Dissipative MHD Boundary- Layer Flow in a Porous ... - CiteSeerX

Applied Mathematical Sciences, Vol. 4, 2010, no. 63, 3133 - 3142

Dissipative MHD Boundary- Layer Flow in a Porous Medium over a Sheet Stretching Nonlinearly in the Presence of Radiation Paresh Vyas Department of Mathematics, University of Rajasthan, Jaipur [email protected] Ashutosh Ranjan Department of Mathematics, University of Rajasthan, Jaipur [email protected] Abstract This communication pertains to the study of effect of thermal radiation and dissipation on heat transfer over stretching sheet placed at the bottom of fluid saturated porous medium. The fluid is assumed to be gray, emitting and absorbing radiation but non scattering medium. Appropriate similarity transformations are employed to transform the governing equations into ordinary differential equations. The momentum equation admits analytic solution of exponential form. This solution is used in energy equation which is then solved numerically by fourth order Runge-Kutta scheme together with Shooting method.

Mathematics Subject Classification: Keywords: Stretching sheet, Dissipation, MHD, Radiation, Porous medium

1

Introduction

The flow and heat transfer of an incompressible viscous fluid over a stretching sheet appear in numerous industrial technological processes, such as aerospace component production metal casting, the aerodynamic extrusion of plastic sheets, glass blowing and spinning are a few of them. The importance of the analysis of such flow problems lies in the fact that the mechanical properties of the final product are deeply influenced by the stretching rate and on the rate of cooling. Wide array of applications and simple geometry have invited rigorous investigations of flow problems due to stretching surface. Crane [14] pioneered

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a simple closed form exponential solution of the two- dimensional flow due to a linearly stretching sheet in an otherwise quiescent incompressible fluid. In the quest of further exact solutions, the problem was extended to more general situations such as power law stretching velocity, application to non-Newtonian fluids, heat transfer (stretching sheet subjected to constant temperature/ variable temperature/ heat flux). Gupta and Gupta [22] extended the work of Crane [14] to study heat and mass transfer over an isothermal stretching sheet with suction and blowing. Grubka and Bobba [10] assumed a power law temperature distribution along a linearly stretching sheet and obtained solution in terms of Kummer’s function. Similarly Chen and Char [2] also obtained their solution in terms of Kummer’s function while considering sheet with prescribed wall temperature and wall heat flux in the presence of suction. The flow due to a sheet stretching with power law velocity has been reported by Afzal and Varshney [19], Kuiken [7] and Banks [31]. Prompted by the fact that the study of magnetohydrodynamic flow of an electrically conducting fluid caused by deformation of the walls of the vessels containing the fluid is of cosiderable importance in metallurgical and metal working processes, the works of Pavlov [12], Chauhan and Vyas [3], Takhar and Nath [8], Vajravelu and Nayfeh [11] and Chiam [29] are worth mentioning. Other physical features such as viscoelasticity of the fluid, micropolar fluid, temperature dependent viscosity, viscous dissipation have been considered by Anderson [6], Troy et al. [30], Boutros et al. [32], Prasad et al. [13], Abel et al. [27] and Veena et al. [21]. All the above investigations were restricted to flow and heat transfer without taking radiation into account. In fact, radiative heat transfer cannot be overlooked in the technological processes involving high temperatures regime and consequently good working knowledge of it helps designing pertinent equipment. Though, radiative transport is often comparable with convective heat transfer but unfortunately the literature is scanty about the effects of radiation on the boundary- layer flow. It is pertinent to record that radiative fluid flow studies are confronted with a few difficulties such as inclusion of radiation term in energy equation makes it highly non- linear; radiation is absorbed/ emitted not only at system boundaries but also in the interior of the system, hence making prediction a difficult task. These physical- computational difficulties pertaining to radiative heat transfer are addressed with reasonable simplifications. A good literature on it can be found in the well presented texts by ¨ Ozisik [16], Sparrow and Cess [4], Seigel and Howell [24]. Following Rosseland approximation Plumb et al. [20] investigated the effect of horizontal cross flow and radiation on natural convection from vertical heated surface in saturated porous media. Ibrahim and Hady [5] investigated mixed convection radiation interaction in boundary- layer flow over a horizontal surface. Hossain et al. [15] determined the effect of radiation on natural convection flow of an optically thick viscous incompressible flow past a heated vertical porous plate.

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Shateyi et al. [28] studied MHD flow past a vertical plate with radiative heat transfer. Duwahiri et al. [9] presented numerical investigations of magnetohydrodynamic natural convection heat transfer from radiate vertical porous surfaces. Raptis [1] studied radiative micropolar fluid flow past a continuously moving plate. Gorla [25], Gorla and Pop [26] investigated the effects of radiation on mixed convection flow over vertical cylinders. Vyas and Srivastava [23] studied radiative MHD flow over a non- isothermal stretching sheet placed at the bottom of a porous medium. We aim in this paper to present a numerical study of forced convective radiative dissipative MHD boundary layar flow arising due to a sheet stretching nonlinearly at the bottom of a fluid saturated porous medium. The sheet is subjected to variable heat flux. Realising the qualitative importance of dissipation in contemporary industrial technology such as polymer industries, lubrication, food processing, instrumentation, it is hoped that the investigations presented here will help peep into analogous real world processes.

2

Mathematical Model

Let us consider the steady two-dimensional MHD radiative boundary layer flow of a viscous, incompressible, electrically conducting fluid in a fluid saturated porous medium. The flow is caused by a heated impermeable stretching sheet placed at the bottom of the porous medium. A cartesian co- ordinate system is used. The x− axis is along the sheet and the y− axis is taken normal to it. Two equal and opposite forces are applied along the sheet so that the position of the origin is unaltered. The stretching velocity varies nonlinearly with the distance from origin. A variable magnetic field B(x) of specified form is applied transverse to the sheet along the y− axis in the opposite direction of gravity. The induced magnetic field is neglected, which is valid for small magnetic Reynolds number. We assume that the wall is subjected to a variable heat flux. Assuming the fluid to be Newtonian, without phase change and gray, we further assume that both the fluid and the porous medium are in local thermal equilibrium. Rosseland approximation [17] is assumed to account for radiating heat flux. The radiative MHD boundary- layer flow taking viscous and Ohmic dissipations into account is given by the following equations:

u

∂u ∂v + = 0, ∂x ∂y

(1)

∂u ∂u ∂ 2u ν σB 2 (x)u +v =ν 2 − u− , ∂x ∂y ∂y κp ρ

(2)

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Paresh Vyas and Ashutosh Ranjan

∂T ∂T K ∂ 2T ν u +v = + 2 ∂x ∂y ρCp ∂y Cp



∂u ∂y

2

1 ∂qr σB02 u2 − + , ρCp ∂y ρCp

(3)

The boundary conditions are: u = axm , v = 0, −K

∂T = qw = E0 xn , at y = 0, ∂y u → 0, T → T∞ as y → ∞.

(4) (5)

where u, v are velocity components in x, y directions respectively, ν is the kinematic viscosity of the fluid, κp is the permeability of porous medium, σ is the electrical conductivity, ρ is the density, T is the temperature, K is the thermal conductivity, B(x) is the applied variable magnetic field, Cp is the specific heat at constant pressure, qr is the radiation heat flux, qw is the rate of heat transfer, E0 is a positive constant, n = 2m is a heat flux parameter and T∞ is the uniform temperature of the ambient fluid. Using Rosseland approximation for radiation [17] we can write: qr = −

4γ ∂T 4 3α ∂y

Where γ , α are Stephan-Boltzmann constant and the mean absorption coefficient respectively. Temperature difference within the flow is assumed to be sufficiently small so that T 4 may be expressed as a linear function of temperature T , using a truncated Taylor series about the free stream temperature T∞ to yield 3 4 T 4 = 4T∞ T − 3T∞ Following Chiam [29], we assume magnetic field of the form B(x) = B0 x(m−1)/2

(6)

where B0 is positive constant and exponent m 6= −1.

3

Solution

Let us introduce the following similarity transformations: (1 + m)U (x) η= 2νx

1/2



1/2



2νxU (x) Ψ(x, y) = 1+m

y,

f (η),

(7)

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Dissipative MHD boundary-layer flow

where Ψ is the stream function. The velocity components are obtained as: u=

∂Ψ = axm f 0 (η), ∂y

r   ∂Ψ νa(1 + m) (m−1)/2 m−1 0 v=− f (η) + =− x ηf (η) . ∂x 2 m+1

(8)

We observe that the equation of continuity (1) is then identically satisfied. Using (6) and (8) the momentum equation (2) reduces to the similarity equation   1 000 00 02 2 f 0 = 0, (9) f + ff − f − M + κ0 with boundary conditions f (0) = 0,

f 0 (0) = 1,

f 0 (∞) = 0.

where a prime denotes differentiation with respect to η, κ0 = 2

σB02 ρa

(10) κp a ν

is the per-

meability parameter and M = is the Magnetic parameter. The equation (9) admits a solution of the form f (η) = A + Be−cη Where constants A, B, c are determined by using the boundary conditions (10) and consequently we get the solution as 1 f (η) = (1 − e−cη ) c q where c = 1 + M 2 + larity variable

1 . κ0

(11)

The energy equation (3), on introducing the simiT − T∞

E0 xn = K

r

ν θ(η) a

(12)

and using (6), (7) and (8), converts to θ00 +

P rf 0 4mP rf 0 2EcP rM 2 f 02 EcP rf 002 θ − θ=− − 1+N (1 + N )(1 + m) (1 + N )(1 + m) (1 + N )

where P r = and N =

µCp K

3 16γT∞

3αK

is the Prandtl number, Ec =

a2√

(13)

is the Eckert number ) is the radiation parameter. The boundary conditions for θ(η) Cp (

E0 K

ν a

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Paresh Vyas and Ashutosh Ranjan

follow from (4), (5) and (12) as θ0 = −1 at η = 0, θ → 0 as η → ∞.

4

(14)

Solution Method

The full nonlinear boundary value problem described by equations (13) and (14) was solved numerically, in double precision, by shooting method using the fourth order Runge- Kutta algorithm with Secant method of iteration to search for the missing θ(0). The essense of the shooting method to solve a boundary value problem (BVP) is to convert it into system of initial value problems (IVP). In the present case the equation (13) is reduced to such system of IVP where missing value of θ(0) for different set of values of parameters having bearing on the phenomena are chosen purely on hit and trial basis such that the boundary condition at the other end i.e. η → ∞, θ(η) → 0 is satisfied. As the appropriate value for η∞ was not known, the integration was first taken up to an arbitrarily chosen value of η∞ . The integration was then repeated with another larger (or smaller, as the case may be) value of η∞ . The values of the initial wall temperature θ(0) were then compared. If they agreed to about 6 significant digits, the last value of η∞ used was considered the appropriate value; otherwise the procedure was repeated until further changes in η∞ did not lead to any more change in the value of θ(0). A grid independent study was conducted to examine the effect of step size. Further, η∞ was computed for every set of values of parameters affecting the phenomena. The step size ∆η = 0.05 has been found to be satisfactory to ensure convergence criterion of 10−6 .

5

Result and Discussion

From Fig- 1, we observe that the non- dimensional temperature θ(η) undergoes a decay with increments in values of permeability parameter κ0 . Fig- 2 exhibits that θ(η) registers an increment for increasing values of Eckert number Ec. The effect of magnetic field has been depicted in Fig- 3, which shows that θ(η) increases with increasing values of magnetic field parameter M . The effect of radiation parameter on θ(η) has been shown in Fig- 4, which demonstrates that with an increase in radiation parameter N , the temperature θ(η) registers a considerable increase. Fig- 5 depicts variation of θ(η) for different values of Prandtl number P r. We find that θ(η) decreases with an increase in P r. Fig6 shows the variations in θ(η) for different values of stretching exponent m. It

Dissipative MHD boundary-layer flow

3139

clearly shows that by increasing m, θ(η) decays.

ACKNOWLEDGEMENT. The first author (P.V.) acknowledges financial support from University Grant Commission, India for carrying out the work vide grant F No. MS- 336/304028/08-09/CRO.

References [1] A. Raptis, Radiation and free convection flow through a porous medium, Int. Comm. Heat Mass Transfer, 25 (1998), 289 - 295. [2] C.K. Chen and M.I. Char, Heat transfer of a continuous stretching surface with suction or blowing, J. Math. And. Appl., 135 (1988), 568 - 580. [3] D.S. Chauhan and P. Vyas, Heat transfer in MHD viscous flow due to stretching of a boundary in the presence of naturally permeable bed, AMSE periodicals, Modelling, Measurement and Control, 60 (1995), 17 - 36. [4] E.M. Sparrow and R.D. Cess, Radiative Heat Transfer, Brooks/Cole Pub. Co., Belmont, California, 1970. [5] F.S. Ibrahiem and F.M. Hady, Mixed convection radiation interaction in boundary layer flow over a horizontal surface, Astrophys. Space Sci., 168 (1990), 263 - 276. [6] H.I. Anderson, MHD flow of a viscoelastic fluid past a stretching surface, Acta Mechanica, 95 (1992), 227 - 230. [7] H.K. Kuiken, On boundary layer in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small, I.M.A. J. Appl. Math., 27 (1981), 387 - 405. [8] H.S. Takhar and G. Nath, Similarity solution of unsteady boundary layer equations with a magnetic field, Meccanica, 32 (1997), 157 - 163. [9] H.M. Duwahiri, R.A. Damesh and B. Tashtoush, Transient nonBoussinesq magnetohydrodynamic free convection flows over a vertical surface, Int. J. Fluid Mech. Res., 33 (2006), 137 - 152. [10] J. Grubka and K.M. Bobba, Heat transfer characteristics of a continuous stretching surface with variable temperature, Trans. ASME, J. Heat Transfer, 107 (1985), 248 - 250. [11] K. Vajravelu and J. Nayfeh, Convective heat transfer at a stretching sheet, Acta Mech, 96 (1993), 47 - 54.

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[12] K.B. Pavlov, Magnetohydrodynamic flow of an incompressible viscous fluid caused by the deformation of a plane surface, Magnitnaya Gidrodinamika, 4 (1974), 146 - 147. [13] K.V. Prasad, S. Abel and S.K. Khan, Study of visco- elastic fluid flow and heat transfer over a stretching sheet with variable viscosity, International Journal of Non- linear Mechanics, 37 (2002), 81 - 88. [14] L.J. Crane, Flow past a stretching plane, Z. Angew Math Phys, 21 (1970), 645 - 647. [15] M.A. Hossain, M.A. Ali and D.A. Rees, The effect of radiation on free convection from a porous vertical plate, Int. J. Heat and Mass Transfer, 42 (1) (1998), 181 - 191. ¨ [16] M.N. Ozisik, Radiative Transfer & Interactions With Conduction & Convection, Werbel & Peck, New York, 1985. [17] M.Q. Brewster, Thermal Radiative Transfer Properties, John Wiley & Sons, New York, 1992. [18] N. Afzal, Heat transfer from a stretching surface, Int. J. Heat Mass Transfer, 20 (1993), 417 - 430. [19] N. Afzal and I.S. Varshney, The cooling of a low heat resistance stretching sheet moving through a fluid, Warme Stoffibertragung, 14 (1980), 289 - 293. [20] O.A. Plumb, J.S. Huenfeld and E.J. Eschbech, The effect of cross flow and radiation on natural convection from vertical heated surfaces in saturated porous media, In: AIAA 16th Thermophysics conference, Palo Alto, CA, USA, (1981), 23 - 25. [21] P.H. Veena, S. Abel, K. Rajgopal and V.K. Pravin, Heat transfer in a visco- elastic fluid past a stretching sheet with viscous dissipation and internal heat generation, ZAMP, 57 (2006), 447 - 463. [22] P.S. Gupta and A.S. Gupta, Heat and mass transfer on a stretching sheet with suction and blowing, Cand. J. Chem. Eng., 55 (1977), 744 - 746. [23] P. Vyas and N. Srivastava, Radiative MHD flow over a non- isothermal stretching sheet in a porous medium, Applied Mathematical Sciences, 4 (50) (2010), 2475 - 2484. [24] R. Seigel and J.R. Howell, Thermal Radiation Heat Transfer, Taylor & Francis, New York, 2002. [25] R.S.R. Gorla, Radiative effect on conjugate forced convection and conductive heat transfer in a circular pin, Int. J. Heat Fluid Flow, 9 (1988), 49 - 51.

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[26] R.S.R. Gorla and I. Pop, Conjugate heat transfer with radiation from vertical circular pin in a non- Newtonian ambient medium, Warmeund Stoffubert- ragung, 28 (1993), 11 - 15. [27] S. Abel, P.H. Veena, K. Rajgopal and V.K. Praveen, Non-Newtonian magnetohydrodynamic flow over a stretching surface with heat and mass transfer, Int. J. Non-linear Mech., 39 (2004), 1067 - 1078. [28] S. Shateyi, P. Sibonda and S.S. Motsa, Magnetohydrodynamic flow past a vertical plate with radiative heat transfer, J. Heat Transfer, 129 (12) (2007), 1708 - 1713. [29] T.C. Chiam, Hydromagnetic flow over a surface stretching with power law velocity, Int. J. Eng. Sci., 33 (1995), 429 - 435. [30] W.C. Troy, E.A. Onermann, G.B. Bremant- Rout and J.P. Kscner, Uniqueness of flow of second order fluid past a stretching sheet, Q. Appl. Math., 44 (1987), 753 - 755. [31] W.H.H. Banks, Similarity solutions of the boundary layer equations for stretching wall, J. Mec. Theor. Appl., 2 (1983), 375 - 392. [32] Y.Z. Boutros, M.B. Abd- el- Malik, N.A. Badran and H.S. Hassan, Liegroup method of solution for steady two- dimensional boundary- layer stagnation point flow towards a heated stretchig sheet placed in a porous medium, Meccanica, 41 (2006), 681 - 691.

10

8

κ0=0.01 κ0=0.03

9 8

κ0=0.08

6

κ0=0.1

7

5

6 Pr=0.71 Ec=0.1 N=1 M=2 m=1

5 4

θ (η)

θ (η)

Ec=0 Ec=0.1 Ec=0.2 Ec=0.3 Ec=0.5

7

κ0=0.05

Pr=0.71 N=1 M=2 κ0=0.1

4 3

m=1

3 2 2 1

1 0

0

10

20

30

40

50 η

60

70

80

90

100

Figure 1: Temperature distribution for varying κ0 Received: April, 2010

0

0

10

20

30

40

50 η

60

70

80

90

100

Figure 2: Temperature distribution for varying Ec

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Paresh Vyas and Ashutosh Ranjan

6

10 M=0 M=1 M=2 M=3 M=4

5

8 7

4 Pr=0.71 Ec=0.1 N=1 κ0=0.1

3

6 θ (η)

θ (η)

N=0.1 N=1 N=2 N=3 N=5

9

m=1

Pr=0.71 Ec=0.1 M=2 κ0=0.1

5

m=1

4 2

3 2

1

1 0

0

5

10

15

20

25 η

30

35

40

45

0

50

Figure 3: Temperature distribution for varying M

40

60

80

100 η

120

140

160

5 Pr=0.71 Pr=1.0 Pr=1.5 Pr=1.75 Pr=2.3

4

180

200

m=1 m=2 m=3 m=4 m=5

4.5 4

3.5

3.5

3

3

Ec=0.1 N=1 M=2 κ0=0.1

2.5

θ (η)

θ (η)

20

Figure 4: Temperature distribution for varying N

5 4.5

m=1

2

Pr=0.71 Ec=0.1 N=1 M=2 κ0=0.1

2.5 2

1.5

1.5

1

1

0.5

0.5

0

0

0

5

10

15

20

25 η

30

35

40

45

50

Figure 5: Temperature distribution for varying P r

0

0

10

20

30 η

40

50

60

Figure 6: Temperature distribution for varying m