viscous-dissipation effects on the heat transfer in a poiseuille flow

PPLIED
A
Applied Energy 83 (2006) 495–512

www.elsevier.com/locate/apenergy

ENERGY

Viscous-dissipation effects on the heat transferin a Poiseuille flow

Orhan Aydın *, Mete Avcı

Karadeniz Technical University, Department of Mechanical Engineering, 61080 Trabzon, Turkey

Received 10 December 2004; received in revised form 11 March 2005; accepted 20 March 2005

Available online 14 June 2005

Abstract

In this study, laminar heat-convection in a Poiseuille flow of a Newtonian fluid with con-

stant properties is analyzed by taking the viscous dissipation into account. At first, both

hydrodynamically and thermally fully-developed flow case is investigated. Then, consideration

is given to thermally-developed laminar forced-convection. The axial heat-conduction in the

fluid is neglected. Two different thermal boundary-conditions are considered: the constant

heat-flux and the constant wall-temperature. Both the hot-wall and the cold-wall cases are

considered. In the literature, the viscous-dissipation effect is commonly represented by the

Brinkman number. Several different definitions of the Brinkman number arise depending on

the thermal boundary conditions. Either for the thermally fully-developed case or the ther-

mally-developing case (the Graetz problem), temperature distributions and the Nusselt num-

bers are analytically determined as functions of the Brinkman number.

� 2005 Elsevier Ltd. All rights reserved.

Keywords: Poiseuille flow; Viscous dissipation; Constant heat-flux; Constant wall-temperature

0306-2619/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apenergy.2005.03.003

* Corresponding author. Tel.: +90 462 377 29 74; fax: +90 462 325 55 26.

E-mail address: [email protected] (O. Aydın).

mailto:[email protected]

Nomenclature

Br Brinkman number, Eq. (8)Bre Brinkman number, Eq. (23)

Brq modified Brinkman number, Eq. (11)

cp specific heat at constant pressure

Gz the Graetz number

k thermal conductivity (W/mK)

Nu Nusselt number

Pr Prandtl number

qw wall�s heat-flux (W/m2)T temperature (K)

u velocity (m/s)

U dimensionless velocity

W channel width (=2w) (m)

y coordinate in vertical direction (m)

Y dimensionless vertical coordinate

z axial direction (m)

Z the reciprocal of Graetz number, dimensionless axial coordinate, Eq.(21)

Greek symbols

a thermal diffusivity (m2/s)

h dimensionless temperature

hq dimensionless temperature modified, Eq. (12)

m kinematic viscosity (m2/s)

l dynamic viscosity (Pa s)q density (kg/m3)

Subscripts

c centreline

e fluids entering

m mean

vd viscous dissipation

w wallwm wall mean

496 O. Aydın, M. Avcı / Applied Energy 83 (2006) 495–512

1. Introduction

Viscous dissipation changes the temperature distributions by playing a role as an

internal heat-generation source. This heat source caused by the shearing of fluid lay-

ers influences temperature distributions and, in the following, heat-transfer rates.

O. Aydın, M. Avcı / Applied Energy 83 (2006) 495–512 497

The merit of the effect of the viscous dissipation depends on whether the duct wall is

hot or cold. A review of the existing literature regarding the effect of the viscous dis-

sipation is given by Aydı́n [1,2].

The work of Brinkman [3] appears to be the first theoretical work dealing with

heat generated by viscous dissipation. The temperature distribution in the en-trance region of a circular pipe at the wall of which was maintained at either

the constant temperature of the entering fluid or constant heat-flux was examined.

The highest temperatures were, not surprisingly, discovered to be localized in the

wall region. Cheng and Wu [4] investigated the effects of viscous dissipation on

the onset of instability for longitudinal vortices in the thermal entrance region

of a horizontal parallel-plate channel with an isothermal, heated lower-plate

and cooled upper-plate. Pinho and Oliveira [5] investigated the forced convection

of a Phan–Thien–Tanner fluid in laminar pipe and channel flows including theeffects of viscous dissipation. It was shown that the beneficial effects of fluid elas-

ticity were enhanced by viscous dissipation. Using a functional analysis method,

Lahjomri [6] analytically studied thermally-developing laminar Hartman flow

through a parallel-plate channel, with a prescribed transversal uniform magnetic

field, including both viscous dissipation, Joule heating and axial heat-conduction

with uniform heat-flux. In a recent study, Nield et al. [7] investigated the thermal

development of forced convection in a parallel plate channel filled by a saturated

porous medium, with walls held at a uniform temperature, and with the effects ofaxial conduction and viscous dissipation included. Davaa et al. [8] studied fully-

developed laminar heat-transfers to non-Newtonian fluids flowing between paral-

lel plates with the axial movement of one of the plates, with an emphasis on the

viscous-dissipation effect.

In the present study, investigated the effect of viscous dissipation on convective

heat-transfer in a Poiseuille flow for a hydrodynamically-developed flow between

plane parallel plates, considering both the thermally fully-developed and developing

cases. The effect of the Brinkman number on the temperature profile and the Nusseltnumber is obtained for the constant wall-heat-flux and the constant wall-temperature

as the thermal boundary conditions. Both the hot-wall and the cold-wall cases are

examined.

2. Analysis

In this study, for the hydrodynamically fully-developed flow, firstly, the thermallyfully-developed flow case is considered (Case A) and it is extended to the thermally-

developing case (Case B).

2.1. Case A (thermally fully-developed)

In this case, the flow is considered to be fully developed, both thermally and

hydrodynamically. Steady, laminar flow, having constant properties (i.e., the thermal

conductivity and the thermal diffusivity of the fluid are considered to be independent


of temperature), is considered. The axial heat conduction in the fluid and through the

wall is assumed to be negligible.

The well-known parabolic velocity-distribution for fully-developed plane laminar

Poiseuille flow is given as follows:

uuc

¼ 1� yw

� �2

. ð1Þ

The conservation of energy, including the effect of the viscous dissipation requires

uoToz

¼ tPr

o2Toy2

þ lqcp

ouoy

� �2

; ð2Þ

where the second term on the right-hand side is the viscous-dissipation term.

Due to axisymmetry at the centre, the thermal boundary condition at y = 0 can be

written as

oToy

��y¼0

¼ 0. ð3Þ

Thermal boundary conditions of constant wall-heat-flux (CHF) and constant

wall-temperature (CWT) at the wall are considered.

2.1.1. CHF Case

The constant heat-flux at the wall

koToy

��y¼w

¼ qw; ð4Þ

where qw is positive when its direction is to the fluid (hot wall); otherwise it is neg-

ative (cold wall).For the uniform wall heat-flux case, the first term on the left-side of Eq. (2) is

oToz

¼ dT w

dz. ð5Þ

By introducing the following non-dimensional quantities:

Y ¼ yW

; h ¼ T w � TT w � T c

. ð6Þ

Eq. (1) can be written as

d2h

dY 2¼ að4Y 2 � 1Þ þ 16BrY 2; ð7Þ

where a ¼ ucW 2

aðTw�T cÞdTw

dz and Br is the Brinkman number given as

Br ¼ lu2ckðT w � T cÞ

. ð8Þ

For the solution of the dimensionless energy-transport equation given in Eq. (7),

the dimensionless boundary-conditions are as follows:


h ¼ 1 ohoY

��Y¼0

¼ 0 at Y ¼ 0;

h ¼ 0 at Y ¼ 0.5.ð9Þ

The solution of Eq. (7) under the thermal boundary-conditions given in Eq. (9) is

hðY Þ ¼ 1� 24

5Y 2 þ 16

5Y 4

� �� 2

5BrðY 2 � 4Y 4Þ. ð10Þ

We can also use the modified Brinkman number, which is:

Brq ¼lu2cWqw

. ð11Þ

In terms of the modified Brinkman number (based on the wall heat-flux) given

above, the temperature distribution is

hq ¼T � T w

Wqwk

¼ � 5

16þ 3

2Y 2 � Y 4

� �þ Brq � 1

2þ 4Y 2 � 8Y 4

� �. ð12Þ

In fully-developed flow, it is usual to utilize the mean fluid-temperature, Tm,

rather than the centre-line temperature when defining the Nusselt number. This

mean or bulk temperature is given by:

Tm ¼RquT dARqudA

. ð13Þ

The dimensionless mean-temperature is obtained as:

hm ¼ Tm � T w

T c � T w

¼ 136

175� 2

175Br. ð14Þ

In terms of Brq defined in Eq. (11), the mean temperature

Tm � T wqwr0k

¼ � 17

70� 24

70Brq. ð15Þ

2.1.2. CWT case

When constant temperatures apply, since dTw/dz = 0, the first term on the left-side

of Eq. (2) is

oToz

¼ T w � TT w � T c

� �dT c

dz. ð16Þ

Substituting this result into Eq. (2) and introducing the dimensionless quantities

given in Eq. (6) gives the following dimensionless equation for the CWT case:

d2h

dY 2¼ ahð4Y 2 � 1Þ þ 64BrY 2; ð17Þ

where a ¼ ucW 2

aðTw�T cÞdT c

dz and Br is the Brinkman number. The boundary conditions gi-

ven in Eq. (9) are also valid for this case. Actually, no simple closed-form solution

can be obtained for this equation. However, the variation of h can be quite easily


obtained, to any required degree of accuracy, by using an iterative procedure [9]. The

temperature profile for the CHF case is used as the first approximation and Eq. (17)

is then integrated to obtain h. This iterative procedure is repeated until an acceptable

convergence is obtained.

The forced convective heat-transfer coefficient is given as follows:

h ¼�k oT

oY

� �Y¼0.5

T w � Tm

;

which is obtained from the Nusselt number:

Nuwm ¼ qwWðT w � TmÞk

; ð18Þ

where Nuwm is the Nusselt number based on the channel width. After performing the

necessary substitutions, we obtain,

Nuwm ¼ 35ð8� BrÞ68� Br

. ð19Þ

In terms of the modified Brinkman number, Brq,

Nuwm ¼ 70

17þ 24Brq. ð20Þ

2.2. Case B (thermally developing)

Taking the assumptions given above into account, this case considers hydrody-namically fully-developed but thermally developing situation. This problem is tradi-

tionally termed as the ‘‘Graetz’’ problem.

2.2.1. CWT case

Introducing the following dimensionless-variables,

U ¼ uum

; h ¼ ðT w � T ÞðT w � T eÞ

; Y ¼ yW

; Z ¼ 1

Gz¼ z=W

Re Pr; ð21Þ

where um is the mean velocity and Re is the Reynolds number based on this mean

velocity and W, which is equal to 2w. The dimensionless variable Z is the reciprocal

of the Graetz number. Then, Eq. (2) becomes

UohoZ

¼ o2h

oY 2� 64BreY 2; ð22Þ

where Bre is the Brinkman number, defined using the fluid�s entry-temperature. Then

Bre ¼lu2c

kðT w � T eÞ. ð23Þ

For the CWT case, the temperature of the wall is kept isothermal in the entrance

region, which is mathematically shown as


For z > 0 : T ¼ T w at y ¼ w. ð24ÞIn dimensionless form, the thermal boundary-conditions, that will be applied in

the solution of the energy equations, are given as:

Y ¼ 0 :ohoY

¼ 0; Y ¼ 0.5 : h ¼ 0. ð25Þ

The local Nusselt number is obtained from

Nuw ¼ ohoY

��Y¼0.5

. ð26Þ

For the mean temperature, rewriting this equation in terms of the dimensionless

variable gives:

T w � Tm

T w � T e

¼ 16

11

Z 0.5

0

UhdY . ð27Þ

The Nusselt number, based on the difference between the wall�s and the mean tem-

peratures, is then given by

Nuwm ¼ NuwT w � T e

T w � Tm

¼ Nuw1611

R 0.50

UhdY. ð28Þ

2.2.2. CHF case

For this case, the following dimensionless temperature is used,

h ¼ ðT � T eÞðqwW =kÞ . ð29Þ

With this definition, the energy equation can be written in dimensionless form as

UohoZ

¼ o2h

oY 2þ 64BrqY 2 ð30Þ

where Brq is the modified Brinkman number, which is given as

Brq ¼lu2cqwW

.

For this case, the Nusselt number is given by,

Nuw ¼ 1

hwð31Þ

and

Nuwm ¼ 1

hw � hm. ð32Þ

For each case, the energy equation has been solved numerically using a finite-difference method. The details of the solution procedure can be found in [9].


As shown above, different thermal boundary-conditions lead to different dimen-

sionless temperatures, which have, in the following, resulted in different definitions

of Brinkman number. Note that it is very important to know the definition or the

content of Brinkman number when generalizing results obtained or comparing them

against those given by others.

3. Results and discussion

In the absence of viscous dissipation, the solution is independent of whether the

wall is hot or cold. However, viscous dissipation always contributes to internal heat-

ing of the fluid; hence the solution will differ according to the process taking place.

The Brinkman number is chosen as the criterion which shows the relative importanceof viscous dissipation to the other terms of the energy-conservation equation. The

thermal boundary-conditions of the constant heat-flux (CHF) and the constant

wall-temperature (CWT) situations are treated and, for each boundary-condition,

both the hot-wall or cold-wall cases are examined.

3.1. Case A (thermally fully-developed)

Remember positive values of Br correspond to the hot-wall (where heat is beingsupplied across the walls into the fluid) case (Tw > Tc), while the opposite is true

for negative values of Br. Viscous dissipation, as an energy source, considerably

influences the temperature profile. For the CHF case, Fig. 1 illustrates the dimen-

sionless temperature distributions with varying Br for the hot wall (a) and cold wall

(b) cases. For lower values of Br, the effect of the Brinkman number is shown to be

negligible. Since the highest shear-rate occurs near the wall, the effect of viscous dis-

sipation is most significant in this region.

The standard way of making temperatures dimensionless based on Eq. (6) is notappropriate for the situation of imposed heat-flux because the temperature scale

DT = Tw � Tc varies with relevant parameters and may cause a misinterpretation

of the corresponding variation of T. In fact, for a given qw, DT is the unknown of

the problem and it is more convenient to define a fixed temperature scale that we

take as qwW/k. As stated earlier, this scale leads to the modified Brinkman-number

definition, Brq. For different values of Brq, Fig. 2 depicts the dimensionless

temperature-profiles. As expected, increasing dissipation increases the bulk temper-

ature of the fluid due to internal heating of the fluid. For the hot-wall case (Fig. 2(a)),this increase in the fluid temperature decreases the temperature difference between

the wall and the fluid, as will be shown later, which is followed with a decrease in

heat transfer. When the cold-wall case is applied, due to the internal-heating effect

of the viscous dissipation on the fluid�s temperature-profile, the temperature differ-

ence is increased with increasing Brq. In fact, the cold-wall case is applied to reduce

the bulk temperature of the fluid, while the effect of the viscous dissipation is increas-

ing the bulk temperature of the fluid. Therefore, the amount of viscous dissipation

may change the overall heat-balance. When the Brq exceeds a certain limiting value,

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.1

0.2

0.3

0.4

0.5

YBr = 5.0Br = 2.0Br = 1.0Br = 0.1Br = 0.0

-0.1 0.1 0.3 0.5 0.7 0.9 1.10.0

0.1

0.2

0.3

0.4

0.5

Y

Br = 0.0Br = -0.1Br = -1.0Br = -2.0Br = -5.0

(a)

(b)

Fig. 1. Dimensionless temperature-distributions for different values of Br for the CHF case: (a) the hot

wall; (b) the cold wall.


the heat generated internally by the viscous dissipation process will overcome the

cooling effect by the cold wall. Similarly, the dimensionless temperature-distribution

for the CWT case is shown in Fig. 3.

-3 -2.5 -2 -1.5 -1 -0.5 0

q

0.0

0.1

0.2

0.3

0.4

0.5

YBrq = 5.0

Brq = 2.0

Brq = 1.0

Brq = 0.1

Brq = 0.0

-0.5 0.0 0.5 1.0 1.5 2.0 2.5

q

0.0

0.1

0.2

0.3

0.4

0.5

Y

Brq = 0.0

Brq = -0.1

Brq = -1.0

Brq = -2.0

Brq = -5.0

(a)

(b)

Fig. 2. Dimensionless temperature-distributions for different values of Brq for the CHF case: (a) the hot

wall, (b) the cold wall.


Table 1 shows the Nusselt number values for different values of the Brinkmannumber for CHF and CWT conditions in the range of �1 < Br < 1. The variation

of the Nusselt number with the modified Brinkman number is shown in Table 2.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5Y

Br = 5.0Br = 2.0Br = 1.0Br = 0.1Br = 0.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

Y

Br = 0.0Br = -0.1Br = -1.0Br = -2.0Br = -5.0

(a)

(b)

Fig. 3. Dimensionless temperature-distributions for different values of Br for the CWT case: (a) the hot

wall; (b) the cold wall.


Table 1

Nusselt number values for different values of Br

Br Nu

CHF CWT

�1 4.565 4.298

�0.1 4.163 3.825

�0.01 4.122 3.776

0 4.118 3.770

0.01 4.113 3.765

0.1 4.072 3.715

1 3.657 3.185


Fig. 4 shows the variation of Nusselt number with the Brinkman number for the

CHF case. As shown, a singularity is observed at Br = 68, which is very clear and the

expected result from Eq. (19). For the hot-wall case, with the increasing value of Br,

Nuwm decreases in the range of 0 < Br < 68. This is because the temperature differ-

ence, which drives the heat transfer, decreases. At Br = 68, the heat supplied by

the wall into the fluid is balanced by the internal heat-generation due to the viscous

heating. For Br > 68, the internally-generated heat by the viscous dissipation over-

comes the wall heat. When Br ! +1, Nuwm reaches an asymptotic value. Whenthe cold-wall case (Br < 0) is applied to reduce the bulk temperature of the fluid,

as explained earlier, the amount of viscous dissipation may change the overall

heat-balance. With increasing value of Br in the negative direction, the Nusselt num-

ber reaches an asymptotic value (when Br ! �1, Nuwm ! 35). As noticed, when Br

goes to infinity for either the hot wall or the cold wall case, the Nusselt number

reaches the same asymptotic value, Nuwm = 35. This is due to fact that the heat gen-

erated internally by viscous dissipation processes will balance the cooling effect of the

cold wall. Fig. 5 illustrates the variation of Nuwm with Brq. The behavior observedcan be explained similarly to that for Br. For Brq = �17/24, a singularity is observed,

as expected from Eq. (20).

3.2. Case B (thermally-developing)

For the CHF case, Fig. 6 depicts the temperature distributions for different Brink-

man numbers at different axial-locations. An increase of Brq significantly affects the

dimensionless temperature-distribution due to the irreversible energy-conversion

Table 2

Nusselt number values for different values of Brq

Brq (CHF) Nu

�1 �10

�0.1 4.795

�0.01 4.177

0 4.118

0.01 4.060

0.1 3.608

1 1.707

50 55 60 65 70 75 80 85 90Br

-3000

-2000

-1000

0

1000

2000

3000N

u wm

Fig. 4. Variation of Nuwm with Br for the CHF case.

-10 -8 -6 -4 -2 0 2 4 6 8 10Brq

-100

-75

-50

-25

0

25

50

75

100

Nu w

m

Fig. 5. Variation of Nuwm with Brq for the CHF case.


0.1 0.2 0.4 0.6 0.8 1.0

θ

0.0

0.1

0.2

0.3

0.4

0.5

0.0

0.1

0.2

0.3

0.4

0.5

Y

0.0

0.1

0.2

0.3

0.4

0.5

Y

Brq = 0.0Brq = 0.01Brq = 0.1Brq = 0.5

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

θ

Y

0.0

0.1

0.2

0.3

0.4

0.5

Y

0.0

0.1

0.2

0.3

0.4

0.5

Y

0.0

0.1

0.2

0.3

0.4

0.5

Y

Brq= 0.0Brq= 0.01Brq= 0.1

Brq= 0.5

-0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30

θ

Brq= 0.0Brq= 0.01Brq= 0.1Brq= 0.5

Z=0.1 Z=0.5

-0.1 0.0 0.1 0.2 0.3 0.4 0.5

θ-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

θ-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

θ

Brq= -0.5Brq= -0.1

Brq= -0.01Brq= 0.0

Brq= -0.5Brq= -0.1

Brq= -0.01Brq= 0.0

Brq= -0.5Brq= -0.1

Brq= -0.01Brq= 0.0

Z=0.3

Fig. 6. Dimensionless temperature-distributions in terms ofBrq for the CHF case: (a) hot wall; (b) cold wall.


originating from viscous dissipation. As seen, the dimensionless temperature distri-

bution is a decreasing/increasing function of Brq for the hot/cold wall case, respec-

tively. This is due to the internal-heating effect of the viscous dissipation on the

fluid�s temperature-profile.

For the CWT case, Fig. 7 illustrates the effect of the Brinkman number on the

developing dimensionless temperature-profiles at various axial positions in the ther-

Fig. 7. Dimensionless temperature-distributions in terms ofBr for the CWT case: (a) hot wall; (b) cold wall.


mal entrance region. The physical mechanisms are shown to be very similar to those

that occurred for CHF condition. The viscous dissipation produces a distribution of

positive heat-sources, and this reinforces the heating effect as the fluid moves

downstream.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Z

3

6

9

12

15

Nu w

m

Brq=0.0

Brq=0.01

Brq=0.1

Brq=0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Z

0

4

8

12

16

20

Nu w

m

Brq = -0.1

Brq = -0.01

Brq = 0.0

Brq = -0.5

(a)

(b)

Fig. 8. Variation of Nuw with Brq for the CHF case: (a) hot wall; (b) cold wall.


For the CHF condition, Fig. 8(a) and (b) represent the downstream variations of

Nuwm with the modified Brinkman number for the hot and cold wall cases, respec-

tively. With the increasing value of Brq, Nuwm decreases for the hot-wall case (Fig.

8(a)), while it increases for the cold-wall case (Fig. 8(b)). For the CWT case, either

for the hot-wall case (Fig. 9(a)) or for the cold-wall case, the downstream variation

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Z

0

5

10

15

20

25

30

Nu w

m

Br=0.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Z

-10

-5

0

5

10

15

20

25

30

Nu w

m

Br=0.01

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Z

-200

0

200

400

600

800

1000

1200

Nu w

m

Br=0.1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Z

-140

-100

-60

-20

20

60

100

140

180

Nu w

m

Br=0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Z

0

5

10

15

20

25

30

Nu w

m

Br = -0.5Br = -0.1Br = -0.01Br = 0.0

(a)

(b)

Fig. 9. Variations of Nuwm with Br for the CWT case: (a) hot wall; (b) cold wall.


of Nuwm with Br is represented. In the downstream variation of Nuwm, very interest-

ing scenarios are observed with the increasing value of Br. For Br = 0, as expected,

Nuwm assumes its usual value, 3.771 [9]. For Br = 0.1, Nuwm decreases up to a critical

point, at which internally-generated heat, due to viscous dissipation (qvd), balances

the heat supplied by the wall (qw). After this critical point qvd suppresses qw. Similarbehaviors are observed for Br = 0.1 and 0.5. With the increasing Br, this critical

point is reached at an earlier point from the entrance. For the cold-wall case (Fig.

9(b)), as expected, an increase of Br resulted in increased heat-transfer rates.

As noted earlier, different dimensionless-temperature definitions, arising from

different thermal boundary conditions applied, result in different definitions of

the Brinkman number. Here, with the CHF thermal boundary-condition, we

came up with the same modified Brinkman number definition both for the ther-

mally-developed and developing cases. And, for the same values of Brq, thesteady-state values of Nuwm for the thermally-developing case are shown to be

identical to those for the thermally-developed case. This finding gives credit to

the validity of the analysis performed in this study. However, for the CWT

thermal boundary-condition, such a comparison is not possible for the ther-

mally-developed and developing cases, since the Brinkman number definitions

are different (see Eqs. (8) and (23)).

4. Conclusions

In this study, the convective heat-transfer problem for the Poiseuille flow between

plane parallel plates has been studied, with an emphasis on the viscous dissipation

effect. For the hydrodynamically fully-developed case, both thermally-developed

and developing cases of the energy equation are examined. Two types of the wall�sthermal boundary-condition have been considered, namely: constant heat-flux

(CHF) and constant wall-temperature (CWT). Both the hot and cold wall casesare examined.

For the thermally fully-developed case, a strong influence of viscous dissipation

on the heat transfer for higher values of Br (Br > 1) has been observed, while this

influence is found to be negligible for lower values of Br. For the CHF case, it is

shown that use of Br is inconvenient to represent the viscous dissipation since it in-

cludes a viscous-dissipation dependent variable (i.e., the centreline temperature, Tc).

Instead, the use of the modified Brinkman-number, Brq is suggested. For the hot-

wall case, with the increasing intensity of the viscous dissipation (with an increaseof Br) the heat transfer decreases up to a critical value of Br. At this critical value,

internally-generated heat due to viscous dissipation balances the heat rate supplied

by the wall. Above this critical value, the heat arising from viscous dissipation sup-

presses the supply of heat from the wall. Similarly, for the cold-wall case, when the

Br number exceeds a critical value, the viscous heating overcomes the heat removed

at the wall and the fluid heats up longitudinally.

For the thermally-developing case, depending on hot or cold wall situations at the

wall, the Brinkman number, Br (for the CWT case) and the modified Brinkman


number, Brq (for CHF case) are shown to play significant roles on the developing

Nusselt number.

References

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hydrodynamically and thermally fully-developed flow. Energ Convers Manage 2005;46:757–69.

[2] Aydı́n O. Effects of viscous dissipation on the heat transfer in a forced pipe flow Part 2. Thermally-

Developing Flow, Energy Conversion and Management, in press.

[3] Brinkman HC. Heat effects in capillary flow I. Appl Sci Res 1951;A2:120–4.

[4] Cheng KC, Wu R-S. Viscous dissipation effects on convective instability and heat transfer in plane

Poiseuille flow heated from below. Appl Sci Res 1976;32:327–46.

[5] Pinho FT, Oliveira PJ. Analysis of forced convection in pipes and channels with the simplified Phan–

Thien–Tanner fluid. Int J Heat Mass Transfer 2000;43:2273–87.

[6] Lahjomri J, Zniber K, Oubarra A. Alemany. Heat transfer by laminar Hartmann�s flow in the thermal

entrance-region with uniform wall heat-flux: the Graetz problem extended. Energ Convers Manage

2003;44:11–34.

[7] Nield DA, Kuznetzov AV, Xiong M. Thermally-developing forced convection in a porous medium:

parallel-plate channel with walls at a uniform temperature, with axial conduction and viscous-

dissipation effects. Int J Heat Mass Transfer 2003;46:643–51.

[8] Davaa G, Shigechi T, Momoki S. Effects of viscous dissipation on fully-developed heat-transfer of

non-Newtonian fluids in plane laminar Poiseuille-Coutte flow. Int Comm Heat Mass Transfer

2004;31(5):663–72.

[9] Oosthuizen PH, Naylor D. Introduction to convective heat-transfer analysis. New York: McGraw-

Hill; 1999.

viscous-dissipation effects on the heat transfer in a poiseuille flow

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