Taylor-Couette flow of viscoplastic fluids

17 th International Couette-Taylor Workshop, Leeds, July 25-27, 2011
Taylor-Couette flow of viscoplastic fluids
B. Alibenyahia 1 , C. Lemaitre 2 , C. Nouar 3
1 University of Blida, Algeria
2 LRGP-ENSIC, 1 rue Grandville, BP 20451, 54001 Nancy Cedex, France
3 LEMTA, UMR 7563 (CNRS), 2 avenue de la forêt de Haye, BP 160, 54504 Vandoeuvre
Cedex, France
The objective of the present communication is to study the stability of a shear-thinning fluid in
Taylor-Couette flow. Few results exist in the literature for nonlinear purely viscous fluids and
they are sometimes in disagreement. Recently, it has been indicated [1] that for highly shearthinning
character, the primary instability would be characterized by longitudinal structures.
This surprising result led us to perform a threedimensional stability analysis.
Problem description
We study the flow of an incompressible fluid between two infinitely long cylinders. The inner
cylinder, of radius ˆR 1 , rotates with an angular velocity ˆΩ 1 and the outer cylinder, of radius ˆR 2 ,
is at rest. The nondimensional equations of motion read
∇.U = 0 , ∂ t U+ Re(U.∇)U =−∇P+∇.τ. (1)
where U is the velocity, P the pressure and τ the deviatoric stress tensor. The Reynolds number
is defined by Re= ˆρ ˆR 1 ˆΩ 1 ˆ d/ ˆµ re f . These equations have been put into a nondimensional form
using the gap thickness ˆ d = ˆR 2 − ˆR 1 , the inner cylinder velocity ˆΩ 1 ˆR 1 , and the reference viscosity
ˆµ re f that will be precised later. We consider shear-thinning fluids, i.e. fluids which viscosity
decreases with increasing shear-stress. We focus on two rheological models, which read in their
nondimensional form
Power law model: τ = µ ˙γ = ˙γ n−1 ˙γ
[ ]
Bingham model: τ = µ ˙γ = 1+ Ḃ γ
˙γ ⇔ τ > B; ˙γ = 0⇔τ ≤ B.
ˆτ 0
The Bingham number is defined by B =
ˆµ p ˆR 1 ˆΩ 1 / dˆ
where ˆτ 0 is the yield stress, ˆµ p is the
plastic viscosity, ˙γ and τ are the second invariant of the strain-rate tensor ˙γ and the deviatoric
√ √
stress tensor τ that are defined as follows: ˙γ = ∇U + ∇U T 1
; ˙γ =
2 ˙γ 1
i j ˙γ i j ;τ =
2 τ i j τ i j . The
reference viscosity ˆµ re f is ˆµ p for the Bingham model and the nominal viscosity ˆK ( ˆR 1 ˆΩ 1 / d ˆ
n−1
,
ˆK being the fluid consistency index, for the power law model.

Re cw
700
600
500
400
300
200
100
η = 0.5
η = 0.4
η = 0.8
η = 0.9
0
0.2 0.4 0.6 0.8 1
n
axial wavenumber, k
5.5
5
4.5
4
3.5
3
η = 0.4
η = 0.5
η = 0.66
η = 0.8
η = 0.9
2.5
0.2 0.4 0.6 0.8 1
n
Figure 1: Critical parameters versus index n for a power law fluid with different radius ratios η.
(o) Our results, (□) Jastrzebski et al.[3]. (Left) Reynolds number based on inner wall viscosity;
(right) critical wavenumber.
The basic flow is assumed to be of the form U b = ( 0,V b (r),0 ) . The expression of V b (r) is
given by Bird (1987) [4] for a power law fluid and by Landry et al. [5] for a Bingham fluid.
Linear stability analysis
An infinitesimal perturbation (εu ′ ,ε p ′ ) is superimposed upon the basic flow and equations
(
(1) are linearized around U b ,P b) to yield:
[( )
∇.u ′ = 0 , ∂ t u ′ + Re U b .∇ u ′ + ( u ′ .∇ ) U b] =−∇p ′ + ∇.τ ′ . (2)
The perturbed shear stress tensor reads τ ′ = 1 { (
τ U b + εu ′) − τ ( U b)} = µ b ˙γ(u ′ )+ ( µ t − µ b) A,
ε
(
where the tangential viscosity µ t has been introduced µ t = µ U b) + ∂ µ (
U b) ˙γ rθ
(U b) . The
∂ ˙γ rθ
components of tensor A are all zero except for A 1,2 = A 2,1 = ˙γ rθ (u ′ ). The solution is seeked under
the shape of normal modes { u ′ , p ′} ={u(r), p(r)}exp[σt+ i(mθ + kz)], where k∈Ris the
axial wavernumber and m∈Z the azimuthal wavenumber. We obtain a generalized eigenvalue
problem where σ is the searched eigenvalue. The numerical resolution is based on a Chebyshev
collocation method [6]. Computations have been carried out with 40 collocation points.
Axisymmetric perturbation: m = 0
Power law fluid: The critical Reynolds number has been computed for different values of
index n and radius ratio η and showed agreement with Jastrzebski et al.[3]. To describe the
effect of the shear-thinning behaviour on the stability threshold, it is more relevent to build
a Reynolds number Re cw with the viscosity at the inner wall µ b ( ˆR 1 ) and no longer ˆµ re f , Fig

3.5
Re c
10 5 B
10 4
10 3
10 2
η=0.883
η=0.5
k c
× (R 0
− R 1
)
3
2.5
η=0.5
η=0.883
10 1
10 0 10 1 10 2
2
0 5 10 15 20
B
Figure 2: (Left) Reynolds number versus Bingham number for two radius ratios: η = 0.5 (−−)
and η = 0.883 (···). (o) Landry et al. [5] for η = 0.5, (□) Landry et al. [5] for η = 0.883, (⋄)
Caton [1] for η = 0.5, (△) Lockett et al. [7] for η = 0.5. (Right) Critical wavenumber versus
Bingham number for two radius ratios, 0.5 and 0.883.
1(left). With this definition, it is observed that shear-thinning delays the appearance of Taylor
vortices. As could have been expected, the delay is larger when η decreases. For η close to
1, the velocity profile is almost linear, and the shear stress and the basic viscosity are almost
constant in the gap. In figure 1(right), the critical axial wavenumber is plot as a function of
n. For a given value of n the wavenumber increases for decreasing η. Thus for η = 0.4 and
n=0.3, the wavenumber is almost twice as high as in the Newtonian case. This may be related
to viscosity stratification in the gap.
Bingham fluid: For a Bingham fluid is observed in figure 2(left) that Re c increases when
viscosity stratification, induced by the Bingham number µ = 1+B/ ˙γ, increases. Our results
confirm those by Landry et al. [5]. The axial size of the Taylor vortices is scaled with the
thickness of the yielded zone ˜k c = k c ×(R 0 − R 1 ), Fig. 2(right). A change in the evolution of
˜k c occurs at B l when a plug zone appears in the flow, attached to the outer wall. Under B l , the
variation is in agreement with that observed for power law fluids. For higher B, the wavenumber
decreases when shear-thinning magnitude grows, which is consistent with the power law for a
small yielded thickness (η close to 1).
Three-dimensional perturbation
A three-dimensional perturbation is now considered. In figure 3 it is shown that the critical
Reynolds number Re c increases with inscreasing azimuthal wavenumber. The minimum is always
achieved in the axisymmetric case, m=0, even for very shear-thinning fluids(n=0.1 or
B=50). These results are in contradiction with those of Caton[1]. The shape of the marginal

stability curve at low axial wavenumbers suggests that Couette flow is linearly stable at k=0.
Further computations have been performed for k=0 and no instability has been found for either
rheological model: B≤100, 0.1≤n