Taylor-Couette Instability of Giesekus Fluids

ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 20, 2012

Mohammad Pourjafar1, and Kayvan Sadeghy2

1 University of Tehran, College of Engineering, School of Mechanical Engineering, Tehran, Iran. 2 University of Tehran, College of Engineering School of Mechanical Engineering, Tehran, Iran.

ABSTRACT inner cylinder). The fact that the critical Taylor-Couette instability of Giesekus Taylor number obtained by Taylor2 closely fluids is investigated at large gaps using a matched the experimental value was temporal, linear instability analysis regarded by many as one of the greatest assuming that the perturbations are achievements of the linear instability axisymmetric. Using normal-mode concept, analysis. In ensuing works, other aspects of an eigenvalue problem is obtained which is the flow (e.g., the effect of the gap size, and solved numerically using pseudo-spectral the effect of the cylinders eccentricity) has collocation method. A fluid’s elasticity is been investigated3. There are also many predicted to have a stabilizing or works addressing the effects of a fluid’s destabilizing effect on the Couette flow non-Newtonian behavior on the instability depending on the Weissenberg number picture of the circular Couette flow4-9. being smaller or larger than a critical value. The interest in studying Taylor-Couette instability for non-Newtonian fluids stems INTRODUCTION mainly from the fact that Couette Tangential flow between two infinitely- rheometers are widely used for long concentric cylinders rotating at characterizing non-Newtonian fluids. But, different angular velocities (the circular the rheological data obtained in Couette Couette flow) loses its stability and switches rheometers are useful only if the gap is free to more complicated laminar flow patterns from secondary flow. Therefore, depending on the Reynolds number1. The determining conditions under which first mode of instability exhibits itself as a secondary flow occurs in the gap is very secondary flow comprising a pair of important in shear rheometry. This is stationary-mode, axisymmetric, counter- perhaps why the literature is so rich when it rotating, toroidal roll cells stacked in the comes to the Taylor-Couette instability of gap. Taylor2 relied on a normal-mode, linear viscoelastic fluids5-9. One can notably instability analysis to show that for mention the numerical work carried out by Newtonian fluids, when the gap spacing is Beris10 in which the critical Taylor number narrow, the secondary flow should emerge has been calculated for the Giesekus fluid under narrow-gap restriction. whenever the parameter Re d R1 (the so- In the present work, we are going to called Taylor number) exceeds 41, where d extend Beris’ work10 to the large-gap is the gap spacing, R is the radius of the 1 situation, to the best of our knowledge, for inner cylinder, and Re is the Reynolds the first time. The case of large gap is more number (based on the angular velocity of the

91 common in shear rheometry, particularly 1  v (rv ) z  0 (1d) when dealing with fluids containing solid rr r  z particles, or for highly-viscous fluids. To that end, we superimpose an infinitesimal perturbation, represented by normal modes, where it has tacitly been assumed that the to the base flow and see what happens to its flow is axisymmetric (i.e., 0 ). In these amplitude in the course of time. An equations,  is the density, vi represents the eigenvalue problem will be obtained this velocity components, and ij are the way in which terms nonlinear in the components of the stress tensor. The stress perturbation quantities will be dropped. To components appearing in these equations ensure high degree of accuracy, spectral need a constitutive equation to be related to method will be used to find the critical the velocity field. In the present work, we conditions for the onset of instability. assume that the fluid of interest obeys the

Giesekus model as its constitutive equation. MATHEMATICAL FORMULATIONS In this viscoelastic fluid model, the extra Taylor-Couette geometry basically stress tensor is given by11, consists of two infinitely long, concentric cylinders of radii R1 and R 2 with the fluid      confined in the annulus between them. In  (.)  V.  t  general, the cylinders are rotating   (2) independently with an angular velocity   1 .V V.T  2D  and  either in the same direction or in  2  opposite direction. For non-Newtonian where 2D is the deformation-rate tensor,  fluids, the equations to start with are the is the mobility,  is the longest relaxation Cauchy equations of motion together with time of the fluid, and  is the fluid’s zero- the continuity equation. Assuming that the shear viscosity. It needs to be mentioned fluid is incompressible, in polar coordinate that, for   0 the Giesekus model reduces system, we have, simply to the upper-convected Maxwell

(UCM) model. On the other hand for vvv 2  v rr r  0 the Giesekus model reduces to (vrz  v)  trrz  (1a) the Newtonian fluid model. 1p  rz The Giesekus model is widely regarded (rrr )   rr r z r as one of the best rheological models for representing polymeric liquids (i.e., polymer

melts and concentrated polymer solutions). vvvvv  The mobility factor, which accounts for (v r  v)  rz anisotropic relaxation of polymer trrz  (1b) 2 macromolecules, profoundly affects the 1 (rrz )  ( ) 2  shear- and extensional-flow behaviour of the rr z fluid. In fact, by increasing  (starting from zero) shear-thinning starts at lower shear rates, and the asymptotic value of the vvvzzz (vrz  v)  extensional viscosity is decreased. This fluid trz (1c) model also correctly predicts the first and 1p(rrz )  ( zz )  second normal stress differences for  rr z z polymeric liquids provided that the mobility factor is selected in the range of 00.5 .

92 BASIC FLOW Figure 1 shows that the velocity profiles As the first step in our instability deviate more from a linear one by an analysis, we need the basic (i.e., steady) increase in the Weissenberg number. flow velocity and stress fields. Porjafar12 has recently shown that for Giesekus fluids, the INSTABILITY ANALYSIS governing equations can be simplified to the Having calculated the basic-flow following ordinary differential equations as velocity and stress fields for the Giesekus 0 fluid, we are now ready to investigate their far as the basic-flow velocity profiles, v , is concerned, vulnerability to instability when perturbed slightly. The perturbations are assumed to be axisymmetric and two-dimensional, so 0 0000 d v rrrrWe    r   (3) that we have.  0 dr r 1(1We)  rr 0 ˆ V(r,z,t) v (r) V(r,z,t), (4) where We R11  d is the Weissenberg 12 number. Porjafar has also shown that in 0 0 (r,z,t) (r)ˆ (r,z,t), (5) this ODE, the basic-flow stress terms rr and 0 0  can be related to r by simple algebraic where Vˆ , and ˆ represent the perturbation equations. As such, a shooting scheme in 0 velocity and stress fields. Assuming that the which r is guessed at the inner wall and perturbations can be represented by normal the velocity profile so-obtained is checked modes we have, to see if it satisfies the no-slip condition at the outer wall can easily be used to find v0 . ikz t  Vˆ  A(r) B(r) C(r) e , (6) Figure 1 shows typical basic-flow velocity profiles obtained this way at a mobility D(r) E(r) F(r) factor of 0.2 for a large-gap radii ratio  ikz t RR  0.65 ˆ E(r) G(r) H(r) e . of 12 when the cylinders are  (7) rotating in the opposite directions. F(r) H(r) Q(r)

1 We=0 1.25 In these equations, k is the 0.5 2.25 (dimensionless) real wavenumber, and

 RI i  is the (dimensionless) complex 0 phase velocity with R 0 signalling the

-0.5 occurrence of instability. 

V Using continuity equation, C(r) can be -1 related to A(r) by,

-1.5 11d  Cr   rAr . (8a) -2 ik r dr 

-2.5 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1  0 r rB r Re 1 A r 1 rD v r          Figure 1. Effect of the Weissenberg number  12rDErr1ikHr         on the basic-flow velocity profiles obtained (8b) 0.65,  0.2 1 at when 21 .

93 00 11  1 e 2We  r Dr  2We  rEr DD D rr  r   r 2 1Ar r  2  00 k  (1 ) 2 1  We rr r D  WeD  rr  r A  r    (8g)   vr0   Gr   2Re B r D Q r  1e Wer 00 Fr   WerHr    rr rr r  0 1  1Werrr 11  1DDr   (1 )ik 1  D D D  A r r 22  krr 11 DD D  (8h) r 2 ik 1 r F r k2   1e Wer 0  Hr   

(8c) 0  1 Wer  r  (1   ) D  F r r 1e We 00 r  r Er    rr     We0 11  1 (1 )ik B r r D D  D   A r 00   22 We(1 ) D  rr r   r D A  r   krr  r  0 (8i) r  r We0 r G r  We D r * ** 2 2 r    1 0  where Dddr and DD d dr. From the (1 ) D  v  r r boundary conditions, one can conclude that: * 0  1 ADA0,andB0.  (1 ) 1  We rr  r D  B r r (8d) NUMERICAL METHOD For solving the instability problem as posed by Eqs. 8a-i, we have decided to rely on the pseudo-spectral, Chebeshev-based, 1e 2We 0 r Gr     collocation method13. The basic idea in all 0 r  spectral methods is to assume that the r 2We E r  unknown function(s) can be approximated  1 0  (1 ) D  v  r by a sum of certain number of base r  functions; that is, 0  1 2We(1 ) r  r D  B r  r N1 0 r  120   A(y) ann (y) 1WeDrWeAr         (9a)  rrr n1 N1 (8e) B(y) bnn (y) (9b) n1 1  1e Q(r)2(1)    D Ar N1 r D(y) dnn (y) (9c) (8f) n1 N1 E(y) enn (y) (9d) n1

94 N1 fast propagation of the boundary errors into F(y) fnn (y) (9e) the integration domain. n1 In the next section, we will report N1 numerical results obtained suing the G(y) gnn (y) (9f) technique described above. As for the error n1 minimization criterion, we are going to force N1 the residue to become exactly equal to zero H(y) h (y)  nn (9g) at N+1 collocation points. The collocation n1 points used in this work are the Gauss- N1 Lobatto points defined by13, Q(y) qnn (y) (9h) n1  yj  cos j ; j 0,1,2,..., N. (11) where n y and n y are the base N functions. In the present work, the base functions will be constructed using In order to find the critical Taylor (or Chebyshev polynomials such that they Reynolds) number, it suffices to search for automatically satisfy the boundary the neutral instability curve. To that end, we conditions. To that end, we write, are going to find the wavenumber for which

we have: max[realcr (k cr ,Re cr ) ] 0. 2n  1 n yT  y  T y  T y We have investigated the effect of the nn1n2 n1n3 n2 (10a) number of collocation points on the results and reached to the conclusion that by setting n3.15 n1.85 N17 our numerical results become yTyTyTy    nn2n 54n 2n 5 n2independent of N. So, all results to be (10b) presented shortly are for this particular number of collocation points. 1 th where T(y) cosncos y is the n n   RESULTS AND DISCUSSIONS Chebyshev polynomials. Since these Having verified the code developed in polynomials are defined in the interval this work using known data for Newtonian 1, 1 we have relied on the transformation fluids, the code was then used to investigate y[2r1  ](1)to map the space the effect of the fluid’s elasticity (as represented by the Weissenberg number) on between the two cylinders to this interval; in the rise of the secondary flow. Instead of r this transformation we have: r  . Now, plotting the results for the critical Taylor R 2 number, we have decided to plot the results by substituting Eqs. 9a-h into Eqs. 8a-i, an in terms of the critical inner Reynolds eigenvalue matrix equation is obtained in number defined by, the form of: AX12 A Xwhere X[aa ,bb ,,qq ] is the Rd 1N11N11N1  Re 11 . (12) 1  eigenvector with A1 and A2 being square matrices. An important aspect of the method is its full satisfaction of the essential Figure 2 shows the effect of the boundary conditions―thanks to the Weissenberg number on the critical particular form of the base functions chosen Reynolds number obtained at a typical in this work. This was done in order to avoid mobility factor of   0.2 for the case in which the outer cylinder is fixed (Re2  0) .

95 From this figure, one can conclude that noted. Also, the rotation of the outer fluid’s elasticity has a stabilizing effect on cylinder is predicted to have a stabilizing the flow as long as it is small. But, at effect on the flow, regardless of its sense of sufficiently high Weissenberg numbers, its rotation, provided that the gap spacing is effect can be destabilizing. This figure also sufficiently small. At large gaps, however, shows that the critical Weissenberg number the smallest critical Reynolds number, Re1, (i.e., the Weissenberg number beyond which occurs when both cylinders are rotating in elasticity starts to destabilize the flow) the same direction. For example, for increases by a decrease in the gap size   0.883 Figure 3 shows that the least suggesting that for vanishingly small gap stable case corresponds to Re 7. spacing, the elasticity can have a stabilizing 2 effect on the flow over a broad range of practical Weissenberg numbers. It is also interesting to note that, at any given 95 Weissenberg number, the critical Reynolds 90 number is larger the smaller the gap size 85 suggesting that working at small gaps is 80 always preferable in Couette viscometers. 75

130 Critical Reynoldas Number(Re1,cr) 60 =0.65 120 0.883 55 =0.65 50 0.883 110 -30 -20 -10 0 10 20 30 Re2 100 Figure 3. Effect of the rotation of the outer 90 cylinder, Re2 , on the critical Reynolds 80 number, Re1 , obtained at 70 Critical Reynolds Number(Re1)  0.2, and We 1.5. 60 CONCLUDING REMARKS 50 Based on the results obtained in this 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 We work, one can conclude that in Taylor- Figure 2. Effect of the Weissenberg number Couette flow of viscoelastic fluids, fluid’s on the critical Reynolds number, Re . elasticity can have a stabilizing or 1 destabilizing effect on the flow depending

on the Weissenberg number being smaller or Figure 3 depicts the effect of the Re (i.e., 2 larger than a critical value. The critical the rotation of the outer cylinder) on the Weissenberg number (i.e., the Weissenberg critical Reynolds number, Re1 , obtained at a number beyond which, elasticity starts to typical Weissenberg number of We = 1.5 destabilize the flow) is increased by a and a typical mobility factor of 0.2 for decrease in the gap size. Also, at any given two different gap spacing of 0.65 and Weissenberg number, the flow becomes 0.883. This figure suggests that by a more stable (i.e., the critical Taylor or decrease in the gap size, the flow becomes Reynolds number is increased) by a progressively more stable, as previously decrease in the gap size.

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