Meccanica (2013) 48:83–89 DOI 10.1007/s11012-012-9585-7

Radiation effects on Marangoni convection boundary layer over a permeable surface N.A. Mat · N.M. Arifin · R. Nazar · F. Ismail · I. Pop

Received: 21 April 2011 / Accepted: 31 July 2012 / Published online: 14 August 2012 © Springer Science+Business Media B.V. 2012

Abstract The problem of Marangoni convection boundary layer flow that can be formed along the interface of two immiscible fluids when the wall is permeable, where there is suction or injection effect, is considered. Similarity equations are obtained through the application of similarity transformation techniques. The effects of suction/injection and radiation parameters on the heat transfer characteristics are numerically studied using the shooting method for a fixed value of the Prandtl number (Pr = 0.7). Numerical results are obtained for the surface temperature gradient or the heat transfer rate as well as the temperature profiles for some values of the governing parameters. Comparisons with known results from the open literature show very good agreements. The results indicate that

N.A. Mat Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia N.M. Arifin () · F. Ismail Department of Mathematics, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia e-mail: [email protected] R. Nazar School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia I. Pop Department of Mathematics, Babe¸s-Bolyai University, 400084 Cluj-Napoca, Romania

the heat transfer rate at the surface decreases as the radiation parameter increases. Further, results show that multiple (dual) solutions exist for a certain range of the governing parameters. Keywords Marangoni convection · Boundary layer · Suction/injection · Radiation

1 Introduction The Marangoni effect is a general term for a family of flows induced by some interfacial tension gradient associated with either chemical or thermal gradients with numerous applications and ramifications (Bahadur et al. [1]). The study of Marangoni convection has attracted the interest of many researchers in recent years because of its vast contributions in various industries. Marangoni convection induced by variation of the surface tension with temperature along a surface influences the crystal growth melts and other processes with liquid-liquid or liquid-gas interfaces. The surface tension gradients for Marangoni convection can be temperature and/or concentration gradients. On the other hand, numerical studies for the Marangoni boundary layers have been discussed in Golia and Viviani [2, 3], Napolitano and Golia [4], Pop et al. [5] and Chamkha et al. [6]. However, an excellent review on the Marangoni flow has been done by Tadmor [7]. Al-Mudhaf and Chamkha [8] studied the

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thermosolutal Marangoni convection along a permeable surface with heat generation or absorption and a first-order chemical reaction effects. Magyari and Chamkha [9] reported exact analytical solutions for the velocity, temperature and concentration fields of steady thermosolutal magnetohydrodynamic (MHD) Marangoni convection. The existence of dual solutions in Marangoni convection boundary layer has been discussed in Arifin et al. [10], which is consistent with the discussion given in Golia and Viviani [2] that for the constant exponent β < 0.5, the solutions are not unique. Detailed results on the dual solutions in Marangoni boundary layer flow can be found in Hamid et al. [11]. The radiative effects have important applications in physics and engineering particularly in space technology and high temperature processes (Mukhopadhyay and Layek [12]). Recently, there is a lot of work on boundary layer flow involving radiation (Chamkha et al. [13], Chaudhary et al. [14], Bataller [15], Ishak [16]). Ishak [17] investigated the thermal boundary layer flow induced by a linearly stretching sheet in a micropolar fluid with radiation effect and he showed that the existence of thermal radiation is to reduce the heat transfer rate at the surface. On the other hand, Chen [18] considered the MHD boundary layer problem for momentum and heat transfer with dissipative energy, thermal radiation and internal heat source/sink in viscoelastic fluid flow over a porous stretching sheet, while Mukhopadhyay and Layek [12] dealt with free convective flow and radiative heat transfer of viscous incompressible fluid having variable viscosity over a stretching porous vertical plate. Further, Ishak et al. [19] studied the steady laminar boundary layer flow over a moving plate in a moving fluid with convective surface boundary condition in the presence of thermal radiation. They showed that the heat transfer rate at the surface decreases and dual solutions are found to exist when the plate and the fluid move in the opposite directions. Very recently, Hamid et al. [20] presented the numerical solutions for the Marangoni flow over a permeable flat surface with the effect of thermal radiation. Motivated by these previous works, we aim to study the Marangoni boundary layer flow that can be formed along the interface of immiscible fluids in the surface driven flows due to an imposed temperature gradient with radiation. The transformed ordinary differential equations are solved numerically for some values of

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the governing parameters. The effects of radiation parameter and the suction and injection parameter on the heat transfer characteristics are presented and discussed. To the best of our knowledge, this specific problem has not been considered before.

2 Mathematical formulation Consider the steady two-dimensional flow along the interface S of two Newtonian immiscible fluids where x and y are the axes of a Cartesian coordinate system. Under the usual boundary layer approximations, the basic governing equations are ∂u ∂v + = 0, ∂x ∂y

(1)

u

∂u ∂u ∂ 2u due +v = ue +ν 2, ∂x ∂y dx ∂y

(2)

u

∂T k ∂ 2T ∂T 1 ∂qr +v = − 2 ∂x ∂y ρcp ∂y ρcp ∂y

(3)

with the boundary conditions, v = −vw , T = T0 (x), ∂u ∂T μ =σ on y = 0, ∂y ∂x u → ue (x),

T → T∞

(4) as y → ∞,

(5)

where u and v are the velocity components along the x and y axes, respectively, ue (x) is the external velocity, ν is the kinematic viscosity, T is the fluid temperature, k is the thermal conductivity, cp is the specific heat of the fluid at a constant pressure, ρ is the fluid density, qr is the component of radiative heat flux, vw is the transpiration (suction or injection) velocity, T0 (x) is the interface temperature distribution, μ is the dynamic viscosity and T∞ is the constant temperature of the external flow. The last condition of (4) represents the Marangoni coupling condition at the interface (balance of the surface tangential momentum), having considered for the surface tension σ the linear relation, see Chamkha et al. [13],   (6) σ = σ0 1 − γ (T − T∞ ) where γ = −(1/σ0 )∂σ/∂T > 0 is the temperature coefficient of the surface tension and σ0 is the constants surface tension at the origin. The directions of the driving actions depend on the orientation of the temperature gradient.

Meccanica (2013) 48:83–89

85

The radiative heat flux, qr is described by the Rosseland approximation [21] such that 4σ1 ∂T 4 (7) qr = − 3αR ∂y where σ1 and αR are the Stefan-Boltzmann constant and the Rosseland mean absorption coefficient, respectively. Following Ishak [16] and others, we assume that the temperature differences within the flow are sufficiently small so that the T 4 can be expressed as a linear function after using Taylor series to expand T 4 about the free stream temperature T∞ and neglecting higher-order terms. This results in the following approximation: 4 4 T 4 ≈ 4T∞ T − 3T∞ .

(8)

Using (7) and (8), we obtain the last term of (3) 3 ∂ 2T 16σ1 T∞ ∂qr =− . ∂y 3αR ∂y 2

(9)

To proceed with the analysis, we assume that ue (x) and T0 (x) have the following form (see Golia and Viviani [2]): ue (x) = u0 x (2β−1)/3 ,

T0 (x) = −h0 x β

(10)

where β is the constant exponent. We look now for a similarity solution of Eqs. (2) and (3) of the following form: u(x, y) = u0 x (2β−1)/3 f  (η), 1 v(x, y) = u0 l0 x (β−2)/3 3  × (2 − β)ηf  (η) − (1 + β)f (η) , T (x, y) = T∞ − h0 x β g(η),

(11)

η = x (β−2)/3 y/ l0

where primes denote differentiation with respect to η and u0 , h0 and l0 are constants. If we take h0 = 1 then u0 and l0 have the following unique values: 1/3 1/3   3 3 u0 = β 2/3 , l0 = β −1/3 . 1+β 1+β (12) Substituting (11) into Eqs. (2) and (3), we get the following ordinary differential equations:  2β − 1  f  + ff  + 1 − f 2 = 0, (13) β +1 (1 + Nr)  3β  θ + f θ − f θ = 0, (14) Pr 1+β along with the boundary conditions f (0) = f0 ,

f  (0) = −1,

θ (0) = 1,

(15)

f  (∞) = 1,

θ (∞) = 0.

(16)

The dimensionless parameters Pr, Nr and f0 appearing in the above equations, represent, respectively, the Prandtl number, thermal radiation parameter and the suction/injection parameter, where f0 > 0 is the constant suction parameter and f0 < 0 is the constant injection parameter.

3 Results and discussion The exact momentum solution provides a general formula for the Marangoni flow (Golia and Viviani [2]). The velocity fields, i.e. the momentum equation solutions, have been discussed in Ref. [11] in details. This present paper focuses on the heat transfer problem with thermal radiation. The nonlinear ordinary differential equations (13) and (14) subject to the boundary conditions (15) and (16) were solved numerically using the shooting method. To validate the accuracy of the present results, the results for the interface velocity f  (0) and the surface temperature gradient −θ  (0) without radiation are compared with those of Hamid et al. [11] as presented in Table 1. They are found to be in very good agreement. Table 2 shows the comparison values of the interface velocity f  (0) and the surface temperature gradient −θ  (0) for various values of the radiation parameter, Nr when Pr = 0.7. Second solutions are given in parenthesis. One can see that the surface temperature gradient, −θ  (0) decreases as Nr increases. This is also illustrated in Fig. 1 for the variation of the surface temperature gradient with Nr. Thus, the presence of radiation will decrease the heat transfer rate at the surface. This result qualitatively agrees with expectations, since the effect of radiation is to decrease the rate of energy transport to the fluid, thereby decreasing the fluid temperature. We also found that, imposition of fluid suction (f0 > 0) at the surface has the tendency to decrease the surface temperature gradient. However, fluid injection (f0 < 0) increases the surface temperature gradient. These are also clearly shown in Fig. 1. Figure 2 presents the variation of the surface temperature gradient with β, when Pr = 0.7, Nr = 2 and with the effect of the parameter f0 . The dashed line refers to the second solution of the problem. From this figure, we can see that the second solutions exist for β < 0.5. This finding is consistent with the discussion given by Hamid et al. [11] without radiation effect. It can be

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Table 1 Comparison values of f  (0) and −θ  (0) with Pr = 0.7. Results in parenthesis are the second (dual) solutions β

0.1

f0

−1 0 1

0.2

−1 0 1

0.3

−1 0 1

0.4

−1 0 1

Hamid et al. [11]

Present

f  (0)

f  (0)

−θ  (0)

β

f0

−θ  (0)

20.42222

2.73049

20.42223

2.73049

(7.52155)

(1.39352)

(7.52156)

(1.39352)

5.63217

1.66616

5.63218

1.66617

(1.13296)

(0.63301)

(1.13296)

(0.63301)

2.18587

1.51789

2.18588

1.51789

(0.32671)

(0.81825)

(0.32671)

(0.81825)

6.88979

1.70167

6.88979

1.70168

(3.22734)

(0.89556)

(3.22734)

(0.89556)

3.14160

1.40729

3.14162

1.40729

(0.96668)

(0.58243)

(0.96668)

(0.58244)

1.87556

1.54931

1.87557

1.54930

(0.23194)

(0.65695)

(0.23195)

(0.65696)

4.09080

1.38372

4.09082

1.38372

(2.49526)

(0.82414)

(2.49526)

(0.82414)

2.44523

1.34800

2.44525

1.34799

(0.97452)

(0.54113)

(0.97453)

(0.54113)

1.73727

1.59029

1.73728

1.59029

(0.24512)

(0.43442)

(0.24513)

(0.43442)

3.09833

1.26054

3.09836

1.26054

(2.2474)

(0.14478)

(2.24744)

(0.14478)

2.14605

1.34240

2.14606

1.34239

(1.08908)

(0.35840)

(1.08909)

(0.35840)

1.65874

1.62937

1.65875

1.62936

(0.34764) (−1.28801)

Table 2 Values of −θ  (0) with Pr = 0.7. Results in parenthesis are the second (dual) solutions

0.1

−1 0 1

0.2

−1 0 1

0.3

−1 0 1

0.4

−1 0 1

Nr = 0

Nr = 1

Nr = 2

−θ  (0)

−θ  (0)

−θ  (0)

2.73049

1.81099

1.38882

(1.39558)

(0.82961)

(0.58107)

1.66617

1.09571

0.84873

(0.63301)

(0.43217)

(0.35075)

1.51789

0.94000

0.71614

(0.81825)

(0.48720)

(0.37642)

1.70168

1.18832

0.94398

(0.89606)

(0.57255)

(0.43586)

1.40729

0.94685

0.74685

(0.58244)

(0.39179)

(0.31795)

1.54930

0.96944

0.74384

(0.65696)

(0.38309)

(0.29963)

1.38372

0.99419

0.80517

(0.82891)

(0.53427)

(0.40999)

1.34799

0.91636

0.72863

(0.54143)

(0.34569)

(0.27591)

1.59029

1.00233

0.77257

(0.43464)

(0.25428)

(0.20734)

1.26054

0.91854

0.75129

(0.14478)

(0.48636)

(0.38143)

1.34239

0.91758

0.73266

(0.35840)

(0.28087)

(0.22714)

1.62936

1.03249

0.79835

(0.12012)

(0.07189)

(−1.28802)

(0.34764) (−1.28802)

0.5 −1 0

2.63924

1.21201

2.63927

1.21200

−1

1.21200

0.88928

0.73112

1.98427

1.35417

1.98428

1.35415

0

1.35415

0.92872

0.74342

1

1.60781

1.66444

1.60781

1.66443

1

1.66443

1.05908

0.82086

1.0 −1 0

1.98027

1.21194

1.98027

1.21193

−1

1.21193

0.90138

0.74144

1.69687

1.44233

1.69688

1.44233

0

1.44233

0.99599

0.80123

1

1.49425

1.78801

1.49425

1.78801

1

1.78801

1.15101

0.89784

2.0 −1 0

1.74213

1.29145

1.74014

1.28971

−1

1.28971

0.90633

0.77979

1.56595

1.55061

1.56595

1.55061

0

1.55061

1.03937

0.86665

1

1.42871

1.90694

1.42871

1.90694

1

1.90694

1.23814

0.97018

5.0 −1 0

1.61621

1.38658

1.61621

1.38658

−1

1.38658

0.99843

0.83830

1.48857

1.65943

1.48857

1.65943

0

1.65943

1.04964

0.84863

1

1.38478

2.01946

1.38478

2.01946

1

2.01946

1.31993

1.03778

0.5

1.0

2.0

5.0

Meccanica (2013) 48:83–89

Fig. 1 Variation of the surface temperature gradient −θ  (0) with Nr when Pr = 0.7 and β = 0.35

87

Fig. 3 Effects of f0 on the temperature profiles when Pr = 0.7, β = 0.35 and Nr = 2

Fig. 2 Variation of the surface temperature gradient −θ  (0) with β when Pr = 0.7 and Nr = 2

observed from Fig. 2 that the effects of suction or injection parameter can be clearly seen when β > 0.4. Figure 3 shows the effects of the suction/injection parameter f0 on the temperature profiles when Pr = 0.7, β = 0.35 and Nr = 2, while Figs. 4, 5, 6 show the effects of the radiation parameter Nr on temperature profiles in the influence of the suction or injection parameter f0 , namely f0 = 0, −1 and 1, respectively. From Figs. 4, 5, 6, we observe that the temperature profiles increase as radiation parameter Nr increases. Thus, radiation can be used to control the thermal boundary

Fig. 4 Effects of Nr on the temperature profiles when Pr = 0.7, β = 0.35 and f0 = 0

layers quite effectively. Figures 3, 4, 5, 6 display the samples of temperature profiles for the dual solutions, namely the first and second solutions. These profiles prove the existence of dual solutions and they satisfy the far field boundary conditions (12) and (13) asymptotically, which support also the numerical results obtained. However, it is worth mentioning that these dual solutions for Eqs. (13) and (14) corresponding to a

88

Meccanica (2013) 48:83–89

cedure for showing the details of the stability analysis has been described by Merkin [22], Weidman et al. [23] and Postelnicu and Pop [24].

4 Conclusions

Fig. 5 Effects of Nr on the temperature profiles when Pr = 0.7, β = 0.35 and f0 = −1

Marangoni convection boundary layer flow that can be formed along the interface of two immiscible fluids in the presence of thermal radiation when the wall is permeable, where there is suction or injection effect, was analyzed numerically. The governing equations were transformed into ordinary differential equations using appropriate transformations, and were then solved numerically by the shooting method. Comparison with previously published work was performed and the results were found to be in excellent agreement. It was found that the solutions for the constant exponent or the similarity parameter β < 0.5 were non-unique. The effects of thermal radiation parameter and the suction or injection parameter on the temperature profiles were presented in graphical form and thoroughly examined. It could be drawn from the present results that when radiation parameter increased, the heat transfer rate at the surface −θ  (0) decreased. It was also shown that the imposition of suction was to decrease the surface temperature gradient, whereas injection showed the opposite effects. Acknowledgements The authors wish to express their thanks to the reviewers for the valuable comments and suggestions. The authors also gratefully acknowledged the financial supports received in the form of a FRGS research grant and a LRGS research grant (LRGS/TD/2011/UKM/ICT/03/02) from the Ministry of Higher Education, Malaysia.

References Fig. 6 Effects of Nr on the temperature profiles when Pr = 0.7, β = 0.35 and f0 = 1

Marangoni flow having upper and lower branch solutions. This is a very important new result for this fluid, which has not been reported in the open literature before. As in similar physical situations, we postulate that the upper branch solutions are physically stable and occur in practice, whilst the lower branch solutions are not physically realizable. This postulate can be verified by performing a stability analysis but this is beyond the scope of the present paper. The pro-

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