Mat et al. Mathematical Sciences 2012, 6:21 http://www.iaumath.com/content/6/1/21

ORIGINAL RESEARCH

Open Access

Radiation effect on Marangoni convection boundary layer flow of a nanofluid Nor Azian Aini Mat1, Norihan M Arifin1*, Roslinda Nazar2 and Fudziah Ismail1

Abstract Purpose: In this paper, we present a mathematical model for Marangoni convection boundary layer flow with radiation and different types of nanoparticles, namely, Cu, Al2O3, and TiO2 in a water-based fluid. Method: The governing equations in the form of partial differential equations have been reduced to a set of ordinary differential equations by applying suitable similarity transformations, which is then solved numerically using the shooting method. Results: 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, namely, the nanoparticle volume fraction φ, the constant exponent β, and thermal radiation parameter Nr. Conclusion: The results indicate that the heat transfer rate at the surface decreases as the thermal radiation parameter Nr increases. MSC: 76N20, fluid mechanics. Keywords: Marangoni convection, Boundary layer, Nanofluid, Thermal radiation

Introduction Recent advances in nanotechnology have allowed development of a new category of fluids termed nanofluids [1]. A nanofluid refers to a fluid that contains particles with dimensions less than 100nm. The base fluid, or dispersing medium, can be aqueous or non-aqueous in nature. Typical nanometer-sized particles are metals, oxides, carbides, nitrides, or carbon nanotubes. Their shapes may be spheres, disks, or rods [2]. Up to the moment this article is written, nanofluids have emerged numerous applications in the industry namely recognized as coolants, lubricants, heat exchangers, and microchannel heat sinks. The convective heat transfer mechanisms of nanofluids are the subject of considerable works and are well understood today. Numerical studies on natural convection heat transfer in nanofluids have been discussed in [3-11].

* Correspondence: [email protected] 1 Institute for Mathematical Research and Department of Mathematics, Universiti Putra Malaysia, UPM Serdang, Selangor 43400, Malaysia Full list of author information is available at the end of the article

Marangoni convection which is induced by the variations of the surface tension gradients has important application in the fields of welding and crystal growth. Moreover, the convection is necessary to stabilize the soap films and drying silicon wafers. Many researchers have investigated Marangoni convection in various geometries such as [12-17]. Recently, Arifin et al. [18] has studied a similar solution for Marangoni boundary layer flow of a nonofluid. They discussed the existence of dual solutions in Marangoni convection boundary layer, which is consistent with the discussion given in Golia and Viviani [12] that for the constant exponent β < 0.5, the solutions are not unique. Detailed results on the dual solutions in Marangoni boundary layer flow also can be found in Hamid et al. [19]. The aim of the present paper is to extend the Marangoni boundary layer problem of nanofluids first considered by Arifin et al. [18], with the effect of thermal radiation. The different types of nanoparticles, namely Cu, Al2O3, and TiO2 in a water-based fluid are tested to investigate the effect of the solid volume fraction parameter φ of the nanofluid on the heat transfer characteristics.

© 2012 Mat et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Problem formulation

Consider the steady two-dimensional laminar Marangoni convection boundary layer flow in a water-based nanofluid containing Cu, Al2O3, and TiO2 nanoparticles. The interface temperature is assumed to be a power-law function of the distance x along the interface. Further, we consider a Cartesian coordinate system (x, y), where x and y are the coordinates measured along the interface S and normal to it as it is shown in Figure 1. The steady boundary layer equations for a nanofluid in non-dimensional form can be written as @u @v þ ¼0 @x @y u u

ð1Þ

@u @u @ue μnf @ 2 u þ þv ¼ ue @x ρnf @y2 @x @y

ð2Þ

@T @T 1 knf =kf @2T 
 þv ¼
@x @y Pr ρCp nf = ρCp f @y2 1 @qr  
ρCp nf @y

ð3Þ

subject to the boundary conditions μnf @u @T ¼ at μf @y @x u ¼ ue ðxÞ; T ¼ Te as y ! 1 v ¼ 0; T ¼ T0 ðxÞ;

y¼0

ð4Þ

Here, u and v are the non-dimensional velocity components along the x and y axes, respectively. T is the non-dimensional temperature of the nanofluid.T0(x) is the interface temperature distribution. ue(x) is the velocity of the external flow. Te is the constant temperature of the external flow. Pr is the Prandtl number. Cp is the specific heat at constant pressure (ρCp)nf is the heat capacitance of the nanofluid. αnf is the thermal diffusivity of the nanofluid. ρnf is the effective density of the nanofluid. knf is the effective thermal conductivity of the

nanofluid, and μnf is the effective viscosity of the nanofluid, which are given by the equation below: knf μf  ; ρ ¼ð1  φÞρf þ φ ρs ; μnf ¼ ρ Cp nf nf ð1  φÞ2:5


 ρ Cp nf ¼ ð1  φÞ ρ Cp f þ φ ρ Cp s ; αnf ¼


knf ðks þ 2 kf Þ  2 φðkf  ks Þ ¼ kf ðks þ 2 kf Þ þ φðkf  ks Þ

ð5Þ

where φ is the solid volume fraction of the nanofluid. ρf is the reference density of the fluid fraction. ρs is the reference density of the solid fraction. μf is the viscosity of the fluid fraction. kf is the thermal conductivity of the fluid, and ks is the thermal conductivity of the solid. The last condition of Equation 4 represents the Marangoni condition at the interface (balance of the surface tangential momentum), having considered for the surface tension σ the linear relation: σ ¼ σ 0 ½1  γ ðT  Te Þ

ð6Þ

where γ = − (1/σ0) @ σ/ @ T > 0 is the temperature coefficient of the surface tension, and σ0 is the constant surface tension at the origin. The directions of the driving actions depend on the orientation of the temperature gradients in nanofluids, rT. We use the Rosseland approximation for radiation of an optically thick boundary layer given by Raptis et al. [20] and Cortel [21] in a simplified radiative heat flux form as qr ¼ 

4 σ  @T 4 3 k  @y

ð7Þ

where σ*and k* are the Stefan-Boltzman constant and Rosseland mean absorption coefficient, respectively. It is assumed that the temperature differences within the flow, T4, may be expressed as linear functions of temperature. This is accomplished by expanding T4 in a Taylor's series about T∞ and neglecting higher-order terms, thus 3 4 T 4  4T1 T  3T1

ð8Þ

Substituting Equations 7 and 8 into Equation 3 in the appropriate form leads to u

@T @T @2T þv ¼ αnf ð1 þ Nr Þ 2 @x @y @y

ð9Þ

where Nr = 16σ * T3∞/(3knfk * ) is the radiation parameter. Following Golia and Viviani [12], we look for a similarity solution of Equations 1 to 3 subject to the boundary conditions (Equation 4). In this respect, we assume that ue(x) and T0(x) have the following form: Figure 1 Physical model and coordinate system.

ue ðxÞ ¼ u0 xð2β1Þ=3 ; T0 ðxÞ ¼ h0 xβ

ð10Þ

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The transformed ordinary differential equations are the following:

Table 1 Thermophysical properties of regular fluid and nanoparticles Physical properties

Fluid phase (water)

Cu

A12O3

TiO2

Cp(J/kgK)

4179

385

765

686.2

ρ(kg/m3)

997.1

8933

3970

4250

k(W/mK)

0.613

400

40

8.9538

2:5


u0 ¼

 β

2=3

; l0 ¼

3 1þβ

1=3 β

1=3

ð13Þ

ð14Þ

along with the boundary conditions 1

f ð0Þ ¼ 0;

where primes denote differentiation with respect to η; u0, h0and l0 are constants. If we take h0 = 1, then u0 and l0 have the following unique values: 1=3

00

ð1 þ Nr Þ knf =kf 00 h ig

 Pr ð1  φÞ þ φ ρ Cp s = ρCp f   3β f 0g ¼ 0 þ fg 0  1þβ

uðx; yÞ ¼ u0 xð2 β1Þ=3 f 0 ðηÞ; 1 vðx; yÞ ¼ u0 l0 xðβ2Þ=3 ½ð2  βÞη f 0 ðηÞ  ð1 þ βÞf ðηÞ 3 Tðx; yÞ ¼ Te  h0 xβ gðηÞ; η ¼ xðβ2Þ=3 y=l0 ð11Þ

3 1þβ

000

 f þ ff

ð1  φÞ 1  φ þ φ ρs =ρf    2β  1  0 2 f 1 ¼0  1þβ

where β is the constant exponent. We look now for a similarity solution of Equations 2 and 3 of the following form:



1

00

2:5

f ð0Þ ¼ 1; gð0Þ ¼ 1

ð15Þ

ð1  φÞ f 0 ð1Þ ¼ 1; gð1Þ ¼ 0

Methods Equations 13 and 14 with the boundary conditions (Equation 15) are solved numerically using the shooting

ð12Þ

Table 2 Values of f0(0) and − g0(0) for different nanoparticles Cu f0(0) β

Arifin et al. [18]

Present

Al2O3 −g0(0)

Arifin et al. [18] Nr = 0

f0(0)

Present

Nr = 0

Nr = 1

0.01 51.7301 51.7301 12.0843 12.0843

8.2527

Arifin et al. [18]

Present

TiO2 −g0(0)

Arifin et al. [18] Nr = 0

f0(0)

Present

Nr = 0

Nr = 1

52.2969 52.2969 12.2781 12.2781

8.4635

Arifin et al. [18]

Present

−g0(0) Arifin et al. [18] Nr = 0

Present

Nr = 0

Nr = 1

52.2568 52.2568 12.4948 12.4948

8.6158

(1.1779) (1.1779) (1.7311) (1.7312) (1.1850) (1.2132) (1.2514) (1.7968) (1.7927) (1.2242) (1.2462) (1.2462) (1.8229) (1.8229) (1.2452) 0.05 10.1729 10.1729

5.7803

5.7803

3.9723

10.4523 10.4523

5.9095

5.9095

4.0905

10.4327 10.4327

6.0103

6.0103

4.1614

(1.0468) (1.0469) (1.7436) (1.7437) (1.1957) (1.1083) (1.1083) (1.7991) (1.7991) (1.2289) (1.1040) (1.1040) (1.8298) (1.8298) (1.2502) 0.1

4.9939

4.9939

4.3621

4.3621

3.0132

5.2150

5.2150

4.4883

4.4883

3.1168

5.1996

5.1996

4.5624

4.5624

3.1690

(0.8970) (0.8972) (1.7111) (1.7113) (1.1725) (0.9544) (0.9544) (1.7678) (1.7678) (1.2053) (0.9504) (0.9504) (1.7983) (1.7983) (1.2261) 0.15

3.3762

3.3762

3.8029

3.8029

2.6355

3.5693

3.5693

3.9330

3.9330

2.7362

3.5559

3.5559

3.9962

3.9962

2.7808

(0.7613) (0.7613) (1.6295) (1.6295) (1.1104) (0.8232) (0.8232) (1.6993) (1.6993) (1.1545) (0.8193) (0.8193) (1.7288) (1.7288) (1.1737) 0.2

2.6529

2.6529

3.5362

3.5362

2.4558

2.8241

2.8241

3.6674

3.6674

2.5542

2.8123

2.8123

3.7256

3.7256

2.5953

(0.6520) (0.6520) (1.5189) (1.5189) (1.0231) (0.7451) (0.7451) (1.6542) (1.6543) (1.0947) (0.7381) (0.7381) (1.6777) (1.6778) (1.1114) 0.25

2.2727

2.2727

3.4078

3.4078

2.3697

2.4255

2.4255

3.5370

3.5370

2.4651

2.4149

2.4149

3.5928

3.5928

2.5045

(0.5884) (0.5884) (1.4250) (1.4250) (0.9339) (0.7025) (0.7025) (1.6312) (1.6312) (1.0513) (0.6939) (0.6939) (1.6507) (1.6507) (1.0652) 0.3

2.0496

2.0496

3.3507

3.3507

2.3323

2.1874

2.1874

3.4765

3.4765

2.4241

2.1778

2.1778

3.5314

3.5314

2.4628

(0.5584) (0.5584) (1.3269) (1.3269) (0.8690) (0.6880) (0.6880)

(1.6260 (1.6261) (1.0483) (0.6816) (0.6816) (1.6523) (1.6523) (1.0603)

0.5

1.6848

1.6848

3.3672

3.3672

2.3487

1.7879

1.7879

3.4802

3.4803

2.4296

1.7806

1.7806

3.5359

3.5359

2.4689

1.0

1.4741

1.4741

3.6070

3.6070

2.5212

1.5500

1.5500

3.7075

3.7075

2.5919

1.5447

1.5447

3.7679

3.7679

2.6345

2.0

1.3818

1.3818

3.8928

3.8928

2.7245

1.4443

1.4443

3.9870

3.9870

2.7903

1.4399

1.4399

4.0528

4.0528

2.8367

5.0

1.3281

1.3281

4.1763

4.1763

2.9257

1.3824

1.3824

4.2671

4.2671

2.9887

1.3786

1.3786

4.3381

4.3381

3.0388

∞

1.2921

1.2921

4.4484

4.4482

3.1164

1.3408

1.3409

4.5370

4.5368

3.1777

1.3374

1.3374

4.6129

4.6127

3.2312

Nr = 0 and 1, Pr = 6.2, and φ = 0.1. Results in parentheses are the second solutions.

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method in the Maple programming language; a commonly used numerical method for the solution of twopoint boundary value problems [22]. The basic idea of this method is to reformulate the problem as a nonlinear parameter estimation problem which requires the solution of a related initial value problem with initial condition chosen to approximate the boundary conditions at the other endpoint. If these boundary conditions are not satisfied to the desired accuracy, the process is repeated with a new set of initial conditions until the desired accuracy is achieved or an iteration limit is reached.

Results and discussion Following Oztop and Abu-Nada [5], we considered the range of nanoparticle volume fraction as 0≤φ≤0:2 . The Prandtl number, Pr, of the based fluid (water) is kept constant at 6.2. Further, it should also be pointed out that the thermophysical properties of fluid and nanoparticles used in this study are given in Table 1. Table 2 presents the comparison for the values of the interface velocity f’(0) and the surface temperature gradient, − g’(0), with those reported by Arifin et al. [18], which shows an excellent agreement for Nr = 0. Table 2 shows the variations of the surface temperature gradient, − g’(0), that represents the heat transfer rate at the surface with β for different types of nanoparticles with ϕ = 0.1 and Nr = 1. Results given in the parentheses are the second (dual) solutions. We found that the values of the surface temperature gradient, − g’(0), decrease as Nr increases. Figures 2 and 3 indicate also that the surface

Figure 2 Variation of − g’(0) with Nr for different types of nanoparticles. φ = 0.1, Pr = 6.2 and β = 1.

Figure 3 Variation of − g’(0) with φ for different Nr and nanoparticles. Pr = 6.2 and β = 1.

temperature gradient, − g’(0), decreases with the increasing values of Nr for all nanoparticles considered. Thus, the heat transfer rate at the surface decreases in the presence of radiation. 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 temperature of the fluid. Figures 4 5 6 show the temperature g(η) profiles for different values of

Figure 4 Temperature profiles of g(η) for Cu nanoparticles. φ = 0.1, β = 0.2 and various values of Nr.

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Al2O3, and TiO2 nanoparticles in a water-based fluid. The governing partial differential equations were transformed into a set of two nonlinear ordinary differential equations using similarity transformation before being solved numerically by the shooting method. Numerical results are obtained for the surface temperature gradient, − g’(0), as well as the temperature profiles for some values of the governing parameters, namely, the solid volume fraction of the nanofluid φ (0≤ϕ≤0:2), the constant exponent β, and the thermal radiation parameter Nr. It was observed that increasing the thermal radiation parameter decreases the heat transfer rate − g(0) at the surface. Competing interests The authors declare that they have no competing interests.

Figure 5 Temperature profiles g(η) for Al2O3 nanoparticles. φ = 0.1, β = 0.2 and various values of Nr.

Nr when ϕ = 0.1 and β = 0.2 with different type of nanoparticles, namely, Cu, Al2O3, and TiO2, respectively. It is seen that the temperature profiles increase as radiation parameter Nr increases for all nanoparticles considered. Thus, the radiation can be used to control the thermal boundary layers quite effectively.

Conclusions In this paper, we studied the effects of thermal radiation on Marangoni convection boundary layer flow for Cu,

Authors' contributions NM participated in the design of the study and performed the analysis of the results. NMA carried out the boundary layer studies and participated in the sequence alignment. RN and FI participated in its design and coordination. All authors read and approved the final manuscript. Authors' information NM obtained her first degree in Mathematics in year 2000 then her Masters degree in Applied Mathematics in year 2002 from Universiti Kebangsaan, Malaysia. Currently, she is doing her PhD in Fluid Dynamics at the Institute for Mathematical Research, Universiti Putra, Malaysia. Acknowledgment The authors gratefully acknowledged the financial support received in the form of a FRGS research grant from the Ministry of Higher Education, Malaysia. Author details 1 Institute for Mathematical Research and Department of Mathematics, Universiti Putra Malaysia, UPM Serdang, Selangor 43400, Malaysia. 2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor 43600, Malaysia. Received: 1 March 2012 Accepted: 20 July 2012 Published: 16 August 2012

Figure 6 Temperature profiles g(η) for TiO2 nanoparticles. φ = 0.1, β = 0.2 and various values of Nr.

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