Heat Mass Transfer (2011) 47:1643–1649 DOI 10.1007/s00231-011-0821-9

ORIGINAL

A similarity solution for the flow and heat transfer over a moving permeable flat plate in a parallel free stream Norfifah Bachok • Mihaela Anghel Jaradat Ioan Pop

•

Received: 27 January 2011 / Accepted: 17 May 2011 / Published online: 14 June 2011 Ó Her Majesty the Queen in Rights of Australia 2011

Abstract This paper presents both a numerical and analytical study in connection with the steady boundary layer flow and heat transfer induced by a moving permeable semi-infinite flat plate in a parallel free stream. Both the velocities of the flat plate and the free stream are proportional to x1/3. The surface temperature is assumed to be constant. The governing partial differential equations are converted into ordinary differential equations by a new similarity transformation. Numerical results for the flow and heat transfer characteristics are obtained for various values of the moving parameter, transpiration parameter and the Prandtl number. Approximate analytical solutions are also obtained when the suction or injection parameter is very large. It is found that dual solutions exist for the case when the fluid and the plate move in the opposite directions. List of symbols f(g) Dimensionless stream function g(g) Dimensionless temperature k Thermal conductivity L Characteristic length Pr Prandtl number qw Heat transfer from the stretching sheet

N. Bachok Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Malaysia M. A. Jaradat Bogdan Voda University, Cluj-Napoca, Romania I. Pop (&) Faculty of Mathematics, University of Cluj, CP 253, 3400 Cluj, Romania e-mail: [email protected]

Re s T Tw T? u, v ue(x) uw(x) U e, U w U? vw(x) x, y

Reynolds number Dimensionless mass flux parameter Fluid temperature Uniform temperature of the stretching sheet Ambient temperature Dimensionless velocity components along the xand y- directions, respectively Dimensionless velocity of the free stream or far from the plate Dimensionless velocity of the moving plate Dimensionless constants Characteristic velocity Dimensionless mass flux velocity Dimensionless Cartesian coordinates along the surface and normal to it, respectively

Greek symbols DT Characteristic temperature g Similarity variable k Dimensionless moving parameter l Dynamic viscosity m Kinematic viscosity h Dimensionless temperature sw Wall skin friction or wall shear stress w Dimensionless stream function

1 Introduction The problem of laminar boundary-layer flows resulting from the flow of an incompressible fluid past a fixed semiinfinite flat plate, first considered by Blasius [1] is of considerable practical interest. Different from Blasius [1], Sakiadis [2–4] published the first papers dealing with boundary layer flow on a continuous moving surface, and

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both approximate and exact solutions for the momentum equation were obtained for laminar and turbulent flows on a surface moving through a stagnant fluid. He found exactly the same equation as Blasius, but the boundary conditions are different. These types of boundary-layer problems are expressed in the form of nonlinear third-order ordinary differential equations which cannot be solved directly in closed form. Accordingly, it is necessary to develop numerical methods capable of providing accurate solutions for problems of this type. As discussed by Abdelhafez [5], the Blasius and Sakiadis two flow problems are physically different and can not be mathematically transformed into one another. Heat transfer from a surface in motion relative to either a stationary or moving fluid occurs in many material processing applications, such as hot rolling, extrusion, drawing, etc. (Jaluria [6]). Examples of processes include continuous casting, plastic forming, bonding, annealing and tempering, heat treatment and many others. In all these processes, the quality of the final product depends on the rate of heat transfer at the moving surface. The first analysis treating simultaneous motion of both fluid and surface was published by Klemp and Acrivos [7, 8]. They studied the flow when the free stream is parallel but in the opposite direction of the moving plate using a similarity method. A critical value of the moving surface to the free stream velocity ratio was found to be 0.3541. The inability to obtain similarity solutions above this value was attributed to boundary layer separation from the moving plate. In contrast to the amount of material published on momentum and heat transfer for laminar flow associated with a surface moving either through a stagnant fluid or for a stream in counter flow, it seems that Bianchi and Viskanta [9] were the first to analyse of the heat transfer for the counterflow situation. Afzal [10] has also considered the heat transfer case of the boundary layer on nonlinear stretching impermeable wall. The heat transfer behaviour in this situation is of importance if temperature has to be controlled. Similar problems with various boundary conditions and in different situations have been considered by Erickson et al. [11], Hussaini et al. [12], Afzal et al. [13], Lin and Huang [14], Chen [15], Riley and Weidman [16], Weidman et al. [17], Fang [18], Fang et al. [19], Ishak et al. [20, 21] and Afzal [22]. Following Magyari and Weidman [23], and Bataller [24], we study in this paper the boundary layer flow past a continuously moving permeable semi-infinite flat plate in a free stream when both the velocities of the moving flat plate and that of the free stream are proportional to x1/3, where x is the coordinate measured along the plate. To the best of the knowledge of the authors, this problem has not been studied before and therefore the results obtained are novel. Examples of practical applications of this problem include the aerodynamic extrusion of plastic sheets, the

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cooling of an infinite metallic plate in a cooling bath, the boundary layers along material handling conveyers and along a liquid film in condensation processes, continuous casting spinning of fibers, etc. It is well known that the skin friction coefficient along a continuous moving flat surface in a quiescent fluid (Sakiadis problem) is about 30% higher than those along a static flat plate in a moving fluid (Blasius problem). Excellent descriptions of the problem of laminar fluid flow which results from the simultaneous motions of a free stream and its bounding surface in the same direction have been examined in detail by Abraham and Sparrow [25], and Sparrow and Abraham [26] using the relative-velocity model, which uses the magnitude of the relative velocity in conjunction with the drag formula for the case in which only one of the participating media is in motion.

2 Problem formulation Consider the steady boundary layer flow of a viscous and incompressible fluid past a permeable moving semi-infinite flat plate in a variable free stream velocity. It is assumed that the velocity of the free stream is ue ð xÞ and that of the flat plate is uw ð xÞ, respectively. It is also assumed that the constant temperature of the moving flat plate is Tw, while the uniform temperature of the ambient fluid is T?. Under these conditions, the boundary layer equations in nondimensional form can be written as ou ov þ ¼0 ox oy

ð1Þ

u

ou ou due o2 u þ v ¼ ue þ ox oy dx oy2

ð2Þ

u

oh oh 1 o2 h þv ¼ ox oy Pr oy2

ð3Þ

where the following non-dimensional boundary layer y=LÞ; u ¼ u=U1 ; variables were used: x ¼ x=L; y ¼ Re1=2 ð v=LÞ u and h ¼ ðT  T1 Þ=DT. Here u and v are v ¼ Re1=2 ð the velocity components along x- and y-axes, U? is the characteristic velocity, L is the characteristic length, DT ¼ Tw  T1 is the characteristic temperature, Re ¼ U1 L=m is the Reynolds number, Pr is the Prandtl number and m is the kinematic viscosity. The boundary conditions of (1)–(3) are taken to be v ¼ vw ðxÞ;

u ¼ uw ðxÞ;

u ! ue ðxÞ; h ! 0

as

h¼1 y!1

at

y¼0

ð4Þ

where vw(x) is the dimensionless mass flux velocity with vw(x) \ 0 for suction and vw(x) [ 0 for injection, respectively.

Heat Mass Transfer (2011) 47:1643–1649

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In order that (1)–(3) admit a similarity solution, we follow Magyari and Weidman [23] or Bataller [24] and assume that uw(x) and ue(x) have the following form uw ðxÞ ¼ Uw x1=3 ; ue ðxÞ ¼ Ue x1=3

ð5Þ

where Uw and Ue([0) are dimensionless constants with Uw [ 0 for a plate moving in the same direction with the external flow and Uw \ 0 for a moving flat plate in the direction opposite to the external flow, respectively. We notice that the same momentum problem with more general plate velocity and free stream conditions uw/Uw = xm and ue/Ue = xm has been studied by Ishak et al. [20] and Afzal [22]. However, the heat transfer problem has not been considered in these papers. Further, it needs to be mentioned that Magyari and Weidman [23] dealt with the shear driven flow over an impermeable semi-infinite flat plate and, consequently, other boundary conditions were used by these authors. As it was mentioned by Bataller [24], this problem studies the boundary layer flow and heat transfer over a flat plate which moves with a nonlinearly velocity in a free stream as a power function of the distance x from the leading edge of the surface, the problem being of considerable engineering interest. Although, these boundary conditions may be difficult to realize in practice, we are carrying out this analysis with an emphasis on the similarity solutions for the boundary layer flow and heat transfer, which constitute the basis for real engineering problems. We introduce now the stream function w which is defined in the usual way as u = qw/qx and v = -qw/qx, and look for a solutions of (2) and (3) of the following form w ¼ Ue1=2 x2=3 f ðgÞ; h ¼ gðgÞ; g ¼ Ue1=2 x1=3 y

ð6Þ

Thus, the velocity components u and m are given by 1 u ¼ Ue x1=3 f 0 ðgÞ; v ¼  Ue1=2 x1=3 ð2f  gf 0 Þ 3

ð7Þ

where primes denote differentiation with respect to g. Thus, we take 2 vw ðxÞ ¼  Ue1=2 x1=3 s 3

ð8Þ

where s is the dimensionless transpiration parameter with s [ 0 for suction and s \ 0 for injection, respectively. Substituting (6) into (2) and (3), we obtain the following ordinary differential equations 3f 000 þ 2ff 00 þ 1  f 02 ¼ 0

ð9Þ

3 00 g þ 2fg0 ¼ 0 Pr

ð10Þ

subject to the boundary conditions (4) which become f ð0Þ ¼ s;

f 0 ð0Þ ¼ k;

f ð1Þ ¼ 1;

gð1Þ ¼ 0

0

gð0Þ ¼ 1

ð11Þ

where k = Uw/Ue is the moving flat plate parameter. It should be mentioned that k [ 0 corresponds to a moving flat plate in the same direction with the free stream, k \ 0 corresponds to a moving flat plate in the opposite direction with the free stream and k = 0 corresponds to a stationary flat plate. Further, we notice that (10) is identical with (17) from the paper by Bataller [24] when the Eckert number Ec = 0, the thermal radiation parameter k0 = 1 and the surface temperature parameter m = 0 (constant surface temperature). Quantities of practical interest are the shear stress at the moving surface sw and the wall heat flux qw, which are given by     o u oT sw ¼ l ; qw ¼ k ð12Þ o y y¼0 o y y¼0 Using the non-dimensional variables (1) and the variables (6), we obtain sw ¼ lðRe1=2 =LÞUe3=2 f 00 ð0Þ; qw ¼ kDTðUe1=2 Re1=2 =LÞx1=3 ½g0 ð0Þ

ð13Þ

3 Results and discussion Numerical solutions to the governing ordinary differential equations (9) and (10) with the boundary conditions (11) are obtained using the shooting method. The equations have been solved for several values of the moving parameter k, mass flux parameter s with the values of the Prandtl number Pr = 0.7 (air), 1.0 and 7.0 (water). Figures 1 and 2 shows the variation of the reduced skin friction f 00 ð0Þ and heat flux from the surface of the flat plate g0 ð0Þ with k. It is seen that there are regions of unique solutions for k C -1.0, dual solutions for kc \ k B -1.0 and no solutions for k \ kc \ -1.0. Therefore, the solutions exist up to the critical value k = kc B -1.0, beyond which the boundary layer separates from the surface and the solution based upon the boundary layer approximations are not possible. Based on our computation, the minimum value of k (say kc) and the corresponding values of the skin friction f 00 ð0Þ and heat flux from the surface of the flat plate g0 ð0Þ for which a solution to the system of ordinary differential equations (9)–(11) exists are given in the Table 1. Table 2 presents the upper and lower branch values of f 00 ð0Þ and g0 ð0Þ denoted by fu00 ð0Þ; fl00 ð0Þ and g0u ð0Þ; g0l ð0Þ for two positive values of s when k = -1.2(opposing flow). Figures 3, 4, 5 and 6 display the variations of velocity and temperature profiles within the boundary layer for different values of s. It is evident from these figures that all curves approach the far field boundary conditions asymptotically, which support the validity of the numerical results obtained, besides supporting the

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2

λ

c1

1.5

λ

3/000 þ 2/00 þ

first solution second solution

s = 1.0

1

f′′ (0)

 1 2//00 þ 1  /02 ¼ 0 2 s 2 Pr 3h00 þ 2 Pr h0 þ 2 /h0 ¼ 0 s

c2

0.5

ð16Þ

with the boundary conditions

0.0

0.5

ð15Þ

-0.5 -1.0

0

/ð0Þ ¼ 0;

/0 ð0Þ ¼ k;

0

hð1Þ ¼ 0

/ ð1Þ ¼ 1;

hð0Þ ¼ 1

ð17Þ

-0.5 s = -1.0, -0.5, 0.0, 0.5, 1.0

-1

-1.5

-1

-0.5

0

0.5

1

1.5

2

λ

Fig. 1 Variation of f 00 ð0Þ with k for various values of s

where primes denote now the differentiation with respect to t. For large suction, / and u can be expressed in power series of 1/s as / ¼ /0 ðtÞ þ

1

λ

c1

λ

c2

0.9

h ¼ h0 ðtÞ þ

1 h1 ðtÞ þ    s2 ð18Þ

Substituting (18) into (15) and (16), and equating the coefficients of successive powers of 1/s2, we obtain two systems of linear differential equations with the corresponding boundary conditions. Solving analytically these equations for Pr = 1, we obtain

s = -1.0, -0.5, 0.0, 0.5, 1.0

0.8 0.7 0.6

-g′(0)

1 / ðtÞ þ    ; s2 1

0.5 0.4 0.3 0.2 first solution second solution

0.1 0

-1.5

-1

-0.5

0

0.5

1

1.5

2

λ

Fig. 2 Variation of g0 ð0Þ with k for various values of s when Pr = 0.7

existence of the dual solutions shown in Table 2 as well as in Figs. 1 and 2. It should be mentioned that the upper brunch solutions fu00 ð0Þand g0u ð0Þ are stable and physically realizable, while the lower brunch solutions fl00 ð0Þ and 0 gl ð0Þare not stable and not physically realizable. The procedure for showing this has been described by Weidman et al. [17], Merkin [27], Harris et al. [28] and Postelnicu and Pop [29], so that we will not repeat it here.

2 /00 ¼ 1  ð1  kÞe2t=3 ; h0 ¼ e2t=3 3   3 0 /1 ¼ ð1  kÞ2 e2t=3  e4t=3 8  1 2 2t=3 þ ð1  kÞ ð1 þ kÞt þ t e 3     3 1 2 2t=3 4t=3 2t=3 h1 ¼ ð1  kÞ e e  kt þ t e 4 3

ð19Þ

From (19), the values of f 00 ð0Þ and g0 ð0Þ can be expressed as 2s ð1  kÞð5 þ 3kÞ 1 ð1  kÞ þ s þ Oðs3 Þ 3 4 2s 1 þ k 1 s þ Oðs3 Þ g0 ð0Þ ¼ þ 3 2 f 00 ð0Þ ¼

ð20Þ ð21Þ

for large s(1). 4.2 Large injection (|s|1)

4 Asymptotic solutions 4.1 Large suction (s  1) Following Watson [30] in solving the general two-dimensional boundary layer flow with a uniform velocity of suction for s  1, we introduce the new variables 1 f ðgÞ ¼ s þ /ðtÞ þ    ; s

gðgÞ ¼ hðtÞ;

t ¼ sg

Substituting (14) into (9) and (10), we obtain

123

ð14Þ

To solve this problem for large injection, Kubota and Fernandez [31] have shown that the boundary layer has to be divided into two regions: an inner region adjacent to the wall where the viscosity plays a minor role and an outer region where the transition occurs from the inner layer to the inviscid flow outside the boundary layer, respectively. A uniformly valid solution can be obtained by matching the solutions of these two regions. In order to compare our solution with exact numerical solutions for large injection, especially, with the values of f 00 ð0Þ and g0 ð0Þ, we follow

Heat Mass Transfer (2011) 47:1643–1649

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s kc

0.5

1.0

-1.2106

-1.5148

0.6321

1.0469

0.1032

0.2278

00

f ð0Þ 0

g ð0Þ

00

f′(η)

Table 1 The minimum values of k (i.e., kc) for which solution to the system of ordinary differential equations (9)–(11) exists for Pr = 1.0

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

00

Table 2 Values of fu ð0Þ; fl ð0Þ and g0u ð0Þ; g0l ð0Þ for different positive values of s when k = -1.2 (opposing flow) and Pr = 0.7 s

fu ð0Þ

00

fl ð0Þ

00

gu ð0Þ

0

gl ð0Þ

0

0.5

0.7632

0.5066

0.1473

0.0624

1.0

1.6957

0.2200

0.4639

0.0005

S = 0.5, 1.0

S = 0.5, 1.0

λ = -1.2

first solution second solution

0

5

10

η

15

Fig. 5 Velocity profiles f 0 ðgÞ for various values of s (suction) when k = -1.2

1 1.5 λ = 1.5

1.45

λ = -1.2

0.8

first solution second solution

0.7 0.6

g(η)

1.4 1.35 1.3

f′(η)

0.9

S = 0.5, 1.0

0.5 0.4 0.3

1.25

0.2

S = -1.0, -0.5, 0, 0.5, 1.0

1.2

0.1

1.15

0

S = 0.5, 1.0

0

5

1.1

10

η

15

1.05 1

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 6 Temperature profiles g(g) for various values of s (suction) and Pr = 0.7 when k = -1.2

η

Fig. 3 Velocity profiles f 0 ðgÞ for various values of s when k = 1.5 2

FðfÞ ¼ ½f 0 ðgÞ ;

GðfÞ ¼ gðgÞ;

1

ð22Þ

0.9

λ = 1.5

0.8

where s ¼ sð [ 0Þ. Then, (9) and (10) become pffiffiffiffi 3e F F 00 þ 2fF 0 þ 2ð1  FÞ ¼ 0 pffiffiffiffi 3 3eFG00 þ 2 Pr F fG0 þ eF 0 G0 ¼ 0 2

Pr = 7.0

0.7

Pr = 0.7

g(η)

0.6 0.5

S = -1.0, -0.5, 0, 0.5, 1.0

0.4

ð23Þ ð24Þ

subject to the boundary conditions

0.3

F ¼ k2 ; G ¼ 1 at f ¼ 1 F ! 1; G ! 0 as f ! 1

0.2 0.1 0

f ðgÞ pffiffi f ¼  ¼ ef ðgÞ s

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

η

Fig. 4 Temperature profiles g(g) for various values of s and Pr = 0.7 when k = 1.5

the method of Katagiri [32] and consider only solutions of an inner region. Thus, it is convenient to look for solutions of (9) and (10) subjected to the boundary conditions (11) by introducing new variables defined as

ð25Þ

To consider an inner (wall) solution, which is valid for fixed f and small eð 1Þ, we expand FðfÞ and GðfÞ in power series of e as F ¼ F0 ðfÞ þ eF1 ðfÞ þ    ð26Þ pffiffi G ¼ G0 ðfÞ þ eGðfÞ þ    Substitution of (26) into (23) and (24), and the boundary conditions (25), we obtain the following two sets of equations with the corresponding boundary conditions

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Heat Mass Transfer (2011) 47:1643–1649

Table 3 Comparison of numerical and approximate 00 0 values of f ð0Þ and g ð0Þ for large values of the suction parameter s when Pr = 1 and k = 1/2

s

00

0

f ð0Þ

g ð0Þ

Numerical

Eq. (20)

0.9251 1.5030

1.072916667 1.536458333

1.6036 2.8330

1.708333333 2.854166667

6

2.1233

2.135416667

4.1177

4.125000000

8

2.7627

2.768229167

5.4238

5.427083333

10

3.4116

3.414583333

6.7399

6.741666667

12

4.0659

4.067708333

8.0615

8.062500000

14

4.7236

4.724702381

9.3863

16

5.3833

5.384114583

10.7131

10.71354167

18

6.0446

6.045138889

12.0414

12.04166667

20

6.7069

6.707291667

13.3706

13.37083333

25

8.3656

8.365833333

16.6966

16.69666667

30

10.0270

10.02708333

20.0249

20.02500000

40

13.3536

13.35364583

26.6854

26.68541667

50

16.6829

16.68291667

33.3483

33.34833333

G00 ¼ 0

ð27Þ f ¼ 1;

F0 ¼ k ; G0 ¼ 1 at pffiffiffiffiffi 2fF10  2F1 þ 3 F0 F000 ¼ 0 pffiffiffiffiffi 3 2 Pr F0 fG01 þ 3F0 G000 þ F00 G00 ¼ 0 2 F1 ¼ 0; G1 at f ¼ 1

G0 ¼ 1; G1  0

f ð0Þ  0; gð0Þ  0

parameter s and the numerical values are compared in Table 3 with the asymptotic approximations (20) and (21). It can be seen from this table that both the numerical and approximate analytical solutions are in very good agreement, especially for large values of the parameter s. Therefore, we are confident that the present results are very accurate. 5 Conclusions

ð29Þ

Thus, we have 00

9.386904762

ð28Þ

The analytical solution of these equations can be expressed as F0 ¼ 1 þ ð1  k2 Þf; F1  0

Eq. (21)

2 4

fF00 þ 1  F0 ¼ 0 2

Numerical

ð30Þ

for large values of the injection parameter |s|1. Therefore, for large values of the transpiration parameter s, the asymptotic analyses yield

New similarity solutions for the problem of the steady boundary layer flow on a permeable moving surface parallel to a free stream with constant (isothermal) surface temperature have been numerically investigated. Both the moving surface and the free stream velocities are assumed proportional to the x1/3. Dual solutions are found to exist when the plate and the free stream move in the opposite directions and the paper highlights the conditions for the existence of similar solutions. The effects of dissipation and buoyancy, which have been neglected in the present paper, are worthy of study, as well.

00

f ð0Þ 8 > < 2s ð1  kÞ þ ð1  kÞð5 þ 3kÞ s1 þ Oðs3 Þ for large suction 3 4 ¼ > : Oðs1 Þ for large injection ð31Þ

8 < 2s þ 1 þ k s1 þ Os3  for large suction 0 3 2 g ð0Þ ¼ : Oðs1 Þ for large injection

ð32Þ Equations (15) and (16) with the boundary conditions (17) are also solved numerically for large values of the suction

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Acknowledgments The authors wish to express their very sincerely thanks to the reviewers for the very good and interesting comments.

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