle aspect ratiosw1 andw2, as defined by (4.14),

are obtained by simple kinematic arguments (see, for example, Bilby and Kolbuszewski, 1977) via ˙ w1=w1( ¯D(2)22 −D¯ (2) 11), w˙2=w2( ¯D33(2)−D¯ (2) 11), (4.39)

where it is noted that the overdot here denotes simple time derivatives (since w1 and w2 depend

only on time). It is also remarked in this context that the components of the tensorial variables associated with the particle phase, here and elsewhere, are referred to the principal axes of the ellipsoidal particle in their current state, as given by the triad{n1,n2,n3}.

Next, the evolution of the orthonormal vectorsn1,n2 andn3, serving to characterize the orien-

tation of particles, are determined by means of the kinematical relations ˙

ni=Ω ni, i= 1,2,3, (4.40)

where Ω is the (antisymmetric) spin tensor of the particle, whose components in the principal coordinate system {n1,n2,n3} are determined by means of the following relations (Ogden, 1997;

Aravas and Ponte Casta˜neda, 2004) Ωij = ¯Wij(2)−

(wi−1)2+ (wj−1)2

(wi−1)2−(wj−1)2

¯

D(2)_ij , i6=j. (4.41) In this notation, wheni or j is equal to 1, we definew1−1 =w0= 1. It should also be noted that

alternative expressions for the evolution of the microstructure can be derived directly in terms of the particle shape tensorsZ, as shown by Goddard and Miller (1967).

As can be seen from relations (4.37), together with (4.33), the calculation of the instantaneous macroscopic stress in the suspension requires knowledge of the average elastic extra stress in the particle phase. As discussed in Appendix C.1, due to the choice ofconstant-per-phase polarization fields, together with the choice of η0 _{= η}(1)_{, the PCW homogenization theory results in uniform}

stress and strain fields in the particle phase. As a consequence of this result, the constitutive relations for the average elastic stress fields in the particle phase take the same form as in the corresponding relations for the local fields. Therefore, for the case of KV particles with a Gent-type elastic stress (characterized by relations (4.7) and (4.8)), the evolution equations for the average elastic extra

stress in the particles is given by ∇ ¯ τ(2)_e = ˙¯τ(2)_e −L¯(2)τ¯(2)_e −τ¯(2)_e (¯L(2))T = 2µD¯(2)+ 2 µ Jmtr h ¯ D(2)τ¯(2) e +µI i ¯ τ(2) e +µI , (4.42) where the material time derivative ˙¯τ(2)_e appearing in the above expression is a simple, time derivative,

due to the fact that the stress field inside the particle is uniform (as already mentioned) and the convective terms hence vanishes. Note that the corresponding “total” form of the constitutive equation is given by ¯ τ(2) e =µ " 1−I¯ (2)₋₃ Jm −1 ¯ B(2)−I # , τ¯(2) v = 2η(2)D¯(2). (4.43)

where ¯I(2) _{= tr( ¯B}(2)_{), ¯B}(2) _{= ¯F}(2)_{( ¯F}(2)₎T _{and ¯F}(2) _{is the average deformation gradient in the}

particles.

It should also be noted that, in the limit as Jm → ∞ (corresponding to KV particles with a

neo-Hookean elastic part), the above evolution equation simplifies to

∇

¯

τ(2)

e = 2µD¯(2). (4.44)

In this context, it is important to emphasize that although the exact solution for the fields in the particles is not uniform, the uniform-field approximation is exact for dilute concentrations of particles

(c <<1), as originally argued by Roscoe (1967) in the context of the steady-state solutions, and by

Gao et al. (2011) for more general time-dependent motions of suspensions of purely elastic particles (η(2) _{= 0). For non-dilute concentrations, it is expected that the approximation of uniform fields}

in the particles will lead to fairly accurate results provided that the concentrations are not large enough to generate strong interactions between the particles.

In summary, for a given macroscopic velocity gradient ¯L = ¯D+ ¯W, the macroscopic stress ¯σ

in the suspension is given by expression (4.37), where ¯D(2) _{is given by expression (4.33). These}

quantities depend on the current values of the microstructural variablesS_{, as defined by expression}

(4.15), and determined by the evolution equations (4.39) and (4.40) from some known initial state, as well as on the current value of the average extra elastic stress ¯τ(2)_e in the particles, as determined by

expression (4.42). Note that the evolution equation for the particle axes (4.40) involves the average vorticity tensor in the particles, which is given in terms of other known variables by expression (4.34).

4.4

Steady-state estimates for the suspensions

It is known from earlier works (Roscoe, 1967; Goddard and Miller, 1967) that an initially spherical particle with Kelvin-Voigt viscoelastic behavior suspended in an infinite Newtonian fluid can admit, under certain conditions, steady-state (SS) solutions, where the particle becomes an ellipsoid with

fixed shape and orientation, while undergoing tank-treading motion with constant stress, strain rate and vorticity. According to the theory developed in Section 4.3, the stress and strain-rate fields are (approximately) uniform inside the particle phase, and steady-state solutions should still be possible for non-dilute concentrations of initially spherical, viscoelastic particles. In this case, existence of a SS solution will depend on flow conditions, as well as on the constitutive properties and volume fraction of the particles. For definiteness, we note that all variables in this section are evaluated at the steady state.

The SS solutions, if they exist, can be determined by setting the terms involving time derivatives equal to zero in the evolution equations for the extra stress tensor inside the particle, as well as in the evolution equations for the particle shape and orientation. The resulting expressions provide a set of algebraic equations to be solved for the six components of the extra stress tensor in the particle, ¯τττ(2)_{, the two aspect ratios,ω}

1,ω2, and the three orientational angles defined by the particle

axes,n1,n2, andn3.

First, making use of the incompressibility constraint in the particle phase (tr( ¯D(2)_{) = 0), together}

with the evolution equation for the aspect ratios (4.39), we deduce that, at the steady state, the normal components of the strain-rate tensor in the particle phase, relative to the principal axesni

of the ellipsoidal particles, are equal to zero: ¯ D11(2)= ¯D (2) 22 = ¯D (2) 33 = 0. (4.45)

Also, at the steady state, the evolution equations for the particle orientation, given by (4.40) and (4.41), imply that the three components of the vorticity tensor in the particle phase are given by

¯ W12(2)= 1 +w2 1 1−w2 1 ¯ D(2)12, W¯ (2) 13 = 1 +w2 2 1−w2 2 ¯ D13(2), W¯ (2) 23 = w2 1+w22 w2 1−w22 ¯ D(2)23. (4.46)

Next, recalling that the principal axes of the Finger tensor ¯B(2)_{= ( ¯V}(2)₎2_{correspond to the Eulerian}

axes of the deformation in the particles, so that the Eulerian axes coincide with the principal axes of the ellipsoidal particles, it follows that, at the steady state, when the particles have reached a fixed orientation, their orientation becomes fixed and is characterized by the triad{n1,n2,n3}. This

implies that, at the steady-state solution, the shear components of the Finger tensor (relative to the particle axes) must all vanish:

¯

B₁₂(2)= ¯B₁₃(2) = ¯B(2)₂₃ = 0. (4.47) Moreover, the normal components of ¯B(2)_{= ( ¯V}(2)₎2_{(again, relative to the particle axes) correspond}

to the principal stretches of the deformation in the particles: ¯ Bii(2)= ¯ λ(2)i 2 , i= 1,2,3 (no sum), (4.48) where ¯λ(2)_i (i = 1,2,3) denote the principal stretches of the deformation in the particles, i.e., the principal values of the left stretch tensor ¯V(2) in the particle. On the other hand, the shape of the

particle is described by the principal stretches as (see Figure 4.1(c) for definitions ofw1and w2) w1= ¯λ(2)2 /λ¯ (2) 1 , w2= ¯λ(2)3 /λ¯ (2) 1 , (4.49)

Making use of the above relations in (4.48), together with the incompressibility constraint in the particle phase ( ¯J(2)_{= det( ¯F}(2)_{) = ¯λ}(2)

1 λ¯ (2) 2 ¯λ (2) 3 = 1), we find that ¯ B₁₁(2)= (w1w2)−2/3, B¯22(2)= (w1)4/3(w2)−2/3, B¯(2)33 = (w1)−2/3(w2)4/3. (4.50)

In this context, it is worth mentioning that the components of the tensor ¯B(2) _{in relations (4.47)}

and (4.50) identically satisfy the evolution equation for the Finger tensor ¯B(2) _{(i.e., B}∇_¯(2) _{= 0)}

at the steady-state solution. Finally, we emphasize that the kinematical equations (4.45)-(4.50) are valid at SS solutions (if they exist) regardless of the constitutive behavior of particles. In the following subsection, we outline the additionalconstitutive equations in SS solutions for the case of KV particles.