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On the fully developed tube flow of a class of non-linear viscoelastic fluids
International Journal of Non-Linear Mechanics 40 (2005) 485 – 493 www.elsevier.com/locate/nlm
On the fully developed tube flow of a class of non-linear viscoelastic fluids Mario F. Leteliera , Dennis A. Siginerb,∗ a Departamento de Ingeniería Mecánica, Universidad de Santiago de Chile, Casilla 3363, Santiago, Chile b Department of Mechanical Engineering, Wichita State University, 1845 Fairmount, Wichita, KS 67260-0133, USA
Received 29 September 2003; received in revised form 28 May 2004
Abstract The fully developed pipe flow of a class of non-linear viscoelastic fluids is investigated. Analytical expressions are derived for the stress components, the friction factor and the velocity field. The friction factor which depends on the Deborah and Reynolds numbers is substantially smaller than the corresponding value for the Newtonian flow field with implications concerning the volume flow rate. We show that non-affine models in the class of constitutive equations considered such as Johnson–Segalman and some versions of the Phan–Thien–Tanner models are not representative of physically realistic flow fields for all Deborah numbers. For a fixed value of the slippage factor they predict physically admissible flow fields only for a limited range of Deborah numbers smaller than a critical Deborah number. The latter is a function of the slippage. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Constitutive equations; Instabilities; Tube flow; Friction factor; Viscoelastic fluids
1. Introduction We consider the class of non-linear viscoelastic fluids described by the following constitutive structure which connects kinematic variables such as deformation measures represented by the rate of deformation tensor D with the viscoelastic contributed stress ∼ tensor : ∼
g(, tr ) + =2m D ∼
◦
•
∼ ◦
∼
= −∇uT − ∇ u +( D + D ), ∼ ∼
∼
∼
∼∼
∼ ∼
= ,t + u ·∇ −(∇u − D ) − (∇uT − D )T .
◦
∼ ∼
through a single relaxation time , a molecular contributed viscosity m and a function g related to the elongation properties of the fluid, both to be defined shortly. Structure (1) represents a class of non-affine constitutive models framed in terms of the ◦ Gordon–Schowalter convected derivative (•),
∼
(1)
∗ Corresponding author. Fax: +1-316-978-3236.
E-mail address: [email protected] (D.A. Siginer). 0020-7462/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2004.07.009
T
∼
∼
∼
∼
∼
∼ ∼
∼
∼
∼
The Gordon–Schowalter derivative can also be cast in terms of the vorticity tensor ,
∼ T = − − −(1 − )( D + D ). ∼ ∼ ∼∼ ∼ ∼ ∼∼ ∼ ∼ ◦
•
486
M.F. Letelier, D.A. Siginer / International Journal of Non-Linear Mechanics 40 (2005) 485 – 493
The deformation of the macroscopic medium is affine by definition, but polymer strands embedded in the medium may slip with respect to the deformation of the macroscopic medium. Thus each strand may transmit only a fraction of its tension to the surrounding continuum as the continuum slips past the strands. This kind of slippage is taken into account in the Gordon–Schowalter convected derivative by the material parameter . When = 0 there is no slippage and the motion becomes affine. The function g (, tr ) in (1) may be defined as ∼ either a linear function of the material parameter related to the elongational behavior of the fluid,
g(, tr ) = 1 + tr ∼ mo ∼
(2)
or as an exponential function of , g(, tr ) = exp tr , ∼ mo ∼ which collapses onto the linear form if the trace of ∼ is small. If g = 1 or equivalently = 0 the stress in steady-state extension becomes,
ii =
coelastic material constants and mo , the relaxation time and the zero-shear rate molecular contributed viscosity. The molecular viscosity m which appears in (1) is defined in terms of the shear rate = 2 tr D 2 , ∼
, and mo , m = mo
1 + (2 − )2 2 (1 + 2 2 )(1−n)/2
,
where n 1 is the power-law index. The zero-shear rate viscosity of the fluid is defined as,
o = No + mo , with No the zero-shear rate viscosity of the Newtonian solvent. In a simple shearing flow the viscometric functions, that is the shear stress (2 ) and the first and the second normal stress differences N1 (2 ) and N2 (k 2 ) implied by the constitutive structure (1) can be deduced from the viscosity function (2 ) and the first and second normal stress coefficients 1 (2 ) and 2 (2 ) widely favored by rheologists,
( ) = N ( ) + 2
g(, tr )m (2 ) ∼
2
m
g 2 (, tr ) + (2 − )2 2
,
∼
•, 1 − 2(1 − ) i •
where the extension rate in the ith direction is i . For • i =1/2(1− ) there is a singularity in the elongational stress and consequently in the elongational viscosity, and stress growth in steady-state extension becomes unbounded. Extensional viscosity may increase with • extension rate , but must be bounded at all extension rates. Experimental evidence indicates that in steady extension the steady-state values of the elongational −
viscosity reach plateaus at high values of the exten• sion rate . In addition, uniaxial extensional data for concentrated polymeric solutions show a maximum • with increasing . With g = 1 both the linear and −
•
exponential forms predict a bounded at high and better yet the exponential form predicts maxima thus leading to a more favorable comparison with uniaxial extension data. In strongly elongational flows setting g (, tr ) = 1 may lead to substantial errors. ∼ The family of constitutive models defined by (1) has two parameters, and , apart from the usual vis-
11 − 22 N1 = 2 2 2 = (2 ) − N (2 ) , g(, tr )
1 (2 ) =
∼
2 (2 ) =
N2 22 − 33 = = − 1 (2 ), 2 2 2
where indices (1) and (2) represent the direction of the shear and the direction perpendicular to the shear planes, respectively. The ratio of the second to the first normal stress coefficient ranges between −0.05 > 2 /1 > − 0.3 in experiments with concentrated solutions which corresponds to 0.6 > > 0.1. For dilute solutions there are no experimental measurements of 2 at zero or at any shear rate. Thus physically realistic values of 2 /1 for dilute solutions are not known. However, various authors have shown theoretically that at very low shear rates 2 /1 = 0.011, Öttinger [1] and Fixman [2]. Note that the ratio 2 /1 is positive which implies that the second normal stress difference is very small and positive at very low shear rates. At high shear
M.F. Letelier, D.A. Siginer / International Journal of Non-Linear Mechanics 40 (2005) 485 – 493
rates 2 goes to zero according to Öttinger [1]. It is well known that dilute solutions can be modeled as suspensions of rigid ellipsoids as a rough approximation of the physical structure as macromolecular coils (polymer strands) become elliptic in shape if the velocity gradients are strong enough. Based on this premise Hinch and Leal [3] have computed the stress components in viscometric flows for suspensions of ellipsoidal particles that are elongated (rod-like), near spherical and disc-like. They find that at very low shear rates 2 /1 = −1/7 for rod-like and near spherical particle suspensions and 2 /1 = −2/7 for a suspension of disc-like particles. These values differ from values based on the theories of Öttinger [1] and Fixman [2] both in sign and order of magnitude and agree with experimentally measured range of 2 /1 for concentrated solutions. In view of the various considerations presented above we will concentrate on results for the parameters and in the range 0.6 > > 0.1 and < 1. We remark that in the class represented by (1) the level of shear thinning is controlled by the parameter if it is much larger than , and the opposite is also true. In particular if = 0 shear thinning is entirely controlled by . We remark for completeness that the family of constitutive equations covered by (1) include the upper convected Maxwell (UCM) equation = = 0, 0 = m = mo ), the Johnson–Segalman model ( = 0, 0 = m = mo ), the Oldroyd-B equation ( = = 0, mo < 0 , 0 < mo /0 < 1, m = mo ), and the various Phan–Thien–Tanner (PTT) constitutive models, the simplified PTT ( = 0, 0 = m = mo ), the original PTT (0 = m = mo ), and the modified PTT (mo = 0 ). To put the following analysis in context we note that the UCM and the Oldroyd-B models yield velocity profiles in pipe flow identical to the linear Newtonian fluid. In this paper we work with the constitutive equation (1) assuming = 0 = and we show that the class of popular non-affine models defined by (1) including the Johnson–Segalman and Phan–Thien–Tanner models lead to instabilities in confined flows, that is a realistic flow field does not exist for all Deborah numbers. When the slippage factor is fixed realistic velocity fields are obtained only for Deborah numbers smaller than a critical Deborah number (De)cr . The latter is a function of the slippage, and does not hold for all
487
values of . The dependence of the friction factor on , and De is also examined. We determine that when the Deborah number is selected and held fixed friction factor increases with decreasing slippage at a fixed value of the constitutive parameter . It should be noted that selecting a Deborah number fixes the allowable range for the slippage factor for realistic velocity fields. These results were presented at the American Society of Mechanical Engineers International Mechanical Engineering Congress & Exposition held in Orlando, Florida in November 2000 and a shorter version was published in the Proceedings (Letelier and Siginer [4]).
2. Mathematical development Linear momentum conservation reads
Du
∼
Dt
=∇ ·T . ∼
With p representing the pressure field the total stress T is given by ∼
T = − p1 +2N D + .
∼
∼
∼
∼
The Newtonian and non-Newtonian contributed stress tensors are 2N D and , respectively. With ∼ ∼ u =w(r) ez and N ∼ 0, that is neglecting the contri∼
∼
bution of the Newtonian solvent to total viscosity we obtain 1 −p,i + (r ir ),r = 0, i = r, z. r
(3)
The constitutive equation (1) yields, g(, ii )rr + w,r zr = 0, g(, ii )zr + w,r zz − 1 − w,r rr 2 2 = w,r m ,
(5)
g(, ii )zz − (2 − )w,r rz = 0.
(6)
(4)
From (4) and (6) we obtain,
rr = −
zz . 2−
(7)
488
M.F. Letelier, D.A. Siginer / International Journal of Non-Linear Mechanics 40 (2005) 485 – 493
Setting the pressure field in (3) equal to, p(r, z) = f (r) − kz + p0 ,
(8)
where p0 is a reference pressure and k a constant we find from the momentum balance (3)
zr = −
kr , 2
(9)
as zr (0, z) ≡ 0. From (5) and (6) using (7) we derive,
(2 − )2zr = ( − zz )zz .
(10)
Further using (9) we obtain a quadratic equation for zz from (10) which yields
{1 − [1 − m2 r 2 ]1/2 }, 2 2 m2 = (2 − ) .
zz =
(11)
(12)
Working with the linear form (2) of the function g(, ii ) and mo = m = , and using (7) we obtain,
g(, ii ) = 1 + (rr + zz ), g(, ii ) = 1 + 2
(1 − ) zz = 1 + zz . (2 − )
(13)
Further developing equation (6) eliminating zz and zr using (11) and (9), respectively, and replacing g (, ii ) with (13) we obtain
jw dw n = = 2 r −1 {1 − [1 − m2 r 2 ]1/2 } + lr jr dr m n=
( + )k ,
l=
(14)
k . 2
m2
is defined in (12). Integrating (14) we obtain after some manipulation, n 1 − m2 r 2 − 1 − m2 a 2 w(r) = 2 m
√ l 1 + 1 − m2 r 2 + (r 2 − a 2 ), − ln √ 2 2 2 1+ 1−m a where a is the tube radius and the no-slip condition at the wall has been implemented. The non-dimensional
form W (R) of the longitudinal velocity w(r) reads as, N 1 − M 2 R2 − 1 − M 2 W (R) = 2 M
√ 1 + 1 − M 2 R2 − ln √ 1 + 1 − M2 L + (R 2 − 1), (15) 2 w(r) ka 2 r , W (R) = , = , a a2 a2 N = n , M 2 = m2 a 2 , L = l . Shear-thinning in the class of constitutive equations defined by (1) is governed by both constitutive parameters and . When is very small the slip factor governs shear-thinning to a large extent and the opposite is also true. Taking the limit of Eq. (15) as → 0 gives the velocity profile in the tube when = 0 = , that is when the motion is affine, R=
W (R) = 0.25(1 − R 2 )[1 + 2De2 (1 + R 2 )],
(15a)
where De = ka/ is the Deborah number. To obtain the velocity profile for the opposite case we set = 0 in (15), −2 W (R) = M 1 − M 2 R2 − 1 − M 2
− ln
√ 1 − M 2 R2 . √ 1 + 1 − M2
1+
(15b)
It is well known that the pressure gradient or the energy loss coefficient k in (8) can be evaluated in terms of a friction factor f and the average velocity U in the cross-section,
U 2 . (16) 4a Integrating (15) we obtain after considerable manipulation,
1 W (R)R dR, U =2 k=f
0
N 2 U= (M − 1)(1 − 1 − M 2) 3M 4 N L + − . 6M 2 4
(17)
M.F. Letelier, D.A. Siginer / International Journal of Non-Linear Mechanics 40 (2005) 485 – 493
Multiplying (17) through with 2a / and incorporating (16) into the resulting equation with
=
ka 2 f = Re2 , 8 2a
M 2 = m2 a 2 = (2 − )
Re =
ka
2aU ,
(18)
2 = (2 − )De2 , (19)
we find for the friction factor f , 1 + 2E 1 1 2 3/2 f Re = − [1 − (1 − M ) ] 8M 2 2 3M 2 −1 E , − 32 in terms of the Reynolds number Re and the Deborah number De defined in (18) and (19), respectively, and a third dimensionless grouping E of the constitutive parameters and , E=
(1 − ) . (2 − )
Expressing the ratio of the friction factor f with its Newtonian counterpart fN we determine, 64 , Re 8(1 + 2E) 1 1 f 2 3/2 = [1 − (1 − M ) ] − fN M2 2 3M 2 −1 . (20) −2E
fN =
3. Discussion It is clear from the longitudinal velocity equation (15), the average velocity and the friction coefficient equations (17) and (20), respectively, that they cease to have physical significance for M 2 > 1. This is a feature of the non-affine constitutive equations in class (1), and would not happen if the parameter which describes the extent of the slip of the strands with respect to the surrounding medium was set to zero, thus reverting back to an affine description of the motion. As the maximum value of M 2 is one to obtain a real
489
valued velocity field any value assigned to the slip factor limits the range of the Deborah number which cannot exceed a critical (De)cr given by (De)cr = [(2 − )]−1/2 > De. For = 0 or very close to zero all De numbers are admissible. The critical Deborah number grows with decreasing values of . For 0.6 > > 0.1 as it is valid for concentrated solutions, De|=0.1 < (De)cr ∼ = 2.29,
(21a)
De|=0.6 < (De)cr ∼ = 1.09.
(21b)
It can be argued that a constitutive equation which breaks down in tube flow for a range of realistic Deborah numbers is not a viable constitutive equation. For a given fluid the relaxation time and the viscosity are set therefore definition (19) of the Deborah number implies that if one sets the pressure gradient k the pipe radius cannot exceed a certain value and vice versa for a flow to exist in the tube according to Eq. (1) with = 0. Obviously, in the real world flow will take place in the tube regardless the values assigned to k and a. The variation of the dimensionless form of the normal stress zz in the longitudinal direction is represented in Fig. 1 in the limiting case M 2 =1. The dimensional form of zz is given in (11). But as M 2 = m2 a 2 where a is the tube radius the dimensionless representation of zz is easily obtained. In particular we note that longitudinal normal stress is zero on the centerline and maximum at the wall. The variation of the dimensionless velocity in the cross-section given in (15) is graphically represented in Figs. 2 and 3 for the critical Deborah numbers defined in (21a) and (21b), respectively. Physically reasonable values for the dimensionless groups N and L have been chosen given the parameters in the definitions of N and L. It is evident from (15) that the dimensionless velocity in the cross-section increases with increasing N and L. But it is less evident that the velocity profile tends to flatten as slippage increases as is shown in Figs. 2 and 3 in the presence of a non-zero constitutive constant which enters the velocity profile (15) through L. An example of the velocity profile
490
M.F. Letelier, D.A. Siginer / International Journal of Non-Linear Mechanics 40 (2005) 485 – 493
Fig. 1. Variation of the dimensionless longitudinal normal stress in the cross-section.
Fig. 2. Dimensionless velocity profiles for De = 2.29 and = 0.1 in the limit M 2 = 1. The parameters N and L in Eq. (15) are assigned the following numerical values: (a) N = 3; L = 1, (b) N = 6; L = 2.5.
in the limiting case of zero slippage (affine model) (15a) is presented in Fig. 4 for a particular value of . The velocity profile (15a) in this case can be conceived
of as the Newtonian velocity profile and an additional term. Thus the velocity at each point in the crosssection is larger than the corresponding Newtonian
M.F. Letelier, D.A. Siginer / International Journal of Non-Linear Mechanics 40 (2005) 485 – 493
491
Fig. 3. Dimensionless velocity profiles for De = 1.09 and = 0.6 in the limit M 2 = 1. The parameters N and L in Eq. (15) are assigned the following numerical values: (a) N = 1.2; L = 0.1, (b) N = 1.5; L = 0.25.
Fig. 4. Dimensionless velocity profiles for = 0 and = 0.5. The Deborah number in Eq. (15a) is assigned the following numerical values: (a) De = 1, (b) De = 5, (c) De = 10.
velocity, leading to the conclusion that the volume flow rate will always be larger than its Newtonian counterpart.
Examples of the velocity profile represented by (15b) in the opposite limiting case of = 0 and = 0 are presented in Fig. 5. If M 2 is set to one the result-
492
M.F. Letelier, D.A. Siginer / International Journal of Non-Linear Mechanics 40 (2005) 485 – 493
Fig. 5. Dimensionless velocity profiles for = 0. The parameters in Eq. (15b) are assigned the following numerical values: (a) for all (De)cr in the limit M 2 = 1, (b) De = 1.15, = 0.5, M 2 = 0.992, (c) De = 1.15, = 0.1 , M 2 = 0.251.
Fig. 6. Variation of the dimensionless friction factor with slippage for several values of in the limit M 2 = 1: (a) = 0, (b) = 0.1, (c) = 0.3, (d) = 0.5.
ing curve represents all the limiting cases for realistic velocity profiles for all (De)cr . That is the velocity profiles for all (De)cr collapse onto a single curve for M 2 ≡ 1. Choosing a De and decreasing the value of
the slippage down from the corresponding ()cr leads to gradually flattened velocity profiles. Finally the variation of the dimensionless friction factor (20) in the general case = 0 = is of
M.F. Letelier, D.A. Siginer / International Journal of Non-Linear Mechanics 40 (2005) 485 – 493
493
Fig. 7. Variation of the dimensionless friction factor with the Deborah number De for several values of in the limit M 2 = 1: (a) = 0, (b) = 0.1, (c) = 0.3, (d) = 0.5.
interest. Representative cases are depicted in Figs. 6 and 7. The slippage factor is determined when De is fixed and M 2 = 1. Conversely the De is determined when is fixed if M 2 =1. All the curves in Figs. 6 and 7 correspond to various limiting cases M 2 = 1 in (20) −1 2 f |M 2 =1 = . (2 + E) fN 3 At any fixed value of the slippage decreasing values of lead to increased friction with the limit, f = 0.75. →0 fN lim
This is in keeping with the fact that largely controls shear-thinning for small values of the slippage or controls it entirely in the absence of it. Smaller values of then lead to less shear-thinning and flattened velocity profiles and therefore to larger values of the friction factor.
Acknowledgements The authors gratefully acknowledge the support of the Chilean Foundation for Research and Development FONDECYT through Grants No. 1010173 and No. 7010173. References [1] H.C. Öttinger, Generalized Zimm model for dilute polymer solutions under theta conditions, J. Chem. Phys. 86 (1987) 3731. [2] M. Fixman, Polymer dynamics: Boson representation and excluded-volume forces, J. Chem. Phys. 45 (1966) 785. [3] E.J. Hinch, L.G. Leal, Time-dependent shear flows of a suspension of particles with weak Brownian rotations, J. Fluid Mech. 57 (1973) 753. [4] M.F. Letelier, D.A. Siginer, Friction effects in pipe flow of Phan–Thien–Tanner fluids, ASME International Mechanical Engineering Congress & Exposition, Orlando, Florida, November 5–10, 2000, ASME FED, vol . 252, pp. 113–117.
On the fully developed tube flow of a class of non-linear viscoelastic fluids Mario F. Leteliera , Dennis A. Siginerb,∗ a Departamento de Ingeniería Mecánica, Universidad de Santiago de Chile, Casilla 3363, Santiago, Chile b Department of Mechanical Engineering, Wichita State University, 1845 Fairmount, Wichita, KS 67260-0133, USA
Received 29 September 2003; received in revised form 28 May 2004
Abstract The fully developed pipe flow of a class of non-linear viscoelastic fluids is investigated. Analytical expressions are derived for the stress components, the friction factor and the velocity field. The friction factor which depends on the Deborah and Reynolds numbers is substantially smaller than the corresponding value for the Newtonian flow field with implications concerning the volume flow rate. We show that non-affine models in the class of constitutive equations considered such as Johnson–Segalman and some versions of the Phan–Thien–Tanner models are not representative of physically realistic flow fields for all Deborah numbers. For a fixed value of the slippage factor they predict physically admissible flow fields only for a limited range of Deborah numbers smaller than a critical Deborah number. The latter is a function of the slippage. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Constitutive equations; Instabilities; Tube flow; Friction factor; Viscoelastic fluids
1. Introduction We consider the class of non-linear viscoelastic fluids described by the following constitutive structure which connects kinematic variables such as deformation measures represented by the rate of deformation tensor D with the viscoelastic contributed stress ∼ tensor : ∼
g(, tr ) + =2m D ∼
◦
•
∼ ◦
∼
= −∇uT − ∇ u +( D + D ), ∼ ∼
∼
∼
∼∼
∼ ∼
= ,t + u ·∇ −(∇u − D ) − (∇uT − D )T .
◦
∼ ∼
through a single relaxation time , a molecular contributed viscosity m and a function g related to the elongation properties of the fluid, both to be defined shortly. Structure (1) represents a class of non-affine constitutive models framed in terms of the ◦ Gordon–Schowalter convected derivative (•),
∼
(1)
∗ Corresponding author. Fax: +1-316-978-3236.
E-mail address: [email protected] (D.A. Siginer). 0020-7462/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2004.07.009
T
∼
∼
∼
∼
∼
∼ ∼
∼
∼
∼
The Gordon–Schowalter derivative can also be cast in terms of the vorticity tensor ,
∼ T = − − −(1 − )( D + D ). ∼ ∼ ∼∼ ∼ ∼ ∼∼ ∼ ∼ ◦
•
486
M.F. Letelier, D.A. Siginer / International Journal of Non-Linear Mechanics 40 (2005) 485 – 493
The deformation of the macroscopic medium is affine by definition, but polymer strands embedded in the medium may slip with respect to the deformation of the macroscopic medium. Thus each strand may transmit only a fraction of its tension to the surrounding continuum as the continuum slips past the strands. This kind of slippage is taken into account in the Gordon–Schowalter convected derivative by the material parameter . When = 0 there is no slippage and the motion becomes affine. The function g (, tr ) in (1) may be defined as ∼ either a linear function of the material parameter related to the elongational behavior of the fluid,
g(, tr ) = 1 + tr ∼ mo ∼
(2)
or as an exponential function of , g(, tr ) = exp tr , ∼ mo ∼ which collapses onto the linear form if the trace of ∼ is small. If g = 1 or equivalently = 0 the stress in steady-state extension becomes,
ii =
coelastic material constants and mo , the relaxation time and the zero-shear rate molecular contributed viscosity. The molecular viscosity m which appears in (1) is defined in terms of the shear rate = 2 tr D 2 , ∼
, and mo , m = mo
1 + (2 − )2 2 (1 + 2 2 )(1−n)/2
,
where n 1 is the power-law index. The zero-shear rate viscosity of the fluid is defined as,
o = No + mo , with No the zero-shear rate viscosity of the Newtonian solvent. In a simple shearing flow the viscometric functions, that is the shear stress (2 ) and the first and the second normal stress differences N1 (2 ) and N2 (k 2 ) implied by the constitutive structure (1) can be deduced from the viscosity function (2 ) and the first and second normal stress coefficients 1 (2 ) and 2 (2 ) widely favored by rheologists,
( ) = N ( ) + 2
g(, tr )m (2 ) ∼
2
m
g 2 (, tr ) + (2 − )2 2
,
∼
•, 1 − 2(1 − ) i •
where the extension rate in the ith direction is i . For • i =1/2(1− ) there is a singularity in the elongational stress and consequently in the elongational viscosity, and stress growth in steady-state extension becomes unbounded. Extensional viscosity may increase with • extension rate , but must be bounded at all extension rates. Experimental evidence indicates that in steady extension the steady-state values of the elongational −
viscosity reach plateaus at high values of the exten• sion rate . In addition, uniaxial extensional data for concentrated polymeric solutions show a maximum • with increasing . With g = 1 both the linear and −
•
exponential forms predict a bounded at high and better yet the exponential form predicts maxima thus leading to a more favorable comparison with uniaxial extension data. In strongly elongational flows setting g (, tr ) = 1 may lead to substantial errors. ∼ The family of constitutive models defined by (1) has two parameters, and , apart from the usual vis-
11 − 22 N1 = 2 2 2 = (2 ) − N (2 ) , g(, tr )
1 (2 ) =
∼
2 (2 ) =
N2 22 − 33 = = − 1 (2 ), 2 2 2
where indices (1) and (2) represent the direction of the shear and the direction perpendicular to the shear planes, respectively. The ratio of the second to the first normal stress coefficient ranges between −0.05 > 2 /1 > − 0.3 in experiments with concentrated solutions which corresponds to 0.6 > > 0.1. For dilute solutions there are no experimental measurements of 2 at zero or at any shear rate. Thus physically realistic values of 2 /1 for dilute solutions are not known. However, various authors have shown theoretically that at very low shear rates 2 /1 = 0.011, Öttinger [1] and Fixman [2]. Note that the ratio 2 /1 is positive which implies that the second normal stress difference is very small and positive at very low shear rates. At high shear
M.F. Letelier, D.A. Siginer / International Journal of Non-Linear Mechanics 40 (2005) 485 – 493
rates 2 goes to zero according to Öttinger [1]. It is well known that dilute solutions can be modeled as suspensions of rigid ellipsoids as a rough approximation of the physical structure as macromolecular coils (polymer strands) become elliptic in shape if the velocity gradients are strong enough. Based on this premise Hinch and Leal [3] have computed the stress components in viscometric flows for suspensions of ellipsoidal particles that are elongated (rod-like), near spherical and disc-like. They find that at very low shear rates 2 /1 = −1/7 for rod-like and near spherical particle suspensions and 2 /1 = −2/7 for a suspension of disc-like particles. These values differ from values based on the theories of Öttinger [1] and Fixman [2] both in sign and order of magnitude and agree with experimentally measured range of 2 /1 for concentrated solutions. In view of the various considerations presented above we will concentrate on results for the parameters and in the range 0.6 > > 0.1 and < 1. We remark that in the class represented by (1) the level of shear thinning is controlled by the parameter if it is much larger than , and the opposite is also true. In particular if = 0 shear thinning is entirely controlled by . We remark for completeness that the family of constitutive equations covered by (1) include the upper convected Maxwell (UCM) equation = = 0, 0 = m = mo ), the Johnson–Segalman model ( = 0, 0 = m = mo ), the Oldroyd-B equation ( = = 0, mo < 0 , 0 < mo /0 < 1, m = mo ), and the various Phan–Thien–Tanner (PTT) constitutive models, the simplified PTT ( = 0, 0 = m = mo ), the original PTT (0 = m = mo ), and the modified PTT (mo = 0 ). To put the following analysis in context we note that the UCM and the Oldroyd-B models yield velocity profiles in pipe flow identical to the linear Newtonian fluid. In this paper we work with the constitutive equation (1) assuming = 0 = and we show that the class of popular non-affine models defined by (1) including the Johnson–Segalman and Phan–Thien–Tanner models lead to instabilities in confined flows, that is a realistic flow field does not exist for all Deborah numbers. When the slippage factor is fixed realistic velocity fields are obtained only for Deborah numbers smaller than a critical Deborah number (De)cr . The latter is a function of the slippage, and does not hold for all
487
values of . The dependence of the friction factor on , and De is also examined. We determine that when the Deborah number is selected and held fixed friction factor increases with decreasing slippage at a fixed value of the constitutive parameter . It should be noted that selecting a Deborah number fixes the allowable range for the slippage factor for realistic velocity fields. These results were presented at the American Society of Mechanical Engineers International Mechanical Engineering Congress & Exposition held in Orlando, Florida in November 2000 and a shorter version was published in the Proceedings (Letelier and Siginer [4]).
2. Mathematical development Linear momentum conservation reads
Du
∼
Dt
=∇ ·T . ∼
With p representing the pressure field the total stress T is given by ∼
T = − p1 +2N D + .
∼
∼
∼
∼
The Newtonian and non-Newtonian contributed stress tensors are 2N D and , respectively. With ∼ ∼ u =w(r) ez and N ∼ 0, that is neglecting the contri∼
∼
bution of the Newtonian solvent to total viscosity we obtain 1 −p,i + (r ir ),r = 0, i = r, z. r
(3)
The constitutive equation (1) yields, g(, ii )rr + w,r zr = 0, g(, ii )zr + w,r zz − 1 − w,r rr 2 2 = w,r m ,
(5)
g(, ii )zz − (2 − )w,r rz = 0.
(6)
(4)
From (4) and (6) we obtain,
rr = −
zz . 2−
(7)
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Setting the pressure field in (3) equal to, p(r, z) = f (r) − kz + p0 ,
(8)
where p0 is a reference pressure and k a constant we find from the momentum balance (3)
zr = −
kr , 2
(9)
as zr (0, z) ≡ 0. From (5) and (6) using (7) we derive,
(2 − )2zr = ( − zz )zz .
(10)
Further using (9) we obtain a quadratic equation for zz from (10) which yields
{1 − [1 − m2 r 2 ]1/2 }, 2 2 m2 = (2 − ) .
zz =
(11)
(12)
Working with the linear form (2) of the function g(, ii ) and mo = m = , and using (7) we obtain,
g(, ii ) = 1 + (rr + zz ), g(, ii ) = 1 + 2
(1 − ) zz = 1 + zz . (2 − )
(13)
Further developing equation (6) eliminating zz and zr using (11) and (9), respectively, and replacing g (, ii ) with (13) we obtain
jw dw n = = 2 r −1 {1 − [1 − m2 r 2 ]1/2 } + lr jr dr m n=
( + )k ,
l=
(14)
k . 2
m2
is defined in (12). Integrating (14) we obtain after some manipulation, n 1 − m2 r 2 − 1 − m2 a 2 w(r) = 2 m
√ l 1 + 1 − m2 r 2 + (r 2 − a 2 ), − ln √ 2 2 2 1+ 1−m a where a is the tube radius and the no-slip condition at the wall has been implemented. The non-dimensional
form W (R) of the longitudinal velocity w(r) reads as, N 1 − M 2 R2 − 1 − M 2 W (R) = 2 M
√ 1 + 1 − M 2 R2 − ln √ 1 + 1 − M2 L + (R 2 − 1), (15) 2 w(r) ka 2 r , W (R) = , = , a a2 a2 N = n , M 2 = m2 a 2 , L = l . Shear-thinning in the class of constitutive equations defined by (1) is governed by both constitutive parameters and . When is very small the slip factor governs shear-thinning to a large extent and the opposite is also true. Taking the limit of Eq. (15) as → 0 gives the velocity profile in the tube when = 0 = , that is when the motion is affine, R=
W (R) = 0.25(1 − R 2 )[1 + 2De2 (1 + R 2 )],
(15a)
where De = ka/ is the Deborah number. To obtain the velocity profile for the opposite case we set = 0 in (15), −2 W (R) = M 1 − M 2 R2 − 1 − M 2
− ln
√ 1 − M 2 R2 . √ 1 + 1 − M2
1+
(15b)
It is well known that the pressure gradient or the energy loss coefficient k in (8) can be evaluated in terms of a friction factor f and the average velocity U in the cross-section,
U 2 . (16) 4a Integrating (15) we obtain after considerable manipulation,
1 W (R)R dR, U =2 k=f
0
N 2 U= (M − 1)(1 − 1 − M 2) 3M 4 N L + − . 6M 2 4
(17)
M.F. Letelier, D.A. Siginer / International Journal of Non-Linear Mechanics 40 (2005) 485 – 493
Multiplying (17) through with 2a / and incorporating (16) into the resulting equation with
=
ka 2 f = Re2 , 8 2a
M 2 = m2 a 2 = (2 − )
Re =
ka
2aU ,
(18)
2 = (2 − )De2 , (19)
we find for the friction factor f , 1 + 2E 1 1 2 3/2 f Re = − [1 − (1 − M ) ] 8M 2 2 3M 2 −1 E , − 32 in terms of the Reynolds number Re and the Deborah number De defined in (18) and (19), respectively, and a third dimensionless grouping E of the constitutive parameters and , E=
(1 − ) . (2 − )
Expressing the ratio of the friction factor f with its Newtonian counterpart fN we determine, 64 , Re 8(1 + 2E) 1 1 f 2 3/2 = [1 − (1 − M ) ] − fN M2 2 3M 2 −1 . (20) −2E
fN =
3. Discussion It is clear from the longitudinal velocity equation (15), the average velocity and the friction coefficient equations (17) and (20), respectively, that they cease to have physical significance for M 2 > 1. This is a feature of the non-affine constitutive equations in class (1), and would not happen if the parameter which describes the extent of the slip of the strands with respect to the surrounding medium was set to zero, thus reverting back to an affine description of the motion. As the maximum value of M 2 is one to obtain a real
489
valued velocity field any value assigned to the slip factor limits the range of the Deborah number which cannot exceed a critical (De)cr given by (De)cr = [(2 − )]−1/2 > De. For = 0 or very close to zero all De numbers are admissible. The critical Deborah number grows with decreasing values of . For 0.6 > > 0.1 as it is valid for concentrated solutions, De|=0.1 < (De)cr ∼ = 2.29,
(21a)
De|=0.6 < (De)cr ∼ = 1.09.
(21b)
It can be argued that a constitutive equation which breaks down in tube flow for a range of realistic Deborah numbers is not a viable constitutive equation. For a given fluid the relaxation time and the viscosity are set therefore definition (19) of the Deborah number implies that if one sets the pressure gradient k the pipe radius cannot exceed a certain value and vice versa for a flow to exist in the tube according to Eq. (1) with = 0. Obviously, in the real world flow will take place in the tube regardless the values assigned to k and a. The variation of the dimensionless form of the normal stress zz in the longitudinal direction is represented in Fig. 1 in the limiting case M 2 =1. The dimensional form of zz is given in (11). But as M 2 = m2 a 2 where a is the tube radius the dimensionless representation of zz is easily obtained. In particular we note that longitudinal normal stress is zero on the centerline and maximum at the wall. The variation of the dimensionless velocity in the cross-section given in (15) is graphically represented in Figs. 2 and 3 for the critical Deborah numbers defined in (21a) and (21b), respectively. Physically reasonable values for the dimensionless groups N and L have been chosen given the parameters in the definitions of N and L. It is evident from (15) that the dimensionless velocity in the cross-section increases with increasing N and L. But it is less evident that the velocity profile tends to flatten as slippage increases as is shown in Figs. 2 and 3 in the presence of a non-zero constitutive constant which enters the velocity profile (15) through L. An example of the velocity profile
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Fig. 1. Variation of the dimensionless longitudinal normal stress in the cross-section.
Fig. 2. Dimensionless velocity profiles for De = 2.29 and = 0.1 in the limit M 2 = 1. The parameters N and L in Eq. (15) are assigned the following numerical values: (a) N = 3; L = 1, (b) N = 6; L = 2.5.
in the limiting case of zero slippage (affine model) (15a) is presented in Fig. 4 for a particular value of . The velocity profile (15a) in this case can be conceived
of as the Newtonian velocity profile and an additional term. Thus the velocity at each point in the crosssection is larger than the corresponding Newtonian
M.F. Letelier, D.A. Siginer / International Journal of Non-Linear Mechanics 40 (2005) 485 – 493
491
Fig. 3. Dimensionless velocity profiles for De = 1.09 and = 0.6 in the limit M 2 = 1. The parameters N and L in Eq. (15) are assigned the following numerical values: (a) N = 1.2; L = 0.1, (b) N = 1.5; L = 0.25.
Fig. 4. Dimensionless velocity profiles for = 0 and = 0.5. The Deborah number in Eq. (15a) is assigned the following numerical values: (a) De = 1, (b) De = 5, (c) De = 10.
velocity, leading to the conclusion that the volume flow rate will always be larger than its Newtonian counterpart.
Examples of the velocity profile represented by (15b) in the opposite limiting case of = 0 and = 0 are presented in Fig. 5. If M 2 is set to one the result-
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Fig. 5. Dimensionless velocity profiles for = 0. The parameters in Eq. (15b) are assigned the following numerical values: (a) for all (De)cr in the limit M 2 = 1, (b) De = 1.15, = 0.5, M 2 = 0.992, (c) De = 1.15, = 0.1 , M 2 = 0.251.
Fig. 6. Variation of the dimensionless friction factor with slippage for several values of in the limit M 2 = 1: (a) = 0, (b) = 0.1, (c) = 0.3, (d) = 0.5.
ing curve represents all the limiting cases for realistic velocity profiles for all (De)cr . That is the velocity profiles for all (De)cr collapse onto a single curve for M 2 ≡ 1. Choosing a De and decreasing the value of
the slippage down from the corresponding ()cr leads to gradually flattened velocity profiles. Finally the variation of the dimensionless friction factor (20) in the general case = 0 = is of
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493
Fig. 7. Variation of the dimensionless friction factor with the Deborah number De for several values of in the limit M 2 = 1: (a) = 0, (b) = 0.1, (c) = 0.3, (d) = 0.5.
interest. Representative cases are depicted in Figs. 6 and 7. The slippage factor is determined when De is fixed and M 2 = 1. Conversely the De is determined when is fixed if M 2 =1. All the curves in Figs. 6 and 7 correspond to various limiting cases M 2 = 1 in (20) −1 2 f |M 2 =1 = . (2 + E) fN 3 At any fixed value of the slippage decreasing values of lead to increased friction with the limit, f = 0.75. →0 fN lim
This is in keeping with the fact that largely controls shear-thinning for small values of the slippage or controls it entirely in the absence of it. Smaller values of then lead to less shear-thinning and flattened velocity profiles and therefore to larger values of the friction factor.
Acknowledgements The authors gratefully acknowledge the support of the Chilean Foundation for Research and Development FONDECYT through Grants No. 1010173 and No. 7010173. References [1] H.C. Öttinger, Generalized Zimm model for dilute polymer solutions under theta conditions, J. Chem. Phys. 86 (1987) 3731. [2] M. Fixman, Polymer dynamics: Boson representation and excluded-volume forces, J. Chem. Phys. 45 (1966) 785. [3] E.J. Hinch, L.G. Leal, Time-dependent shear flows of a suspension of particles with weak Brownian rotations, J. Fluid Mech. 57 (1973) 753. [4] M.F. Letelier, D.A. Siginer, Friction effects in pipe flow of Phan–Thien–Tanner fluids, ASME International Mechanical Engineering Congress & Exposition, Orlando, Florida, November 5–10, 2000, ASME FED, vol . 252, pp. 113–117.