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Fluid Flow and Heat Transfer Analyses in Curvilinear Microchannels

419

velocity near wall, the temperature jump decreases the Nusselt number by increasing the

absolute difference of the wall temperature and mean gas temperature. In this work, by

choosing

=0.9, σ

=0.9 and the specified geometry, it is observed that the effect of

temperature jump is dominant.

Furthermore, in the convergent region, due to increasing of the average velocity and

especially slip velocity, there is a jump in local Nusselt number. As the fluid flow

approaches the developed region, no change in local Nusselt number is observed and local

Nusselt number converges to its fully developed value.

Fig. 15. Variation of local Nusselt number along the constricted microchannel with Knudsen

number at Re=2 and a=0.15

Fig. 16. Variation of local Nusselt number along the constricted microchannel with

ampilitude of the wave at Kn=0.075 and Re=2

Fig. 16 illustrates the variation of Nusselt number as a function of amplitude of the wave,

while keeping the Reynolds number and Knudsen number constant. The effect of

Evaporation, Condensation and Heat Transfer

420

constriction on Nusselt number is more sensible for lesser throttle area. In other word, by

decreasing the throttle area and consequently increasing the average velocity, Nusselt

number experiences much larger jump. For instance, by variation of amplitude of the wave

from 0.05 to 0.15, Nusselt number increases 15% in the throttle region.

Fig. 17. Variation of local Nusselt number with Eckert number at Re=2, Kn=0.075 and a=0.15

Fig. 18. Variation of developed C

.Re and Nusselt number with Knudsen number at Re=2

and a=0.15

A comparison is carried out in Fig. 17 to investigate the effect of viscous dissipation on local

Nusselt number. As already stated, viscous dissipation cause a singular point in local

Nusselt number. In addition, taking viscous dissipation into account causes an increase in

Nusselt number. Viscous dissipation increases the average temperature and accordingly in

the region of θ>1 increases the difference of average temperature and wall temperature.

Therefore, the heat transfer rate in this region is enhanced. Also, in Eq. (11), wall

Fluid Flow and Heat Transfer Analyses in Curvilinear Microchannels

421

temperature gradient increases by involving viscous dissipation and this increase is

dominant. Therefore, it causes the Nusselt number to increase. Furthermore, as it is

depicted, by increasing the Eckert number from 0.05 to 0.2, the fully developed Nusselt

number does not vary, but the location of singular point changes. In Fact, as Eckert number

increases, the viscous dissipation effect and consequently the region of θ>1 increases and the

singular point occurs at closer distances to the entrance of channel.

In Fig. 18, the developed Nusselt number and C

.Re at the end of constricted microchannel

verses Knudsen number are reported. The decreasing effect of Knudsen number on

developed Nusselt number and C

.Re is obvious in this figure. Moreover, by variation of

Knudsen number from 0.01 to 0.1, the Nu

∞

and C

.Re

∞

declines 57% and 38% respectively.

8. Conclusion

The thermally/ hydrodynamically fluid flow through a constricted microchannel is studied

by taking the effect of viscous dissipation into account. The numerical method which was

served to solve governing equations is a finite volume method (SIMPLE) in curvilinear

coordinate. In this study, effects of Knudsen number and geometry on C

.Re and Nusselt

number are investigated. The results show that viscous dissipation significantly affects the

thermal behavior of fluid flow. However, hydrodynamic flow field is slightly affected by

viscous dissipation.

The main results obtained can be summarized as follows:

The rarefaction effect intensifies the temperature jump and slip velocity at the solid walls.

By increasing the Knudsen number, Nusselt number and C

.Re decrease. For instance,

Nusselt number and C

.Re decrease 57% and 38% respectively at the end of the

microchannel in the range of 0.01<Kn< 0.1.

Convergent region makes C

.Re to decrease rapidly and amplitude of wave intensifies

this decrease.

Nusselt number experiences a rapid jump in the convergent part and this jump is more

considerable for lower Knudsen numbers and higher amplitude of wave.

The effect of viscous dissipation is more significant by increasing Knudsen number.

Also, viscous dissipation leads a singular point in Nusselt number profiles.

Viscous dissipation increases local Nusselt number. However, it does not significantly

affect C

.Re.

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FLUID FLOW AND HEAT TRANSFER SIMULATION IN A CONSTRICTED

MICROCHANNEL: EFFECTS OF RAREFACTION, GEOMETRY, AND VISCOUS

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Thermal Sciences, 48, pp. 271–281.

Effects of Fluid Viscoelasticity in

Non-Isothermal Flows

Tirivanhu Chinyoka

University of Cape Town

South Africa

1. Introduction

The study of non-Newtonian ﬂuids is of fundamental importance in practically all branches

of science and engineering that deal with incompressible ﬂuid ﬂow. Blood rheology,

food processing, petroleum engineering, polymer blending and pharmaceutical product

development are only but a few areas in which non-Newtonian ﬂuids play a major role.

Inelastic ﬂuids with shear rate dependent viscosities are an example of non-Newtonian ﬂuids,

such ﬂuids are called Generalized Newtonian ﬂuids. Fluids that exhibit elastic effects are

another example of non-Newtonian ﬂuids, these ﬂuids are called Viscoelastic ﬂuids. We

focus attention on viscoelastic ﬂuids whose viscosities are either independent of applied

shear-rates (Boger ﬂuids) or whose viscosities are shear-rate dependent (e.g. the Generalized

Oldryd-B ﬂuids). In either case the ﬂuid viscosity will be considered temperature dependent

and our investigations will focus on the ﬂuids’ heat transfer characteristics in simple ﬂows.

As in Chinyoka (2008; 2009a;b; 2010; 2011) the viscoelastic ﬂuid behavior is compared to

that for corresponding inelastic (Newtonian and/or Generalized Newtonian) ﬂuids and it is

demonstrated that depending on the physical application, viscoelasticity may or may not be

favorable. For a comprehensive overview of non-Newtonian ﬂows in general and viscoelastic

ﬂuid phenomena in particular, we refer to the excellent treatises of Bird et al. (1987); Ferry

(1981).

Investigations of heat transfer in ﬂuid ﬂow have mostly been conducted for inelastic ﬂuids.

Temperature dependent ﬂows of viscoelastic ﬂuids have been largely limited by the slow

development of the relevant universally accepted non-isothermal constitutive models. The

mathematical discussion of the constitutive modeling of non-isothermal effects in the ﬂow

of viscoelastic ﬂuids is still underway and the references Dressler et al. (1999); Hütter et al.

(2009); Peters & Baaijens (1997); Sugend et al. (1987); Wapperom & Hulsen (1998) provide a

clear picture as to the current developments. What is now beyond doubt, among these

representative cited works, is that temperature changes in such ﬂowing polymeric systems

should at the very minimum capture the effects of the three processes; conductive heat

transfer effects, entropic effects due to stress work and energetic effects due to the changes

in the polymer orientations. Secondary effects, say due radiation and chemical reactions

can be included or neglected depending on the exact nature of the physical situation. The

major difference between the most recent work Hütter et al. (2009) and previous works is

the realization in Hütter et al. (2009) that the usual modeling of energetic effects using the

conformational tensor may fail to capture those energetic effects that may arise from fast

deformation/relaxation processes due to microscopic changes, say, resulting from continual

2 Will-be-set-by-IN-TECH

changes of orientation of adjacent atoms in a polymer chain. A local fast variable is thus

used in Hütter et al. (2009) as opposed to the slow tensor used in Dressler et al. (1999);

Peters & Baaijens (1997); Wapperom & Hulsen (1998) and all the similar related works.

Microscopically-based models might thus currently not be our best choice since their energy

equations are still an area of very active discussion. In the current work, we will thus focus

on conformational tensor based models for the energetic effects. We however leave open the

possibility of revisiting this in the future, once the results of Hütter et al. (2009) have been fully

tested experimentally and hence once the numerical values of the local variables needed are

readily available for actual polymeric systems.

Simple ﬂows of liquids undergoing exothermic reactions is a model problem in industrial

processes involving chemically reactive lubricants and most petroleum products. In

lubrication applications it is of paramount importance to use ﬂuids with delayed susceptibility

to thermal runaway phenomena else the lubricants are easily degraded and expensive

material exposed to wear. Similarly it would be important to understand and be able

to anticipate and hopefully delay the onset of such thermal runaway phenomena in the

pipeline transport of reactive petroleum products in order that the end-products remain

usable. Investigations into reactive lubricants subjected to shear ﬂow were carried out, say,

in Chinyoka (2008) where it was demonstrated that viscoelastic (Oldroyd-B) liquids can

withstand higher values of the reaction parameter without undergoing thermal runaway as

opposed to corresponding Newtonian lubricants. A similar investigation in Chinyoka (2010)

also showed that the linear Phan-Thien-Tanner ﬂuids also display better thermal loading

properties compared to corresponding Newtonian ﬂuids. A question thus naturally arises;

is it always the case that viscoelastic ﬂuids show better thermal loading properties (and hence

delayed thermal runaway) under exothermic reactions than corresponding Newtonian ﬂuids?

Our current investigations with Johnson-Segalman ﬂuids show that the answer is negative.

The viscoelastic models used in Chinyoka (2008; 2009a;b; 2010; 2011) exhibit a monotonic shear

stress/shear rate relationship in simple shear ﬂows. The Johnson-Segalman ﬂuid, allows for a

non-monotonic relationship between the shear stress and rate of shear. For certain values of

the viscoelastic material parameters, the Johnson-Segalman model exhibits a non-monotonic

stress-strain relationship. Under certain values of such parameters, only weak solutions are

admissible in pressure driven channel ﬂow of Johnson-Segalman liquids. Physically, the

weak solutions manifest as shear banding, where ﬂuid near to the walls moves rapidly with

high shear rates whereas the bulk ﬂuid within the channel largely exhibits plug ﬂow with

very low shear rates. This leads to jumps in the shear rates leading to zones of markedly

different shear-rates within the ﬂuid and hence the name shear-banding. The shear banding

phenomenon has been observed experimentally in the ﬂow of certain classes of viscoelastic

ﬂuids. The phrase “Johnson-Segalman ﬂuid” refers to a class of ﬂuids that exhibit shear

banding in experiment. The presence of high shear rate regions in the pressure driven ﬂow of

Johnson-Segalman ﬂuids can lead to larger increases in the ﬂuid temperature and hence easy

susceptibility to thermal runways phenomena (in case of reacting ﬂows) compared to even

the inelastic ﬂuids.

The main objective of this work is thus to demonstrate conditions under which ﬂuid

viscoelasticity may or may not enhance the thermal loading properties in real ﬂuid ﬂow

situations. The question of whether solutions of partial differential equations exist globally

in time or develop singularities in ﬁnite time remains a focus of attention in the scientiﬁc

community. The fact that problems of industrial and engineering signiﬁcance are governed

by these equations makes the discussion all the more important. For a comprehensive

overview of the typical examples where ﬁnite-time blow-up, or at least very rapid growth,

occurs in mechanical systems and in particular those of thermal-ﬂuid mechanics refer to

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Evaporation, Condensation and Heat Transfer

Effects of Fluid Viscoelasticity in

Non-Isothermal Flows 3

Straughan (1998). In chemical kinetics, such ﬁnite time blow up of physical temperatures

is commonly referred to as thermal runaway and is closely connected with the so-called

Kamenetskii parameter, named after the pioneering work of Frank-Kamenetskii (1969). The

need to understand and control such phenomena as thermal runaway provides the impetus

for investigations like those chronicled in this work.

The work in this chapter is organized as follows. Section (2) summarizes the constitutive

and mathematical models that are relevant in non-isothermal ﬂow of polymeric (viscoelastic)

ﬂuids. In section (3), we look at the relative signiﬁcance of the energy elastic effects in ﬂow of

polymeric ﬂuids, basically summarizing the works in Chinyoka (2010); Hütter et al. (2009). In

sections (4,5,6), we give some applications to reacting ﬂows in lubrication, heat exchangers

and convection driven ﬂows respectively. The issues related to the pressure driven ﬂow

of Johnson-Segalman ﬂuids are summarized in section (7) in which we also highlight some

current investigations involving non-isothermal ﬂow of viscoelastic ﬂuids with shear-rate

dependent viscosity. Concluding remarks follow in section (8).

2. Mathematical modeling

The dimensionless governing equations for the velocity u, temperature T and extra stress

components τ

are,

∇·u = 0, (1)

= −Re ∇p + ∇·σ + F,(2)

Re Pr

= ∇

T + Br Q

+ δ,(3)

+ We



τ −τ

(1 + αT)



+ f (τ)=2 βμ

S.(4)

Here σ

= τ + 2μ

(1 − β) S is the extra stress tensor, S =(∇u +(∇u)

)/2 is the deformation

rate tensor and τ

is the polymeric tensor, F represents body forces for the momentum equation,

is the dissipation function that takes into account both entropic and energetic effects, δ

represents source terms for the energy equation, f

(τ) is a nonlinear function of the polymer

stress tensor, Re is the Reynolds number, Pr is the Prandtl number, Br is the Brinkman number,

We is the Weissenberg number, β is the ratio of polymer to solvent viscosity. The time

derivatives are deﬁned by:

∂A

∂t

+(u ·∇)A,(5)

and

τ=(1 −

)

∇

τ +



τ,0≤ ξ ≤ 2, (6)

where

∇

τ=

−∇u · τ − τ · (∇u)



τ =

+ τ ·∇u +(∇u)

· τ.(7)

The Johnson Segalman model has 0

< ξ < 2, the lower convected Maxwell model has ξ = 2

and all the other viscoelastic ﬂuid models considered in this work have ξ

= 0. Polymeric

ﬂuids have 0

< β ≤ 1sothatβ = 0 corresponds to inelastic ﬂuids. The expressions for

the dimensionless dissipation function Q

depends on the particular ﬂuid model and will be

speciﬁed independently for each case.

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Effects of Fluid Viscoelasticity in Non-Isothermal Flows

4 Will-be-set-by-IN-TECH

The temperature dependence of the solvent viscosity, polymer viscosity and relaxation time is

given by μ

, μ

and

λ respectively. These will be speciﬁed in terms of William-Landel-Ferry

(WLF), Arrhenius or Nahme shift factors.

3. Energy elastic effects

It is concluded in Hütter et al. (2009) that the usual modeling of energetic effects using the

conformational tensor may fail to capture those energetic effects that may arise from fast

deformation/relaxation processes due to microscopic changes, say, resulting from continual

changes of orientation of adjacent atoms in a polymer chain. We illustrate these issues using

the work in Chinyoka (2010). In this case, the dissipation function takes the form,

= γ( τ : S)+(1 − γ)

Tr(τ)

2We

,(8)

where the parameter γ signiﬁes the ability of viscoelastic ﬂuids to store energy due to their

elastic behavior. In particular, 0

≤ γ ≤ 1, where γ = 0 corresponds to the case of pure

energy elasticity and γ

= 1 corresponds to pure entropy. Allowance for exothermic reactions

is modeled via Arrhenius kinetics,

= δ

exp



1 + αT



,(9)

where the reaction parameter, δ

is also called the Frank-Kamenetskii parameter. The

nonlinear polymer stress function for the Phan-Thien-Tanner (PTT) model is given by,

(τ)=

(τ)τ, (10)

where ε is a dimensionless quantity depending on the ﬂuid. For the temperature dependence

of the viscosities and relaxation time respectively, we use the Nahme and WLF models. In

particular, the solvent viscosity is modeled via a Nahme-type law:

(T)=exp(−εαT), (11)

and the polymer viscosity and relaxation time are modeled via the WLF equation,

(T)=exp



−

αc

+ αT



, (12)

(T)=exp



−

αc

+ αT



. (13)

The parameters c

and c

are respectively assigned the values 15 and 50 corresponding to

the case where the initial ﬂuid temperature is close to the glass transition temperature and

hence the polymer viscosity and relaxation times are strongly temperature dependent. The

parameter α is an activation energy parameter and isothermal ﬂows have α

= 0. We consider

both shear and pressure driven channel ﬂow and consider no-slip conditions for velocity on

the walls. The channel wall temperatures are also kept constant (and uniform) and continuity

of the stress tensors at the walls is enforced. We employ semi-implicit ﬁnite difference schemes

for the solution process of the highly nonlinear and transient problems. Such schemes were

426

Evaporation, Condensation and Heat Transfer

Effects of Fluid Viscoelasticity in

Non-Isothermal Flows 5

given in, say, Chinyoka et al. (2005) for isothermal viscoelastic ﬂow and then modiﬁed and

extended to the energy equation in Chinyoka (2008; 2009a;b; 2010; 2011).

3.1 Poisseuille ﬂow results

Fig. 1. shows the temperature distribution in a pressure driven channel ﬂow of a PTT ﬂuid

for a range of the parameter γ ranging from the case of pure entropy to pure energy elasticity.

0 0.005 0.01 0.015

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature

γ = 0.0

γ = 0.2

γ = 0.4

γ = 0.8

γ = 1.0

Fig. 1. Non-isothermal Poiseuille ﬂow of PTT ﬂuid in a channel

γ 0.0 0.2 0.4 0.8 1.0

max

0.0137 0.0139 0.0141 0.0146 0.0148

Table 1. Maximum temperatures for various choices of γ.

The results show that ﬂuid elasticity can be used to reduce the growth of ﬂuid temperature.

We next show a comparison of the results of Fig. 1. and Table 1. with results for inelastic

(Newtonian) ﬂuids. Fig. 2. shows the temperature distribution in a pressure driven channel

ﬂow of a PTT and a Newtonian ﬂuid for the values of the parameter γ corresponding to pure

entropy to pure energy elasticity. We notice that the highest temperatures are recorded for

the Newtonian ﬂuid and that for the viscoelastic ﬂuids, attainable temperatures increase with

increasing γ.

As with the conclusions of Peters & Baaijens (1997), for the non-isothermal ﬂow of a PTT ﬂuid

around a cylinder, the difference in maximum attainable temperatures for the cases of pure

entropy and pure energy are relatively small at the given parameter values. If we increase the

shear-rates, by increasing the driving pressure gradient see Fig. 3. and Table 3., we notice that

the aforementioned differences in the maximum temperatures increase signiﬁcantly in line

with the conclusions of Hütter et al. (2009). We similarly notice that the highest temperatures,

besides being much higher than before, are also recorded for the Newtonian ﬂuid. We thus

conclude that energy elastic effects may be neglected at low shear rate ﬂows. In practice,

pressure driven channel ﬂows usually obtain at relatively low shear rates.

427

Effects of Fluid Viscoelasticity in Non-Isothermal Flows

6 Will-be-set-by-IN-TECH

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature

Newt

PTT, γ = 0

PTT, γ = 1

Fig. 2. Non-isothermal Poiseuille ﬂow of PTT & Newtonian ﬂuids

Newtonian PTT, γ = 0 PTT, γ = 1

max

0.0180 0.0137 0.0148

Table 2. Maximum temperatures for various choices of γ.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature

Newt

PTT, γ = 0

PTT, γ = 1

Fig. 3. Non-isothermal Poiseuille ﬂow of PTT & Newtonian ﬂuids (Higher Pressure Gradient)

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Evaporation, Condensation and Heat Transfer