Numerical investigation of transient contraction flows for worm-like micellar systems using Bautista–Manero models

J. Non-Newtonian Fluid Mech. 166 (2011) 102–117

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Numerical investigation of transient contraction flows for worm-like micellar systems using Bautista–Manero models H.R. Tamaddon Jahromi a , M.F. Webster a,∗ , J.P. Aguayo b , O. Manero c a b c

Institute of Non-Newtonian Fluid Mechanics, College of Engineering, Swansea University, SA2 8PP, United Kingdom Facultad de Química, Universidad Nacional Autónoma de México (UNAM), Coyoacán, Mexico City, 04510, Mexico Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México (UNAM), Coyoacán, Mexico City, 04510, Mexico

a r t i c l e

i n f o

Article history: Received 15 June 2010 Received in revised form 3 November 2010 Accepted 4 November 2010

Keywords: Worm-like micellar systems Computational prediction Modified Bautista–Manero model Finite element and finite volume Viscoelastic surfactants Contraction flow dynamics

a b s t r a c t This study is concerned with the numerical modelling of the Modified Bautista–Manero (MBM) model, for both steady-state and transient solutions in planar 4:1 contraction flow. This model was proposed to represent the structured composition and behaviour of worm-like micellar systems which have importance in industrial oil-reservoir recovery applications. A parameter sensitivity analysis for the rheology of this model is presented in both transient and steady response, covering pure shear and uniaxial extension. In addition, some features in evolutionary flow-structure are demonstrated in contraction flows due to the influence and imposition of start-up transient boundary conditions. The different effects of various model parameter choices are described through transient field response, stress and viscosity fields in the contraction flow setting. Distinction may be drawn between fluid response in the strong/moderate extension hardening regimes by matching both steady-state and transient shear and extensional viscosity peaks, contrasting between micellar (MBM) models against network-based counterparts Phan-Thien/Tanner (PTT). Simulations are performed with a hybrid finite volume/element algorithm. The momentum and continuity equations are solved by a Taylor–Galerkin/pressure-correction finite element method, whilst the constitutive equation is dealt with by a cell-vertex finite volume algorithm. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved.

1. Introduction This study is motivated by the need to model worm-like micellar systems that arise in typical oil-recovery situations. Such systems exhibit Maxwellian-type behaviour in small-amplitude oscillatory shear and the saturation of shear stress in steady simple shear flow. Nevertheless, according to Manero and co-workers [1–3], their adequate representation through suitable rheological constitutive equations remains short of that desired. Good representation of the extensional properties of such viscoelastic fluid systems is also necessary if one is to sensibly predict some important phenomena that arise in porous-media flow. Here we have firmly in mind, the prediction and understanding of enhanced pressure-drop in viscoelastic systems, which is by far and away at variance from that anticipated due to the shear viscosity of the system alone, and hence, its purely viscous equivalent system. Colloidal and polymeric liquid systems exhibit a variety of rheological response, some of which have direct application in the food, oil-extraction, cosmetics and coating industries. The complexity of viscoelastic micellar systems and their applications have

∗ Corresponding author. E-mail address: [email protected] (M.F. Webster).

greatly motivated rheological research on these types of fluids; for example, Castelletto et al. [4] studied phase transitions of micellar fluids through linear viscoelastic measurements. Akers and Belmonte [5] conducted a study of impact dynamics of spheres falling into wormlike micellar fluids. Yesilata et al. [6] studied the rheology of solutions of erucyl bis(2-hydroxyethyl) methyl ammonium chloride (EHAC – surfactant) and ammonium chloride (NH4Cl – salt) in water. These authors found that for linear viscoelastic experiments, a single relaxation time for the Maxwell model describes the response well at low frequencies, with significant departures as frequency increases. Hysteresis was observed for stepped shear flow experiments when increasing/decreasing the applied shear-stress, this being related to the onset of shear-banding. For regions outside this range/phenomenon, flow curves coincide. The Giesekus model was successfully used to represent shear and extensional flows, although with different relaxation times, depending upon the flow. Miller and Rothstein [7] designed a Couette cell to capture shearbanding features of solutions of cetylpyridinium (CPy+ – surfactant) and sodium salicylate (NaSal – salt) in water. In their research, they were able to measure velocity in the rheometer gap. From the resulting velocity profiles, multiple shear-bands were experimentally observed for the first time. No hysteresis was detected in their rheological tests. Additional transient measurements help to clarify shear-banding dynamics. A nonlinear dynamics study on

0377-0257/$ – see front matter. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2010.11.002

H.R. Tamaddon Jahromi et al. / J. Non-Newtonian Fluid Mech. 166 (2011) 102–117

these surfactant systems was performed by Ganapathy and Sood [8]; who showed that transient shear response in stress exhibits overshoots and undershoots, and can be followed by some chaotic behaviour. Specifically in oil-recovery processes, there is need for fluids capable of transforming from low-viscosity, Newtonian type to highly viscoelastic gels, and returning to low-viscosity liquid form, in response to change in prevailing flow conditions (rheological modification). These adjustments can be induced at various stages of the oil-recovery process. Viscoelastic surfactants systems (VES), with worm-like micelles formation, fall into this category and, as such, are increasingly employed as reservoir stimulation fluids due to their assembly and disassembly properties. Hydraulic stimulation of oil-wells is a technique to increase well-productivity when reservoir permeability is low. Fracturing treatment using viscoelastic gels is one such technique. This consists of transmitting hydraulic pressure to the rock to provoke fractures in the formation. Such a gel also contains particles (proppant), ceramic or sand, the whole constituting a porous proppant pack. The aim is for the gel to transport the pack into place and for the proppant to hold the fractures open, once fluid pressure is removed via gel degradation. Polymeric fluids, used as fracturing liquids, present the inconvenience that they cannot be totally degraded and their particle size is sufficient to block the pores of the fracture, limiting the hydraulic conductivity of the pack. The use of viscoelastic surfactants systems as fracturing fluids, displaying worm-like micelles, has been a relatively new innovation. Once the proppant pack, armed with these fluids, comes into contact with hydrocarbon produced by the fracture, the internal structure reverts into small spherical micelles or microemulsions. As a consequence, the viscosity and elasticity of the fluid residues fall dramatically, allowing the residues to easily flow from the pack. Very little pore blockage is therefore encountered, so that practically full fracture flow-back efficiency is achieved, see Boek et al. [9]. Investigating the rheological response of worm-like micelles, Manero et al. [1] proposed a model similar in structure to the Oldroyd-B constitutive equation for the evolution of stress, coupled with a Fredrickson (kinetic) equation. Such a model accounts for construction and destruction of micelles in solution, which results in a constitutive formulation capable of reproducing shear-thinning and strain-hardening/softening phenomena. The model is reported to be in excellent agreement with experimental data from steady shear and small amplitude oscillatory measurements for the EHAC surfactant. Nevertheless, the extensional viscosity of the model gives rise to unbounded response (discontinuous structure) at finite deformation rates, and as such, presents some abnormal behaviour. Since typical extension-rates experienced in porous-media flow enter this range of uncertainty, it is important to rectify this deficiency. This motivates the modifications developed below by Boek et al. [10]. Bautista et al. [11] employed a kinetic equation with a shear rate-dependent destruction term to describe shear-banding flow in wormlike micellar solutions. These authors confirmed that at low shear-rates, the flow is Newtonian. It becomes shear thinning at moderate shear-rates up to a critical level, whereupon the flow becomes unstable and manifests shear-banding. Miller and Rothstein [7] also performed a series of experiments to further investigate the phenomenon of shear-banding in surfactant solutions. Another recent attempt to represent wormlike micellar systems is via the model presented by Vasquez et al. [12]. This is a model in which chain rupture depends on local values of deformationrate. After rupture, two short chains can recombine to form a long chain. Such a model can well-describe the storage and loss moduli at a wide interval of frequencies; it also represents shear-thinning and various degrees of extension-hardening. Additionally, Manero et al. [13] recently improved the Bautista–Manero model with irre-

103

versible thermodynamics concepts. This improved model variation is able to reproduce thickening and thinning of shear viscosity, whilst the stress plateau is also remarkably well described. In the present study, we specifically compare and contrast flow response for these micellar MBM models against their networkbased PTT counterparts in planar 4:1 rounded-corner contraction flow. By matching on extensional viscosity, shear viscosity and Trouton ratio peaks across these two classes of models under steady-state conditions, we have been able to differentiate between the role and influence that these quantities have upon their respective numerical solutions in complex contraction flows. Then this comparison is taken further forward in the transient context, where we distinguish between moderate and strong hardening deformation regimes. This is intended to demonstrate the impact of the transient functionality of the worm-like micellar systems (with their assembly and disassembly properties) under complex deformation; such transient description is absent in the PTT-network models. Notably, here transient rheological differences do become apparent in both shear and extensional regimes, but to different extents under each and dependent on the deformation-rate range. As such, the moderate-hardening context bears out the most provocative findings. 2. Governing equations and numerical method We begin with the specification of the original Bautista–Manero (BM) model: extra stress evolution for (␶): ␶+

 ∇ ␶2 G0



∇



D + J D

,

(2.1)

kinetic equation for (1/): d −1 1 [ ] = dt S

1

−

0

1 



+k

 1 ∞

construction

−

1 



␶ : D,

(2.2)

destruction

where S is the structural relaxation time (construction of the micellar network), J the retardation time (related to the solvent viscosity), k the kinetic constant for structural breakdown (network destruction), 0 the viscosity constant at zero shear-rate, ∞ the viscosity constant at high shear-rates, and G0 is the elastic modulus. In order to overcome the unbounded extensional viscosity response of the original BM-model, Boek et al. (see [10]), proposed a modified Bautista–Manero (MBM) model, where the constant solvent, s , and polymeric, p , viscosity contributions are split. In this reformulation, the coefficient k/∞ is treated as a single parameter and remains finite. The MBM constitutive equation provides a continuous extensional viscosity response, which yields the possibility of supporting physically realistic strain-hardening/softening properties (see Eqs. (2.3)–(2.6)). The reformulated system may be expressed as follows: total stress (␶): ␶ = ␶p + ␶s ,

(2.3)

viscoelastic stress evolution (␶p ): ␶p +

p ∇ ␶p = 2p D, G0

kinetic equation (1/p ): d dt



1 p



1 = S



1 1 − 0 p

(2.4)

   k +

∞

␶p : D,

(2.5)

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H.R. Tamaddon Jahromi et al. / J. Non-Newtonian Fluid Mech. 166 (2011) 102–117

Fig. 1. Rheometrical response in pure shear and uniaxial extension, varying 0 .

solvent contribution (␶s ):

kinetic equation:

␶s = 2s D.

(2.6)

In Eq. (2.5), 0 corresponds to p (˙ → 0). Non-dimensional group numbers are defined via Reynolds (Re) and Weissenberg number (We), with reference to the following material parameters: Re = 

UL , 

(t, x) =

We =

p (t, x) , 

 U , G0 L ω = S

ˇ= U , L

s , 

=

0 =

 k  ∞

0 , 

s

U , L

(2.7)

where 0 is the zero-shear viscosity,  is the polymeric viscosity in terms of time and space, and ω and are the parameters governing construction and destruction within the model, respectively. Note that in the BM model, 0 is the total zero shear-rate viscosity, whilst in the modified version, it only accounts for the viscoelastic contribution. Then, in definitions (2.7),  = 0 for the BM model and for the modified MBM version,  = 0 + s . In this manner, the flow equations and MBM model may be expressed in equivalent non-dimensional form: momentum-continuity −∇ p + ∇ · (␶p + 2ˇD),

∇ · u = 0,

equations:(2.8a)Re

∂

∂t

u + u · ∇u =

stress evolution: ∇

1 

(2.9)

=

1 ω

1 0

−

1 

   +

ˇ

␶p : D.

(2.10)

Here, we employ our previously established numerical algorithm, a novel hybrid finite volume/element scheme (hy fV). This scheme has been shown to be second-order accurate [14,15]. In brief, the hybrid scheme consists of a Taylor–Galerkin (predictor–corrector) finite element discretisation [16,17], in conjunction with and a cell-vertex fluctuation-distribution finite volume stencil. These two procedures are coupled on each timestep to solve the associated evolving parabolic/hyperbolic system of partial differential equations. The finite element approximation is applied to the momentum-continuity set of equations, whilst the hyperbolic constitutive equation is treated via the finite volume discretisation. A three-stage algorithmic structure emerges per time-step cycle, which can be expressed in discrete algebraic form (see [18]) as: stage 1a :

Aau (U n+1/2 − U n ) = ba (pn , pn−1 , U n , T n , Dn ),

2 A (T n+1/2 − T n ) = b (U n , T n , Dn ), t stage 1b : Abu (U ∗ − U n ) = bb (pn , pn−1 , U n , U n+1/2 , T n+1/2 , Dn+1/2 )

stage 2 :

1 A (T n+1 − T n ) = b (U n+1/2 , T n+1/2 , Dn+1/2 ), t A2 (pn+1 − pn ) = b2 (U ∗ ),

stage 3 :

A3 (U n+1 − U ∗ ) = b3 (pn , pn+1 ),



(2.8b)

␶p + We␶p = 2D,

d dt

,

(2.11)

where the superscript n represents the time level, t the time step and U, U*, p, T and D are the nodal velocity, non-solenoidal velocity, pressure, elastic stress and deformation gradient, respectively. At stage 1, the velocity matrix Aau for stage 1a is given by Aau = (2Re/t)M + (s /2)S, and Abu for stage 1b is given by Abu =

H.R. Tamaddon Jahromi et al. / J. Non-Newtonian Fluid Mech. 166 (2011) 102–117

105

Fig. 2. Rheometrical response in pure shear and uniaxial extension, varying k/∞ .

(Re/t)M + (s /2)S, where M and S are the consistent mass and diffusive matrices. The stress matrix A equates to the identity matrix for the FV method, whilst under the finite element implementation would take a mass matrix form. The pressure matrix A2 is the standard stiffness matrix and A3 is the mass matrix. The equation right hand side terms bau , bbu , b2 and b3 are defined as: ba = {F 1 − [2 S + Re N(U)]U − B1 T }n + LT {P n + 1 (P n − P n−1 )}, bb = {F 1 − [2 S + Re N(U)]U − B1 T }n+1/2 + LT {P n + 1 (P n − P n−1 )}, b2 =

2 Re LU ∗ ,

t

(2.12)

0 ≤ 1 ≤ 1,

b3 = − LT (P n+1 − P n ),

= 0.5,

where N and L are the consistent advection and pressure gradient matrices, respectively. More detailed discussion on these issues can be found elsewhere in [14,19,20]. 3. Material functions for MBM model In order to establish the influence of the various model parameters in viscometric flows, plots are presented varying 0 , (k/∞ ) and  in Figs. 1–3, respectively, using the reference value of ˇ of 10−3 . We may first look to variation in (0 ). Shear viscosity shifts from an almost constant level (0 = 0.01), approaching the OldroydB response, to a case with extreme shear-thinning where the second plateau is reached at high shear-rates. There is no visible effect on N1 when the plot is presented in dimensionless form (Fig. 1). Increasing the zero shear-rate viscosity produces an increment in the degree of strain-hardening and reduction in the level of the second plateau of extensional viscosity.

The influence of variation in (k/∞ ) can be gathered from Fig. 2. For shear viscosity, an increase in this parameter is reflected in a decrease of the shear-rate, affecting where the thinning starts and the second plateau is reached. The levels of asymptotic plateaux are independent of (k/∞ ). From k/∞ = 0.001 to 100, the final limiting value of the first normal stress difference decreases by more than five decades (N1 /G0 ≈ 7 × 104 to 7 × 10−2 ) and is attained at much lower shear-rates. A similar increment in k/∞ lowers the degree of strain-hardening from 3 × 103 to zero, so that for the fluid with k/∞ = 100, extensional viscosity decreases monotonically, softening even at very low deformation rates. Referring to Fig. 3, the response of the model to a change in  somewhat replicates that dealt with under k/∞ . We can appreciate that the shear viscosity enters the thinning regime at lower deformations rates, and N1 is decreased by about four decades when  varies from 0.01 to 100. As above, an increase in  produces a considerable decrease in the degree of strain-hardening. Once the strain-rate transcends that associated with the peak in Trouton ratio (Tr), the second plateau in extensional viscosity is approached as strain-rates elevate further. Lower values of  are observed to generate larger peaks in extensional viscosity and steeper softening regimes. Note that, for ( ≥ 10), no strain-hardening is observed and so no peak in extensional viscosity survives. It is of interest to note that for even smaller values of solvent fraction ˇ of 10−4 (more difficult to handle numerically), some significant differences in transient material functions can be identified in the product of (k/∞ ) at values of unity and above. This shows up, for example, in two different fluid compositions of equal product value but in separate values of independent parameters (destruction, k/∞ ) and (construction, ). This may be demonstrated, for example, in shear viscosity with two choices of {k/∞ , } of {1.0, 1.0} and {0.01, 102 }, two different MBM-fluids of equal

106

H.R. Tamaddon Jahromi et al. / J. Non-Newtonian Fluid Mech. 166 (2011) 102–117

Fig. 3. Rheometrical response in pure shear and uniaxial extension, varying .

(k/∞ ) of unity, see Fig. 4. So, at a shear-rate of unity, these two fluids generate essentially identical shear viscosity up to times of 0.2 s, departing thereafter until steady-state is established at much longer times of O(102 ) s. In this figure, transient values of material functions for (k/∞ = 0.01,  = 102 ) display considerably larger values than for (k/∞ = 1,  = 1), although trends are similar; note that each case exhibits one large overshoot.

4. Discussion of results 4.1. Steady planar 4:1 contraction flow results The present study extends beyond our preliminary work on shear flow in a channel (see [21]) into more generalised flow configurations. Hence, we turn to the more complex flow through a

Fig. 4. Transient material functions for (k/∞ = 0.01,  = 100)-continuous and (k/∞ = 1,  = 1)-dashed, in both cases the product is (k/∞ ) = 1.

H.R. Tamaddon Jahromi et al. / J. Non-Newtonian Fluid Mech. 166 (2011) 102–117

107

Fig. 5. (a) Rm1, (b) Rm2 and (c) Rm3, planar 4:1 rounded-corner domain: section of contraction-flow mesh.

Table 1 Mesh characteristics, 4:1 rounded-corner geometry.

4:1 contraction, which offers a mix of shear and extensional deformation. Predictions have been obtained for the MBM-model in a 4:1 rounded-corner planar contraction flow, under inertia less conditions (Re = 0), with solvent/total viscosity ratio of ˇ = 1/9 (as benchmark), and zero shear viscosity of 0 = 8/9. The meshing used in this study is illustrated in Fig. 5b, with characteristics of 2593 quadratic elements, 5652 nodes (mesh Rm2 of Table 1), and a minimum mesh spacing (incircle of smallest element) of 0.01 located around the corner. To demonstrate solution mesh-independence,

Mesh

Elements

Nodes

Rm1 Rm2 Rm3

1626 2693 4751

3433 5652 9790

Strong Hardening

a

10

Extensional

10

c

EPTTε=0.25 MBM mh

η/η 0 shear

η/η 0 shear

10

0

Shear

-1

10

10

-2

0

λPTT

2

2

10 γ or λPTT ε

10 10 λ0 γ or λ0 ε

10

1

10

0

Extensional

10

4

-1 -2

10

Normalized Trouton ratio

d

Shear

10

0

2

10 λPTT γ or λPTT ε

2

10 10 λ0 γ or λ0 ε

Tr / 3

Tr / 3

EPTTε=0.25 MBM mh

1

0

3

10

2

10

1

10

0

0

2

10 λ PTT ε

10 λ0 ε

10

10 -2 10

4

f

EPTT ε=0.02 MBM sh

10

2

10

1

10

-1

10 -2 10

1

0

2

10 λPTT ε

10 λ0 ε

10

4

N1

N1

10

10

0

10 -2 10

e

4

Normalized Trouton ratio

EPTTε=0.02 MBM sh

10

0.0170 0.0097 0.0060

2

EPTTε=0.02 MBM sh

1

Rmin

Shear and extensional viscosities

b

2

10

18,069 29,740 51,470

Moderate Hardening

Shear and extensional viscosities 10

Degrees of freedom (u,p,␶)

10

-1

10

0

10

1

10

2

λ PTT γ or λ 0 γ

10

3

10

4

0

10 -1 -2 10

EPTT ε=0.25 MBM mh

10

-1

10

0

10

1

10

2

10

3

10

4

λ PTT γ or λ 0 γ

Fig. 6. Shear and uniaxial extensional viscosities, Trouton ratio, and N1 for both models; (a, c and e) strong hardening (MBM sh); (b, d and f) moderate hardening (MBM mh).

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H.R. Tamaddon Jahromi et al. / J. Non-Newtonian Fluid Mech. 166 (2011) 102–117

Ψmin EPTT (ε = 0.02)

a

Ψint

-ψ min*10 -3

MBM (ω = 0.28, ξ = 0.1) 10

1

10

0

L

X

-1

10 -1 10

10

0

10

We

1

10

2

X 5

EPTT (ε = 0.02)

4

MBM (ω = 0.28, ξ = 0.1)

c

EPTT (ε = 0.02)

5

MBM (ω = 0.28, ξ = 0.1)

4

3

3

2

2

L

X

b

L

1

1

10

-1

10

0

10

1

10

2

We

10

-1

10

0

10

1

10

2

We

Fig. 7. Salient-corner: (a) vortex intensity (×10−3 ); (b) horizontal (X); (c) vertical (L) cell-size, strong hardening [EPTT (ε = 0.25), MBM (ω = 0.28, = 0.1)], Tol = 10−6 .

we have also performed simulations on two additional meshes, Rm1 (Fig. 5a) and Rm3 (Fig. 5c), the characteristics for which are also supplied in Table 1. Furthermore, we may comment that extensive mesh refinement for this problem has already been conducted and appears elsewhere in Aboubacar et al. [23]. The lengths of upstream and downstream sections are 27.5L and 49L, respectively, and typical time-steps employed are of the order O(10−3 ). Time-step termination is ensured when the L2 -norm relative maximum difference between solution approximations over two successive time steps falls below a set threshold, (as in [24], Tol = 10−5 ). Simulations are initiated from a (We = 0.1)-solution state and continuation in We parameter-space is employed until a critical level of Weissenberg number is reached (Wecrit ), beyond which either oscillation or numerical divergence is encountered. Here, we study two separate trial cases, matching peaks in extensional viscosity corresponding to instances of strong hardening and moderate hardening, respectively. The comparable data (see Fig. 6 for material functions, steady scenario) was drawn from that exhibited by a shear-thinning exponential PhanThien/Tanner (EPTT) fluid with εPTT = 0.02 (strong strain-hardening) and εPTT = 0.25 (moderate hardening), with which we have past experience. Here, and in contrast to the situation for pom-pom models [22], both extensional and shear viscosity MBM peaks can be fitted to PTT-models with only one set of parameters to cover various regimes of hardening. It can be observed that both models display the same response at low shear-rates. In shear viscosity, the agreement of fit is excellent throughout a wide range of deformation rates. Under moderate deformations rates, the MBM model softens more rapidly, reaching its plateau earlier than that for the corresponding EPTT fluid. At high deformation rates, the moderate

hardening model tends to its limiting extensional viscosity level by rates ∼O(103 ), whilst this is delayed to rates ∼O(104 ) for the more strongly strain-hardening form. Such properties are then mirrored in Trouton ratios. Notably for the two sets of parameters chosen, the MBM-fluid with less degree of hardening reaches a larger critical Weissenberg number (Wecrit = 22) when compared to that for the strongly hardening scenario (Wecrit = 14). This reduction in critical We level attainable is a common feature observed when increasing the degree of strain-hardening involved, see Aboubacar et al. [23,24].

4.1.1. Strong-hardening: MBM (ω = 0.28, = 0.1) and EPTT (ε = 0.02) Next, we turn to consider flow structure upon the field, particularly in vortices generated, through streamlines patterns. Fig. 7 illustrates the salient-corner vortex intensity ( min ), and cell-size (both horizontal (X) and vertical (L) length measures, normalised by the downstream channel width) for this strain-hardening case; see also [24] for further streamlines pattern comparison with this EPTT-model. Both models (MBM (ω = 0.28, = 0.1) and EPTT (ε = 0.02)) display increase in vortex intensity, X and L at higher We; whilst for low We and up to a level around 3, both models manifest vortex and cell-size reduction. Note particularly, that vortex cell-size increases subsequently (We > 3) for EPTT (ε = 0.02), whilst it decreases up to We = 10 for MBM (ω = 0.28, = 0.1); and increases thereafter up to Wecrit = 14. The delayed departure in vortex behaviour from reduction-to-enhancement observed in MBM-solutions up to We = 10 may be viewed as a consequence of the more rapid strain-softening behaviour of the MBM-model, which is reflected also in Trouton Ratio (see Fig. 6a and c).

H.R. Tamaddon Jahromi et al. / J. Non-Newtonian Fluid Mech. 166 (2011) 102–117

Ψint

Ψmin

a

L X

EPTT (ε = 0.25)

3 2.4 1.8

109

MBM (ω = 4.0, ξ = 0.1)

-ψ min*10 -3

1.2 0.6

10

-1

10

0

10

We

1

X

b

10

2

L

3

c

EPTT (ε = 0.25) MBM (ω = 4.0, ξ = 0.1)

2.5

3

EPTT (ε = 0.25) MBM (ω = 4.0, ξ = 0.1)

2.5 2

1.5

1.5

X

L

2

1

1

0.5 -1 10

10

0

We

10

1

10

2

0.5 -1 10

10

0

We

10

1

10

2

Fig. 8. Salient-corner: (a) vortex intensity (×10−3 ); (b) horizontal (X); (c) vertical (L) cell-size, moderate hardening [EPTT (ε = 0.25), MBM (ω = 4.0, = 0.1)], Tol = 10−6 .

4.1.2. Moderate-hardening: MBM (ω = 4.0, = 0.1) and EPTT (ε = 0.25) Moreover, vortex-enhancement (mainly intensity) is apparent in the moderate-hardening case (Fig. 8) within the low deformation-rate regime (up to We = 1), as opposed to the initial vortex-inhibition for the strongly hardening scenario (see above, Fig. 7). We can associate this modest vortex enhancement with the mild initial strain-hardening of these model options (MBM (ω = 4.0, = 0.1) and EPTT (ε = 0.25)). Another striking feature is the significant vortex intensity reduction apparent in the range 5 ≤ We ≤ 20 for EPTT (ε = 0.25), in comparison with solutions for MBM (ω = 4.0, = 0.1). Note that the EPTT (ε = 0.25)-model encounters a sharp decrease in extensional viscosity at deformation-rates between 5 and 20, see Fig. 6b. The cell-size, X and L, of the EPTT (ε = 0.25) model remains practically static up to We = 20; then decreases subsequently. The corresponding MBM (ω = 4.0, = 0.1) cell-size measures sustain consistently lower levels than those for EPTT against We-increase. Again, this can be attributed to the more rapid strain-softening behaviour of this MBM version, see Fig. 6b. Consistency with mesh refinement has been established by cross-checking solutions through a series of meshes (Rm1, Rm2 and Rm3) for this MBM (ω = 4.0, = 0.1) moderate hardening case. Quantitative information regarding mesh convergence of the salient-corner vortex intensity ( min ), vertical (L) and horizontal (X) cell-size is provided in Table 2 covering the range 0.1 ≤ We ≤ 22. The information in this table demonstrates that convergence with mesh refinement has been achieved for the range of parameters considered. Here, differences in vortex intensity and size reflect a decline in consistency to within 0.7% (maximum) relating from mesh Rm2 to mesh Rm3, whilst standing at within 4% between mesh Rm1 to mesh Rm2.

4.1.3. Couette correction: MBM and EPTT In Fig. 9, we chart the Couette correction (C) vs We for the two models and settings of strong and moderate-hardening (MBM and EPTT). The Couette correction C, may be defined as C=

pen , 2 w

where pen = (p–pu Lu –pd Ld ).

In the above identity, p is the total pressure difference between the inlet and outlet (measured typically by transducers), pu is the fully developed pressure gradient in the upstream section, pd is the fully developed pressure gradient in the downstream section, Lu and Ld are their respective lengths, and w is the fully developed wall shear stress in the downstream channel. The Couette correction is found to be strongly influenced by variation in extensional behaviour. Indeed, though both MBM and PTT models considered are shear-thinning, trends are distinctly disparate according to chosen model parameter sets, and re-echo the differences in extensional properties. For the strong

Table 2 Mesh convergence data: salient-corner: vortex intensity (−␺min × 10−3 ); vertical (L); horizontal (X) cell-size, MBM (ω = 4.0, = 0.1), Tol = 10−5 . We

0.1 1.0 5.0 10.0 20.0 22.0

L

min

X

Rm1

Rm2

Rm3

Rm1

Rm2

Rm3

Rm1

Rm2

Rm3

0.531 0.572 0.330 0.348 0.506 0.407

0.541 0.591 0.345 0.359 0.513 0.414

0.545 0.593 0.344 0.358 0.514 0.415

1.40 1.47 1.18 1.28 1.27 1.20

1.44 1.50 1.22 1.33 1.33 1.26

1.44 1.51 1.21 1.32 1.34 1.27

1.19 1.22 1.0 0.98 0.98 0.92

1.22 1.27 1.01 1.0 1.0 0.95

1.22 1.28 1.01 1.0 1.01 0.95

110

H.R. Tamaddon Jahromi et al. / J. Non-Newtonian Fluid Mech. 166 (2011) 102–117

6

Transient planar extensional viscosity 103

MBM ( ω = 0.28, ξ= 0.1) MBM ( ω = 4.0, ξ= 0.1) EPTT ( ε = 0.02) EPTT ( ε = 0.25)

2

η+Planar / η0 shear

4

2 EPTT ( ε = 0.25)

C

0

MBM, ω = 4.0, ξ = 0.1 EPTT, ε=0.25

10

101

dε/dt=1 dε/dt=10

100

-2

10-1 10-2

EPTT ( ε = 0.02)

-4

MBM ( ω = 0.28, ξ= 0.1)

0

10

MBM ( ω = 4.0, ξ= 0.1)

20

We

30

101 40

Fig. 9. Couette correction vs We, strong hardening, MBM (ω = 0.28, = 0.1), EPTT (ε = 0.02); moderate-hardening, MBM (ω = 4.0, = 0.1), EPTT (ε = 0.25).

strain-hardening case, the Couette correction for the MBM (ω = 0.28, = 0.1) fluid follows its EPTT (ε = 0.02) counterpart up to We = 2, cf. Fig. 9. Beyond this point, a sharp decrease is observed in Couette correction with this MBM variant. This outcome can be attributed to the more rapid strain-softening behaviour of the MBM compared to the EPTT model. Moreover, for the moderatehardening scenario, the EPTT (ε = 0.25) case and beyond We = 1, there is a rising Couette correction trend (positive values) observed, as opposed to a declining trend into negative values with its MBM (ω = 4.0, = 0.1) counterpart (itself smaller in magnitude than that for the MBM strong hardening result). For the MBM model, the entry correction pen decreases as the level of We increases (as occurs with the Oldroyd-B model); this is strongly influenced by solvent fraction, extensional viscosity, N1 and shearthinning influence, see [25–27]. Therefore, for the MBM model both the entry correction and (flow-exit) wall shear-stress scaling (always above unity and tends to limiting constant plateau) are responsible for the continuous decrease in Couette correction. This is true for both MBM cases and We ≥ 2, with strong hardening of MBM (ω = 0.28, = 0.1) and moderate-hardening of MBM (ω = 4.0, = 0.1). Note, that the Couette correction result for the strong-hardening MBM case lies between that of EPTT (0.02) and Oldroyd-B model [24]. Following our findings in [25–27] for pressure-drops in axisymmetric problems, one may suspect that these trends are governed by N1 behaviour. Nevertheless in this planar context, data on N1 in Fig. 6e and f indicates that this is hardly the case. One suspects that the limiting level of We for MBM is more constrained for this model (as is deformation rate) than PTT, due to the strong link to Oldroyd-B structure in its derivation. It is conspicuous that there is only one instance amongst the Couette correction solutions that displays a rising trend against increasing We, that of EPTT (ε = 0.25) moderate-hardening. In this case, the rising Couette correction result is argued to be primarily due to declining wall shear stress and scaling; in fact there is little actual change in pressure drop itself. In contrast and for the EPTT (ε = 0.02) strong-hardening instance, the Couette correction is declining. Here, the pressure drop itself is negative but rising beyond We = 5, yet wall shear stress effects reverse this trend to generate negative but falling Couette correction. This finding is due once more to sign and scaling magnitude of wall shear stress. Such a position and explanation may be extracted from the additional detailed evidence provided in Appendix A for both cases separately, of moderate and strong hardening. There the two contributing com-

100

101

t / λ0b

102

Transient shear viscosity

50

η+Shear / η0 shear

-6

10-1

MBM, ω = 4.0, ξ = 0.1 EPTT,ε=0.25 dγ/dt=1

0

10

dγ/dt=10

10-1 -2 10

10-1

100

t / λ0b

101

102

Fig. 10. Transient shear and planar extensional viscosities, moderate hardening, MBM (ω = 4.0, = 0.1), EPTT (ε = 0.25).

ponents to Couette Correction, of pressure drop and wall shear stress, are shown comparatively to avoid clutter in the quantity of data (to be interpreted against Fig. 9). 4.2. Transient planar 4:1 contraction flow results A principal goal of this study has been to compare and contrast the dynamic response in complex flows of both MBM and PTT models. Hence, this is now related in terms of evolution in vortex characteristics, stress and viscosity fields. Here, the same set of parameters is chosen as under steady conditions of Section 4.1 with the same models (MBM and EPTT). For the corresponding transient 4:1 contraction flow problem, time-dependent inlet flow conditions may be chosen as either transient flow-rate controlled driving boundary conditions (BC) (fixed flow-rate, impulsive instigation), or force-controlled (p) boundary conditions (fixed force, impulsive instigation). The fixed flow-rate condition equates to the sudden imposition of a steady-state flow-rate (fully developed inlet velocity profile, Dirichlet conditions-fixed in time – a shocked condition). Both forms of these conditions have been implemented and contrasted elsewhere for Oldroyd and pom-pom models [28]. The force-control boundary conditions are more physically realisable and suitable with the MBM model (hence, pursued below), since an exact transient analytical BC is not readily available. Then, the level of pressure-difference for the force-control case is selected as that equivalent to the steady-state pressure-drop under flow-rate control, hence maintaining parity in steady-state flow-rate. With respect to temporal discretisation and convergence in time, we refer to our earlier detailed studies on this same problem, in Webster et al. [15], where we analysed temporal order. There, we employed a structured mesh and hierarchical reduction in time-step (t), via three different sizes of t (0.005, 0.0025 and 0.001). Importantly, temporal convergence rates were estab-

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111

Fig. 11. Transient development of velocity, stress, pressure and viscosity around contraction, We = 1, MBM (ω = 4.0, = 0.1), EPTT (ε = 0.25).

lished as being above first-order and tending to second-order, lying between O(t = 1.29) and O(t = 1.89). 4.2.1. Moderate-hardening: MBM (ω = 4.0, = 0.1) and EPTT (ε = 0.25) The transient viscometric data is provided first for the moderate hardening case as this provides more stark differences. Data in Fig. 10 contrasts response in shear and planar extensional viscosities for the MBM (ω = 4.0, = 0.1) fluid against that of EPTT (ε = 0.25), at two different levels of deformation rate of O(1) and O(10). Notably, here transient rheological differences are apparent in both shear and extensional regimes. The MBM (ω = 4.0, = 0.1) model in extension demonstrates overshoots and undershoots, both at low and moderate strain-rates, with peak value in extensional viscosity of around 20 at an extension-rate of O(10) (see below for the strong-hardening case). In shear, there are no significant differences apparent at lower shear-rates between MBM and EPTT behaviour. However, these are beginning to appear by shear-rates of O(10). Graphical data by way of MBM and EPTT transient profiles is provided in Fig. 11 for this moderate hardening case (We = 1), sampled around the corner, just beyond its rounding above the downstream wall. There are significant differences observed in the transient development of velocity, stress, pressure and viscosity profiles, as anticipated from the transient viscometric data. Here, overshoots are detected in xx , xy and pressure with the MBM model; whilst

Ux transient growth in MBM is distinctly suppressed below that of the EPTT model. However, such localised pressure changes are not detectable in measured overall entry-exit pressure drops. Viscosity (p ) decreases continually in time, rapidly to t = 2 units, and reaches its plateau (of ∼O(0.3)) thereafter at the protracted time of t = 13 units. Steady-state conditions for EPTT (ε = 0.25)-solutions are attained around t = 5 units, some three times faster than for the MBM (ω = 4.0, = 0.1) model, as borne out from the transient material response in Fig. 10. The transient streamline growth patterns for MBM (ω = 4.0, = 0.1), EPTT (ε = 0.25) are presented in Fig. 12. This moderate hardening case at We = 1, in contrast to that of strong hardening (below), reflects clear transient differences across the models, in magnitude, size and vortex intensity. This is most stark between times, t = 1.0 and t = 4.0 units, corresponding to detectable differences in xx , xy and pressure sustained throughout this period. Here, the more rapid early vortex growth is visible in MBM-solutions between times, t = 1.0 and t = 2.0, which retards by t = 4.0 units. These times correspond to noted transient extensional viscosity overshoots illustrated in Fig. 10. Alongside streamline patterns it is also instructive to consider counterpart stress components through N1 with vector field superimposed in black-grey, Fig. 13. Amongst the various stress field components, it is interesting to note that the N2 -fields ( yy ), left as a residue of N1 in the recess corner, provide a signature for vortex development in the corner, and hence, differentiate between model

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MBM ω = 4.0, ξ = 0.1

EPTT εptt= 0.25

t=0.5

t=1.0

t=2.0

t=3.0

t=4.0

t=5.0

Fig. 12. Streamline development in time, We = 1, moderate hardening, MBM (ω = 4.0, = 0.1), EPTT (ε = 0.25).

solutions over the time period t = 3 to t = 5 units (see superimpose vector field on plots). N2 -minima, along and around the centreline gradually disperse downstream (not shown); this taking three times as long to occur for the MBM model than for EPTT model. Max-

ima are firmly localised to the corner cap region, where they have developed relatively rapidly by the time, t = 3 units. In addition, transient development of the first normal stress difference (N1 ) field, with MBM (ω = 4.0, = 0.1) in Fig. 13, shows an increase at t = 3 to a decrease at t = 5 in maximum value attained (from 5.81 to 4.50). This finding may be associated with the MBM extensional viscosity overshoot around t = 3 and undershoot around t = 5 at extension rate of O(1), cf. Fig. 10. Notably in the EPTT (ε = 0.25) solutions, an increasing trend is observed in maximum value of N1 from an early stage of time development (t = 1) up to (t = 5) (N1 -values from 2.01 to 4.17). This coincides with the increasing trend noted in EPTT (ε = 0.25) transient extensional viscosity at extension rate of O(1), see Fig. 10. Fig. 14 provides a typical snapshot of deformation rate fields for the two moderate hardening model cases at an advanced time of t = 4 units. At the early time t = 1.0 (not shown), stretch rates are of order unity and shear rates are of order 3, and there are no apparent differences in deformation rate fields observable between the MBM and EPTT solutions. At the later time, t = 4.0, some variation has developed and is captured in localised peak deformation rates between the model solutions, with lower values corresponding to the MBM-case. Typically, stretch-rate minima (around corner) vary between orders, 1.1 and 1.5, and shear-rate maxima (along downstream wall) vary likewise between orders, 3.8 and 5.3 for MBM and EPTT models, respectively. For this moderate hardening case and in contrast to that of strong hardening below (cf. Fig. 10 vs Fig. 16), clear new features are apparent across solutions for the models chosen, localised to the shear boundary layer along the downstream

Fig. 13. Transient development of first normal stress difference (N1 ) ( yy ) field, We = 1, moderate hardening, MBM (ω = 4.0, = 0.1), EPTT (ε = 0.25).

H.R. Tamaddon Jahromi et al. / J. Non-Newtonian Fluid Mech. 166 (2011) 102–117

113

Transient planar extensional viscosity 3

10

2

η+Planar / η0 shear

10

MBM, ω = 0.28, ξ = 0.1 EPTT, ε = 0.02 dε/dt=1

1

dε/dt=10

10

0

10

-1

10

10-2

1

10-1

100 t / λ0b

101

102

Transient shear viscosity

Fig. 14. Extensional (du/dx) and shear (du/dy) rate fields, We = 1, moderate hardening, t = 4.0, MBM (ω = 4.0, = 0.1), EPTT (ε = 0.25).

wall (shear-rates vary in magnitude and size; their values for MBM half of EPTT). There are also some differences, but lesser, captured between models in stretch-rate (negative values) around the corner. Hence, as noted, both shear and extensional differences are prominent for this parametric range and models. Switching attention to the transient development of the viscosity (p ) field, the MBM (ω = 4.0, = 0.1) model displays large change in viscosity minimum band (blue) values (decline from 0.87 to 0.26 by t = 13 units), which spreads out across the channel (from the downstream wall to centreline) as time evolves from rest to t = 13 units. This is of course one of the principal reasons to devise and utilise such model fluids (temporal rheological modification

η+Shear / η0 shear

10

MBM, ω = 0.28, ξ = 0.1 EPTT, ε = 0.02 dγ/dt=1

0

10

dγ/dt=10

10-1 -2 10

10-1

100 t / λ0b

101

102

Fig. 16. Transient shear and planar extensional viscosities, strong hardening, MBM (ω = 0.28, = 0.1), EPTT (ε = 0.02).

Fig. 15. Transient development of viscosity (p ) field, We = 1, moderate hardening, MBM (ω = 4.0, = 0.1).

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Fig. 17. Transient development of velocity, stress, pressure and viscosity around contraction, We = 1, MBM (ω = 0.28, = 0.1), EPTT (ε = 0.02).

subject to adjustment in deformation). The minimum band in viscosity shown in Fig. 15, has spread out from the corner to reach the centreline during the time period between t = 5 and t = 10 units. These field developments may be primarily associated with shear influences, being driven by those generated in the shear boundary layer. For the more severe strong-hardening responsive models (MBM (ω = 0.28, = 0.1)), such viscosity changes are far less dramatic (drop from 0.89 to 0.65 by t = 5 units), with the viscosity minimum band restricted to the downstream wall shear boundary layer. 4.2.2. Strong-hardening: MBM (ω = 0.28, = 0.1) and EPTT (ε = 0.02) Here again under the strongly hardening scenario, the same set of model parameters is employed as under steady-state analysis, to investigate transient solution evolution for the MBM and EPTT models. Transient shear and planar extensional viscosities are shown in Fig. 16 for MBM (ω = 0.28, = 0.1) and EPTT (ε = 0.02) models at levels of deformation rate of O(1) and O(10). Now in shear, there is little difference in response detected in shear viscosity between MBM (ω = 0.28, = 0.1) and EPTT (ε = 0.02). Nevertheless, shear viscosity values at larger rates and longer times are slightly larger for the MBM, in comparison

to the EPTT model. In extensional data and at low strain-rates of O(1), there is close parity between MBM and EPTT response; whilst differences begin to appear in the transient with undershoots/overshoots at larger strain-rates of around O(10). One notes, that corresponding predicted field solutions (not shown) for the contraction flow cover the deformation rates around the order of unity. Fig. 17 demonstrates the corresponding transient development of velocity, stress, pressure and viscosity profiles around contraction for the strong-hardening setting (MBM (ω = 0.28, = 0.1), EPTT (ε = 0.02)) at We = 1. This position lies in distinct contrast to the moderate hardening case (see above, Figs. 11, 10 and 16), where there are significant solution profile differences to note between the two model variants (MBM (ω = 4.0, = 0.1) and EPTT (ε = 0.25)). With strong-hardening property, there is no apparent difference noted between models in shear stress, localised pressure, or entry-exit pressure drop (in time). Conspicuously, velocity transient solution profiles for the MBM-model undershoot those for the EPTT-model, which also leads to normal stress ( xx and yy ) undershoot. Furthermore, for this strong hardening regime at We = 1 (not shown), and in contrast to the moderate hardening instance, only minor differences in transient vortex development are

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115

Fig. 18. Transient development of viscosity (p ) field, strong hardening, MBM (ω = 0.28, = 0.1).

detected (magnitude, size and intensity) across the models, MBM (ω = 0.28, = 0.1) to EPTT (ε = 0.02). In Fig. 18 and for MBM model, the viscosity (p ) profile around the contraction decreases to a minimum around t = 2 units, before gradually recovering to approach its ultimate plateau limit in time, which is reached in comparable fashion between times t = 4 and t = 5 units. In contrast and considering larger elasticity levels with We = 2, the impact of longer time scales now begins to reveal itself through differences in transient vortex structure development. This is amply demonstrated in Fig. 19, where one generally observes significantly more pronounced transient vortex formation with the MBM-model over its EPTT counterpart between t = 2 and t = 4 units – a provocative and interesting finding (yet without a positive Couette Correction result, see above comments). A further solution comparison between the two strong-hardening model fluids lies in the transient development of first-normal stress difference field for We = 2. Significant differences between N1 (MBM) and N1 (EPTT) fields occur at time t = 4 units: MBM N1 -max = 27.66 vs EPTT N1 -max = 20.95 (not shown, greater disparity than noted above in Fig. 13). This is a consequence of the shear boundary effect, with slightly larger shear-rates being ultimately developed by the MBM model. The converse is found in extension along and around the symmetry line upstream of the contraction plane. Here, larger values of N1 are experienced by the EPTT model, with greater spread downstream (see also Fig. 13).

MBM ω = 0.28, ξ = 0.1

EPTT εptt= 0.02

t=0.5

t=1.0

t=2.0 t=3.0

t=4.0

t=10.0

t=6.0

Fig. 19. Streamline development in time, We = 2, strong hardening, MBM (ω = 0.28, = 0.1), EPTT (ε = 0.02).

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a

4

Moderate Hardening C = Δ pen / 2τw

2

τw (MBM) τw (EPTT)

0

Δ pen (EPTT)

-2 -4 -6 Δ pen(MBM)

-8

0

10

20

30

40

50

We

b

Strong Hardening

4

C = Δpen / 2τw τw (MBM)

2

τw (EPTT)

0 Δ pen (EPTT)

-2 -4 -6 Δ pen (MBM)

-8

0

5

10

15

20

25

30

We Fig. A. Couette correction components vs We: (a) moderate-hardening, MBM (ω = 4.0, = 0.1), EPTT (ε = 0.25) and (b) strong hardening, MBM (ω = 0.28, = 0.1), EPTT (ε = 0.02).

to EPTT models for two different settings of strong strain-hardening and moderate-hardening. For the strong-hardening case, similar trends in solution response may be extracted in terms of vortex enhancement or inhibition at higher We and for low We up to the level of 3. The cell-size of the salient-corner vortices is found to increase subsequently for EPTT, whilst it decreases up to We = 10 for the MBM model. Under moderate-hardening properties, MBM cellsizes sustain consistently lower levels than those for counterpart EPTT fluids against We-increase. Furthermore, we have also investigated the dynamic response in complex flows of both MBM and PTT models in terms of evolution in vortex characteristics, stress and viscosity fields. In this respect and for the MBM model, the transient viscometric data for the moderate hardening case presents more stark differences than those gathered from the strong-hardening regime, with clear apparent differences in both shear and extensional regimes for moderate hardening, that are suppressed in shear under strong-hardening. Moreover in extension, the MBM (ω = 4.0, = 0.1) model (moderate hardening) demonstrates overshoots and undershoots, both at low and moderate strain-rates, whilst overshoots/undershoots are only visible at higher rates for the MBM (ω = 0.28, = 0.1) case (stronghardening). Hence, there are significant differences observed at We = 1 in the transient development of velocity, stress, pressure and viscosity profiles around the contraction zone for the MBM (ω = 4.0, = 0.1) fluid against that of EPTT (ε = 0.25). This is a direct consequence of differences in their transient viscometric data. Larger elasticity levels of We = 2 are required for the strong-hardening MBM (ω = 0.28, = 0.1) fluid to detect differences in transient vortex structure development over its EPTT (ε = 0.02) counterpart. Furthermore, the transient development of the viscosity field for the moderate hardening MBM (ω = 4.0, = 0.1) model represents large change in minimum band viscosity values, which spreads out across the channel as time evolves. For this model the minimum band in viscosity has spread out from the corner to reach the centreline during the time period between t = 5 and t = 10 units. This dynamic response is consistent with the underlying derivation theory and can be associated with shear influences, being driven by those generated in the shear boundary layer. For the more severe stronghardening responsive model MBM (ω = 0.28, = 0.1), such viscosity changes are much less dramatic (suppression), with the viscosity minimum band restricted to the downstream wall shear boundary layer.

5. Concluding remarks

Acknowledgements

Phenomenological representation of worm-like micellar systems, of some importance in industrial oil-recovery situations, was originally proposed via the Bautista–Manero (BM) model. Reflecting some common properties with polymeric systems, micellar systems are described through mechanisms of build-up and decay of system structure. Unfortunately, the BM model was found to exhibit unphysical response in extensional viscosity that demanded some modification. This has given rise to an amended form, the modified Bautista–Manero model (MBM), introduced by Boek et al. [10], which has rendered it possible to model uniaxial flow successfully. Material functions for this new MBM version have been fitted to match two different configurations of the exponential version of the Phan-Thien/Tanner model (networkbased). Here, both extensional and shear viscosity MBM peaks can be fitted to PTT-models with only one set of parameters to cover various regimes of hardening. In this regard, steady-state and transient solutions have been generated for the complex flow configuration of the 4:1 contraction flow, using idealized solvent viscosity-fraction parameter settings (non-industrial levels). We have sought to compare and contrast steady extensional response and vortex characteristics across model types, from MBM

The authors acknowledge financial support for the present work from Schlumberger Cambridge, EPSRC-UK, Ministry of Education (SEP), Mexico, and Project 100195 from CONACYT during the course of this research. Appendix A. Couette correction separate components See Fig. A. References [1] O. Manero, F. Bautista, J.F.A. Soltero, J.E. Puig, Dynamics of worm-like micelles: the Cox-Merz rule, J. Non-Newton. Fluid Mech. 106 (2002) 1–15. [2] F. Bautista, J.F.A. Soltero, J.H. Perez-Lopez, J.E. Puig, O. Manero, On the shear banding flow of elongated micellar solutions, J. Non-Newton. Fluid Mech. 94 (2000) 57–66. [3] F. Bautista, J.F.A. Soltero, J.E. Puig, O. Manero, Understanding thixotropic and antithixotropic behavior of viscoelastic micellar solutions and liquid crystalline dispersions. I. The model, J. Non-Newton. Fluid Mech. 80 (1999) 93–113. [4] V. Castelletto, I.W. Hamley, R.J. English, W. Mingvanish, SANS and rheology study of aqueous solutions and gels containing highly swollen diblock copolymer micelles, Langmuir 19 (2003) 3229–3235. [5] B. Akers, A. Belmonte, Impact dynamics of a solid sphere falling into a viscoelastic micellar fluid, J. Non-Newton. Fluid Mech. 135 (2006) 97–108.

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