magnetohydrodynamic mixed convection flow and ...

Magnetohydrodynamic Mixed Convection Flow and BoundaryCONVECTION Layer Control of A Nanofluid With Heat MAGNETOHYDRODYNAMIC MIXED FLOW AND Generation/Absorption Effects, M.Chandrasekar, M.S.Kasiviswanathan, Journal Impact Factor (2015): 8.8293 BOUNDARY LAYER CONTROL OF A NANOFLUID WITH HEAT Calculated by GISI (www.jifactor.Com) GENERATION/ABSORPTION EFFECTS

M.Chandrasekar1,

M.S.Kasiviswanathan2

Volume 6, Issue 6, June (2015), pp. 18-32 Article ID: 30120150606003 International Journal of Mechanical Engineering and Technology © IAEME: http://www.iaeme.com/IJMET.asp ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online)

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IJMET

©IAEME

Department of Mathematics, Anna University, Chennai-600 025, India, Department of Mathematics, Anna University, Chennai-600 025, India,

ABSTRACT This research work is focused on the numerical solution of steady MHD mixed convection boundary layer flow of a nanofluid over a semi-infinite flat plate with heat generation/absorption and viscous dissipation effects in the presence of suction and injection. Gyarmati’s variational principle developed on the thermodynamic theory of irreversible processes is employed to solve the problem numerically. The governing boundary layer equations are approximated as simple polynomial functions, and the functional of the variational principle is constructed. The Euler-Lagrange equations are reduced to simple polynomial equations in terms of momentum and thermal boundary layer thicknesses. The velocity, temperature profiles as well as skin friction and heat transfer rates are solvable for any given values of Prandtl number Pr, magnetic parameter ξ, heat source/sink parameter Q, buoyancy parameter Ri, suction/injection parameter H and viscous dissipation parameter Ec. The obtained results are compared with known numerical solutions and the comparison is found to be satisfactory. Keywords: Boundary Layer, Gyarmati’s Variational Principle, Heat Source/Sink, Mixed Convection, Nanofluid 1. INTRODUCTION The prime objective of this work is to study the heat transfer enhancement in mixed convection nanofluid flow over a flat plate with heat source/sink and magneto hydrodynamic effects using a genuine variational principle developed by Gyarmati. Recently in many industrial applications nanofluids are used as heat carriers in heat transfer equipment instead of conventional fluids due to its relatively higher thermal conductivity. The potential benefits of nanofluids are theoretically and experimentally investigated by many researchers in the past two decades. Buongiorno [1] explained the seven slip mechanisms as reasons for the heat transfer enhancement observed in nanofluids. Due to the great potential and characteristics of nanofluid still more research work to be done to study heat transfer enhancement mechanism. Khan and Pop [2] solved the numerical solution of a nanofluid flow over a stretching sheet. The analysis on free convection nanofluid flow over a vertical plate with different boundary conditions on the nanoparticle volume fraction was investigated by Kuznetsov and Nield [3, 4]. Iaeme.com/ijmet.asp

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Magnetohydrodynamic Mixed Convection Flow and Boundary Layer Control of A Nanofluid With Heat Generation/Absorption Effects, M.Chandrasekar, M.S.Kasiviswanathan, Journal Impact Factor (2015): 8.8293 Calculated by GISI (www.jifactor.Com)

Chamkha and Aly [5] considered the boundary layer equations for natural convection flow of an electrically conducting nanofluid past a plate in the presence of heat generation and absorption effects. The same problem without Brownian motion and thermophoresis effects was analyzed by Hamad et al. [6]. Rana and Bhargava [7] presented an analysis on mixed convective boundary layer flow of nanofluid over a vertical flat plate with temperature dependent heat source/sink. The stagnation point flow of unsteady case in a nanofluid was described by Bachok et al. [8]. The boundary layer solution for forced convection flow of alumina-water nanofluid over a flat plate in the presence of magnetic effect was studied by Hatami et al. [9]. Vajravelu et al. [10] observed the effects of variable viscosity and viscous dissipation on the forced convection flow of water based nanofluids. The stagnation point flow of nanofluid towards a stretching sheet in the presence of transverse magnetic field was studied by Ibrahim et al. [11]. By considering all the above facts, in this study non similar mixed convection flow of water based nanofluid containing one of the nanoparticles Copper (Cu), Silver (Ag), Alumina(Al2O3) with the volume fraction range 0-4% over a semi-infinite flat plate in the presence of constant magnetic flux density, heat source/sink, suction/injection and viscous dissipation effects were analyzed. The Gyarmati’s variational technique has been employed to solve the non-similar boundary layer equations. The computational results are given for velocity profile temperature profile, the coefficient of skin friction (shear stress) and local Nusselt number (heat transfer) for various values of heat generation/absorption parameter Q, magnetic parameter ξ and buoyancy parameter Ri. The results obtained by the present analysis are compared with the numerical solution of Rana and Bhargava [7] and the comparison establishes the fact that the accuracy is remarkable. The main intention of this investigation is to justify that, the Gyarmati’s variational technique is one of the most general and exact variational techniques in solving flow and heat transfer problems. Chandrasekar [12, 13], Chandrasekar and Baskaran [14], Chandrasekar and Kasiviswanathan [15] already applied Gyarmati’s variational technique for steady and unsteady heat transfer and boundary layer flow problems. 2. THE GOVERNING EQUATIONS OF THE SYSTEM The system of steady, two dimensional, incompressible and laminar boundary layer flow of nanofluid over a semi-infinite flat plate with suction and injection is considered. The leading edge of the plate is at x = 0, the plate is parallel to the x-axis and infinitely long downstream. In this study it is assumed that the flow is with free stream velocity U∞ and the ambient temperature T∞ which are parallel to x-axis. And the temperature of the plate is held at a constant temperature T0 which is greater than the ambient temperature T∞. A uniform magnetic field of strength B0 is applied normal to the x-axis and assumed that the induced magnetic field, the imposed electric field intensity and the electric field due to the polarization of charges are negligible. By Boussinesq-boundary layer approximations and with the assumption that all fluid properties are constants, the governing boundary layer equations for the present system are as follows, see Aydin and Kaya [16]

ux + vy = 0 uux + vuy =

(1) 1

µnf uyy +κ B02 (U∞ − u) + (ρβ )nf g(T −T∞ ) ρnf

1 knf Tyy + µnf (uy )2 −κ B02u(U∞ − u) + Q0 (T −T∞ ) (ρCp )nf subject to the boundary conditions uTx + vTy =

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(2) (3)

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y = 0; u = 0, v = v0 , T = T0, y → ∞; u=U∞= constant, T = T∞

(4)

Here u, v, v0 , T, κ, B0 , Q0 and g are velocity of the fluid in x-direction, velocity of the fluid in y-direction, suction/injection velocity, temperature of the fluid, electric conductivity, externally imposed magnetic field in the y-direction, heat generation/absorption coefficient and acceleration due to gravity respectively. The thermophysical properties of nanofluid namely density, dynamic viscosity, thermal diffusivity, volumetric expansion coefficient, heat capacity and thermal conductivity are denoted by respectively ρnf, µnf, αnf, (ρβ)nf, (ρCp)nf, knf and have been calculated as functions of thermophysical properties of nanoparticle (spherical shaped) and base fluid as follows, ρ nf = (1 − φ ) ρ f + φρ s µ nf = α nf =

µf (1 − φ ) 2.5 k nf

(5)

( ρ C p ) nf

( ρβ ) nf = (1 − φ )( ρβ ) f + φ ( ρβ ) s ( ρ C p ) nf = (1 − φ )( ρ C p ) f + φ ( ρ C p ) s k nf

and

kf

=

k s + 2 k f − 2φ ( k f − k s ) ks + 2k f + φ (k f − ks )

Here φ is the particle volume fraction. The thermophysical properties of base fluid and nanoparticle are distinguished by subscripts f and s respectively. 3. VARIATIONAL FORMULATION OF THE PROBLEM The purpose of this analysis is to obtain the approximate numerical solution of irreversible thermodynamics problem by a variational technique. Gyarmati [17, 18] developed a variational principle known as “Governing Principle of Dissipative Processes” (GPDP) which is given in its universal form

δ ∫ (σ − ψ − Φ ) dV = 0.

(6)

V

The principle (6) describes the evaluation of linear, quasi linear and some nonlinear irreversible processes at any instant of time and space under constraints that the balance equations

ρ a&i + ∇ ⋅ J i = σ i ,

(i = 1, 2,3,L f )

(7)

are satisfied. In Equation (6), δ is the variational symbol, σ is the entropy production, ψ and Φ are dissipation potentials and V is the total volume of the thermodynamic system. In Equation (7), ρ is the mass density and a& i , Ji, σi are respectively substantial variation, flux and source density of the ith extensive transport quantity ai. The entropy production σ per unit volume and unit time can always be written in the bilinear form Iaeme.com/ijmet.asp

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Magnetohydrodynamic Mixed Convection Flow and Boundary Layer Control of A Nanofluid With Heat Generation/Absorption Effects, M.Chandrasekar, M.S.Kasiviswanathan, Journal Impact Factor (2015): 8.8293 Calculated by GISI (www.jifactor.Com) f

σ = ∑ Ji ⋅ Xi ≥ 0

(8)

i =1

where Ji and Xi are fluxes and forces respectively. According to Onsager’s linear theory the fluxes are linear functions of forces, that is

[19, 20]

f

J i = ∑ Lik X k ,

(i = 1, 2,3,L f )

(9)

(i = 1, 2,3,L f )

(10)

k =1

or alternatively f

Xi = ∑ Rik J k , k =1

The constants Lik and Rik are conductivities and resistances respectively satisfying the reciprocal relations [19, 20] (11) Lik = Lki and Rik = Rki, (i, k = 1,2,3,…f ) The matrices of Lik and Rik are mutually reciprocals and they are symmetric, that is f

f

m =1

m =1

∑ Lim Rmk = ∑ Lmk Rim = δ ik ,

(i, k = 1, 2,3,L f )

(12)

where δik is the Kronecker delta. The local dissipation potentials ψ and Φ are defined [19, 20] as, f

ψ( X, X) = (1 / 2) ∑ Lik Xi ⋅ Xk ≥ 0

(13)

i , k =1

f

Φ(J , J ) = (1 / 2) ∑ Rik J i ⋅ J k ≥ 0

(14)

i , k =1

In the case of transport processes, the forces Xi can be generated as gradients of certain “Γ” variables and can be written as Xi =∇Γi (15) The principle (6) with the help of Equations (8), (13), (14) and (15), takes the form f f  f  δ ∫  ∑ J i ⋅∇Γi − (1/ 2) ∑ Lik ∇Γi ⋅∇Γ k − (1/ 2) ∑ Rik J i ⋅ J k  dV = 0 (16) i , k =1 i , k =1  V  i =1 This variational principle has been already applied for various dissipative systems and was established as the most general and exact variational principle of macroscopic continuum physics. Many other variational principles have already been shown as partial forms of Gyarmati’s principle. The balance equations of the system play a central role in the formulation of Gyarmati’s variational principle and hence the governing boundary layer Equations (1-3) are written in the balance form as ∇ ⋅ V = 0, Iaeme.com/ijmet.asp

(V = iu + jv)

(17) 21

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ρ nf (V ⋅ ∇)V + ∇ ⋅ P = (κ B02 )[U ∞ − (i ⋅ V )] + ( ρβ ) nf g (T − T∞ )i

(18)

( ρ C p ) nf (V ⋅∇)T + ∇ ⋅ J q = µnf (u y2 ) − (κ B02 )(i ⋅ V )[U ∞ − (i ⋅ V )] + Q0 (T − T∞ )

(19)

These equations represent the mass, momentum and energy balances respectively. Here i and j being unit vectors in the directions of x and y axes respectively. In Equation (18) P denotes the pressure tensor which can be decomposed [17] as o

P = pδ + P vs

(20) o

where p is the hydrostatic pressure, δ is the unit tensor and P vs is the symmetrical part of the viscous pressure tensor, whose trace is zero. In the study of heat transfer and fluid flow problems, the energy picture of Gyarmati’s principle is always advantageous over entropy picture. Therefore, the energy dissipation Tσ is used instead of entropy production σ. The energy dissipation for the present system is given [17] by, T σ = − J q (∂lnT / ∂y ) − P12 (∂u / ∂y )

(21) o

where Jq is the heat flux and P12 is the only component of momentum flux P vs , satisfy the constitutive relations connecting the independent fluxes and forces as J q = − Lλ (∂lnT / ∂y )

and

P12 = − Ls (∂u / ∂y )

(22)

Here Lλ = λT and Ls = µ, where λ and µ are the thermal conductivity and viscosity respectively. With the help of Equations (22) the dissipation potentials in energy picture are found as follows T ψ = (1/ 2)  Lλ (∂lnT / ∂y)2 + Ls (∂u / ∂y ) 2 

(23)

T Φ = (1/ 2)  Rλ J q2 + Rs P122 

(24)

where Lλ = Rλ−1 and Ls = Rs−1 . Using Equations (21-24), Gyarmati’s variational principle (6) is formulated in the following form

 − J q (∂lnT ∂y) − P12 (∂u ∂y) − ( Lλ 2)(∂lnT ∂y)2  dydx = 0 , δ ∫∫  2 2 2 − ( L 2)( ∂ u ∂ y ) − ( R 2) J − ( R 2) P  0 0 s q s 12 λ  in which l is the representative length of the surface. l ∞

(25)

4. METHOD OF SOLUTION It is assumed that the trial functions for velocity and temperature fields inside the respective boundary layers are as follows

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u U ∞ = 2 y d1 − 2 y 3 d13 + y 4 d14 u = U∞

( y < d1 )

( y ≥ d1 )

(T − T∞ ) (T0 − T∞ ) = θ

(26)

= 1 − 2 y d 2 + 2 y 3 d 23 − y 4 d 24 T = T∞

( y < d2 )

( y ≥ d2 )

where d1, d2 are the velocity and temperature boundary layer thicknesses which are to be determined from the variational procedure. The trial functions (26) satisfy the following compatibility conditions, y = 0 ; u = 0, v = v0 , T = T0, ∂ 2T/∂ y2 = 0 y = d1 ; u = U∞ = constant, ∂ u /∂ y = 0 (smooth fit), ∂ 2u /∂ y2 = 0 y = d2 ;T = T∞, ∂ T/∂ y = 0 (smooth fit), ∂ 2T/∂ y2 = 0

(27)

The smooth fit conditions ∂ u /∂ y = 0 and ∂ T/∂ y = 0 correspond to P12 = 0 and Jq = 0 at their respective edges of the boundary layer. Using the boundary conditions (27), the transverse velocity component v is obtained from the mass balance equation (17) as v = U ∞ (4 y 5 / 5d15 − 3 y 4 / 2d14 + y 2 / d12 )d1' + v0 ,

(28)

where v0 is the suction/injection velocity. The velocity and temperature functions (26) and the boundary conditions (27) are used in the governing boundary layer Equations (17-19) and on direct integration with respect to y with the help of their corresponding smooth fit conditions uy = 0 and Ty = 0, the momentum flux P12 and energy flux Jq are obtained. The momentum flux P12 remains the same for any Prandtl number Pr but the energy flux Jq has different expressions for Pr ≤ 1 and Pr ≥ 1. When Pr ≤ 1 the expression for Jq in the range d1 ≤ y ≤ d2 is obtained first and the expression for Jq in the range 0 ≤ y ≤ d1 is determined subsequently by matching the expressions of the two regions at the interface. The expressions for momentum and the energy fluxes P12 and Jq are obtained respectively as follows,

−P12 / Ls =(U∞2d1′ /υnf )(−4y9 45d19 +2y8 5d18 −3y7 7d17 −11y6 15d16 +7y5 5d15 4 4 3 3 −2y3 3d13 +101/1800)+(vU 0 ∞ υnf )(y d1 −2y d1 +2y d1 −7/10) 2 5 4 4 3 2 +(κBU 0 ∞ µnf )(y 5d1 − y 2d1 + y d1 −y+7d1 30)+U∞ d1

−βnf gT ( 0 −T∞)/υnf (−y5 5d24 +y4 2d23 −y2 d2 +y+d15 30d24 −d14 10d23 +d12 3d2 −d1 2)

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(29)

(0≤ y ≤d1)

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− J q Lλ = U∞ (T0 − T∞ ) αnf  d2′ (4 y9 9d14d25 − y8 d13d25 − 3 y8 4d14d24 + 12 y 7 7d13d24  6 5 6 + 4 y 3d1 d2 + y 3d14d22 −12 y5 5d1 d24 − 4 y5 5d13d22 + 4 y3 3d1 d22 + d15 45d25 − 9d14 140d24 + 2d12 15d22 − 3 /10) + d1′ (−16 y9 45d15d24 + 3 y8 5d15d23 + 3 y8 4d14d24 − 9 y 7 7d14d23 − 2 y 6 3d12d24 − 4 y 6 15d15d2 + 3 y 5 5d14d2 + 6 y5 5d12d23 − 2 y3 3d12d2 + 49d14 180d24 − 18d13 35d23 + d1 3d2 )

(30)

+(v0 U∞ )(− y 4 d24 +2 y3 d23 −2 y d2 + d14 d24 − 2d13 d23 + 2d1 d2 )  + (U∞2 µnf α nf ( ρC p )nf )(−16 y 7 7d18 + 8 y 6 d17 − 36 y5 5d16 − 4 y 4 d15 + 8 y3 d14 − 4 y d12 + 52 35d1 ) + (κ B02 U∞2 α nf ( ρC p )nf )(− y 9 9d18 + y8 2d17 − 4 y 7 7d16 − 2 y 6 3d15 + 9 y5 5d14 − y 4 2d13 − 4 y3 3d12 + y 2 d1 − 37d1 315) + (Q0 (T0 − T∞ ) α nf ( ρC p )nf )( y5 5d24 − y 4 2d23 + y 2 d2 − y + 3d2 10)

(0 ≤ y ≤ d1 ); ( Pr ≤ 1)

−Jq Lλ = U∞ (T0 −T∞ )d2′ αnf  (4y5 5d25 − 3y4 2d24 + y2 d22 −3/10)   + (Q0 (T0 −T∞ ) αnf (ρCp )nf )( y5 5d24 − y4 2d23 + y2 d2 −y + 3d2 10)

(31)

(d1 ≤ y ≤ d2 ); (Pr ≤1) −Jq Lλ = U∞ (T0 −T∞ ) αnf  d2′(4 y9 9d14d25 − y8 d13d25 − 3y8 4d14d24 +12y7 7d13d24  + 4y6 3d1 d25 + y6 3d14d22 −12y5 5d1 d24 − 4 y5 5d13d22 + 4y3 3d1 d22 − d24 36d14 + 3d23 35d13 − 4d2 15d1 ) + d1′(−16y9 45d15d24 +3y8 5d15d23 + 3y8 4d14d24 − 9y7 7d14d23 − 2y6 3d12d24 − 4y6 15d15d2 + 3y5 5d14d2 + 6y5 5d12d23 − 2y3 3d12d2 + d25 45d15 − 9d24 140d14 + 2d22 15d12 ) +(v0 U∞ )(− y4 d24 +2y3 d23 −2y d2 +1) − (U∞2 µnf αnf (ρCp )nf ) (16y7 7d18 − 8 y6 d17 + 36 y5 5d16 + 4 y4 d15 − 8y3 d14 + 4 y d12 7 2

8 1

6 2

7 1

5 2

6 1

4 2

5 1

3 2

(32)

4 1

−16 d 7d + 8d d − 36d 5d − 4d d + 8d d

− 4d2 d12 ) + (κ B02 U∞2 αnf (ρCp )nf )(− y9 9d18 + y8 2d17 − 4y7 7d16 − 2y6 3d15 + 9y5 5d14 − y4 2d13 − 4 y3 3d12 + y2 d1 + d29 9d18 − d28 2d17 + 4 d27 7d16 + 2d26 3d15 − 9d25 5d14 + d24 2d13 + 4d23 3d12 − d22 d1 ) + (Q0 (T0 −T∞ ) αnf (ρCp )nf )( y5 5d24 − y4 2d23 + y2 d2 − y + 3d2 10) (0 ≤ y ≤ d2 ); (Pr ≥ 1)

The prime indicates differentiation with respect to x. Using the expressions of P12 and Jq together with velocity and temperature functions (26), the variational principle (25) is formulated independently for Pr ≤ 1 and Pr ≥ 1 cases. After performing the integration with respect to y, one can obtain the variational principle in the following forms, l

δ ∫ L1[d1 , d 2 , d1′, d 2′ ]dx = 0, ( Pr ≤ 1)

(33)

0

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Magnetohydrodynamic Mixed Convection Flow and Boundary Layer Control of A Nanofluid With Heat Generation/Absorption Effects, M.Chandrasekar, M.S.Kasiviswanathan, Journal Impact Factor (2015): 8.8293 Calculated by GISI (www.jifactor.Com) l

δ ∫ L2 [d1 , d 2 , d1′, d 2′ ]dx = 0, ( Pr ≥ 1)

(34)

0

where L1, L2 are the Lagrangian densities of the principle. The variation is carried out with respect to the independent parameters d1 and d2. These variational principles (33), (34) are found identical when d1=d2. The Euler-Lagrange equations corresponding to these variational parameters are (∂L1,2 ∂d1 ) − (d dx)(∂L1,2 ∂d1′) = 0 (∂L1,2 ∂d 2 ) − (d dx)(∂L1,2 ∂d 2′ ) = 0,

(35) ( Pr ≤ 1, Pr ≥ 1)

(36)

where L1,2 represents the Lagrangian densities L1 and L2 respectively. These Equations (35) and (36) are second order ordinary differential equations in terms of d1 and d2. The procedure for solving Equations (35) and (36) can be considerably simplified by introducing the non-dimensional boundary layer thicknesses d1* , d 2* and are given by d1 = d1* υ f x / U ∞

and

d 2 = d 2* υ f x / U ∞

(37)

These variational principles (33) and (34) are subject to the transformations (37). The resulting Euler-Lagrange equations are obtained as simple polynomial equations, ∂L1,2 ∂d1* = 0 (38)

∂L1,2 ∂d 2* = 0, ( Pr ≤ 1, Pr ≥ 1)

(39)

The coefficients of these Equations (38) and (39) dependent on the independent parameters Pr, ξ, Q, Ri, H and Ec where Pr = υ f α f (Prandtl number), ξ = κ B02 x ρU ∞ (magnetic parameter), Q = Q0 x / U ∞ ( ρ C p ) f (heat generation/ absorption parameter) Ri=Gr/Re2 (Richardson number),

Gr = g β f (T0 − T∞ ) x3 υ 2f (Grashof number), Re = U ∞ x υ f (Reynolds number), H = v0 x υ f U ∞ (suction/injection parameter) and Ec = U ∞2 C p (T0 − T∞ ) (Eckert number). In the present analysis heat generation and absorption are presented by Q > 0 and Q < 0 respectively and the suction and injection are represented by H < 0 and H > 0 respectively. Equations (38) and (39) are simple coupled polynomial equations and it can be solved for any values of Pr, ξ, Q, Ri, H and Ec and it is found that the obtained simultaneous solutions d1* and d 2* are as the only one set of positive real roots. After obtaining the values of d1* and d 2* for given Pr, ξ, Q, Ri, H and Ec the values of velocity, temperature profiles, skin friction (shear stress) and heat transfer (local Nusselt number) are calculated with the help of the following expressions,

η = y U∞ υ f x

(40)

τ w = υ f x U ∞3 (− P12 Ls ) y =0

(41)

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Magnetohydrodynamic Mixed Convection Flow and Boundary Layer Control of A Nanofluid With Heat Generation/Absorption Effects, M.Chandrasekar, M.S.Kasiviswanathan, Journal Impact Factor (2015): 8.8293 Calculated by GISI (www.jifactor.Com) 2

and Nul = υ f x U ∞ (T0 − T∞ ) ( J q Lλ ) y =0 .

(42)

5. RESULTS AND DISCUSSION

The main and important characteristics of the problem analyzed are skin friction and heat transfer values. The energy equation has been solved for two cases d1* ≤ d 2* ( Pr ≤ 1) and d1* ≥ d 2* ( Pr ≥ 1) . These two independent analyses yield solutions and it is matching at Pr =1. It is found that both the analyses lead to satisfactory results in the respective ranges of Pr. The thermophysical properties of water and nanoparticles given in Table 1 are used to compute each case of nanofluid. Table 1: Thermophysical properties of water and nanoparticles.

ρ (kgm-3)

Cp (Jkg-1K-1)

k (Wm-1K-1)

β×10-5 (K-1)

H2O

997.1

4179

0.613

21

Al2O3

3970

765

40

0.85

Cu

8933

385

401

1.67

Ag

10500

235

429

1.89

It is customary that when a new mathematical method is applied to a problem, the obtained results are compared with the available solution in order to determine the accuracy of the results involved in the present technique. In Table 2, the heat transfer values of regular fluid for various values of Pr (Pr ≤ 1 and Pr ≥ 1) when ξ = Q = Ri = H = Ec = 0 are obtained by the present variational technique. From this table it is evidently clear that the present results are in good agreement with Chamkha et al. [21], Aydin and Kaya [16], Rana and Bhargava [7]. It is also observed that the heat transfer increases with the values of Prandtl number. Since the higher Prandtl number has very low thermal conductivity, the local Nusselt number increases rapidly. This means that the variation of the heat transfer rate is more sensitive to the larger Prandtl number than the smaller one. Table 2: Local Nusselt number for various values of Pr when ξ = Q = Ri = H = Ec = φ =0. Rana & Bhargava [7] Present Results Chamkha et al. [21] Aydin & Kaya [16] Pr Nul Nul Nul Nul 0.0596 0.01 0.054742313 0.051830 0.051437 0.1579 0.1 0.147754551 0.142003 0.148123 0.3319 1 0.334277544 0.332173 0.332000 0.7278 10 0.738452128 0.728310 0.727801 1.5721 100 1.599967934 1.572180 1.573141

Figs. 1-4, represent the effects of buoyancy parameter Ri on the velocity profile, temperature profile, local Nusselt number and skin friction respectively. These results are obtained for Pr = 6.2, Q = 0.05 corresponding to pure water and copper-water nanofluid with volume fraction φ = 0.04.

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Fig. 1: Velocity profile for different values of Ri with Pr = 6.2 and Q = 0.05 when ξ = H= Ec = 0.

Fig. 2: Temperature profile for different values of Ri with Pr = 6.2 and Q = 0.05 when ξ = H = Ec = 0.

From Figs. 1 and 2, it can be easily observed that as buoyancy force increases accordingly, the non-dimensional velocity also increases and the temperature profile decreases. In addition, the effect of Cu/H2O nanofluid on velocity and temperature profiles is depicted that nanofluid makes an increase in temperature profile also it causes decrease in velocity profile as compared to pure water. Figs. 3 and 4 represent respectively the local Nusselt number and skin friction values as a function of magnetic parameter ξ, for different values of Ri. From these two figures it is observed that both local Nusselt number and skin friction increases with buoyancy parameter Ri and due to the higher thermal conductivity of nanofluid, heat transfer as well as skin friction increase when compared to the pure water.

Fig. 3: Variation of local Nusselt number as a function of ξ for different values of Ri with Pr = 6.2 and Q = 0.05 when H = Ec = 0.

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Fig. 4: Skin friction values as a function of ξ for different values of Ri with Pr = 6.2 and Q = 0.05 when H = Ec = 0.

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Figs. 5-8 present the effects of three different types of nanofluids containing the nanoparticles, namely Copper (Cu), Alumina (Al2O3) and Silver (Ag) on velocity, temperature, Nusselt number and skin friction respectively. These results are obtained by considering Pr = 6.2, Ri = 1, Q = 0.05 and the volume fraction as φ = 4%. The velocity profile increases from Al2O3 to Ag and the trend reverses in thermal boundary layers as shown in Figs. 5 and 6.

Fig. 5: Velocity profile for different nanoparticles with φ = 0.04, Pr = 6.2, Ri = 1 and Q = 0.05 when ξ = H = Ec = 0.

Fig. 6: Temperature profile for different nanoparticles with φ = 0.04, Pr = 6.2, Ri = 1 and Q = 0.05 when ξ = H = Ec = 0.

From Figs. 7 and 8, it is found that the nanofluid has higher values in heat transfer rates and skin friction when it is compared with pure water. In addition, the heat transfer rate in Cu/H2O nanofluid is higher than Ag/H2O nanofluid even though Ag has higher thermal conductivity than that of Cu and also the skin friction increases from Al2O3/H2O nanofluid to Ag/H2O nanofluid.

Fig. 7: Variation of local Nusselt number as a Fig. 8: Skin friction values as a function of a function of ξ for different nanoparticles with ξ for different nanoparticles with φ = 0.04, φ = 0.04, Pr = 6.2, Ri = 1 and Q = 0.05 when Pr = 6.2, Ri = 1 and Q = 0.05 when H = Ec = 0. H = Ec = 0.

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Fig. 9: Velocity profile for different volume Fig. 10: Temperature profile for different fraction of Cu-Water nanofluid Pr = 6.2, volume fraction of Cu-Water nanofluid Ri = 1 and Q = 0.05 when ξ = H = Ec = 0. Pr = 6.2, Ri = 1 and Q = 0.05 when ξ = H = Ec = 0.

In Figs. 9-12, the effects of the volume fraction (φ) on velocity, temperature, Nusselt number and skin friction are presented respectively. The numerical results are obtained by considering Pr = 6.2, Q = 0.05 and Ri = 1. For increasing volume fraction, the velocity profile decreases but the increase is not in significant level. The thermal boundary layer increases with volume fraction. These studies show that thermal conductivity of the fluid-particle system increases when volume fraction increases. Hence heat transfer increases with volume fraction as shown in Fig. 11 and the skin friction also follows the same trend as in Fig. 12.

Fig. 11: Variation of local Nusselt number as a Fig. 12: Skin friction values as a function function of ξ for different volume fraction of of ξ for different volume fraction of Cu-Water nanofluid with Pr = 6.2, Ri = 1 and Cu-Water nanofluid with Pr = 6.2, Ri = 1 Q = 0.05 when H = Ec = 0. and Q = 0.05 when H = Ec = 0.

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Fig. 13: Velocity profile for different values of Fig. 14: Temperature profile for different heat source/sink parameter Q with Pr = 6.2 and values of heat source/sink parameter Q Ri = 1 when ξ = H = Ec = 0. with Pr = 6.2 and Ri =1 when ξ=H=Ec =0.

Figs. 13-16 describe the effects of heat generation or absorption (Q) on velocity, temperature, Nusselt number and skin friction respectively. The results obtained for Pr = 6.2, Ri = 1 corresponding to pure water and copper-water nanofluid with volume fraction φ = 0.04. It is observed that increasing of heat generation or adsorption (Q) increases both velocity and temperature profiles.

Fig. 15: Variation of the local Nusselt number as a function of ξ for different values of heat source/sink parameter Q with Pr = 6.2 and Ri = 1 when H = Ec = 0.

Fig. 16: Skin friction values as a function of ξ for different values of heat source/sink parameter Q with Pr = 6.2 and Ri = 1 when H = Ec = 0.

From Figs. 15 and 16, it is observed that the local Nusselt number decreases as Q increases. Contrarily skin friction increases with the increasing values of Q. From this analysis it is evidently clear that heat transfer rate with in the boundary layer is enhanced by a nanofluid when we compared to the conventional fluid. Iaeme.com/ijmet.asp

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6. CONCLUSION

This work deals with the effects of heat generation/absorption, buoyancy parameter, volume fraction, different types of nanofluids, skin friction and surface heat transfer over a semi infinite flat plate. The governing partial differential equations are reduced to simple polynomial equations whose coefficients are of independent parameters Pr, ξ, Q, Ri, H and Ec. These equations offer a practicing engineer a rapid way of obtaining shear stress and heat transfer for any combinations of Pr, ξ, Q, Ri, H and Ec. The great advantage involved in the present technique is that the results are obtained with high order of accuracy and the amount of calculation is certainly less when compared with more conventional methods. Hence the practicing engineers and scientists can employ this unique approximate technique as a powerful tool for solving boundary layer flow and heat transfer problems. Further, the work can be extended by considering Brownian motion and thermophoresis effects in the nanofluid flow model. 7. REFERENCES

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