Flow of a generalized second grade non-Newtonian fluid with variable viscosity

Digital Object Identifier (DOI) 10.1007/s00161-004-0178-0Continuum Mech. Thermodyn. (2004) 16: 529–538

Original article

Flow of a generalized second grade non-Newtonian fluidwith variable viscosity

M. Massoudi, T.X. Phuoc

U.S. Department of Energy, National Energy Technology Laboratory, P.O. Box 10940, Pittsburgh, PA 15236, USA

Received August 28, 2003 / Accepted March 3, 2004Published online June 25, 2004 – © Springer-Verlag 2004Communicated by B. Straughan, Durham

Dedicated to Professor Brian Straughan

Abstract. A modified constitutive equation for a second grade fluid is proposed so that the modelwould be suitable for studies where shear-thinning (or shear-thickening) may occur. In addition,the dependence of viscosity on the temperature follows the Reynolds equation. In this paper, wepropose a constitutive relation, (18), which has the basic structure of a second grade fluid, wherethe viscosity is now a function of temperature, shear rate, and concentration. As a special case,we solve the fully developed flow of a non-Newtonian fluid given by (11), where the effects ofconcentration are neglected.

Key words: generalized second grade fluids, shear-thinning or shear-thickening non-Newtonianfluids, pipe flow, temperature dependent viscosity

Introduction

Multicomponent flows whether occurring in nature, such as debris flows, avalanches, and mud slides, or in in-dustrial applications, such as fluidized beds, solids transport and many other chemical and agricultural processes,present a formidable challenge to engineers and scientists. To model and study the flow and behaviour of suchcomplex fluids, one can use either statistical theories or continuum theories, in addition to the phenomenologi-cal/experimental approaches. Within the continuum mechanics approach, there are two distinct ways of studyingthese flows. In certain applications such as fluidization or gasification, one may represent the mixture as a mix-ture of two or more interacting continua [24, 29]. In this approach, more specific information such as particledistribution, particle velocity, and temperature can be obtained. This is not the approach that we will take inthis study. Alternatively one can model the complex mixture with a constitutive relation for the suspension as awhole. In this approach one only works with the equations of motion for the suspension and as a result one onlygets information about the global characteristics of the mixture. This is the approach that we take in this paper.

Traditionally, in fossil fuel combustion processes, coal-water slurry is prepared and in some cases heatedprior to testing and use. Its rheological properties are determined using a basic type of viscometer. Coal slurriesexhibit non-Newtonian flow characteristics; it is likely that they also exhibit normal stress effects [10]. In disperseflows with high concentration of the dispersed particle, the viscosity of the suspension can depend also on the

Correspondence to: M. Massoudi (e-mail: [email protected])

530 M. Massoudi, T.X. Phuoc

particle-particle interaction (see [53]) volume fraction of the particles ( [27,50,51]) or particle size ( [2,36]). Afull review of these issues is given in a review article by Gupta [15]. In addition, the predictability of handlingand injecting coal-water or coal-oil slurries is an important combustion issue. It has been demonstrated [49] thatsubstantial performance benefits can be obtained if the coal-water mixture is preheated, as there is significantchange in the properties of the fluid. There are many other applications such as injection moulding or oil-welldrilling where there are significant variations in the viscosity of the fluid, caused primarily by shear-rate andtemperature ( [34]). Furthermore, in most of these applications the dissipative source term, which appears in theenergy equation, cannot be ignored a priori (see [6,52]).

Researchers investigating the flow of non-Newtonian fluids often use the power-law constitutive relation.Although the power-law model adequately fits the shear stress and the shear rate measurements for many non-Newtonian fluids, it cannot be used to accurately describe phenomena such as “die-swelling” and “rod-climbing”which are manifestations of the stresses that develop orthogonally to planes of shear in the flow of these complexfluids. The power-law model does not predict these normal stress effects, although it can predict some of the usualcharacteristics of non-Newtonian fluids such as shear-thinning and shear-thickening. One of the recent advancesin the theoretical studies in rheology is the development of generalized differential grade models. The simplicity oftheir form and the fact that these modified constitutive relations can be used to study shear-thinning/thickening,the decrease/increase in viscosity with increasing/decreasing shear rate, as well as predicting normal stressdifferences, have opened the way for the solution to a series of engineering problems (see [14,21,25]).

In this paper we will first present the basic governing equations. Then we discuss constitutive relations fora class of second grade fluids which have been proposed in the last few years, and suggest a new one, (18). Inthe final section, we present the numerical solution to the flow of a modified second grade fluid in a pipe. Theviscosity of this particular model is assumed to be dependent on temperature and shear rate. We will also discussthe effect of viscous dissipation.

Governing equations

The governing equations of motion are the conservation of mass, linear momentum, and energy equation. Theseare

Conservation of mass:∂ρ

∂t+ div (ρu) = 0 (1)

where ρ is the density of the fluid, ∂/∂t is the partial derivative with respect to time, and u is the velocity vector.For an isochoric motion we have

div u = 0 . (1b)

Conservation of linear momentum:

ρdudt

= divT + ρb. (2)

where b is the body force vector, T is the stress tensor, and d/dt is the total time derivative, given byd(.)dt

=∂(.)∂t

+ [grad(.)]u . (3)

Conservation of energy:

ρdε

dt= T · L − divq + ρr (4)

where ε is the specific internal energy, L is the gradient of velocity, q is the heat flux vector, and r is the radiantheating. Thermodynamical considerations require the application of the second law of thermodynamics or theentropy inequality. The various forms of this law are known as the Clausius-Duhem inequality, or the principleof dissipation [48, p. 295] which states that:

1ρtr(TD) + θη̇ − ε̇ ≥ 0 (5)

where θ is the temperature field and η is the specific entropy.We can see that constitutive relations are required for T, q, r, ε, and η. In this paper, we will focus our

attention on modeling T. By not using the balance of angular momentum, we have tacitly assumed that the stresstensor is symmetric.

Flow of a generalized second grade non-Newtonian fluid with variable viscosity 531

Constutive relations

The non-linear time-dependent response of complex fluids constitute an important area of mathematical modelingof non-Newtonian fluids. The most widely used model, the power–law model is deficient in many ways. It cannotpredict the normal stress differences or yield stresses, nor can it capture the memory or history effects. However,the power-law model has been used for a variety of applications where the shear viscosity is not constant,and especially for the cases where µ is a function of (an appropriately frame invariant form) the velocitygradient [5,19,41]. Perhaps the simplest model which can capture the normal stress effects (which could leadto phenomena such as ‘die-swell’ and ‘rod-climbing’, which are manifestations of the stresses that developorthogonally to planes of shear) is the second grade fluid, or the Rivlin-Ericksen fluid of grade two ( [35,48]).This model has also been used and studied extensively ( [8]) and is a special case for fluids of differential type( [9]). For a second grade fluid the stress is given by:

T = −p1 + µA1 + α1A2 + α2A21 (6)

where p is the indeterminate part of the stress due to the constraint of incompressibility, µ is the coefficient ofviscosity, and α1 and α2 are material moduli which are commonly referred to as the normal stress coefficients.The kinematical tensors A1 and A2 are defined through

A1 = L + LT (7a)

A2 =dA1

dt+ A1L + (L)T A1 (7b)

L = gradu . (7c)

The thermodynamics and stability of fluids of second grade have been studied in detail by Dunn and Fosdick [8].They show that if the fluid is to be thermodynamically consistent in the sense that all motions of the fluid meet theClausius-Duhem inequality and that the specific Helmholtz free energy of the fluid be a minimum in equilibrium,then

µ ≥ 0,

α1 � 0,

α1 + α2 = 0.

(8)

For many practical engineering cases, where materials such as paint, slurries, etc. are used, the shear viscositycan be a function of one or all of the following:

– Shear rate– Concentration– Temperature– Pressure

Thus in general,µ = µ(π, θ, φ, p) (9)

where π is some measure of the shear rate, θ the temperature, φ the concentration, and p the pressure. Of course,in certain materials or under certain conditions, the dependence of one or more of these can be dropped. In thispaper we do not consider the effects of pressure on viscosity. This, however, does not mean that this dependencecan always be ignored; in fact, a recent study by Hron et al. [17] points to the significant influence of pressureon the viscosity, and as a result on the flows of a Navier-Stokes fluid.

In an effort to obtain a model that does exhibit both normal stress effects and shear-thinning/thickening, Manand Sun [20], and Man [21] modified the constitutive equation for a second grade fluid by allowing the viscositycoefficient to depend upon the rate of deformation. The two proposed models are:

T = −p1 + µΠm/2A1 + α1A2 + α2A21 (10)

T = −p1 + Πm/2 (µA1 + α1A2 + α2A2

1)

(11)


whereΠ = 1

2 trA21 (12)

is the second invariant of the symmetric part of the velocity gradient, and m is a material parameter. Whenm < 0, the fluid is shear-thinning, and if m > 0, the fluid is shear-thickening. Both models are special cases ofcomplexity 2. In (10) only the shear viscosity depends on the shear rate, whereas in (11) the viscosity and thenormal stress coefficients are dependent upon shear rate. Both constitutive equations are capable of predictingshear-thinning as well as shear-thickening. Szeri and Rajagopal [47] have also proposed a model which canpredict the shear-thickening phenomenon. Recent additional studies into the flow of second and third gradefluids have been carried out by Siddiqui et al. [40], Asghar et al. [1], Mollica and Rajagopal [26], and Ayub etal. [3].

A subclass of models given by (10) is the generalized power-law model, which can be obtained from (10) bysetting α1 = α2 = 0. That is

T = −p1 + µ0(trA2

1)m

A1 . (13)

Later, Gupta and Massoudi [13,14] generalized the model given by (10), by allowing the shear viscosity tobe a function of temperature. That is:

T = −p1 + µ(θ)Πm/2A1 + α1A2 + α2A21 (14)

where µ(θ) was assumed to obey the Reynolds viscosity model,

µ(θ) = µ0e−Mθ (15)

whereM = n (θ2 − θ1) . (16)

This model can find applications in many processes where preheating of the fuel is used as a means to enhanceheat transfer effects. In addition, for many fluids such as lubricants, polymers, and coal slurries where viscousdissipation is substantial, an appropriate constitutive relation where viscosity is a function of temperature shouldbe used. Viscosity changes due to temperature also have significant effects on the stability of flows such asBenard convection, which is the cellular motion whenever a fluid layer is heated from below. The influence oftemperature in the linear and nonlinear stability of flows for a class of fluids has been studied in the recent yearsby Straughan [42–45], Franchi and Straughan [11], and Capone and Gentile [7].

Interestingly, materials which apparently have nothing in common can be expressed rheologically in a similarmanner. For example, many studies indicate that for lava [12], or coal slurries [49] the viscosity is a function oftemperature, volume fraction, and to some extent on the size and shape of the particles. In many applications, forexample, melt fraction [32] and basaltic lavas, the apparent viscosity is assumed to follow the Einstein–Roscoerelation [37,38]:

η(θ, φ) = η0

(1 − φ

φmax

)−2.5

eγ(θ0−θ) (17)

where φmax is the maximum crystal fraction that flow can occur, θo and ηo are reference values, and γ is aconstant. We now make the following assumptions:

1. For a suspension of coal mixtures, the material exhibits normal stress effects.2. The shear viscosity depends on volume fraction, temperature, and shear rate.

Based on these two conjectures, we propose the following simple constitutive relation:

T = −p1 + µ(θ, φ)Πm/2A1 + α1A2 + α2A21 (18)

where µ(θ,φ) is given by an equation similar to (17).This model is a general frame invariant model, suitable for flows of non-Newtonian fluids such as coal slurries,

melt fraction, polymers etc., where the viscosity might be a function of temperature, concentration, and shearrate. In addition, this model exhibits both normal stress differences. Although we have proposed this modelwhich is based on various experimental observations, in this paper we are not using this equation. Instead, weuse a modified version of (11) which was originally proposed by Man [21]. Obviously, the methodology thatwe have presented here is not very rigorous. For example, the thermodynamics and stability treatments for theseconstitutive relations, except for the standard second grade fluid, are not available and studies done on these


models have been limited to a class of boundary value problems. Recently, however, a thermodynamic framework,using the Multiple Natural Configurations Theory has been devised by Rajagopal and Srinivasa [30,31], whichin theory is capable of deriving these constitutive relations in a rigorous way.

In this work, we study the behavior of suspensions such as coal slurries and assume that they can be representedby a constitutive equation similar to that of generalized second grade fluid, and consider the viscosity to be afunction of temperature. Specially, we use a modified form of (11), where viscosity is a function of temperature.It should be emphasized that (6) and (10) are also sub-classes of (18) proposed above. Previous studies of secondand third grade fluids in various other geometries have been performed by Straughan [42], Rajagopal and Na [28],Szeri and Rajagopal [47], Massoudi and Christie [22], Gupta and Massoudi [14], and Massoudi and Phuoc [25].

Flow in a pipe

Let us consider the steady laminar fully developed flow of a suspension characterized by (11) through a (heated)circular horizontal pipe, driven by a pressure gradient along the length of a pipe. Obviously, unsteady problemswill give rise to many interesting issues, as discussed in previous studies on the second and third grade fluids(see for example, [4,16]). Velocity and temperature fields are assumed to be given by

v = v(r)ez (19a)

andθ = θ(r) . (19b)

Substituting (11) into the balance of linear momentum and using the constraint that the fluid can only undergoisochoric motion (incompressibility constraint), we obtain:

−∂p̂

∂r+

1r

∂

∂r

[r (2α1 + α2)

(v′2)(m/2)+1

]= 0 (20)

−∂p̂

∂θ= 0, (21)

−∂p̂

∂z+

1r

∂

∂r

[rµ

(v′2)m/2

v′]

= 0. (22)

where p̂ = p + φ. In the derivation of the above equations, it is assumed that the body force b is conservative(b = −grad φ). It is also assumed that the z-differential of pressure is constant since the flow is due to aprescribed pressure gradient. From (22) the velocity field is determined. Next, the pressure field is determinedfrom (20). In this model, the modified pressure field depends upon the power index m. Equation (22) is similarto the equation for the power-law model; the influence of the normal stress coefficients α1 and α2 is only on thepressure field. Once the flow field is determined, the actual pressure field p̂ can be obtained from the modifiedpressure field p. The exact form of (22) depends upon the form of the temperature dependent viscosity, µ(θ).

Recalling that the balance of energy is given by

ρdε

dt= T · L + ρr − div q (23)

where the heat flux vector q is assumed to be given by Fourier’s conduction law

q = −k grad θ (24)

where k is the thermal conductivity, and is assumed to be constant. The specific radiant energy is assumednegligible, and the specific internal energy, ε, related to the specific Helmholtz free energy, is given [8] by

ε = Ψ + θη = ε (θ,A1,A2) = ε̂ (r) (25)

where η is the specific entropy. Strictly speaking this equation is not valid for the modified second grade fluidused in our analysis. For example, the internal energy may now in some ways depend on the parameter m.Nevertheless, as we shall see, due to the kinematical assumptions made about the velocity and temperaturefields, the term corresponding to the derivative of ε, disappears from the energy equation. Thus, it follows that

dε

dt= 0 (26)


and the energy equation reduces to:

µ(v′2)m/2

v′2 +k

r

∂

∂r

(r∂θ

∂r

)= 0 . (27)

Two viscosity models are considered. The simplest one is µ = µ0 = a constant. The other is the Reynoldsmodel [46], where

µ = µ0 exp (−nθ) . (28)

Although we do not consider the Vogel’s model in this paper, just as Yurusoy and Pakdemirli [54] indicate,very interesting behavior is observed for the flow of a third grade fluid in a pipe where the Vogel’s model is used.Nondimensionalization is performed and the variables r∗, µ∗, v∗, z∗, p∗, and θ∗ are created by scaling with R,µ0, V0, L, ρV 2, and θw, respectively (a pipe with radius R, length L, and fixed wall temperature). For simplicity,the asterisk will be dropped and all variables are assumed nondimensionalized. The balance of energy becomes

1r

d

dr

(rdθ

dr

)= Γµ |v′|m+2 (29)

where

Γ = − µ0

kθw

(V0

R

)m+2

R2 . (30)

Note that Γ is a negative number. Equation (29) is solved numerically subject to the boundary conditions ofradial symmetry and fixed wall temperature,

θ(1) = 1; anddθ

dr

∣∣∣∣r=0

= 0 (31)

In the present study, we consider the effect of viscous dissipation, Γ [22,47]. For many non-Newtonian fluids,such as polymer processing at very high temperatures, viscous dissipation is very important. It is interestingto note that this term cannot be neglected even in non-Newtonian fluids encountered in glacier physics [18].Since the viscosity is temperature dependent, the equation of motion depends upon the precise structure of theviscosity. The behavior for two models is now examined. In the nondimensional form, the constant viscosityµ = 1 and the Reynolds viscosity models are written

µ = exp (−Mθ) ; M = Nθw. (32)

The velocity equation may be written as

|v′|m v′ = − C

2µr (33)

where

C = −ρV0D

µ0

(R

V0

)m 12L

dp

dz. (34)

Note that C is positive, because the pressure gradient must be negative for the flow to occur. Equation (33) issolved subject to the boundary conditions for no-slip and symmetry,

v(1) = 0; anddv

dr

∣∣∣∣r=0

= 0 . (35)

Equations (29) and (33), subject to their boundary conditions, and (31) and (35) are solved numerically usingthe finite difference technique. The resulting system of equations has a tridiagonal coefficient matrix. Finergrids are used near the boundaries. The velocity, with constant temperature distribution, is solved first. Then atemperature distribution is calculated. This is repeated until a certain tolerance is met. Then, a new viscosity iscalculated and the entire process is repeated until the velocity, temperature, and viscosity differences betweentwo successive iterations fall within specified tolerance limits.


Fig. 1. Dimensionless velocity profiles for different values of m (non-Newtonian parameter) and C (pressure drop parameter), whenM = 0.5 (variable viscosity) and no viscous dissipation Γ = 0

Fig. 2. Dimensionless velocity profiles for different values of m (non-Newtonian parameter) and C (pressure drop parameter), whenM = 0.5 (variable viscosity) and viscous dissipation Γ = −2.25

Results and discussion

Velocity v and temperature θ are plotted as functions of the pipe radius. The point y = 0 on Figs. 1–4 refersto the center of symmetry, and the points y = 1 and y = −1 correspond to the pipe walls. The parameters m,C, M , and Γ were varied. The results are best presented by comparing curves for different values of M , eitherM = 0.5 or M = 0. For each M , the viscous dissipation term Γ was varied; results for Γ = 0 and Γ = −2.25were obtained. On each graph where a set of (M, Γ ) are presented, results of five combinations of m and C areplotted. The curves of material properties m = −0.5, 0, 0.5 represent shear thinning, second grade, and shearthickening fluids, respectively. The parameter C is related to the pressure drop; numerical results using valuesof 2.0 and 2.5 are presented.

When M = 0.5, the differences in the curves with respect to viscous dissipation are pronounced.A comparisonof the curves in Figs. 1 and 2 indicates that as the absolute value of Γ increases, the average velocity increases.The largest changes occur for the case when C = 2.5 and m = 0 (second grade fluid). Figure 3 depicts thethermal profiles which correspond to the cases shown in Figs. 1 and 2. When Γ = 0, the thermal profile is givenby a vertical line. The largest differences in the magnitude of the thermal gradients occur with the second gradefluid, when neither shear thinning or thickening occur. When M = 0, the curves for various values of the viscousdissipation, Γ , are similar to each other, and are given in Fig. 4. The thermal profiles, shown in Fig. 5, indicatemuch smaller gradients compared to those with M = 0.5.


Fig. 3. Dimensionless temperature profiles for different values of m (non-Newtonian parameter) and C (pressure drop parameter),when M = 0.5 (variable viscosity) and viscous dissipation Γ = −2.25

Fig. 4. Dimensionless velocity profiles for different values of m (non-Newtonian parameter) and C (pressure drop parameter), whenM = 0.0 (constant viscosity) and no viscous dissipation Γ = 0

Conclusion

The effectiveness of handling, and transporting of coal-water slurries is one of the important combustion-relatedissues in using these fuels. In general, coal slurries exhibit non-Newtonian flow characteristics. At the same time,the viscosity of many suspensions depends on the temperature and the concentration. The fully developed flow of asuspension of rigid particles in a pipe, where the mixture is modeled as a second grade fluid with a temperature andshear dependent viscosity is studied. The results in terms of dimensionless velocity and temperature profiles arepresented for various dimensionless numbers. The coupled system of non-linear ordinary differential equations issolved numerically using a standard finite difference scheme. The important parameters considered in this studyare the power-law index (indicating a shear thinning or shear thickening fluid), an exponent M which representsthe variation of viscosity with temperature, a term C, representing the driving force, i.e., the pressure gradient,and a term Γ , representing the viscous dissipation. A Reynolds viscosity model is assumed for the temperaturedependence, and it is also observed that the viscous dissipation plays an important role.


Fig. 5. Dimensionless temperature profiles for different values of m (non-Newtonian parameter) and C (pressure drop parameter),when M = 0.0 (constant viscosity) and viscous dissipation Γ = −2.00

Acknowledgements. We thank the referee for his helpful comments. We also appreciate Dr. G. Gupta’s help in the numerics. Thispaper is dedicated to Professor Brian Straughan for his support and encouragements. His contributions to Continuum Mechanicshave been very influential in our research activities.

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Flow of a generalized second grade non-Newtonian fluid with variable viscosity

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