A STUDY ON THE MIXED CONVECTION BOUNDARY LAYER FLOW AND HEAT TRANSFER OVER A VERTICAL SLENDER CYLINDER by

Rahmat Ellahi1 ;2;3 , Arshad Riaz3 , Saeid Abbasbandy4 , Tasawar Hayat5;6 and Kambiz Vafai2 , 2 Department of Mechanical Engineering Bourns Hall Riverside, CA USA 3 Department of Mathematics & Statistics, FBAS, IIU, Islamabad, Pakistan 4 Department of Mathematics, SRB, Islamic Azad University, Tehran, Iran 5 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan 6 Department of Mathematics, College of Science, KSU, Riyadh, Saudi Arabia

In this investigation, the series solutions of mixed convection boundary layer ‡ow over a vertical permeable cylinder are constructed. Two types of series as well numerical solutions are presented by choosing exponential and rational bases. The resulting differential system are solved by employing homotopy analysis method (HAM) and Pade technique which have been proven to be successful in tackling nonlinear problems. We o¤er various veri…cations of the solutions by comparing to existing, documented results and also mathematically, through reduction of sundry parameters. The convergence of the series solutions have been discussed explicitly. Comparison with existing results reveal that the series solutions are not only valid for large (aiding ‡ow) but also for small values (opposing ‡ow) of and the dual solutions do not obtain in both cases. Key words: Mixed convection; Series solutions; Homotopy analysis method; Pade technique; Porous cylinder; Steady ‡ow

Introduction The mixed convection ‡ows under boundary layer analysis are of fundamental theoretical and practical interest. Heat exchangers, electric transformers, refrigeration coils, solar collectors, study of movement of natural gas, oil, water through oil reservoirs and nuclear reactors are few examples in this direction. The ‡ows of non-Newtonian ‡uids through a porous medium [1 6] are also quite prevalent in nature. Examples of these applications are …ltration processes, biomechanics, packed bed, ceramic geothermal engineering, insulation systems, ceramic processing, chromatography, and geophysical phenomena. Over the last 1 Author’s

e-mail: [email protected]

1

several decades a large number of research work related to the mixed convection ‡ow of a viscous and incompressible ‡uid over a vertical ‡at plate was conducted by a number of investigators. For instance, a critical review of mixed convection ‡ows has been presented by Pop and Ingham [7]: Mahmood and Merkin [8] have presented dual similarity solutions for the axisymmetric mixed convection boundary layer ‡ow along a vertical cylinder in the case of opposing directions only. Later on Ridha [9] has reported that when the buoyancy force acts in the direction of ‡ow then the dual solutions for aiding ‡ow also exist. Ishak et al [10] have also reported a numerical simulation by Keller-box method for boundary layer ‡ow and heat transfer over a vertical slender cylinder. In this study, the authors concluded that for aiding/assisting ‡ow dual solutions exist but for opposing ‡ow there are either dual solutions, unique solutions or no solution exist. The purpose of the present investigation is to revisit the problem formulated in references [7 10] for the analytic solutions. The main stream velocity and wall surface temperature depend upon the axial distance along the cylinder surface. The homotopy analysis method [11] has been applied to obtain series solutions of velocity and temperature …elds by making choices of two base functions. Convergence of the obtained series solutions is taken care properly. Furthermore, the skin friction and Nusselt number are analyzed. Comparison with the previous relevant studies is made. To the best of our knowledge, the series solutions for this particular model have not been presented in the past. It is worth mentioning that in both cases, the dual solutions do not exist, therefore, the comparison with existing results reveal that our series solutions are valid for > 0 (heated cylinder) and for < 0 (cooled cylinder) as well. It is shown that like several existing studies [12 26]; the HAM in the present paper is accepted as an elegant tool for e¤ective solutions for a number of complicated ‡uid problems.

Problem formulation Here, we analyze the convection ‡ow and heat transfer characteristic along a vertical permeable slender cylinder with radius a . We choose the cylindrical coordinates (x; r) such that x and r-axes are along the cylinder surface (vertically) and radial directions, respectively. Symmetric nature of the ‡ow is assumed with respect to the transverse coordinate. Furthermore, cylinder is kept in an incompressible viscous ‡uid of uniform ambient temperature T1 and constant density 1 . Using continuity, momentum and energy equations, one obtains the following boundary layer equations [27; 28] @ @ (ru) + (rw) = 0; @x @r u

@u dU @u +w =U + @x @r dx u

@ 2 u 1 @u + @r2 r @r

(1) + g (T

@2T 1 @T + @r2 r @r

@T @T +w = @x @r 2

:

T1 );

(2) (3)

In above equations, u and w indicate the velocity components, U (x) the mainstream velocity, T the ‡uid temperature, g the gravitational acceleration, the thermal di¤usivity, the thermal expansion coe¢ cient and the kinematic viscosity. The subjected boundary conditions are [10] u = 0; w = V; T = Tw (x); at r = a;

(4)

u ! U (x); T ! T1 ; as r ! 1;

(5)

where V (> 0) and V (< 0) correspond to the injection and suction velocities, respectively. Further, U (x) and temperature of the cylinder surface Tw (x) are x ; Tw (x) = T1 + `

U (x) = U1

T

x ; `

(6)

in which ` denotes characteristic length and T is the characteristic temperature. Note that T > 0 corresponds to a heated surface and T < 0 for a cooled surface. De…ning =

r2

a2

U x

2a

0:5

= (U x)0:5 af ( );

;

a r

u = U f 0 ( ); w =

( )=

T Tw

T1 ; T1

(7)

0:5

U1 `

f ( );

(8)

where the stream function can be de…ned by u = r 1 @ [email protected] and w = 1 r @ [email protected] Moreover prime denotes di¤erentiation with respect to . Invoking equations (7) and (8), Eq. (1) is identically satis…ed, and equations (2)and (3) yield (1 + 2

)f 000 ( ) + 2 f 00 ( ) + f ( )f 00 ( ) + 1

(1 + 2

)

00

( )+2

and the curvature parameter =

` U1 a2

0

f 02 ( ) +

( ) + P r f ( ) 0( )

( ) = 0;

f 0 ( ) ( ) = 0;

(9) (10)

and V are 0:5

; V =

a r

U1 `

0:5

f0 ;

where f0 = f (0). It is noted that f0 < 0 is for mass injection and f0 > 0 is for mass suction. The buoyancy or mixed convection parameter is =

g ` T : 2 U1

Here > 0 and < 0 are for aiding ‡ow (heated cylinder) and for opposing ‡ow (cooled cylinder), respectively. For = 0, one has pure forced convection ‡ow without buoyancy force. 3

The boundary conditions (4) and (5) reduce in the following form f (0) = f0 ; f 0 (0) = 0; f 0 (1) = 1; (0) = 1; (1) = 0:

(11)

Now the skin friction coe¢ cient (Cf ) and the local Nusselt number (N ux ) are Cf =

w ; U 2 =2

N ux =

xqw k(Tw

T1 )

where = 1 [1 (T T1 )] and the skin friction from the plate qw can be written as follows w

=

@u @r

; qw =

w

@T @r

k

r=a

;

(12)

and the heat transfer

; r=a

in which and k are the dynamic viscosity and thermal conductivity, respectively. Making use by Eq. (7) one obtains 1 Cf (Rex )0:5 = f 00 (0); N ux =(Rex )0:5 = 2

0

(0):

(13)

In above expression Rex = U x= denotes the local Reynold number [10].

Solution of the problem Here we o¤er two types of series solutions by choosing exponential and rational bases. Exponential bases According to equations (9) and (10) and the boundary conditions (11), we write the solution in the following form f ( ) = a0 + +

1 +1 X X

q

aq;m

m

e

;

(14)

m=1 q=0

( )=

+1 X 1 X

bq;m

q

e

m

;

(15)

m=1 q=0

where a0 , aq;m and bq;m are coe¢ cients to be determined and is a spatial-scale parameter. By rule of solution expression denoted by equations (14) and (15) and the boundary conditions (11), it is natural to choose 1

f0 ( ) = f0 + 0(

)=e

4

exp(

;

)

;

(16) (17)

as the initial approximation to f ( ) and ( ), respectively. We use the method of higher order di¤erential mapping, [29] to choose the auxiliary linear operators L1 and L2 by @2 @3 + ( ; ; p); (18) L1 [ ( ; ; p)] = @ 3 @ 2 @ @2 + @ 2 @

L2 [ ( ; ; p)] =

( ; ; p);

(19)

which satisfy the properties L1 [C1 + C2 + C3 e

] = 0; L2 [C4 + C5 e

] = 0;

(20)

where Ci , i = 1; 2; : : : ; 5 are constants. This choice of L1 and L2 is motivated by equations (14) and (15), respectively, and from boundary conditions (11), we have C2 = C4 = 0. From (9) and (10) we de…ne nonlinear operators as follows N1 [ ( ; ; p); ( ; ; p)] = (1+2

)

N2 [ ( ; ; p); ( ; ; p)] = (1 + 2

@2 @2 @3 +2 + +1 @ 3 @ 2 @ 2 )

@2 @ +2 @ 2 @

+ Pr

@ @ @ @

2

+ @ @

; (21)

; (22)

and then construct the homotopy H1 [ ( ; ; p); ( ; ; p)] = (1

p)L1 [

H2 [ ( ; ; p); ( ; ; p)] = (1

p)L2 [

f0 ( )]

0(

)]

~1 pH1 ( )N1 [ ; ];

(23)

~2 pH2 ( )N2 [ ; ];

(24)

where ~1 6= 0 and ~2 6= 0 are the convergence-control parameters [30]; H1 ( ) and H2 ( ) are auxiliary functions. Setting Hi [ ( ; ; p); ( ; ; p)] = 0, for i = 1; 2, we have the following zero-order deformation problems (1

p)L1 [

(1

p)L2 [

f0 ( )] = ~1 pH1 ( )N1 [ ; ]; 0(

(25)

)] = ~2 pH2 ( )N2 [ ; ];

(26)

subject to conditions (0; ; p) = f0 ;

@ ( ; ; p) @

= 0; =0

(0; ; p) = 1;

@ ( ; ; p) @

= 1; =1

(1; ; p) = 0;

in which p 2 [0; 1] is an embedding parameter. When the parameter p increases from 0 to 1, the solution ( ; ; p) varies from f0 ( ) to f ( ) and the solution ( ; ; p) varies from 0 ( ) to ( ). If these continuous variations are smooth 5

enough, the Maclaurin’s series with respect to p can be constructed for ( ; ; p) and ( ; ; p), respectively, and further, if these series are convergent at p = 1, we have +1 +1 X X f ( ) = f0 ( ) + fm ( ) = (27) m ( ; ; ~1 ); m=1

( )=

0(

)+

m=0

+1 X

m(

)=

m=1

where fm ( ) =

1 @ m ( ; ; p) m! @pm

+1 X

m(

; ; ~2 );

(28)

m=0

;

m(

)=

p=0

1 @ m ( ; ; p) m! @pm

: p=0

To calculate fm ( ), we now di¤erentiate Eqs. (25) and (26) and related conditions m times with respect to p, then set p = 0, and …nally divide by m!. The resulting mth-order deformation problems are L1 [fm ( )

m fm 1 (

)] = ~1 H1 ( )R1;m ( );

(m = 1; 2; 3; : : :);

(29)

L2 [

m m 1(

)] = ~2 H2 ( )R2;m ( );

(m = 1; 2; 3; : : :);

(30)

m(

)

0 fm (0)

fm (0) = 0;

= 0;

0 fm (1)

= 0;

m (0)

= 0;

m (1)

= 0;

(31)

where R1;m ( ) and R2;m ( ) are given by R1;m ( ) = (1+2

000 00 )fm 1 +2 fm

m X1

1+

00 fi fm

i 1

0 fi0 fm

i 1

+

m 1 +(1

m );

i=0

R2;m ( ) = (1 + 2

)

00 m 1

+2

0 m 1

+ Pr

m X1

0 m i 1

fi

0 i fm i 1

;

i=0

where prime denotes di¤erentiation with respect to m

=

0; 1;

and

m 1; m > 1:

The general solution of Eqs. (29) and (30) are fm ( ) = f^m ( ) + C1 + C2 + C3 e m(

) = ^m ( ) + C4 + C5 e

; ;

(32) (33)

where Ci for i = 1; : : : ; 5 are constants, f^m ( ) and ^m ( ) are particular solutions of Eqs. (29) and (30), respectively. For simplicity, here we take H1 ( ) = H2 ( ) = 1. According to the rule of solution expression denoted by equations (14) and (15), C2 = C4 = 0. The other unknowns are governed by 0 f^m (0) + C1 + C3 = 0; f^m (0)

6

C3 = 0; ^m (0) + C5 = 0;

and according to our algorithm, the other boundary conditions are ful…lled. In this way, we derive fm ( ) and m ( ) for m = 1; 2; 3; : : :, successively for every . At the N th-order approximation, we have the analytic solution of Eqs. (9) and (10), namely f( )

FN ( ) =

N X

fi ( );

( )

=

N(

)=

i=0

N X

i(

):

(34)

i=0

For simplicity, here we take ~1 = ~2 = ~. The auxiliary parameter ~ is useful to adjust the convergence region of the series (34) in the homotopy analysis solution. By plotting ~-curve, it is straightforward to choose an appropriate range for ~ which ensures the convergence of the solution series. As pointed out by Liao [11], the appropriate region for ~ is a horizontal line segment.

Rational bases Invoking equations (9) and (10) and the boundary conditions (11), one can write +1 m X X2 f ( ) = d0 + + dq;m q (1 + ) m ; (35) m=1 q=0

( )=

+1 m X X2

eq;m

q

(1 + )

m

;

(36)

m=1 q=0

where d0 , dq;m and eq;m are coe¢ cients to be determined. According to the rule of solution expression denoted by equations (35) and (36) and the boundary conditions (11), the initial approximations of f ( ) and ( ) are selected as follows 1 ; (37) f0 ( ) = f0 1 + + 1+ 0(

)=

1 : 1+

(38)

The auxiliary linear operators L1 and L2 are L1 [ ( ; p)] =

@3 3 @2 + 3 @ 1+ @ 2

( ; p);

(39)

L2 [ ( ; p)] =

@2 2 @ + 2 @ 1+ @

( ; p);

(40)

L1 [D1 + D2 + D3 (1 + )

1

] = 0; L2 [D4 + D5 (1 + )

1

] = 0;

(41)

where Di , i = 1; 2; : : : ; 5 are constants. This choice of L1 and L2 is motivated by equations (35) and (36), respectively, and from boundary conditions (11), we have D2 = D4 = 0. 7

In this case, the nonlinear operator Ni [ ; ], the homotopy Hi [ ; ], Ri;m ( ) for i = 1; 2, and the zero-order deformation equations, the mth-order deformation equations are designed as in the previous case without the parameter . The general solution of mth-order deformation equations here are fm ( ) = f^m ( ) + D1 + D2 + D3 (1 + ) m(

) = ^m ( ) + D4 + D5 (1 + )

1

1

;

;

(42) (43)

where Di for i = 1; : : : ; 5 are constants, f^m ( ) and ^m ( ) are particular solutions of mth-order deformation equations. By rule of solution expressions denoted by (35) and (36) and from mth-order deformation equations, the auxiliary functions H1 ( ) and H2 ( ) are chosen in the form H1 ( ) = (1 + ) 1 ; H2 ( ) = (1 + ) 2 : It is found that when 1 < 4 and 2 < 3 the term log(1 + ) appears in the solution expression of fm ( ) and m ( ), which disobeys the rule of solution expression denoted by (35) and (36), respectively. In addition, when 1 > 4 and 2 > 3 we omit some terms in solution expression. This uniquely determines the corresponding auxiliary functions H1 ( ) = (1 + )

4

; H2 ( ) = (1 + )

3

:

According to (35) and (36), D2 = D4 = 0. The other unknowns are governed by 0 f^m (0) + D1 + D3 = 0; f^m (0)

D3 = 0; ^m (0) + D5 = 0;

and according to our algorithm, the other boundary conditions are satis…ed. In this way, we derive fm ( ) and m ( ) for m = 1; 2; 3; : : :, successively. Like previous case, we will take here ~1 = ~2 = ~.

Numerical results and discussion We use the widely applied symbolic computation software MATHEMATICA to solve equations (29) and (30). By means of the so-called ~-curve, it is straightforward to choose an appropriate range for ~ which ensures the convergence of the solution series. As pointed out by Liao [11], the appropriate region for ~ is a horizontal line segment. We can investigate the in‡uence of ~ on the 0 convergence of f 00 (0) and (0), by plotting the curve of it versus ~, as shown in Figs. 1 4 (for > 0) and Figs. 5 8 (for < 0) are the examples of two cases. By considering the ~ curve we can obtain the reasonable interval for ~ in each case. Our computations show that in …rst case it is better to we choose 2. Also, by computing the error of norm 2 for two successive approximation of FN ( ) or N ( ) of (34), we can obtain the best value for ~ in each case. Figs. 9 10 show the error of F20 ( ) for 2 [0; 5] in …rst case and we obtain that the 8

best value of ~ is 0:209 with error 7:01 10 6 (for > 0) and ~ is 0:218 with error 6:02 10 5 (for < 0) respectively. With the same contrast Figs. 11 and 12 show the error of 20 ( ) in second case and we obtain that the best value of ~ are 1:327 and 1:95 with error 3:47 10 7 and 3:17 10 8 for < 0 and > 0 respectively.

Fig. 1: The ~-curve of the f 00 (0) versus ~ for the 20th-order approximation and P r = 0:7, = 2, = 0, = 1, f0 = 0 by exponential bases.

Fig. 2: The ~-curve of the and P r = 0:7, = 2, = 0,

0

(0) versus ~ for the 20th-order approximation = 1, f0 = 0 by exponential bases.

9

Fig. 3: The ~-curve of the f 00 (0) versus ~ for the 20th-order approximation and P r = 0:7, = 1, = 1, f0 = 1 by rational bases.

0 Fig. 4: The ~-curve of the (0) versus ~ for the 20th-order approximation and P r = 0:7, = 1, = 1, f0 = 1 by rational bases.

10

Fig. 5: The ~-curve of the f 00 (0) versus ~ for the 20th-order approximation and P r = 0:7, = 2, = 0, = 1, f0 = 0 by exponential bases.

Fig. 6: The ~-curve of the and P r = 0:7, = 2, = 0,

0

=

(0) versus ~ for the 20th-order approximation 1, f0 = 0 by exponential bases.

11

Fig. 7: The ~-curve of the f 00 (0) versus ~ for the 20th-order approximation and P r = 0:7, = 1, = 1, f0 = 1 by rational bases.

0 Fig. 8: The ~-curve of the (0) versus ~ for the 20th-order approximation and P r = 0:7, = 1, = 1, f0 = 1 by rational bases.

12

Fig. 9: The norm 2 error of F20 ( ) versus ~ with P r = 0:7, f0 = 0 by exponential bases.

= 2,

Fig. 10: The norm 2 error of F20 ( ) versus ~ with P r = 0:7, = 1, f0 = 0 by exponential bases.

13

= 0,

= 2,

= 1,

= 0,

Fig. 11: The norm 2 error of f0 = 1 by rational bases.

20 (

) versus ~ with P r = 0:7,

= 1,

= 1,

Fig. 12: The norm 2 error of 20 ( ) versus ~ with P r = 0:7, = 1, = 1, f0 = 1 by rational bases. The so-called homotopy-Padé technique (see [11]) is employed, which greatly accelerates the convergence. The [r; s] homotopy-Padé approximations of f 00 (0) 0 (or Cf ), and (0) (or N ux ) in equation (13), according to equations (27) and (28) are formulated by Pr Pr 0 00 (0; ~) (0; ~) Psk=0 00k Ps k=0 0 k ; ; 1 + k=1 n+k+1 (0; ~) 1 k=1 n+k+1 (0; ~) 14

respectively. In many cases, the [r; r] homotopy-Padé approximation does not depend upon the auxiliary parameter ~. To verify the accuracy of HAM, com0 parisons of f 00 (0) and (0) with those reported by Ramachandran et al. [31], Hassanien and Gorla [32], Lok et al. [33] and Ishak et al. [10] (upper branch values) are given in Tables 1 and 2 for = 1 and Tables 3 and 4 for = 1 when f0 = = 0 and di¤erent values of P r, respectively. Table 1: Results for [15; 15] Homotopy-Padé approach for f 00 (0) for Exponential bases when > 0 Pr 0.7 1 7 10 20 40 50 60 80 100

Ramachandran et al. [23] 1.7063 1.5179 1.4485 1.4101 1.3903 1.3774 1.3680

Hassanien and Gorla [25] 1.70632 1.49284 1.40686 1.38471

Lok et al. [26] 1.7064 1.5180 1.4486 1.4102 1.3903 1.3773 1.3677

Ishak et al. [2] 1.7063 1.6754 1.5179 1.4928 1.4485 1.4101 1.3989 1.3903 1.3774 1.3680

Table 2: Results for [15; 15] Homotopy-Padé approach for bases when > 0 Pr 0.7 1 7 10 20 40 50 60 80 100

Ramachandran et al. [23] 0.7641 1.7224 2.4576 3.1011 3.5514 3.9095 4.2116

Hassanien and Gorla [25] 0.76406 1.94461 3.34882 4.23372

Lok et al. [26] 0.7641 1.7226 2.4577 3.1023 3.5560 3.9195 4.2289

0

Ishak et al. [2] 0.7641 0.8708 1.7224 1.9446 2.4576 3.1011 3.3415 3.5514 3.9095 4.2116

HAM Case 1 Case 2 1.70633 1.70632 1.67543 1.67547 1.51504 1.51787 1.48361 1.49281 1.41955 1.44847 1.34401 1.41376 1.31251 1.40384 1.28181 1.38853 1.21908 1.37220 1.15091 1.36176 (0) for Rational HAM Case 1 Case 2 0.76406 0.76406 0.87079 0.87074 1.71284 1.72236 1.90718 1.94462 2.83443 2.46210 3.34308 3.09395 3.56039 3.33583 3.73733 3.54544 4.00633 3.90255 4.20065 4.19654

Table 3: Results for [15; 15] Homotopy-Padé approach forf 00 (0) and

15

0

(0)

for Exponential bases when

< 0:

Pr

f 00 (0)

0.7 1 7 10 20 40 50 60 80 100

0.69166 0.73141 0.93929 0.92896 0.98383 1.03355 1.05328 1.07171 1.10605 1.13784

0

(0)

0.63325 0.73141 1.53467 1.71625 2.94287 3.28208 3.50010 3.67989 3.95540 4.15545

Table 4: Results for [15; 15] Homotopy-Padé approach forf 00 (0) and for Rational bases when < 0: Pr

f 00 (0)

0.7 1 7 10 20 40 50 60 80 100

0.69149 0.73124 0.92339 0.95258 1.00325 1.04527 1.04887 1.06128 1.07903 1.08650

0

0

(0)

(0)

0.63319 0.73132 1.54611 1.76342 2.26820 2.89948 3.13856 3.34669 3.70261 4.00749

Conclusions Here we have applied the homotopy analysis method which has been proven to be successful in tackling nonlinear problems to compute the in‡uence of suction/injection on the mixed convection ‡ows along a vertical cylinder. It is interesting to note that when = 0 then one recovers the ‡at plate case [34]. The problem reduces to the case of impermeable cylinder for f0 = 0 [35]. The case of arbitrary surface temperature can also be recovered by = f0 = 0 [31]: It is further revealed that usage of rational base is easier, because it has one auxiliary parameter less than the exponential case ( ). It is worth mentioning that in both cases, the dual solutions do not obtain, therefore, the comparison with existing results reveal that our series solutions are valid for all values (R 0):

16

Acknowledgements R. Ellahi thanks to United State Education Foundation Pakistan and CIES USA to honored him by Fulbright Scholar Award for the year 2011-2012. RE and Arshad Riaz are also grateful to the Higher Education Commission for NRPU and …nancial support. Nomenclature T Tw T1 k u; w x; y U1

N ux Rex

Dynamic viscosity, [N sm 2 ] Fluid temperature, [K] Surface temperature, [K] Ambient temperature, [K] Fluid density, [kgm 3 ] Kinematic viscosity, [m2 s 1 ] Thermal conductivity, [W m 1 K Velocity components, [ms 1 ] Cartesian components, [m] Free stream velocity, [Ls 1 ] Dimensionless temperature, [ ] Similarity variable, [m] Stream function, [m2 s 1 ] Nusselt number, [ ] Local Reynolds number, [ ]

1

]

References [1] Vafai, K., Handbook of Porous Media (Second Edition), Taylor & Francis, USA, 2005 [2] Vafai, K., Porous Media: Applications in Biological Systems and Biotechnology, Taylor & Francis, USA, 2010. [3] Tan, W. C., Masuoka, T., Stokes’ …rst problem for a second grade ‡uid in a porous half space with heated boundary, Int. J. Nonlinear Mech. 40 (2005), pp. 512–522 [4] Tan, W. C., Masuoka, T., Stability analysis of a Maxwell ‡uid in a porous medium heated from below. Phys. Lett. A, 360 (2007), pp. 454–460 [5] Fetecau, C., Hayat, T., MHD transient ‡ows in a channel of rectangular cross-section with porous medium, Physics letters A, 369 (2007). pp. 4454 [6] Hayat, T., Mambili-Mamboundou, H., Mahomed, F. M., Unsteady Solutions in a Third-Grade Fluid Filling the Porous Space, Mathematical Problems in Engineering, 2008 (2008), pp. 139560-13 17

[7] Pop, I., Ingham, D. B., Convective Heat Transfer, Pergamon, Amsterdam, 2001 [8] Mahmood, T., Merkin, J. H., Similarity solutions in axisymmetric mixed convection boundary-layer ‡ow, J. Eng. Math. 22 (1988), pp. 73-92 [9] Ridha, A., Aiding ‡ows non-unique similarity solutions of mixed convection boundary-layer equations, J. Appl. Math. Phys. 47 (1996), pp. 341-352 [10] Ishak, A., Nazar, R., Pop, I., The e¤ects of transpiration on the boundary layer ‡ow and heat transfer over a vertical slender cylinder, Int. J. NonLinear Mech. 42 (2007), 1010-1017 [11] Liao, S. J., Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press, 2003 [12] Xu, H., Liao, S. J., Pop, I., Series solutions of unsteady boundary layer ‡ow of a micropolar ‡uid near the forward stagnation point of a plane surface, Acta Mech. 184 (2006), pp. 87-101. [13] Hayat, T., Ellahi, R., Asghar, S., Modelling of ‡ow and heat transfer in a generalized second grade ‡uid, International Journal of Applied Mechanics. & Engineering, 13 (2008), pp. 101-121 [14] Ellahi, R., Ariel, P. D., Hayat, T., Asghar, S., E¤ect of heat transfer on a third grade ‡uid in a ‡at channel, International Journal of Numerical Method in Fluids, 63 (2010), pp. 847-859 [15] Ellahi, R., Riaz, A., Analytical solution for MHD ‡ow in a third grade ‡uid with variable viscosity, Mathematical and Computer Modelling, 52 (2010), pp. 1783-1793 [16] Ellahi, R., Afzal, S., E¤ect of variable viscosity in a third grade ‡uid with porous medium. An analytical solution. Commun Nonlinear Sci Numer Simulation 14 (2009), 2056-2072 [17] Nadeem, S., Hayat, T., Abbasbandy, S., Ali, M., E¤ects of partial slip on a fourth-grade ‡uid with variable viscosity: An analytic solution, Nonlinear Analysis: Real World Applications, 11 (2010), pp. 856-868 [18] Abbasbandy, S., Yürüsoy, M., Pakdemirli, M., The analysis approach of boundary layer equations of power-law ‡uids of second grade, Z. Naturforsch. A 63(a) (2008) 564-570. [19] J. Cheng, S. J. Liao, R. N. Mohapatra, K. Vajravelu, Series solutions of nano boundary layer ‡ows by means of the homotopy analysis method, J. Math. Anal. Appl. 343 (2008), pp. 233-245

18

[20] Abbasbandy, S., Homotopy analysis method for generalized BenjaminBona-Mahony equation, Z. Angew. Math. Phys. (ZAMP) 59 (2008), pp. 51-62. [21] Ellahi, R., Abbasbandy, S., Hayat, T., Zeeshan, A., On comparison of series and numerical solutions for second Painlevé equation, Numerical Methods for Partial Di¤erential Equations, 26 (2010), pp. 1070-1078 [22] Abbasbandy, S., Hayat, T., M. Mahomed, F. M., Ellahi, R., On comparison of exact and series solutions for thin …lm ‡ow of a third-grade ‡uid, International Journal for Numerical Methods in Fluids, 61, (2009), pp. 987–994 [23] Xu, H., Liao, S. J., Pop, I., Series solutions of unsteady free convection ‡ow in the stagnation-point region of a three-dimensional body, Int. J. Therm. Sci. 47 (2008), pp. 600-608 [24] Ellahi, R., Zeeshan, A., A study of pressure distribution of a slider bearing lubricated with second grade ‡uid, Numerical Methods for Partial Di¤erential Equations, 27 (2011), pp. 1231-1241 [25] Liao, S. J., A general approach to get series solution of non-similarity boundary-layer ‡ows, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009), pp 2144-2159 [26] Ellahi, R., E¤ects of the slip boundary condition on non-Newtonian ‡ows in a channel, Communication in Nonlinear Science and Numerical Simulations, 14 (2009), pp. 1377-1384 [27] Gebhart, B., Jaluria, Y., Mahajan, R. L., Samakia, B., Buoyancy Induced Flows and Transport, Hemisphere, New York, 1988 [28] Rajagopal, K. R., Ruzicka, M., Srinivasa, A. R., On the Oberbeck-Boussinesq approximation, Math. Models Methods Appl. Sci. 6 (1996), pp. 11571167 [29] Van Gorder, R. A., Vajravelu, K., On the selection of auxiliary functions, operators, and con-vergence control parameters in the application of the Homotopy Analysis Method to nonlinear di¤erential equations: A general approach, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), pp. 40784089 [30] Liao, S. J., Notes on the homotopy analysis: Some de…nitions and theorems, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009), pp. 983-997 [31] Ramachandran, N., Chen, T. S., Armaly, B. F., Mixed convection in stagnation ‡ows adjacent to a vertical surface, ASME J. Heat Transfer 110 (1988), pp. 373-377

19

[32] Hassanien, I. A., Gorla, R. S., Combined forced and free convection in stagnation ‡ows of micropolar ‡uids over vertical non-isothermal surface, Int. J. Eng. Sci. 28 (1990), pp. 783-792. [33] Lok, Y. Y., Amin, N., Pop, I., Unsteady mixed convection ‡ow of a micropolar ‡uid near the stagnation point on a vertical surface, Int. J. Therm. Sci. 45 (2006), pp. 1149-1157 [34] Ishak, A., Nazar, R., Pop, I., Dual solutions in mixed convection ‡ow near a stagnation point on a vertical porous plate, Int. J. Thermal Sci. 47 (2008), pp. 417-422. [35] Mahmood, T., Merkin, J. H., Similarity solutions in axisymmetric mixedconvection boundary-layer ‡ow, J. Eng. Math. 22 (1988), pp. 73-92.

20

Rahmat Ellahi1 ;2;3 , Arshad Riaz3 , Saeid Abbasbandy4 , Tasawar Hayat5;6 and Kambiz Vafai2 , 2 Department of Mechanical Engineering Bourns Hall Riverside, CA USA 3 Department of Mathematics & Statistics, FBAS, IIU, Islamabad, Pakistan 4 Department of Mathematics, SRB, Islamic Azad University, Tehran, Iran 5 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan 6 Department of Mathematics, College of Science, KSU, Riyadh, Saudi Arabia

In this investigation, the series solutions of mixed convection boundary layer ‡ow over a vertical permeable cylinder are constructed. Two types of series as well numerical solutions are presented by choosing exponential and rational bases. The resulting differential system are solved by employing homotopy analysis method (HAM) and Pade technique which have been proven to be successful in tackling nonlinear problems. We o¤er various veri…cations of the solutions by comparing to existing, documented results and also mathematically, through reduction of sundry parameters. The convergence of the series solutions have been discussed explicitly. Comparison with existing results reveal that the series solutions are not only valid for large (aiding ‡ow) but also for small values (opposing ‡ow) of and the dual solutions do not obtain in both cases. Key words: Mixed convection; Series solutions; Homotopy analysis method; Pade technique; Porous cylinder; Steady ‡ow

Introduction The mixed convection ‡ows under boundary layer analysis are of fundamental theoretical and practical interest. Heat exchangers, electric transformers, refrigeration coils, solar collectors, study of movement of natural gas, oil, water through oil reservoirs and nuclear reactors are few examples in this direction. The ‡ows of non-Newtonian ‡uids through a porous medium [1 6] are also quite prevalent in nature. Examples of these applications are …ltration processes, biomechanics, packed bed, ceramic geothermal engineering, insulation systems, ceramic processing, chromatography, and geophysical phenomena. Over the last 1 Author’s

e-mail: [email protected]

1

several decades a large number of research work related to the mixed convection ‡ow of a viscous and incompressible ‡uid over a vertical ‡at plate was conducted by a number of investigators. For instance, a critical review of mixed convection ‡ows has been presented by Pop and Ingham [7]: Mahmood and Merkin [8] have presented dual similarity solutions for the axisymmetric mixed convection boundary layer ‡ow along a vertical cylinder in the case of opposing directions only. Later on Ridha [9] has reported that when the buoyancy force acts in the direction of ‡ow then the dual solutions for aiding ‡ow also exist. Ishak et al [10] have also reported a numerical simulation by Keller-box method for boundary layer ‡ow and heat transfer over a vertical slender cylinder. In this study, the authors concluded that for aiding/assisting ‡ow dual solutions exist but for opposing ‡ow there are either dual solutions, unique solutions or no solution exist. The purpose of the present investigation is to revisit the problem formulated in references [7 10] for the analytic solutions. The main stream velocity and wall surface temperature depend upon the axial distance along the cylinder surface. The homotopy analysis method [11] has been applied to obtain series solutions of velocity and temperature …elds by making choices of two base functions. Convergence of the obtained series solutions is taken care properly. Furthermore, the skin friction and Nusselt number are analyzed. Comparison with the previous relevant studies is made. To the best of our knowledge, the series solutions for this particular model have not been presented in the past. It is worth mentioning that in both cases, the dual solutions do not exist, therefore, the comparison with existing results reveal that our series solutions are valid for > 0 (heated cylinder) and for < 0 (cooled cylinder) as well. It is shown that like several existing studies [12 26]; the HAM in the present paper is accepted as an elegant tool for e¤ective solutions for a number of complicated ‡uid problems.

Problem formulation Here, we analyze the convection ‡ow and heat transfer characteristic along a vertical permeable slender cylinder with radius a . We choose the cylindrical coordinates (x; r) such that x and r-axes are along the cylinder surface (vertically) and radial directions, respectively. Symmetric nature of the ‡ow is assumed with respect to the transverse coordinate. Furthermore, cylinder is kept in an incompressible viscous ‡uid of uniform ambient temperature T1 and constant density 1 . Using continuity, momentum and energy equations, one obtains the following boundary layer equations [27; 28] @ @ (ru) + (rw) = 0; @x @r u

@u dU @u +w =U + @x @r dx u

@ 2 u 1 @u + @r2 r @r

(1) + g (T

@2T 1 @T + @r2 r @r

@T @T +w = @x @r 2

:

T1 );

(2) (3)

In above equations, u and w indicate the velocity components, U (x) the mainstream velocity, T the ‡uid temperature, g the gravitational acceleration, the thermal di¤usivity, the thermal expansion coe¢ cient and the kinematic viscosity. The subjected boundary conditions are [10] u = 0; w = V; T = Tw (x); at r = a;

(4)

u ! U (x); T ! T1 ; as r ! 1;

(5)

where V (> 0) and V (< 0) correspond to the injection and suction velocities, respectively. Further, U (x) and temperature of the cylinder surface Tw (x) are x ; Tw (x) = T1 + `

U (x) = U1

T

x ; `

(6)

in which ` denotes characteristic length and T is the characteristic temperature. Note that T > 0 corresponds to a heated surface and T < 0 for a cooled surface. De…ning =

r2

a2

U x

2a

0:5

= (U x)0:5 af ( );

;

a r

u = U f 0 ( ); w =

( )=

T Tw

T1 ; T1

(7)

0:5

U1 `

f ( );

(8)

where the stream function can be de…ned by u = r 1 @ [email protected] and w = 1 r @ [email protected] Moreover prime denotes di¤erentiation with respect to . Invoking equations (7) and (8), Eq. (1) is identically satis…ed, and equations (2)and (3) yield (1 + 2

)f 000 ( ) + 2 f 00 ( ) + f ( )f 00 ( ) + 1

(1 + 2

)

00

( )+2

and the curvature parameter =

` U1 a2

0

f 02 ( ) +

( ) + P r f ( ) 0( )

( ) = 0;

f 0 ( ) ( ) = 0;

(9) (10)

and V are 0:5

; V =

a r

U1 `

0:5

f0 ;

where f0 = f (0). It is noted that f0 < 0 is for mass injection and f0 > 0 is for mass suction. The buoyancy or mixed convection parameter is =

g ` T : 2 U1

Here > 0 and < 0 are for aiding ‡ow (heated cylinder) and for opposing ‡ow (cooled cylinder), respectively. For = 0, one has pure forced convection ‡ow without buoyancy force. 3

The boundary conditions (4) and (5) reduce in the following form f (0) = f0 ; f 0 (0) = 0; f 0 (1) = 1; (0) = 1; (1) = 0:

(11)

Now the skin friction coe¢ cient (Cf ) and the local Nusselt number (N ux ) are Cf =

w ; U 2 =2

N ux =

xqw k(Tw

T1 )

where = 1 [1 (T T1 )] and the skin friction from the plate qw can be written as follows w

=

@u @r

; qw =

w

@T @r

k

r=a

;

(12)

and the heat transfer

; r=a

in which and k are the dynamic viscosity and thermal conductivity, respectively. Making use by Eq. (7) one obtains 1 Cf (Rex )0:5 = f 00 (0); N ux =(Rex )0:5 = 2

0

(0):

(13)

In above expression Rex = U x= denotes the local Reynold number [10].

Solution of the problem Here we o¤er two types of series solutions by choosing exponential and rational bases. Exponential bases According to equations (9) and (10) and the boundary conditions (11), we write the solution in the following form f ( ) = a0 + +

1 +1 X X

q

aq;m

m

e

;

(14)

m=1 q=0

( )=

+1 X 1 X

bq;m

q

e

m

;

(15)

m=1 q=0

where a0 , aq;m and bq;m are coe¢ cients to be determined and is a spatial-scale parameter. By rule of solution expression denoted by equations (14) and (15) and the boundary conditions (11), it is natural to choose 1

f0 ( ) = f0 + 0(

)=e

4

exp(

;

)

;

(16) (17)

as the initial approximation to f ( ) and ( ), respectively. We use the method of higher order di¤erential mapping, [29] to choose the auxiliary linear operators L1 and L2 by @2 @3 + ( ; ; p); (18) L1 [ ( ; ; p)] = @ 3 @ 2 @ @2 + @ 2 @

L2 [ ( ; ; p)] =

( ; ; p);

(19)

which satisfy the properties L1 [C1 + C2 + C3 e

] = 0; L2 [C4 + C5 e

] = 0;

(20)

where Ci , i = 1; 2; : : : ; 5 are constants. This choice of L1 and L2 is motivated by equations (14) and (15), respectively, and from boundary conditions (11), we have C2 = C4 = 0. From (9) and (10) we de…ne nonlinear operators as follows N1 [ ( ; ; p); ( ; ; p)] = (1+2

)

N2 [ ( ; ; p); ( ; ; p)] = (1 + 2

@2 @2 @3 +2 + +1 @ 3 @ 2 @ 2 )

@2 @ +2 @ 2 @

+ Pr

@ @ @ @

2

+ @ @

; (21)

; (22)

and then construct the homotopy H1 [ ( ; ; p); ( ; ; p)] = (1

p)L1 [

H2 [ ( ; ; p); ( ; ; p)] = (1

p)L2 [

f0 ( )]

0(

)]

~1 pH1 ( )N1 [ ; ];

(23)

~2 pH2 ( )N2 [ ; ];

(24)

where ~1 6= 0 and ~2 6= 0 are the convergence-control parameters [30]; H1 ( ) and H2 ( ) are auxiliary functions. Setting Hi [ ( ; ; p); ( ; ; p)] = 0, for i = 1; 2, we have the following zero-order deformation problems (1

p)L1 [

(1

p)L2 [

f0 ( )] = ~1 pH1 ( )N1 [ ; ]; 0(

(25)

)] = ~2 pH2 ( )N2 [ ; ];

(26)

subject to conditions (0; ; p) = f0 ;

@ ( ; ; p) @

= 0; =0

(0; ; p) = 1;

@ ( ; ; p) @

= 1; =1

(1; ; p) = 0;

in which p 2 [0; 1] is an embedding parameter. When the parameter p increases from 0 to 1, the solution ( ; ; p) varies from f0 ( ) to f ( ) and the solution ( ; ; p) varies from 0 ( ) to ( ). If these continuous variations are smooth 5

enough, the Maclaurin’s series with respect to p can be constructed for ( ; ; p) and ( ; ; p), respectively, and further, if these series are convergent at p = 1, we have +1 +1 X X f ( ) = f0 ( ) + fm ( ) = (27) m ( ; ; ~1 ); m=1

( )=

0(

)+

m=0

+1 X

m(

)=

m=1

where fm ( ) =

1 @ m ( ; ; p) m! @pm

+1 X

m(

; ; ~2 );

(28)

m=0

;

m(

)=

p=0

1 @ m ( ; ; p) m! @pm

: p=0

To calculate fm ( ), we now di¤erentiate Eqs. (25) and (26) and related conditions m times with respect to p, then set p = 0, and …nally divide by m!. The resulting mth-order deformation problems are L1 [fm ( )

m fm 1 (

)] = ~1 H1 ( )R1;m ( );

(m = 1; 2; 3; : : :);

(29)

L2 [

m m 1(

)] = ~2 H2 ( )R2;m ( );

(m = 1; 2; 3; : : :);

(30)

m(

)

0 fm (0)

fm (0) = 0;

= 0;

0 fm (1)

= 0;

m (0)

= 0;

m (1)

= 0;

(31)

where R1;m ( ) and R2;m ( ) are given by R1;m ( ) = (1+2

000 00 )fm 1 +2 fm

m X1

1+

00 fi fm

i 1

0 fi0 fm

i 1

+

m 1 +(1

m );

i=0

R2;m ( ) = (1 + 2

)

00 m 1

+2

0 m 1

+ Pr

m X1

0 m i 1

fi

0 i fm i 1

;

i=0

where prime denotes di¤erentiation with respect to m

=

0; 1;

and

m 1; m > 1:

The general solution of Eqs. (29) and (30) are fm ( ) = f^m ( ) + C1 + C2 + C3 e m(

) = ^m ( ) + C4 + C5 e

; ;

(32) (33)

where Ci for i = 1; : : : ; 5 are constants, f^m ( ) and ^m ( ) are particular solutions of Eqs. (29) and (30), respectively. For simplicity, here we take H1 ( ) = H2 ( ) = 1. According to the rule of solution expression denoted by equations (14) and (15), C2 = C4 = 0. The other unknowns are governed by 0 f^m (0) + C1 + C3 = 0; f^m (0)

6

C3 = 0; ^m (0) + C5 = 0;

and according to our algorithm, the other boundary conditions are ful…lled. In this way, we derive fm ( ) and m ( ) for m = 1; 2; 3; : : :, successively for every . At the N th-order approximation, we have the analytic solution of Eqs. (9) and (10), namely f( )

FN ( ) =

N X

fi ( );

( )

=

N(

)=

i=0

N X

i(

):

(34)

i=0

For simplicity, here we take ~1 = ~2 = ~. The auxiliary parameter ~ is useful to adjust the convergence region of the series (34) in the homotopy analysis solution. By plotting ~-curve, it is straightforward to choose an appropriate range for ~ which ensures the convergence of the solution series. As pointed out by Liao [11], the appropriate region for ~ is a horizontal line segment.

Rational bases Invoking equations (9) and (10) and the boundary conditions (11), one can write +1 m X X2 f ( ) = d0 + + dq;m q (1 + ) m ; (35) m=1 q=0

( )=

+1 m X X2

eq;m

q

(1 + )

m

;

(36)

m=1 q=0

where d0 , dq;m and eq;m are coe¢ cients to be determined. According to the rule of solution expression denoted by equations (35) and (36) and the boundary conditions (11), the initial approximations of f ( ) and ( ) are selected as follows 1 ; (37) f0 ( ) = f0 1 + + 1+ 0(

)=

1 : 1+

(38)

The auxiliary linear operators L1 and L2 are L1 [ ( ; p)] =

@3 3 @2 + 3 @ 1+ @ 2

( ; p);

(39)

L2 [ ( ; p)] =

@2 2 @ + 2 @ 1+ @

( ; p);

(40)

L1 [D1 + D2 + D3 (1 + )

1

] = 0; L2 [D4 + D5 (1 + )

1

] = 0;

(41)

where Di , i = 1; 2; : : : ; 5 are constants. This choice of L1 and L2 is motivated by equations (35) and (36), respectively, and from boundary conditions (11), we have D2 = D4 = 0. 7

In this case, the nonlinear operator Ni [ ; ], the homotopy Hi [ ; ], Ri;m ( ) for i = 1; 2, and the zero-order deformation equations, the mth-order deformation equations are designed as in the previous case without the parameter . The general solution of mth-order deformation equations here are fm ( ) = f^m ( ) + D1 + D2 + D3 (1 + ) m(

) = ^m ( ) + D4 + D5 (1 + )

1

1

;

;

(42) (43)

where Di for i = 1; : : : ; 5 are constants, f^m ( ) and ^m ( ) are particular solutions of mth-order deformation equations. By rule of solution expressions denoted by (35) and (36) and from mth-order deformation equations, the auxiliary functions H1 ( ) and H2 ( ) are chosen in the form H1 ( ) = (1 + ) 1 ; H2 ( ) = (1 + ) 2 : It is found that when 1 < 4 and 2 < 3 the term log(1 + ) appears in the solution expression of fm ( ) and m ( ), which disobeys the rule of solution expression denoted by (35) and (36), respectively. In addition, when 1 > 4 and 2 > 3 we omit some terms in solution expression. This uniquely determines the corresponding auxiliary functions H1 ( ) = (1 + )

4

; H2 ( ) = (1 + )

3

:

According to (35) and (36), D2 = D4 = 0. The other unknowns are governed by 0 f^m (0) + D1 + D3 = 0; f^m (0)

D3 = 0; ^m (0) + D5 = 0;

and according to our algorithm, the other boundary conditions are satis…ed. In this way, we derive fm ( ) and m ( ) for m = 1; 2; 3; : : :, successively. Like previous case, we will take here ~1 = ~2 = ~.

Numerical results and discussion We use the widely applied symbolic computation software MATHEMATICA to solve equations (29) and (30). By means of the so-called ~-curve, it is straightforward to choose an appropriate range for ~ which ensures the convergence of the solution series. As pointed out by Liao [11], the appropriate region for ~ is a horizontal line segment. We can investigate the in‡uence of ~ on the 0 convergence of f 00 (0) and (0), by plotting the curve of it versus ~, as shown in Figs. 1 4 (for > 0) and Figs. 5 8 (for < 0) are the examples of two cases. By considering the ~ curve we can obtain the reasonable interval for ~ in each case. Our computations show that in …rst case it is better to we choose 2. Also, by computing the error of norm 2 for two successive approximation of FN ( ) or N ( ) of (34), we can obtain the best value for ~ in each case. Figs. 9 10 show the error of F20 ( ) for 2 [0; 5] in …rst case and we obtain that the 8

best value of ~ is 0:209 with error 7:01 10 6 (for > 0) and ~ is 0:218 with error 6:02 10 5 (for < 0) respectively. With the same contrast Figs. 11 and 12 show the error of 20 ( ) in second case and we obtain that the best value of ~ are 1:327 and 1:95 with error 3:47 10 7 and 3:17 10 8 for < 0 and > 0 respectively.

Fig. 1: The ~-curve of the f 00 (0) versus ~ for the 20th-order approximation and P r = 0:7, = 2, = 0, = 1, f0 = 0 by exponential bases.

Fig. 2: The ~-curve of the and P r = 0:7, = 2, = 0,

0

(0) versus ~ for the 20th-order approximation = 1, f0 = 0 by exponential bases.

9

Fig. 3: The ~-curve of the f 00 (0) versus ~ for the 20th-order approximation and P r = 0:7, = 1, = 1, f0 = 1 by rational bases.

0 Fig. 4: The ~-curve of the (0) versus ~ for the 20th-order approximation and P r = 0:7, = 1, = 1, f0 = 1 by rational bases.

10

Fig. 5: The ~-curve of the f 00 (0) versus ~ for the 20th-order approximation and P r = 0:7, = 2, = 0, = 1, f0 = 0 by exponential bases.

Fig. 6: The ~-curve of the and P r = 0:7, = 2, = 0,

0

=

(0) versus ~ for the 20th-order approximation 1, f0 = 0 by exponential bases.

11

Fig. 7: The ~-curve of the f 00 (0) versus ~ for the 20th-order approximation and P r = 0:7, = 1, = 1, f0 = 1 by rational bases.

0 Fig. 8: The ~-curve of the (0) versus ~ for the 20th-order approximation and P r = 0:7, = 1, = 1, f0 = 1 by rational bases.

12

Fig. 9: The norm 2 error of F20 ( ) versus ~ with P r = 0:7, f0 = 0 by exponential bases.

= 2,

Fig. 10: The norm 2 error of F20 ( ) versus ~ with P r = 0:7, = 1, f0 = 0 by exponential bases.

13

= 0,

= 2,

= 1,

= 0,

Fig. 11: The norm 2 error of f0 = 1 by rational bases.

20 (

) versus ~ with P r = 0:7,

= 1,

= 1,

Fig. 12: The norm 2 error of 20 ( ) versus ~ with P r = 0:7, = 1, = 1, f0 = 1 by rational bases. The so-called homotopy-Padé technique (see [11]) is employed, which greatly accelerates the convergence. The [r; s] homotopy-Padé approximations of f 00 (0) 0 (or Cf ), and (0) (or N ux ) in equation (13), according to equations (27) and (28) are formulated by Pr Pr 0 00 (0; ~) (0; ~) Psk=0 00k Ps k=0 0 k ; ; 1 + k=1 n+k+1 (0; ~) 1 k=1 n+k+1 (0; ~) 14

respectively. In many cases, the [r; r] homotopy-Padé approximation does not depend upon the auxiliary parameter ~. To verify the accuracy of HAM, com0 parisons of f 00 (0) and (0) with those reported by Ramachandran et al. [31], Hassanien and Gorla [32], Lok et al. [33] and Ishak et al. [10] (upper branch values) are given in Tables 1 and 2 for = 1 and Tables 3 and 4 for = 1 when f0 = = 0 and di¤erent values of P r, respectively. Table 1: Results for [15; 15] Homotopy-Padé approach for f 00 (0) for Exponential bases when > 0 Pr 0.7 1 7 10 20 40 50 60 80 100

Ramachandran et al. [23] 1.7063 1.5179 1.4485 1.4101 1.3903 1.3774 1.3680

Hassanien and Gorla [25] 1.70632 1.49284 1.40686 1.38471

Lok et al. [26] 1.7064 1.5180 1.4486 1.4102 1.3903 1.3773 1.3677

Ishak et al. [2] 1.7063 1.6754 1.5179 1.4928 1.4485 1.4101 1.3989 1.3903 1.3774 1.3680

Table 2: Results for [15; 15] Homotopy-Padé approach for bases when > 0 Pr 0.7 1 7 10 20 40 50 60 80 100

Ramachandran et al. [23] 0.7641 1.7224 2.4576 3.1011 3.5514 3.9095 4.2116

Hassanien and Gorla [25] 0.76406 1.94461 3.34882 4.23372

Lok et al. [26] 0.7641 1.7226 2.4577 3.1023 3.5560 3.9195 4.2289

0

Ishak et al. [2] 0.7641 0.8708 1.7224 1.9446 2.4576 3.1011 3.3415 3.5514 3.9095 4.2116

HAM Case 1 Case 2 1.70633 1.70632 1.67543 1.67547 1.51504 1.51787 1.48361 1.49281 1.41955 1.44847 1.34401 1.41376 1.31251 1.40384 1.28181 1.38853 1.21908 1.37220 1.15091 1.36176 (0) for Rational HAM Case 1 Case 2 0.76406 0.76406 0.87079 0.87074 1.71284 1.72236 1.90718 1.94462 2.83443 2.46210 3.34308 3.09395 3.56039 3.33583 3.73733 3.54544 4.00633 3.90255 4.20065 4.19654

Table 3: Results for [15; 15] Homotopy-Padé approach forf 00 (0) and

15

0

(0)

for Exponential bases when

< 0:

Pr

f 00 (0)

0.7 1 7 10 20 40 50 60 80 100

0.69166 0.73141 0.93929 0.92896 0.98383 1.03355 1.05328 1.07171 1.10605 1.13784

0

(0)

0.63325 0.73141 1.53467 1.71625 2.94287 3.28208 3.50010 3.67989 3.95540 4.15545

Table 4: Results for [15; 15] Homotopy-Padé approach forf 00 (0) and for Rational bases when < 0: Pr

f 00 (0)

0.7 1 7 10 20 40 50 60 80 100

0.69149 0.73124 0.92339 0.95258 1.00325 1.04527 1.04887 1.06128 1.07903 1.08650

0

0

(0)

(0)

0.63319 0.73132 1.54611 1.76342 2.26820 2.89948 3.13856 3.34669 3.70261 4.00749

Conclusions Here we have applied the homotopy analysis method which has been proven to be successful in tackling nonlinear problems to compute the in‡uence of suction/injection on the mixed convection ‡ows along a vertical cylinder. It is interesting to note that when = 0 then one recovers the ‡at plate case [34]. The problem reduces to the case of impermeable cylinder for f0 = 0 [35]. The case of arbitrary surface temperature can also be recovered by = f0 = 0 [31]: It is further revealed that usage of rational base is easier, because it has one auxiliary parameter less than the exponential case ( ). It is worth mentioning that in both cases, the dual solutions do not obtain, therefore, the comparison with existing results reveal that our series solutions are valid for all values (R 0):

16

Acknowledgements R. Ellahi thanks to United State Education Foundation Pakistan and CIES USA to honored him by Fulbright Scholar Award for the year 2011-2012. RE and Arshad Riaz are also grateful to the Higher Education Commission for NRPU and …nancial support. Nomenclature T Tw T1 k u; w x; y U1

N ux Rex

Dynamic viscosity, [N sm 2 ] Fluid temperature, [K] Surface temperature, [K] Ambient temperature, [K] Fluid density, [kgm 3 ] Kinematic viscosity, [m2 s 1 ] Thermal conductivity, [W m 1 K Velocity components, [ms 1 ] Cartesian components, [m] Free stream velocity, [Ls 1 ] Dimensionless temperature, [ ] Similarity variable, [m] Stream function, [m2 s 1 ] Nusselt number, [ ] Local Reynolds number, [ ]

1

]

References [1] Vafai, K., Handbook of Porous Media (Second Edition), Taylor & Francis, USA, 2005 [2] Vafai, K., Porous Media: Applications in Biological Systems and Biotechnology, Taylor & Francis, USA, 2010. [3] Tan, W. C., Masuoka, T., Stokes’ …rst problem for a second grade ‡uid in a porous half space with heated boundary, Int. J. Nonlinear Mech. 40 (2005), pp. 512–522 [4] Tan, W. C., Masuoka, T., Stability analysis of a Maxwell ‡uid in a porous medium heated from below. Phys. Lett. A, 360 (2007), pp. 454–460 [5] Fetecau, C., Hayat, T., MHD transient ‡ows in a channel of rectangular cross-section with porous medium, Physics letters A, 369 (2007). pp. 4454 [6] Hayat, T., Mambili-Mamboundou, H., Mahomed, F. M., Unsteady Solutions in a Third-Grade Fluid Filling the Porous Space, Mathematical Problems in Engineering, 2008 (2008), pp. 139560-13 17

[7] Pop, I., Ingham, D. B., Convective Heat Transfer, Pergamon, Amsterdam, 2001 [8] Mahmood, T., Merkin, J. H., Similarity solutions in axisymmetric mixed convection boundary-layer ‡ow, J. Eng. Math. 22 (1988), pp. 73-92 [9] Ridha, A., Aiding ‡ows non-unique similarity solutions of mixed convection boundary-layer equations, J. Appl. Math. Phys. 47 (1996), pp. 341-352 [10] Ishak, A., Nazar, R., Pop, I., The e¤ects of transpiration on the boundary layer ‡ow and heat transfer over a vertical slender cylinder, Int. J. NonLinear Mech. 42 (2007), 1010-1017 [11] Liao, S. J., Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press, 2003 [12] Xu, H., Liao, S. J., Pop, I., Series solutions of unsteady boundary layer ‡ow of a micropolar ‡uid near the forward stagnation point of a plane surface, Acta Mech. 184 (2006), pp. 87-101. [13] Hayat, T., Ellahi, R., Asghar, S., Modelling of ‡ow and heat transfer in a generalized second grade ‡uid, International Journal of Applied Mechanics. & Engineering, 13 (2008), pp. 101-121 [14] Ellahi, R., Ariel, P. D., Hayat, T., Asghar, S., E¤ect of heat transfer on a third grade ‡uid in a ‡at channel, International Journal of Numerical Method in Fluids, 63 (2010), pp. 847-859 [15] Ellahi, R., Riaz, A., Analytical solution for MHD ‡ow in a third grade ‡uid with variable viscosity, Mathematical and Computer Modelling, 52 (2010), pp. 1783-1793 [16] Ellahi, R., Afzal, S., E¤ect of variable viscosity in a third grade ‡uid with porous medium. An analytical solution. Commun Nonlinear Sci Numer Simulation 14 (2009), 2056-2072 [17] Nadeem, S., Hayat, T., Abbasbandy, S., Ali, M., E¤ects of partial slip on a fourth-grade ‡uid with variable viscosity: An analytic solution, Nonlinear Analysis: Real World Applications, 11 (2010), pp. 856-868 [18] Abbasbandy, S., Yürüsoy, M., Pakdemirli, M., The analysis approach of boundary layer equations of power-law ‡uids of second grade, Z. Naturforsch. A 63(a) (2008) 564-570. [19] J. Cheng, S. J. Liao, R. N. Mohapatra, K. Vajravelu, Series solutions of nano boundary layer ‡ows by means of the homotopy analysis method, J. Math. Anal. Appl. 343 (2008), pp. 233-245

18

[20] Abbasbandy, S., Homotopy analysis method for generalized BenjaminBona-Mahony equation, Z. Angew. Math. Phys. (ZAMP) 59 (2008), pp. 51-62. [21] Ellahi, R., Abbasbandy, S., Hayat, T., Zeeshan, A., On comparison of series and numerical solutions for second Painlevé equation, Numerical Methods for Partial Di¤erential Equations, 26 (2010), pp. 1070-1078 [22] Abbasbandy, S., Hayat, T., M. Mahomed, F. M., Ellahi, R., On comparison of exact and series solutions for thin …lm ‡ow of a third-grade ‡uid, International Journal for Numerical Methods in Fluids, 61, (2009), pp. 987–994 [23] Xu, H., Liao, S. J., Pop, I., Series solutions of unsteady free convection ‡ow in the stagnation-point region of a three-dimensional body, Int. J. Therm. Sci. 47 (2008), pp. 600-608 [24] Ellahi, R., Zeeshan, A., A study of pressure distribution of a slider bearing lubricated with second grade ‡uid, Numerical Methods for Partial Di¤erential Equations, 27 (2011), pp. 1231-1241 [25] Liao, S. J., A general approach to get series solution of non-similarity boundary-layer ‡ows, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009), pp 2144-2159 [26] Ellahi, R., E¤ects of the slip boundary condition on non-Newtonian ‡ows in a channel, Communication in Nonlinear Science and Numerical Simulations, 14 (2009), pp. 1377-1384 [27] Gebhart, B., Jaluria, Y., Mahajan, R. L., Samakia, B., Buoyancy Induced Flows and Transport, Hemisphere, New York, 1988 [28] Rajagopal, K. R., Ruzicka, M., Srinivasa, A. R., On the Oberbeck-Boussinesq approximation, Math. Models Methods Appl. Sci. 6 (1996), pp. 11571167 [29] Van Gorder, R. A., Vajravelu, K., On the selection of auxiliary functions, operators, and con-vergence control parameters in the application of the Homotopy Analysis Method to nonlinear di¤erential equations: A general approach, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), pp. 40784089 [30] Liao, S. J., Notes on the homotopy analysis: Some de…nitions and theorems, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009), pp. 983-997 [31] Ramachandran, N., Chen, T. S., Armaly, B. F., Mixed convection in stagnation ‡ows adjacent to a vertical surface, ASME J. Heat Transfer 110 (1988), pp. 373-377

19

[32] Hassanien, I. A., Gorla, R. S., Combined forced and free convection in stagnation ‡ows of micropolar ‡uids over vertical non-isothermal surface, Int. J. Eng. Sci. 28 (1990), pp. 783-792. [33] Lok, Y. Y., Amin, N., Pop, I., Unsteady mixed convection ‡ow of a micropolar ‡uid near the stagnation point on a vertical surface, Int. J. Therm. Sci. 45 (2006), pp. 1149-1157 [34] Ishak, A., Nazar, R., Pop, I., Dual solutions in mixed convection ‡ow near a stagnation point on a vertical porous plate, Int. J. Thermal Sci. 47 (2008), pp. 417-422. [35] Mahmood, T., Merkin, J. H., Similarity solutions in axisymmetric mixedconvection boundary-layer ‡ow, J. Eng. Math. 22 (1988), pp. 73-92.

20