Acre Mechanica 138, 21-30 (1999)

ACTA MECHANICA 9 Springer-Verlag 1999

Theoretical magnetohydrodynamic analysis of mixed convection boundary-layer flow over a wedge with uniform suction or injection N. D. Nanousis, Patras, Greece (Received September 22, 1998; revised October 26, 1998)

Summary. The purpose of this work is to study the effects of an applied magnetic field on the mixed convection boundary-layer flow over a wedge with suction or injection. The fluid is assumed to be viscous, incompressible and electrically conducting, and the magnetic field is applied transversally to the direction of the flow. Such a flow model has great significance not only of its own theoretical interest, but also for applications to aerodynamics and engineering. The governing partial differential equations of this prob!em, subjected to their boundary conditions, are solved numerically by applying an efficient solution scheme for local nonsimilarity boundary-layer analysis. Numerical calculations are carried out for different values of the dimensionless parameters of the problem, and the analysis of the obtained results showed that the flow field is appreciably influenced by the applied magnetic field.

1 ]Introduction A flow situation where both forced and free convection effects are of comparable order is called mixed convection. The study of such a forced and free flow with suction or injection finds application in the boundary-layer control. The simplest physical model of such a flow is the two-dimensional laminar mixed convection flow along a vertical flat plate, and extensive studies have been conducted on this type of problems [1]-[5]. In these problems the dimensionless pressure-gradient parameter is m - - 0 and the bouyancy parameter A = 1. The effect of a transverse magnetic field on the flow past a porous wedge placed symmetrically with respect to the flow direction (ra # 0 and A = 0) and with a transverse magnetic field was studied by H a d y and Hassanien [6] whereas Elghabaty and Rahm a n [7] studied the magnetohydrodynamic viscous boundary-layer flow of a non-Newtonian fluid past a wedge using the method of successive approximations. Watanabe [8] investigated the behavior of the boundary layer over a wedge with suction or injection in forced flow, whereas W a t a n a b e et al. [9] presented an analysis of the mixed convection boundary-layer flow over a wedge (m # 0 and A r 0) with suction or injection. Finally, Kafoussias and Nanousis [10] studied the magnetic flow over a wedge with suction or injection. The aim of this work is to study the effects of an applied magnetic field on the mixed convect;ion boundary-layer flow over a wedge with uniform suction or injection. The fluid is assumed to be incompressible and electrically conducting, and the magnetic field is applied transversely to the direction of the flow. The system of partial differential equations, subjected to their boundary condition, is solved numerically by applying an efficient numerical solution scheme for local nonsimilarity boundary-layer analysis [11] [13].

22

N.D. Nanousis

Numerical calculations were carried out for different values of the various dimensionless parameters governing the problem. The results are shown graphically and are discussed in both a quantitative and a qualitative manner. In this ensuing discussion, emphasis has been placed on the influence of the dimensionless parameters A (buoyancy parameter), m (pressure gradient parameter) and Mp (magnetic parameter).

2 Mathematical analysis We consider a magnetohydrodynamic forced and free mixed convection boundary-layer flow over a wedge with uniform suction or injection. The fluid is assumed to be incompressible, viscous and electrically conducting, and a constant magnetic field Bo is applied transversely to the direction of the flow. The coordinate system is shown in Fig. 1 with the x-axis parallel to the wall and the y-axis normal to it. The magnetic Reynolds number of the flow was considered to be small enough, so that the induced magnetic field can be neglected and the wall was assumed to be electrically nonconducting. The fluid properties are also assumed to be constant in a limited temperature range, Under these assumptions, the basic equations relevant to the problem are:

Ou Ou 02u udU + f2 ~Bo2u , ~ + v ~ = ~ , ~ + -dTx gg(T- T~)cos ~ --~ , Ou

Ou

Ou Ov Ox

oy

u 02T

(1)

orB02 2

= o,

(a)

ao

L

i

TTTTT o,x, Fig. 1. Flow field and coordinate system

Theoretical magnetohydrodynamic analysis

23

whereas the boundary conditions are y=0:

u=0,

v=v0(const),

T=T~, (4)

y-+ oo : u = U(x) , T = Too,

where u, v are the fluid velocity components along and perpendicular to the wall, U is the flow velocity at the outer edge of the boundary layer, z~ the kinematic viscosity, cr the electric conductivity of the fluid, L) the fluid density, B0 the applied magnetic field strength, T the temperature, Pr the Prandtl number and/3 the coefficient of thermal axpansion. In the problem under consideration the potential flow velocity U can be written as:

U(x)=cx ~,

_> 0,

m-2_7

(5)

where c is a constant and 7 is the Hartree pressure-gradient parameter that corresponds to 7 = f2/rr for a total angle {2 of the wedge. So, the following dimensionless transformations will be introduced making use of the stream function ~b(x, y):

( ~ f ( x , 7) -- t

2

) 1/2 .~:u

T-Too e(x, 7)

(6)

T~ - Too '

._ ( m _ ~ l

U , */2

Then, the components of the fluid velocity can be expressed as:

= u~,

(r) l x dU . Of Or] . Of'~ ~,2fq 2 U d x f - C - X N O x - ~ X O z ) "

v'U~ 1/2//1 v---

m+l

x-J

(8)

Substituting Eqs. (7) and (8) in Eqs. (1) and (2), the following differential equations are obtained:

ov3 ~ 0 n 2 + ~ ; 7 2

1

x f . O2f Of


Of O 2 f ,

~'

o-Bo2 Of],

(9)

J cot7 + Pr f

= Pr

m + 1x

-~z cotl

Ox N

oU(x)

cp(T~ - Too) \&lJ

'

(10)

where A is the buoyancy parameter, Gr the Grashof number and Re the Reynolds number. They are expressed as follows: A = G r / R e 2 = c*x 1-2"~ ,

Gr = gfl(Tw - T ~ ) x a / v 2 .,

Re = U x / v .

(11)

24

N.D. Nanousis

The boundary conditions (4) can also be written as:

of

r/=0:

&/

1 1 z dU Of (rn @ I 2g "~1/2 2 f + 2 -J ~z Z + z oz = - v~ - w ~-UJ '

0,

of

r/--+ oo : ~ = 1 ,

0=1, (12)

0=0,

where v0 is the velocity of suction or injection, when v0 < 0 or vo > 0, respectively. On introducing the dimensionless parameters s=-v0C

n§

x~l/s ~-0~7 = _4_zx(a 2~)/2

2

(suction/injection parameter)

orB02

(magnetic parameter)

Mp = Ocz2

(13)

C2

Ec =

(Eckert number)

~(T~ - T~) (~)~/(~-~)

the equations of the problem (9), (10) and the boundary conditions (12) can be written as:

1 rn ~[02f Of Of 02f] 2 Of --l+rn [O~Or1or; O~ Orl2J+l+rnMp~20~

(14)

020

(15)

-

~

0~ 2 ~-Prf

1-rn

of

77=0:

[O00f

= P r ~

~0r/

l(l+rn)f

--=0, 0r~

OfO0]

2Pr MpEc~2(l+~)/(1-rn)(Of~, \Or;/

O~

+ 1 - rn

l+rn

Of

0=1, (16)

of

r/--+ oo : ~ = 1 ,

0=0,

where s is the suction/injection parameter when either s > 0 or s < 0, respectively, and = xx( 1-2"~)/2 is the dimensionless distance along the wedge. The system of Eqs. (14) and (15) can also be written as:

f,, + f f,

2m f,2

1;m

2rn

~

2

l+m

Mp~2f,

1 -rn [~Off ~,

'

Of ,,]

-

O'+PrfO'

(1-m)Pr -

i-+Tn-

[~00 f [ 0~

-~

Ofo,] 0~ J

2

A0eos g ~' 2Pr MpEc~20+~)/(l_r~)f,2

(17) (18)

l+rn

where primes denote partial differentiation with respect to 7.

3 N u m e r i c a l solution

In order to solve the system of Eqs. (17) and (18) subject to the boundary conditions (16), we apply an efficient numerical solution scheme, for local nonsimilarity boundary-layer analysis. It consists in proceeding in the ~-direction, i.e., calculate unknown profiles at ~i+1 when the

Theoretical magnetohydrodynamic analysis

25

same profiles are known at ~i position. The process starts at ~i = 0, where Eqs. (17) and (18) reduce to

f

,,,+

ff

,,

2m fl2 _ l+m

2ra l+ra

2 l+m

2 1+~

@" + Pr f@' = 0,

AO c o s -X? 2 '

(t9) (20)

subject to the boundary conditions r~=0:

f'=0,

~/-+ oc : f ' = l ,

1 ~ ( l + m ) f = s,

0=1

(2i)

0=0.

To proceed from #i to #i+1, the equations are discretized at ~i+89 rb with central differences for first- and second-order r/-derivatives averaged at ~i and ~i+1, and backward differences for first-order ~-derivatives. Once an estimate of unknown f~+l,j is available, Eq. (18) provides an estimate for @~+1 using a tridiagonal scheme. This profile is fed into the m o m e n t u m equation (17) which is solved by iterating on a tridiagonal equation, and a new estimate of f~+l,j is produced. The process is repeated until the velocity profile at ~+1 does not exhibit appreciable variations. Numerical infinity for r/, viz. r/oo, as well as steps A~ and A t / a r e experimentally adjusted until the results obtained are not numerically sensitive beyond the desired accuracy. Derivatives are calculated via a forward differences Newton scheme. In the numerical scheme, a step of length Ar I = 0,02 and value of r/~ = 3 were found to be adequate to satisfy a convergence criterion of 10 5 for the physically important quantities f"(~, 0) and - O ' ( ~ , 0). The above numerical scheme is similar to that described in [11], [12] and [13] and has been proved to be quite simple and efficient.

4 Results and discussion 4.1 Velocity and temperature field The velocity distributions are presented in Fig. 2, as a function of ~7for various values of pressure-gradient parameter m. Referring to a specific distance x along the wedge and keeping Isl = 0.3, an increase in the values of the pressure-gradient parameter m leads to an increase in the values of the parameter k, and this affects the values of the magnetic parameter 3/@, Eckert number Ec and the buoyancy parameter A. So, in both cases of suction (s > 0) or injection (s < 0), the velocity of the fluid decreases as the pressure-gradient parameter ra increases. Figure 3 shows the temperature distributions 0 for various values of pressure-gradient parameter m. In both cases of suction (s > 0) or injection (s < 0) the temperature increases as the values of the pressure-gradient parameter m increase. The effects of the applied magnetic field and buoyancy parameter A on the velocity and temperature field are shown in Fig. 4 and Tables 1 and 2, respectively. It can be seen that at a specific point inside the boundary layer over the wedge, the fluid velocity decreases as the values of Mp increase, and the fluid velocity increases as the values of A increase. To quantify this'statement, in the case of air (Pr = 0,71) and for Ec = 0.001 and m = 0.0909 with ~] = 1.0 and s = 0.3, increasing the value of Mp from 0.20 to 0.80 decreases f ( 0 . 3 , 1) by 1.24%. In this case keeping M p = 0.80, increasing the value of A from 2,0 to 3,0 increases f ( 0 . 3 , 1) by 14.4%.

26

N . D . Nanousis

2 ~o

o.. 1,5

1 s +0.3 +0.3 +0.3 -0.3 -0.3 -0.3

1 2 3 4 5 6

0,5

m 0.0141 0.0909 0.3333 0.0141 0.0909 0.3333

Pr=0.71 Mp 0.05154 0.05000 0.04528 0.05154 0.05000 0.04528

Ec A 0.00062 5.931786 0.00100 5.573626 0.00830 4.578981 0.00062 5.931786 0.00100 5.573626 0.00830 4.578981

i

0,5

|

1

1,5

2

2,5

3

Dimensionless distance rl Fig. 2. Variation of the velocity profiles

1

X

0,9

0,8 Q..

,

~K~

o,7

~%,\ kkk\\ \\\\\\

m

Pr=~71

Ec

+0.3 0.0141

2 3 4

+0.3 0.3333 0.04528 0 . 0 0 8 3 0 4 . 5 7 8 9 8 1 - 0 . 3 o . o 1 4 1 0 . 0 5 1 5 4 0.00062 5 . 9 3 1 7 8 6

§

0.05154

0.00062

A

1

5.931786

0.0909 0.05000 0.00100 5.573626

~~~~~,x ~ :~:~~:~ ~:~ ~:~ ~:~

o) 0,6

0,5 0,4 0,3 0,2 0,1

0

0,5

1

1,5

2

2,5

Dimensionless distance r 1

Fig. 3. Variation of the temperature profiles

Theoretical magnetohydrodynamic analysis

27

1,6

~1,4 ~,1,2 o #

P ~ o . ~ m7o:o9o9,

0,8

c=0.001

0,6

1 2 3 4 5 6

0,4 0,2

0,5

1

1,5

2

s +0.3 +0.3 +0.3 -0.3 -0.3 -0.3

Mp 0.2 0.8 0.8 0.2 0.8 0.8

A 2~ 2.0 3.0 2.0 2.0 3.0

2,5

Dimensionless distance q Fig. 4. Variation of the velocity profiles

Table 1. Values of 0 for Pr = 0.71 m=0.0909 A=2 S=0.3 Ec=0.001

m=0.0909 A=3 S=0.3 Mp=0.2

r1

Mp = 0.2

Mp = 0.8

Ec = 0.001

Ec

0.0 0.5 1.0 1.5 2.0 2.5 3.0

1.000 000 0.670 675 0.389 900 0.192296 0.080 265 0.028 448 0.008 576

1.000 000 0.672 741 0.393 379 0.195903 0.082 957 0.029 966 0.009 280

1.000 000 0.649 028 0.357 205 0.163844 0.063 004 0.020 488 0.005 657

1.000 000 0.649 034 0.357 213 0.163852 0.063 013 0.020 496 0.005 665

0.002

Table 2. Values of 0 for Pr = 0.71 m=0.0909 A=2 S=-0.3 Ec=0.001

rr~=0.0909 A = 3 S=-0.3 Mp=0.2

r]

Mp = 0.2

Mp = 0.8

Ec = 0.001

Ec

0.0 0.5 1.0 1.5 2.0 2.5 3.0

1.000 000 0.710 509 0.436 887 0.227149 0.099 604 0.037004 0.011 682

1.000 000 0.712 479 0.440 421 0.231049 O. 102 689 0.038867 0.012 576

1.000 000 0.686 404 0.398 203 0.191511 0.076 853 0.026013 0.007 467

1.000 000 0.686 409 0.398 212 0.191519 0.076 862 0.026021 0.007 475

0.002

3

28

N.D. Nanousis

On the contrary, the influence of these parameters on the temperature field has the opposite effect. Quantitatively, it can be seen from Table 1 that, when Mp increases from 0,20 to 0,80, there is a 0,89% increase in 0(0.3, 1), and when A increases from 2,0 to 3,0 there is a 8,38% decrease in 0(0.3, 1).

4.2 Skin friction and rate of heat transfer and The skin-friction coefficient and the Nusselt number are defined by Cf - -~U2/2 Nu

-

(22)

zq~, z*(Tw -Too) '

respectively, where x* is the thermal conductivity of the fluid, and the skin friction on the wedge r~ and the rate of heat transfer q~ are given by % = / ~

and y=0

y=0

respectively. Using the expressions (6) and (7), the quantities (22) can be written as =

f"(O) and

N u x (1+.~)/2 = [-0'(0)].

(24)

Figures 5 and 6 show the variations of the dimensionless skin-friction coefficient i f ( 0 ) and rate of heat transfer - 0 ' ( 0 ) , respectively, with the magnetic parameter Mp for different values of the buoyancy parameter A. It is observed that in all the above cases (s > 0 or s < 0) if(0) as well as - 0 ' ( 0 ) decrease as the magnetic parameter Mp increases. Also, the skin-friction coefficient if(0) and the Nusselt number increase as the buoyancy parameter A increases. In the case of suction, when s = 0.3 and m = 0.090 9 and A increases from 2 to 4, the corresponding percentage increases in if(0) and - 0 ' ( 0 ) are 57.5% and 10.49%, respectively.

Pr=0.71, Ec=0.001, m=0.0909 4,5

2

!4 8

.o =

4

3,5

S

i. 2.

A

+0.3 2.0 +0.3 4.0

s

3. 4.

A

-0.3 2.0 -0.3 4,0

3

2,5 2

0,2

0,4

0,6

0,8

MagneticParameter Mp Fig. 5. Variation of the skin-friction coefficient

Theoretical magnetohydrodynamic analysis

29

0,95 .~

0,85

o

0,75

o

Pr=0.71, Ec=0. 001, m=0. 0909

2

T--S

0,65 4

0,55

/

0,45 ---------

/

i. 2. 3. 4.

A

+0.32.0 +0.34.0 -0.32.0 -0.34.0

3

0,35 0

0,2

0,4

0,6 0,8 MagneticParameterMp

1

Fig. 6. Variation of the heat-transfer coefficient

5 Concluding remarks

From the results of the above analysis, it is concluded that the flow field is noticeable influenced by the presence of the applied magnetic field. More precisely: (i) The magnetic field decelerates the fluid motion. (ii) The temperature of the fluid increases in the presence of a magnetic field. (iii) The frictional drag from the limiting surface increases when the strength of the applied magnetic field increases. (iv) The heat transfer from the limiting surface to the fluid increases when the strength of the applied magnetic field increases.

References [1] Merkin, J. H.: The effect of buoyancy forces on the boundary-layer flow over semi-infinite vertical flat plate in a uniform free stream. J. Fluid Mech. 35, 439 450 (1969). [2] Lloyd, J. R., Sparrow, E. M.: Combined forced and free convection flow on vertical surfaces. Int. J. Heat Mass Transf. 13, 434-438 (1970). [3] Wilks, G.: Combined forced and free convection flow on vertical surfaces. Int. J. Heat Mass Transl. 16, 1958-1964 (1973). [4] Tingwei, G., Bachrum, R., Daguent, M.: Influence de la convection naturele le sur la convection forcee an dessus d'une surface plane verticale voumise a un flux de rayonnement. Int. J. Heat Mass Transf. 25, 1061 - 1065 (1982). [5] Rajn, M. S., Lin, X. R., Law, C. K.: A formulation of combined forced and free convection past horizontal and vertical surfaces. Int. J. Heat Mass Transf. 27, 2215-2224 (1984). [6] Hady, F. M., Hassanien, I. A.: Effects of a transverse magnetic field and porosity on the FalknerSkan flows of a non-Newtonian fluid. Astrophys. Space Sci. 112, 381 -391 (1985).

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N.D. Nanousis: Theoretical magnetohydrodynamic analysis

[7] Elghabayy, S. S., Rahman, G. M.: Magnetohydrodynamic boundary-layer flow for a non-Newtonian fluid past a wedge. Astrophys. Space Sci. 141, 9 - 1 9 (1988). [8] Watanabe, T.: Thermal boundary layers over a wedge with uniform suction or injection in forced flow. Acta Mech. 83, 119-126 (1990). [9] Watanabe, T., Funazaki, K., Taniguchi, H.: Theoretical analysis on mixed convection boundarylayer flow over a wedge with uniform suction or injection. Acta Mech. 105, 133 - 141 (1994). [10] Kafoussias, N. G., Nanousis, N. D.: Magnetohydrodynamic laminar boundary-layer flow over a wedge with suction or injection. Can. J. Phys. 75, 733- 745 (1997). [11] Kafoussias, N. G., Daskalakis, J. E.: Free-forced convective boundary-layer flow past a vertical flat plate with temperature dependent viscosity. In: Proceedings of the 4 th Greek National Congress on Mechanics, Xanthi, June 26-29, 1995, vol. II, pp. 997-1008. [12] Daskalakis, J. E.: Free convection effects in the boundary layer along a vertically stretching flat surface. Can. J. Phys. 70, 1252-1260 (1992). [13] Daskalakis, J. E.: Mixed free and forced convection in the incompressible boundary layer along a rotating vertical cylinder with fluid injection. Int. J. Energy Res. 17, 689-695 (1993). Author's address: N. D. Nanousis, Technological Education Institute, Mechanical Engineering Depart-

ment, Koukouli, Patras, Greece