International Journal of Heat and Mass Transfer 53 (2010) 5017–5021

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Technical Note

Heat transfer on a power law stretching sheet subjected to free streampressure gradient

Noor Afzal *, M. Maqbool WarisFaculty of Engineering, Aligarh Muslim University, Aligarh 202 002, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 10 June 2008Received in revised form 30 April 2009Accepted 2 January 2010

Keywords:Thermal boundary layerStretching surfacePressure gradientContinuous strip motion along productionlineSimilarity solution

0017-9310/$ - see front matter Crown Copyright � 2doi:10.1016/j.ijheatmasstransfer.2010.06.024

* Corresponding author.E-mail address: noor.afzal@yahoo.com (N. Afzal).

The present work deals with the numerical study of temperature distribution in the laminar boundarylayer driven by the stretching boundary surface subjected to pressure gradient. The similarity transfor-mation obeying the same power law based on composite reference velocity (union of velocities of thestretching boundary and free stream) has been employed that leads to a single set of equations, irrespec-tive of the condition whether Uw > U1 or Uw < U1, containing three parameters: b measuring the stretchrate of the moving boundary, e is the ratio of free stream velocity to composite reference velocity and Pr isthe Prandtl number of the ambient fluid. The numerical solutions of the thermal boundary layer equa-tions are obtained for three Prandtl numbers 0.7, 1.0 and 10 for 0 6 e 6 1 and for 0 6 b 6 2. The heattransfer coefficient show appreciable dependence on the ratio of free stream velocity to union of veloc-ities of the stretching surface boundary and free stream.

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1. Introduction reference velocity leading to single set of equations irrespective of

Many industrial processes require the cooling of continuousstrips or filaments, while drawing in the industries along the pro-duction line. In these processes after drawing, the filament ispassed through a fluid to cool it to a desirable temperature. Thereare many examples of such types of processes, the spinning of arti-ficial fiber, drawing, annealing and tinning of copper wires, produc-tion of coaxial television cables and plastic films, etc. where bothUw and U1 are parallel to each other.

The boundary layer flow on a continuous flat surface were pre-sented by Abdelhafez [1] and Chappidi and Gunnerson [2] for uni-form wall with zero pressure gradient considering two casesseparately, where Uw is greater or less than U1. The similarity solu-tion of velocity fields in laminar boundary layer driven by thestretching surface boundary with pressure gradient, each propor-tional to the same power law of the downstream coordinate, basedon composite reference velocity has been formulated by single setof equations by Afzal [4]. The self-similar boundary layer flow on anon-linear power law stretching sheet was first studied by Afzaland Varshney [5], and Afzal [6] (see [7]). Afzal [8] studied thenon-isothermal stretching sheet for temperature exponent n =�1, 0, 1, 2, 3, and 4 and Afzal [9] presented analytical solutions.Chappidi and Gunnerson [2] carried out the numerical and analyt-ical investigations for thermal transport for a uniform surface mov-ing through a flowing fluid, while Afzal et al. [3] studied composite

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whether the plate is moving faster or slower than the free streamfor 0.1 6 Pr 6 100. Heat transfer on an isothermal flat surface mov-ing in a parallel free stream was also studied by Abdelhafez [1] andthe work was further extended by Chen [10] for non-isothermalwall. The effects of buoyancy have been neglected.

Present work deals with the numerical study of the temperaturefields following the similarity solution of velocity fields by Afzal [4]for power law stretching sheet moving in the positive x directionsubjected to free stream pressure gradient by considering compos-ite reference velocity for 0 6 b 6 2 and three Prandtl numbers 0.7,1 and 10.

2. Analysis

The governing continuity, momentum and energy equationsafter boundary layer approximations are:

ouoxþ ov

oy¼ 0 ð1Þ

uouoxþ v ou

oy¼ � 1

qopoxþ m

o2uoy2 ð2Þ

uoToxþ v oT

oy¼ m

Pro2Toy2 ð3Þ

As shown in Fig. 1, the stretching surface moves as a twodimensional sheet at a velocity Uw(x) based on the power lawthrough the fluid environment, subjected to pressure gradient op/ox = �qU1 dU1/dx, where U1 is the velocity at the edge of bound-

ights reserved.

Nomenclature

Cf coefficient of skin frictionf non-dimensional stream functiong non-dimensional temperature = (T � T1)/(Tw � T1)hx local heat transfer coefficient at wallk thermal conductivity of the fluidm power index of velocity at the edge of momentum

boundary layern power index of temperature difference between wall

and edge of thermal boundary layerNux local Nusselt numberop/ox static pressure gradient = �qU1 dU1/dxPr molecular Prandtl number of the fluid = m/aRex local Reynolds number = U x/mT temperatureu velocity in x-directionv velocity in y-directionU(x) composite reference velocity = Uw(x) + U1(x)Uw(x) velocity of the stretching surfaceU1(x) velocity of the oncoming streamx Cartesian coordinate along the surfacey Cartesian coordinate normal to the surface

Greek symbolsa thermal diffusivity of the fluidb stretching parameter = [2m/(m + 1)]e ratio of the free stream velocity to the union of veloci-

ties of the stretching boundary and free stream =U1=ðU1 þ UwÞ½ �, Eq. (10)

g non-dimensional similarity variable, Eq. (7b)q density of the fluidw stream functionl absolute viscositym kinematic viscosity

Subscriptsw condition at the stretching surface1 condition at the free stream

SuperscriptPrime derivative with respect to g

5018 N. Afzal, M. Maqbool Waris / International Journal of Heat and Mass Transfer 53 (2010) 5017–5021

U∞ = U0∞xm

Uw = U0wxm

T∞

x

y

Tw

Thermal Boundary layer

Monentum Boundary layer

Fig. 1. Schematic representation of a momentum and thermal boundary layers over a stretching surface subjected to an oncoming free stream velocity.

ary layer. The free stream velocity, at the edge of the boundarylayer is also based on the power law and is parallel to the movingstretching surface. The boundary conditions at the stretching sur-face and at the edge of boundary layer are:

y ¼ 0; u ¼ UwðxÞ; v ¼ 0; T ¼ TwðxÞ ð4Þyd!1; u ¼ U1ðxÞ; T ¼ T1 ð5Þ

The stretching sheet and free stream velocities and temperaturedifference are:

UwðxÞ ¼ U0wxm; U1ðxÞ ¼ U01xm; TwðxÞ � T1 ¼ Cxn ð6Þ

The similarity transformation for stream function and tempera-ture, based on composite reference velocity U(x) = Uw(x) + U1(x)introduced by Afzal [4], are

Table 1Comparison of g0(0) for e = 0 and for various values of b and Pr.

b Pr Jacobi [11] Grubka and Bobba [1

0.0 0.7 �0.4938 –0.0 1.0 �0.6276 –0.0 10.0 �2.3745 –1.0 0.7 – –1.0 1.0 – �0.58201.0 10.0 – �2.30801.5 1.0 – –1.5 10.0 – –

w ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mxUðxÞmþ 1

rf ðgÞ; T ¼ T1 þ ðTwðxÞ � T1ÞgðgÞ ð7aÞ

g ¼ y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUðxÞð1þmÞ

2mx

r; UðxÞ ¼ U1ðxÞ þ UwðxÞ ð7bÞ

u ¼ UðxÞf 0ðgÞ; v ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimUðxÞð1þmÞ

2x

rf þ ðb� 1Þgf 0f g ð7cÞ

The momentum boundary layer equations reduce to

f 000 þ ff 00 þ b e2 � f 02� �

¼ 0 ð8Þ

f ð0Þ ¼ 0; f 0ð0Þ ¼ 1� e; f 0ð1Þ ¼ e ð9a;b; cÞ

2] Ali [13] Ali [14] Present results

�0.4916 – �0.4920749�0.6245 �0.6247 �0.6246141�2.3636 �2.3638 �2.3346567�0.45255 – �0.4523918�0.59988 �0.5801 �0.5794469�2.29589 �2.2960 �2.2687035– �0.5633 �0.5622802– �2.2695 �2.2427879

Table 2Comparison of g0(0) for e = 1, b = 0 and Pr = 0.7.

Afzal [8] Present result

�0.4122 �0.4126316

N. Afzal, M. Maqbool Waris / International Journal of Heat and Mass Transfer 53 (2010) 5017–5021 5019

where

e ¼ U1U1 þ Uw

; b ¼ 2mmþ 1

ð10Þ

The thermal boundary layer equations reduce to

g00 þ Pr fg0 � nð2� bÞf 0g� �

¼ 0 ð11Þ

gð0Þ ¼ 1; gð1Þ ¼ 0 ð12a;bÞ

0

0.2

0.4

0.6

0.8

1

ε = 0, 0.2, 0.5, 0.8, 1

Pr = 0.7

Pr = 10

0

0.2

0.4

0.6

0.8

1

Pr = 0.7

Pr = 10

ε = 0, 0.2, 0.5, 0.8, 1

0

0.2

0.4

0.6

0.8

1

0 1 2

0 1 2

0 1 2

Pr = 0.7

Pr = 10

ε = 0, 0.2, 0.5, 0.8, 1

g (η

)g

(η)

g (η

)

Fig. 2. Temperature profile g(g) against g for Prandtl number Pr = 0.7 and 10 for non-dimb = 0, 1 and 2.

Here both the free stream and the stretching wall are moving inthe positive x-direction. For e 6 0.5 the stretching surface is mov-ing faster than the free stream and for e P 0.5 the free stream ismoving faster than the stretching surface.

The solutions of momentum equations (8) and (9a,b,c) for vari-ous values of b and e are described by Afzal [4]. The numerical solu-tions of temperature fields represented by Eqs. (11) and (12) arepresented for Prandtl numbers 0.7, 1 and 10 for 0 6 b 6 2,0 6 e 6 1 and for n = 0.

The relations for coefficient of friction Cf and Nusselt numberNux are given by

Cf ¼1ffiffiffiffiffiffiffiRexp

ffiffiffiffiffiffiffiffiffiffiffiffiffimþ 1

2

rf 00ð0Þ ð13Þ

Nux ¼ �ffiffiffiffiffiffiffiRex

p ffiffiffiffiffiffiffiffiffiffiffiffiffimþ 1

2

rg0ð0Þ ð14Þ

ε = 1, 0.8, 0.5, 0.2, 0

β = 0

β = 1

ε = 1, 0.8, 0.5, 0.2, 0

3 4 5 6

3 4 5 6

3 4 5 6

β = 2

ε = 1, 0.8, 0.5, 0.2, 0

η

η

η

ensional velocity parameter 0 6 e 6 1 and for various values of stretching parameter

0.55

εε1

0.80.9

Pr = 0.7

5020 N. Afzal, M. Maqbool Waris / International Journal of Heat and Mass Transfer 53 (2010) 5017–5021

where Rex = U x/m is the local Reynold’s number. It shows that f00(0)is the measure of the coefficient of friction or velocity gradient atthe surface and g0(0) represents the heat transfer coefficient or sur-face temperature gradient.

0.4

0.45

0.5

0 0.5 1 1.5 2

0.70.6 0.50.4 0.3 0.20.10

β

-g' (

0)

Fig. 4. Temperature gradient at the stretching surface g0(0) for Prandtl numberPr = 0.7 against b for various values of velocity parameter e = 0 (0.1) 1.

3. Results and discussion

The numerical solutions of momentum and thermal boundarylayer equations are obtained using fourth order Runge–Kuttamethod employing shooting method, with a step size ofDg = 0.005 for g1 = 15. Solutions of thermal boundary layer Eqs.(11) and (12) have been obtained and the values of g0(0) have beencalculated numerically for various values of the stretching param-eter 0 6 b 6 2 and molecular Prandtl number Pr = 0.7, 1, and 10 for0 6 e 6 1. For accuracy and validity of present numerical analysis,the present results for certain special cases are compared withavailable data in the literature. Table 1 presents a comparison ofresults with Jacobi [11], Grubka and Bobba [12] and Ali [13,14]for the heat transfer coefficient for e = 0 and for various values ofb and Pr. The present data are found to be quiet accurate. Compar-ison is also made in Table 2 with Afzal [8] for the heat transfer coef-

0.3

0.4

0.5

0.6

0.7

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

Pr = 0.7

β21.510.5

0

0

0.51

2

β

1.5

Pr = 1

0.5

1

1.5

2

2.5

1.5

0

1

2

β

0.5

Pr = 10

-g' (

0)

-g' (

0)

-g' (

0)

ε

0 0.2 0.4 0.6 0.8 1ε

0 0.2 0.4 0.6 0.8 1ε

Fig. 3. Temperature gradient at the stretching surface g0(0) against e for variousvalues of Prandtl number Pr = 0.7, 1 and 10 for stretching parameter b = 0, 0.5, 1, 1.5and 2.

ficient for e = 1, b = 0 and Pr = 0.7 and found to agree within 0.1%accuracy. The good agreement between the previously publisheddata and present results encourages a further study on thermalboundary layer for non-isothermal stretching sheet. Suction/injec-tion can also lead to further scope of the present work. The temper-ature profile g(g) against g is displayed in Fig. 2 for Pr = 0.7 and 10for stretching parameter b = 0, 1, 2 for non-dimensional parametere = 0, 0.2, 0.5, 0.8 and 1. The boundary layer thickness decreaseswith b if e and Pr are held fixed.

The heat transfer coefficient at the wall g0(0) against e for vari-ous values of stretching parameter b is displayed in Fig. 3 forPr = 0.7, 1 and 10. The magnitude of heat transfer coefficient de-creases with stretching rate for same e when stretching surface ismoving faster than the free stream (0 6 e 6 0.5) while the heattransfer rate increases with stretching rate when free stream ismoving faster than stretching surface (0.5 6 e 6 1). The magnitudeof heat transfer rate increases with increase in Prandtl number forall b and e.

For low value of Pr = 0.7, the magnitude of heat transfer coeffi-cient decreases when stretching is not considered as free stream isimposed on the stretching surface, remains almost constant forb = 0.5 and increases for higher values of stretching rates. As Pra-ndtl number increases more curves of heat transfer coefficientsagainst e for different stretching rates tend to slope downwards.For high Prandtl number (Pr = 10) all the curves of heat transfercoefficients are sloping downwards as e increases. From Fig. 4,one very interesting result can be seen that for Pr = 0.7 and bapproximately equal to 0.5 there is no effect of changing the veloc-ity parameter e on the heat transfer coefficient, for b 6 0.5 the heattransfer coefficient decreases as e increases and vice versa forb P 0.5

One very important conclusion can be made from Figs. 3 and 4that the heat transfer coefficient is larger for lower values ofstretching parameter when stretching surface is moving fasterthan the free stream whereas it is larger for greater values ofstretching parameter when free stream is moving faster than thestretching surface. These results are very important from applica-tion point of view. If more heat transfer rate is preferred for thecase in which the stretching surface is moving faster than the freestream the fluid having high Prandtl number should be chosen andthe stretching rate must be lower. Other similar argument can bemade from the present results.

4. Conclusion

In the present work, the heat transfer characteristics of the ther-mal boundary layer driven by the stretching surface boundary sub-jected to pressure gradient are studied numerically. Heat transfer

N. Afzal, M. Maqbool Waris / International Journal of Heat and Mass Transfer 53 (2010) 5017–5021 5021

results are obtained for isothermal sheet obeying power law. Thesimilarity transformation is based on composite reference velocity(union of velocities of the stretching boundary and free stream).The dimensionless temperature profiles and heat transfer coeffi-cient are presented for various values of e (ratio of free streamvelocity and the composite reference velocity), stretching parame-ter and Prandtl number. The temperature profiles show that forlow Prandtl number the boundary layer thickness decreases as eincreases. But for large Prandtl number it increases with e. It hasalso been noticed that for low Prandtl number the magnitude ofheat transfer coefficient decreases as e increases for low values ofstretching parameter and increases for higher values. The magni-tude of heat transfer coefficient increases with Prandtl number.The effect of e on heat transfer coefficient is accountable for samestretching parameter and Prandtl number. The selection of e,stretching parameter and Prandtl number can affect the heat trans-fer characteristics considerably, and therefore decide the proper-ties and quality of final products obtained from various industrialprocesses like drawing, production of cables, etc.

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[5] N. Afzal, I.S. Varshney, The cooling of a low heat resistance stretching sheetmoving through a fluid, Heat Mass Transfer 14 (1980) 289–293.

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