APPLIED

Applied Energy 83 (2006) 856–867

www.elsevier.com/locate/apenergy

ENERGY

Laminar forced convection with viscous dissipationin a Couette–Poiseuille flow between parallel plates

Orhan Aydın *, Mete Avcı

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

Received 22 April 2005; received in revised form 27 June 2005; accepted 13 August 2005Available online 13 February 2006

Abstract

In this study, analytical solutions are obtained to predict laminar heat-convection in a Couette–Poiseuille flow between two plane parallel plates with a simultaneous pressure gradient and an axialmovement of the upper plate. A Newtonian fluid with constant properties is considered with anemphasis on the viscous-dissipation effect. Both hydrodynamically and thermally fully-developedflow cases are investigated. The axial heat-conduction in the fluid is neglected. Two different orien-tations of the thermal boundary-conditions are considered: the constant heat-flux at the upper platewith an adiabatic lower plate (Case A) and the constant heat-flux at the lower plate with an adiabaticupper plate (Case B). For different values of the relative velocity of the upper plate, the effect of themodified Brinkman number on the temperature distribution and the Nusselt number are discussed.Comparison of the present analytical results for a special case with those available in the literatureindicates an excellent agreement.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Couette; Poiseuille flow; Parallel plates; Viscous dissipation

1. Introduction

The flow and heat transfers in the combined form of the Couette and Poiseuille flowsbetween parallel plates are important in many materials-processing applications, such asextrusion, metal forming, continuous casting, as well as wire and glass fiber drawing.

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

doi:10.1016/j.apenergy.2005.08.005

* Corresponding author. Tel.: +90 462 377 29 74; fax: +90 462 325 55 26.E-mail address: oaydin@ktu.edu.tr (O. Aydın).

Nomenclature

Brq modified Brinkman-number, Eq. (10)cp specific heat at constant pressurek thermal conductivity (W/mK)Nu Nusselt numberP pressure (N/m2)Pr Prandtl numberqw wall heat-flux (W/m2)T temperature (K)u velocity (m/s)u* dimensionless velocity Eq. (4)U axial velocity of the moving plate (m/s)U* dimensionless velocity, Eq. (4)w half distance between plates (m)W distance between plates (m)y vertical coordinateY dimensionless vertical-coordinatez axial coordinate

Greek symbols

a thermal diffusivity (m2/s)l dynamic viscosity (Pa s)q density (kg/m3)t kinematic viscosity (m2/s)h dimensionless temperature, Eq. (8)

Subscripts

m meanw wall

O. Aydın, M. Avcı / Applied Energy 83 (2006) 856–867 857

Despite these broad applications, these flows have received less research interest thaneither the Couette flow only or the Poiseuille flow only.

Lin [1] studied laminar heat-transfer to a non-Newtonian Couette flow with pressuregradient using the power-law model. The effects of pressure gradient and viscous dissipa-tion on the heat transfer were discussed. Davaa et al. [2] numerically studied fully-devel-oped laminar heat-transfer to non-Newtonian fluids flowing between parallel plates withthe axial movement of one of the plates with an emphasis on the viscous-dissipation effect.Increasing the Brinkman number increased the heat-transfer rates at the heated wall whenthe movement direction of the upper plate was the same as the direction of the main flow,while the opposite is true for the movement of the upper plate in the opposite direction.Hashemabadi et al. [3] obtained an analytical solution to predict the fully-developed,steady and laminar heat-transfer of viscoelastic fluids between parallel plates. Their resultsemphasized the significant effect of viscous heating on the Nusselt number.

858 O. Aydın, M. Avcı / Applied Energy 83 (2006) 856–867

Viscous dissipation plays a role like an internal heat-generation source in the energytransfer, which, in the following, affects temperature distributions and heat-transfer rates.This heat source is caused by the shearing of fluid layers. The merit of the effect of the vis-cous dissipation depends on whether the duct wall is hot or cold. A review of the existingliterature regarding the effect of the viscous dissipation is given by Aydın [4,5]. In a recentstudy, Aydın and Avcı [6] analytically examined laminar heat-convection in a Poiseuilleflow of a Newtonian fluid with constant properties by taking the viscous dissipation intoaccount.

The aim of the present study is to investigate analytically the effect of viscous dissipa-tion on steady-state laminar heat transfer in a Couette–Poiseuille flow between plane par-allel plates with a simultaneous pressure gradient and the axial movement of the upperplate. The effect of modified Brinkman number on the temperature profile and, in the fol-lowing, the Nusselt number is obtained for two different configurations of the thermalboundary-conditions.

2. Analysis

Consider steady, hydrodynamically and thermally fully-developed, laminar flow of anincompressible fluid between two parallel plates (Fig. 1). The thermal conductivity andthe thermal diffusivity of the fluid are considered to be independent of temperature. Theupper plate is assumed to move at a constant velocity, while the lower one is stationary.The axial heat-conductions in the fluid and in the wall are neglected. The momentumequation in the z-direction is described as

d2udy2¼ 1

ldPdz

. ð1Þ

The conservation of energy, including the effect of the viscous dissipation, can be writtenas follows:

uoToz¼ t

Pro2Toy2þ l

qcp

ouoy

� �2

; ð2Þ

where the second term on the right-hand side is the viscous-dissipation term.Under the following boundary conditions:

Y ¼ 0; u ¼ 0;

Y ¼ 1; u ¼ Uð3Þ

u

U

Stationary plate

zy

W

Moving plate

Fig. 1. Schematic diagram of the flow domain.

O. Aydın, M. Avcı / Applied Energy 83 (2006) 856–867 859

and using the following dimensionless parameters:

u� ¼ uum

; U � ¼ Uum

; Y ¼ yW; ð4Þ

Eq. (1) is solved to give the dimensionless velocity-distributions as

u� ¼ ð3U � � 6ÞðY 2 � Y Þ þ U �Y . ð5ÞThe constant heat-flux is assumed at the wall, which states that

koToy

����y¼w

¼ qw; ð6Þ

where qw is positive when its direction is to the fluid (the hot wall), otherwise it is negative(the cold wall).

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

oToz¼ dT w

dz. ð7Þ

Introduction of the following non-dimensional temperature:

h ¼ T � T wqww

k

ð8Þ

modifies Eq. (2) into the following dimensionless form:

d2h

dY 2¼ a½ð3U � � 6ÞðY 2 � Y Þ þ U �Y � � 2Brq ð3U � � 6Þð2Y � 1Þ þ U �½ �2; ð9Þ

where a ¼ umkWPrqwt

dT w

dz and Brq represents the modified Brinkman number, which is in termsof qw while the original Brinkman number is defined in terms of the temperature difference,and defined as

Brq ¼lu2

m

2Wqw

. ð10Þ

Two different forms of the thermal boundary-conditions are applied according to [2],which are shown in Fig. 2. In the following, we treat these two different cases separately.

U

qw

y

z

insulated

FlowFlow

insulated

qw

U

z

y

(a) (b)

Fig. 2. Thermal boundary-conditions considered in this study: (a) Case A; (b) Case B.

860 O. Aydın, M. Avcı / Applied Energy 83 (2006) 856–867

For Case A, the thermal boundary-conditions in the dimensionless form are written as

h ¼ 0ohoY

��Y¼1¼ 1 at Y ¼ 1;

ohoY

��Y¼0¼ 0 at Y ¼ 0.

(ð11Þ

The solution of Eq. (9) under the thermal boundary-conditions given above in Eq. (11) isobtained as

hðY Þ ¼ ½1þ 8Brqð3� 3U � þ U �2Þ� ð3U � � 6Þ Y 4

12� Y 3

6

� �þ U �

Y 3

6þ U �

12� 1

2

� �

� 2Brq½2ðU � � 3Þ2Y 2 � 4ð6� 5U � þ U �2ÞY 3 þ 3ðU � � 2Þ2Y 4 � 6

þ U �ð4� U �Þ�. ð12Þ

In Case B, the dimensionless-type thermal boundary conditions are as in the following:

h ¼ 0ohoY

��Y¼0¼ �1 at Y ¼ 0;

ohoY

��Y¼1¼ 0 at Y ¼ 1.

(ð13Þ

Similarly, the solution of Eq. (9) under the conditions of Eq. (13) gives

hðY Þ ¼ ½1þ 8Brqð3� 3U � þ U �2Þ� ð3U � � 6Þ Y 4

12� Y 3

6

� �þ U �

Y 3

6

� �

� 2Brq½2ðU � � 3Þ2Y 2 � 4ð6� 5U � þ U �2ÞY 3 þ 3ðU � � 2Þ2Y 4� � Y . ð14Þ

In the fully-developed flow, it is usual to utilize the mean fluid temperature, Tm, ratherthan the center line temperature when defining the Nusselt number. This mean or bulktemperature is given by

T m ¼R

quT dARqu dA

. ð15Þ

The dimensionless mean-temperature is obtained for Cases A and B, respectively, as

hm ¼T m � T w

qwWk

¼ � 39þ 2BrqðU � � 3Þ2ð4U �2 � 23U � þ 9Þ þ ðU �2 � 11U �Þ

105; ð16Þ

hm ¼T m � T w

qwWk

¼ � 78þ 4Brqð9þ 3U � � 2U �2Þ2 þ ð13U � þ 2U �

2Þ210

. ð17Þ

The forced convective heat transfer coefficient is given as follows:

h ¼ qw

T w � T m

; ð18Þ

which is obtained from Nusselt number that is defined as

Nu ¼ qwWðT w � T mÞk

¼ � 2

hm

. ð19Þ

After performing necessary substitutions, the Nusselt numbers for Cases A and B areobtained, respectively, as

O. Aydın, M. Avcı / Applied Energy 83 (2006) 856–867 861

Nu ¼ 210

39þ 2BrqðU � � 3Þ2ð4U �2 � 23U � þ 9Þ þ ðU �2 � 11U �Þ

; ð20Þ

Nu ¼ 420

78þ 4Brqð9þ 3U � � 2U �2Þ2 þ ð13U � þ 2U �

2Þ. ð21Þ

3. Results and discussion

Here, we study the Couette–Poiseuille flow between two plane parallel plates with asimultaneous pressure gradient and the axial movement of the upper plate. As stated ear-lier, the problem is steady, laminar, and hydrodynamically and thermally fully developed.Three different conditions of the upper plate are considered: (i) stationary; (ii) moving inthe positive z-direction; (iii) moving in the negative z-direction. In order to validate thepresent analysis, we compare our results with those of [2], which is given in Table 1. Asseen, an excellent agreement is obtained.

For the case without the influence of the viscous dissipation (Brq = 0), Fig. 3(a) and (b)depicts the dimensionless temperature-distributions for different magnitudes of the upperplate velocity for the Cases A and B, respectively. Note that its positive velocity representsthe movement in the positive z-direction, while the negative velocity is in the oppositedirection. For the three different conditions of the upper plate mentioned above, Fig. 4illustrates the variations of the temperate distributions depending on the modified Brink-man number for the Cases A (Fig. 4(a)–(c)) and B (Fig. 4(d)–(f)). For each case, both posi-tive and negative values of the heat fluxes at the plates are considered. For the positiveheat-flux case, the purpose is to heat the bulk fluid whereas, for the negative heat flux case,the purpose is cooling the bulk fluid. For each case, as stated earlier, the viscous dissipa-tion behaves like an energy source increasing the temperature of the bulk fluid. For thepositive heat flux case at the wall, the increasing viscous dissipation will result in decreas-ing temperature-differences between the wall and the bulk fluid or in decreasing tempera-ture gradient at the wall. However, for the negative heat-flux case at the wall, it willincrease temperature differences between the wall and the bulk fluid or to increase temper-ature gradients at the wall.

Table 1Nusselt number values for different values of Brq and U*

U* Brq Nu, Case A Nu, Case B

Present Ref. [2] Present Ref. [2]

�1.0 0 4.1176 4.118 6.2688 6.2690.01 3.3589 3.359 6.2093 6.2090.1 1.2635 1.264 5.7221 5.722

0 0 5.3846 5.385 5.3846 5.3850.01 5.1699 5.170 5.1700 5.1700.1 3.8043 3.804 3.8043 3.804

1.0 0 7.2414 7.241 4.5161 4.5160.01 7.4468 7.747 4.3299 4.3300.1 10.000 10.00 3.1579 3.158

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1θ

0.0

0.2

0.4

0.6

0.8

1.0

Y

Brq=0.0

U*=-1.0

U*= 0.0

U*= 1.0

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1θ

0.0

0.2

0.4

0.6

0.8

1.0

Y

U*=-1.0

U*= 0.0

U*= 1.0

Brq=0.0

(a)

(b)

Fig. 3. Dimensionless temperature-distributions at different values of U* at Brq = 0.0: (a) Case A; (b) Case B.

862 O. Aydın, M. Avcı / Applied Energy 83 (2006) 856–867

From the engineering point-of-view, the heat-transfer rates are of important, which canpredicted from the Nusselt number. Fig. 5(a) and (b) illustrates the variation of the Nusseltnumber with the dimensionless relative velocity of the upper plate for different modifiedBrinkman numbers or Cases A and B, respectively. For Case A, where a constant heat fluxis applied at the upper plate, the movement of the upper plate in the positive directionenhances the heat transfer, while that in the negative direction decreases it. The effect ofthe viscous dissipation expressed by the modified Brinkman number can be best explainedin terms of the energy balance. At U* = 0, for positive values of Brq, i.e. the positive heatfluxes at the wall, Nu decreases with an increase in Brq. This is due to decreasingtemperature differences between the wall and the bulk fluid or the decreasingtemperature-gradient at the wall, as stated earlier. And, as expected, negative Brq values

1.0

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0

θ

0.0

0.2

0.4

0.6

0.8

U*=-1.0

Brq= 0.1

Brq= 0.01

Brq= 0.0

Brq=-0.01

Brq=-0.1

-0.8 -0.6 -0.4 -0.2 0θ

0.0

0.2

0.4

0.6

0.8

1.0

YY

U*=0.0

Brq= 0.1

Brq= 0.01

Brq= 0.0

Brq=-0.01

Brq=-0.1

-0.8 -0.6 -0.4 -0.2 0θ

0.0

0.2

0.4

0.6

0.8

1.0

Y

Brq=-0.1

Brq=-0.01

Brq= 0.0

Brq= 0.01

Brq= 0.1

U*=1.0

(a)

(b)

(c)

Fig. 4. Dimensionless temperature-distributions at different values of Brq: (a), (b), (c) Case A; (d), (e) and(f) Case B.

O. Aydın, M. Avcı / Applied Energy 83 (2006) 856–867 863

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0θ

0.0

0.2

0.4

0.6

0.8

1.0

Y

Brq=-0.1

Brq=-0.01

Brq= 0.0

Brq= 0.01

Brq= 0.1

U*=-1.0

-0.8 -0.6 -0.4 -0.2 0.0 0.2θ

0.0

0.2

0.4

0.6

0.8

1.0

Y

Brq=-0.1

Brq=-0.01

Brq= 0.0

Brq= 0.01

Brq= 0.1

U*=0.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2θ

0.0

0.2

0.4

0.6

0.8

1.0

Y

U*=1.0

Brq=-0.1

Brq=-0.01

Brq= 0.0

Brq= 0.01

Brq= 0.1

(d)

(e)

(f)

Fig. 4 (continued)

864 O. Aydın, M. Avcı / Applied Energy 83 (2006) 856–867

-1.0 -0.5 0.0 0.5 1.0

U*

0

2

4

6

8

10

Nu

Brq=-0.01Brq= 0.0Brq= 0.01Brq= 0.1

-1.0 -0.5 0.0 0.5 1.0

U*

2

4

6

8

10

Nu

Brq=-0.1

Brq=-0.01

Brq= 0.0

Brq= 0.01

Brq= 0.1

(a)

(b)

Fig. 5. The variations of the Nusselt number with the U* at different values of the modified Brinkman number:(a) Case A; (b) Case B.

O. Aydın, M. Avcı / Applied Energy 83 (2006) 856–867 865

representing the negative heat-flux values at the wall will increase the Nusselt number as aresult of the increasing temperature gradient at the wall. For the movement of the upperplate in the negative direction (U* < 0), this trend of Nu versus Brq amplifies with theincreasing U*. However, for the movement of the upper plate in the positive direction(U* > 0), an interesting scenario arises. Up to U* = 0.422, with an increase in U*, thedependency of Nu on Brq gradually decreases and at U* = 0.422, the Nusselt numbersfor all the modified Brinkman numbers considered become identical. Beyond this value,U* > 0.422, the viscous dissipation becomes considerable again with an increase in U*.However, as observed from the figure, interestingly, the positive modified Brinkman

-3 -2 -1

0 1 2 3 4

0 1 2 3 4

5

Brq

Brq

-30

-10

10

30

Nu

U*= -1.0

U*= 0.0

U*= 1.0

-5 -4 -3 -2 -1-50

-30

-10

10

30

50

Nu

U*= -1.0

U*= 0.0

U*= 1.0

(a)

(b)

Fig. 6. Variations of Nu with Brq: (a) Case A; (b) Case B.

866 O. Aydın, M. Avcı / Applied Energy 83 (2006) 856–867

numbers suggest higher values than Brq = 0, while the negative ones suggest lower values.We may attribute the above observations to the energy balance. For the Couette flow case,the shear is generated by the movement of the upper plate. For the Poiseuille flow case(U* = 0), the pressure gradient applied generates the shear. In our case, the Couette–Poiseuille flow case, the movement of the upper plate and the pressure gradient appliedare together responsible for the shear occurring. The direction of the heat flow dependson the relative magnitudes of the heat flux supplied by the wall and the heat generatedby the shear. The variation of Nu with U* for different values of Brq for Case B(Fig. 5(b)) can be explained similarly. In this case, increasing U* in the negative directionenhances the heat transfer, while the opposite is true for increasing U* in the positive

O. Aydın, M. Avcı / Applied Energy 83 (2006) 856–867 867

direction. The effect of increasing Brq is to increase Nu for its negative values, while it is todecrease for its positive values.

For a larger range of Brq, Fig. 6(a) and (b) shows the variation of the Nusselt numberwith the modified Brinkman number at different values of the dimensionless relative veloc-ity of the upper plate. For each case, singularities are obtained in the behavior. We attri-bute these singularities to the energy balance between the heat flux supplied by the walland the viscous dissipation. At the singular point, the heat supplied by the wall is balancedby the shear heat. Beyond this point, the heat transfer changes its direction.

4. Conclusions

In this study, an analytical solution has been derived for the laminar, steady, convectiveheat-transfer problem in a Couette–Poiseuille flow between plane parallel plates with asimultaneous pressure-gradient and the movement of the upper plate. Interest has beenfocused on the influence of the viscous dissipation. Two different orientations of the wallthermal boundary-conditions have been considered, namely: the constant heat-flux at theupper moving plate with the adiabatic stationary lower wall and the constant heat flux atthe stationary lower wall with an adiabatic moving upper wall. For different relative veloc-ities of the upper plate, either in positive- or negative z-direction, the effect of the modifiedBrinkman number on the Nusselt number has been discussed in terms of the energybalance.

References

[1] Lin SH. Heat transfer to plane non-Newtonian Couette flow. Int J Heat Mass Transfer 1979;22:1117–23.[2] Davaa G, Shigechi T, Momoki S. Effect of viscous dissipation on fully-developed heat transfer of non-

Newtonian fluids in plane laminar Poiseuille–Couette flow. Int Comm Heat Mass Transfer 2004;31(5):663–72.[3] Hashemabadi SH, Etemad SGh, Thibault J. Forced-convection heat-transfer of Couette–Poiseuille flow of

non-linear viscoelastic fluids between parallel plates 2004;47:3985–91.[4] Aydın O. Effects of viscous dissipation on the heat transfer in a forced pipe-flow. Part 1: Both

hydrodynamically and thermally fully-developed flow. Energy Convers Manage 2005;46:757–69.[5] Aydın O. Effects of viscous dissipation on the heat transfer in a forced pipe-flow Part 2. Thermally-developing

flow. Energy Convers Manage 2005;46:3091–102.[6] Aydın O, Avcı M. Viscous dissipation effects on the heat transfer in a Poiseuille flow. Appl Energy, in press,

doi:10.1016/j.apenenergy.2005.03.003.