Laminar boundary layer flow over a horizontal permeable flat plate

Applied Mathematics and Computation 161 (2005) 229–240

www.elsevier.com/locate/amc

Laminar boundary layer flow overa horizontal permeable flat plate

Orhan Aydın *, Ahmet Kaya

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

Abstract

An analysis is performed to study the laminar boundary layer flow over a porous flat

plate with injection or suction imposed at the wall. Two different analyses techniques are

used for the solution of boundary layer equations: similarity solution and numerical

solution. The effect of uniform suction/injection on the heat transfer is discussed. The

constant surface temperature thermal boundary condition is used for the horizontal flat

plate. The effect of Prandtl number on heat transfer is also investigated. A scale analysis

is performed to get more insight into the Prandtl effect. Friction coefficients and Nusselt

numbers are calculated for constant fluid injection/suction along the plate. The results

indicate that the suction enhances the heat transfer coefficient while injection causes a

decrease in heat transfer.

� 2003 Elsevier Inc. All rights reserved.

Keywords: Boundary layer; Horizontal; Similarity; Numerical; Suction; Injection; Porous

1. Introduction

The analysis of laminar, two-dimensional flow past heated or cooled bodies

with porous walls is of interest in different engineering branches. Examples

include boundary layer control on airfoils, transpiration cooling of turbine

blades, lubrication of ceramic machine parts, food processing, electronics

cooling, the extraction of geothermal energy, nuclear reactor cooling system,

filtration process, etc.

* Corresponding author.

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

0096-3003/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2003.12.021

mail to: [email protected]

230 O. Aydın, A. Kaya / Appl. Math. Comput. 161 (2005) 229–240

Watanabe [1–3] carried out theoretical studies on the investigation of forced,

free or mixed convection flows past heated and cooled bodies such as plate,edge and cone with suction or injection. Chandran et al. [4] studied the effect of

magnetic field on the flow and heat transfer past a continuously moving porous

plate in a stationary fluid. Murthy and Singh [5] studied the effect of surface

mass flux on the non-Darcy natural convection over a horizontal flat plate in a

saturated porous medium. Hsu et al. [6] studied the combined effects of the

shape factor, suction/injection rates and viscoelasticity on the flow and tem-

perature fields of the flow past a wedge. Chamkha [7] obtained similarity

solutions for the laminar boundary-layer equations describing steady, hydro-magnetic two-dimensional flow and heat transfer in an electrically conducting

and heat-generating fluid driven by a continuously moving porous surface

immersed in a fluid-saturated porous medium. Rao et al. [8] analyzed the

momentum and heat transfer in the laminar boundary layer of a non-New-

tonian power-law fluid flowing over a flat plate, which is moving in the

direction opposite to uniform main stream, and with arbitrary fluid injection/

suction along the plate surface. Magyari and Keller [9] obtained exact solutions

for the two-dimensional self-similar boundary-layer flows induced by perme-able stretching surfaces. In a recent study, Magyari et al. [10] studied mixed

convection boundary layer flow past a horizontal permeable flat plate. El-

bashbeshy [11] studied the heat transfer characteristics of laminar mixed con-

vective boundary layer over a semi-infinite horizontal flat plate embedded in

porous medium.

Kelson and Farrell [12] studied self-similar boundary layer flow a micro-

polar fluid driven by a porous stretching sheet and obtained analytical results

for the shear stress and the microrotation at the surface for the limiting cases oflarge suction or injection.

The purpose of the present study is to analyze the effect of the surface mass

flux (i.e. the injection/suction parameter) on the momentum and heat transfer

about a horizontal porous plate with injection or suction at the wall.

2. Analysis

2.1. Mathematical formulation

Consider steady, incompressible, laminar two-dimensional, boundary layer

flow over a permeable plate. Far above the plate, the velocity and the tem-

perature of the uniform main stream are U1 and T1, respectively. The x-coordinate is measured from the leading edge of the plate and y-coordinate ismeasured normal to the plate. The corresponding velocity components in the xand y directions are u and v, respectively and surface mass flux, vw aligned

normal to a uniform free stream velocity, U1. The surface mass flux is assumed

U∞, T∞y, v

x, uvw

Fig. 1. The schematic of the problem.

O. Aydın, A. Kaya / Appl. Math. Comput. 161 (2005) 229–240 231

to be constant with vw > 0 for injection and vw < 0 for suction. The entire

surface of the plate is maintained at a uniform temperature of Tw. This situa-tion is shown in Fig. 1. All the thermophysical properties are assumed to be

constant. It is also assumed that the magnitude of the injection velocity is not

large enough to significantly alter the inviscid flow field outside the boundary

layer.

Assuming that the Reynolds number is large enough for the boundary layer

assumptions to be applicable, the governing boundary layer equations of theproblem which are based on the balance laws of mass, momentum and energy

are:

ouox

þ ovoy

¼ 0; ð1Þ

uouox

þ vouoy

¼ to2uoy2

; ð2Þ

uoTox

þ voToy

¼ tPr

� � o2Toy2

: ð3Þ

The appropriate boundary conditions for the velocity and temperature of this

problem are:

y ¼ 0; u ¼ 0; v ¼ �vw; T ¼ Tw ¼ constant;

y large u ! U1; T ! T1:ð4Þ

Here, U1 and T1 are the free stream velocity and temperature, respectively.

Vw represents the suction/injection velocity.

2.2. Similarity solution

In the similarity solution, the governing partial differential equations arereduced to a set of ordinary differential equations by means of a suitable

transformation. A useful similarity transformation is possible for this problem.

The following similarity variables are used [13]:

uU1

¼ f 0ðgÞ; g ¼ yx

ffiffiffiffiffiffiffiRex

p; h ¼ Tw T

Tw T1: ð5Þ


Substitution of the above similarity variables into Eqs. (1)–(4) yields:

2f 000 þ ff 00 ¼ 0; ð6Þ

h00 þ Pr2

h0f ¼ 0; ð7Þ

f ð0Þ ¼ fw; f 0ð0Þ ¼ 1; hð0Þ ¼ 1; f 0ð1Þ ¼ 0; hð1Þ ¼ 0; ð8Þ

where fw ¼ 2vwU1


p.

Consequently, the velocity and temperature distributions are obtained by

solving Eqs. (6) and (7) via the fourth-order Runge–Kutta integration and theshooting method.

Of special significance for this type of flow and heat transfer situation are the

skin-friction coefficient and the Nusselt number. These can be defined as fol-

lows:

cf ¼ sw12qU 2

1¼ 2ffiffiffiffiffiffiffi

Rexp f 00jg¼0 and Nux ¼


ph0jg¼0: ð9Þ

2.3. Numerical solution

Similarity solutions for the boundary layer equations have been introduced

in the previous section. Another way of solving the boundary layer equations

involves approximating the governing partial differential equations by alge-

braic finite-difference equations [14].

In most cases, it is convenient to write the equations in dimensionless formbefore deriving the finite difference approximations to them. The following

dimensionless variables are used:

U ¼ uU1

; V ¼ vffiffiffiffiffiffiffiffiReL

p

U1; X ¼ x

L; Y ¼ y

ffiffiffiffiffiffiffiffiReL

p

L; h ¼ Tw T

Tw T1; ð10Þ

where ReL is the Reynolds number based on L. In terms of these variables Eqs.(1)–(3) become

oUoX

þ oVoY

¼ 0; ð11Þ

UoUoX

þ VoUoY

¼ o2UoY 2

; ð12Þ

UohoX

þ VohoY

¼ 1

Pr

� �o2hoY 2

ð13Þ


and the boundary conditions given in Eq. (4) become:

Y ¼ 0 : U ¼ 0; V ¼ fw2

ffiffiffiffiX

p ; h ¼ 1;

Y large : U ! 1; h ! 0:

ð14Þ

Again, the skin-friction coefficient and the Nusselt number can be predicted

as follows:

cf ¼ 2ffiffiffiffiX

pffiffiffiffiffiffiffiRex

p oUoY

��Y¼0

and Nux ¼ ffiffiffiffiffiffiffiRex

p ffiffiffiffiX

p ohoY

��Y¼0

: ð15Þ

2.4. Scale analyses

A scale analysis is performed in order to predict the flow and heat transferphenomena based on the driving mechanisms of the momentum and heat

transfer for high-Pr ðPr � 1Þ fluids and low-Pr fluids ðPr 1Þ, respectively [15–17].

2.4.1. Pr � 1 case

For the high-Pr fluids such as oils, the rough geometry of the hydrodynamicand thermal boundary layers that occur is given in Fig. 2. The momentum

equation (Eq. (2)) suggest a balance between inertia and viscous forces. In

terms of scales, this balance can be written as:

FI � FV ;

U1U1

L� t

U1

d2: ð16Þ

Also, the energy equation implies a balance between enthalpy flow and con-

duction:

QH � QK ;

U∞

δU

δθ

∆T

Fig. 2. Pr � 1 case.


UhDTL

� aDT

d2h: ð17Þ

Dividing Eq. (16) by Eq. (17) leads:

U1

Uh� Pr

dh

d

� �2

: ð18Þ

Using the similarity in Fig. 2, one obtains:

U1

Uh� d

dh: ð19Þ

Combining Eqs. (18) and (19), we will obtain

ddh

� Pr1=3: ð20Þ

The Nusselt number is defined as:

Nu ¼ hLk: ð21Þ

From the energy balance at the wall, one obtains:

kDTdh

� hDT : ð22Þ

Recalling Eq. (20) and carrying h in Eq. (22) into Eq. (21), finally, we reach[16]:

Nu � Re1=2Pr1=3: ð23Þ

2.4.2. Pr 1 case

For the low-Pr fluids such as liquid metals, the rough geometry of the

hydrodynamic and thermal boundary layers that occur is given in Fig. 3. If the

δθ

δ

U∞ ∆T

Fig. 3. Pr 1 case.


geometry is carefully examined, similar to the case for high-Pr fluid, force andenergy balance lead, respectively:

U1

L� t

d2;

U1DTL

� aDT

d2h: ð24Þ

Combining these above two relations gives the following:

ddh

� Pr1=2: ð25Þ

Using this relation, the Nusselt number can be obtained as:

Nu � Re1=2Pr1=2: ð26Þ

For the impermeable wall case, it is just a matter of several computations to

predict the coefficients for the relations given in Eqs. (23) and (26). But for

the permeable wall case, the effect of the wall mass flux parameter, fw on

these relations will be questioned for both the injection and suction cases in

Section 3.

3. Results and discussion

In order to study the effect of suction or injection, the following values of the

suction/injection parameter, fw are used: )0.5, )0.2, )0.1, 0, 0.1, 0.2 and 0.5.Fig. 4 depicts representative profiles of either f 0 or h for different values of fw,

f’θ

η0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14

fw Pr = 1.0

-0.5-0.2-0.100.10.20.5

Fig. 4. Effect of fw on velocity (or temperature, as Pr ¼ 1) profile.

1.0

0.0

0.2

0.4

0.6

0.8

0 2 4 6 8 10 12 14

0.01

0.1

0.7

1.0

fw =-0.2

θ

η

η

η

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14

0.01

0.1

0.7

1.010

Pr=50

fw = 0.0

θ

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14

0.01

0.10.7

10

Pr=50

1.0

fw =0.2

θ

Pr=50

10

Pr=50

(a)

(b)

(c)

Fig. 5. Effect of Pr on temperature profiles for different wall conditions: (a) injection, (b) imper-meable and (c) suction.



respectively, since the distributions of f 0 and h are identical for Pr ¼ 1. It

should be noted that fw > 0 corresponds to suction, fw < 0 corresponds toinjection and fw ¼ 0 represents the flow over an impermeable surface. As might

be expected, injecting fluid into the boundary layer broadens the velocity dis-

tribution and increases the hydrodynamic boundary layer thicknesses as shown

in Fig. 4. Also, the wall shear stress would be increased with the application of

suction whereas injection tends to decrease wall shear stress. This can be ex-

plained by the fact that the wall velocity gradient is increased with the

increasing value of fw. Fig. 4 also aims to explore the effects of the injectionparameter on the temperature profile. The effect of injection is found tobroaden the temperature distribution, decrease the wall temperature gradient,

and hence reduce the heat transfer rate. On the other hand, the thermal

boundary layer becomes thinner and the wall temperature gradient becomes

larger when suction is applied.

The influence of the Prandtl number, Pr on the heat transfer has been al-ready explained through the scale analysis. Since the flow problem is uncoupled

from the thermal problem, changes in the values of Pr will not affect the fluidvelocity. For this reason, velocity profiles for this case are not shown.Increasing the Prandtl number tends to reduce the thermal boundary layer

thickness along the plate. This yields a reduction in the fluid temperature. As

the Prandtl number is increased, the suction or injection at the porous wall

appreciably affects the temperature variation. The boundary layer becomes

thicker for the injection case (Fig. 5a) when compared to that of the imper-

meable wall (Fig. 5b), while it becomes thinner for the suction case (Fig. 5c).

From the engineering point of view, the skin-friction coefficient and the

Nusselt number are the two most important parameters, which lead the pres-sure loss and the heat transfer rate, respectively. Fig. 6 shows the variation of

fw

Cf.R

e x1/

2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

-0.50 -0.25 0.00 0.25 0.50

similaritynumerical

Fig. 6. Variation of the skin-friction coefficient with fw.


the friction coefficient with the injection/suction parameter, fw. As seen, thesimilarity results and numerical results are found to be in an agreement, whichgives a credit to the validity of each solution technique. As inferred from the

results of velocity profiles given in Fig. 4, decreased hydrodynamic boundary

layer thicknesses by the increasing fw resulted in an increase at cf . That meansthat the application of the suction at the wall enhances the momentum transfer,

while the injection of the fluid decreasing it.

In the scale analysis, the physics of the problem leads to Nu � f ðRe; PrÞrelations in different forms for high- and low-Pr fluids. For Pr � 1, we obtain

Nu � Re1=2Pr1=3, while Nu � Re1=2Pr1=2 is obtained for Pr 1. In view of thesescale analysis results, the effect of the wall mass flux is examined for both

high- and low-Pr regions. Fig. 7a shows the effect of fw on Nu=Re1=2Pr1=3 for

fw

Nu/

Re x

1/2 P

r1/2

0.0

0.2

0.4

0.6

0.8

-0.50 -0.25 0.00

(a)

0.25 0.50

Pr=0.01

0.1

(b)

fw

Nu/

Re x

1/2 P

r1/3

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-0.50 -0.25 0.00 0.25 0.50

Pr=50

107

1

Fig. 7. Effect of fw on the Nusselt number for (a) low-Pr and (b) high-Pr fluids.


the high-Pr fluids. As shown, increasing fw resulted an increase in the heat

transfer. This can be explained by the increasing temperature gradient ordecreasing thermal boundary layer thickness with the increasing fw (Fig. 4).But, as seen from the figure, although we have included the effect of Prandtl

number in the heat transfer term, Nu=Re1=2Pr1=3, the effect of fw is influencedby the Prandtl number. As seen, for the higher values of the Prandtl number,

the effect of fw becomes much more important for the suction region ðfw > 0Þcomparing that for the injection region ðfw < 0Þ. However, for lower values ofPrandtl number, the effect of fw is nearly the same for both the injection

(fw < 0) and suction ðfw > 0Þ cases, fw � Nu=Re1=2Pr1=2 variation suggesting alinear variation (Fig. 7b). As already disclosed, increasing fw enhances the

heat transfer. This behavior can be explained from the fact a fluid with a

larger Prandtl number possesses a larger heat capacity that and enhance the

heat transfer.

4. Conclusions

The problem of steady, two-dimensional laminar boundary layer flow about

a porous plate is analyzed using the similarity solution and the numerical

solution. Fluid suction or blowing at the surface is considered, which has been

found to have a considerable influence on the heat transfer mechanism. Suction

of fluid at the wall increases both the skin-friction coefficient and the Nusselt

number; whereas, injection causes a decrease in both. The effect of the Prandtl

number on flow and heat transfer is also examined for low- and high-Pr fluids.The effect of the surface mass flux, fw on heat transfer in the low-Pr region isfound to be different to that in the high-Pr region.

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Documents

Laminar boundary layer flow over a horizontal permeable flat plate