Magnetohydrodynamic mixed convection flow and boundary layer control of a nanofluid with heat generation absorption effects

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)

Iaeme.com/ijmet.asp 18 [email protected]

1Department of Mathematics, Anna University, Chennai-600 025, India,

2Department 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].

MAGNETOHYDRODYNAMIC MIXED CONVECTION FLOW AND

BOUNDARY LAYER CONTROL OF A NANOFLUID WITH HEAT

GENERATION/ABSORPTION EFFECTS

M.Chandrasekar1, M.S.Kasiviswanathan

2

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)

IJMET

© I A E M E





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]

2

0

2 2

0 0

0 (1)

1( ) ( ) ( ) (2)

1( ) ( ) ( ) (3)

( )

x y

x y nf yy nf

nf

x y nf yy nf y

p nf

u v

uu vu u B U u g T T

uT vT k T u B u U u Q T TC

µ κ ρβρ

µ κρ

∞ ∞

∞ ∞

+ =

+ = + − + −

+ = + − − + −

subject to the boundary conditions





y = 0; u = 0, v = 0v , T = T0,

y → ∞; u=U∞= constant, T = T∞ (4)

Here u, v, 0v , T, κ, 0B , 0Q 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,

2.5

(1 )

(1 )

( )

( ) (1 )( ) ( )

( ) (1 )( ) ( )

2 2 ( )and

2 ( )

nf f s

f

nf

nf

nf

p nf

nf f s

p nf p f p s

nf s f f s

f s f f s

k

C

C C C

k k k k k

k k k k k

ρ φ ρ φρ

µµ

φ

αρ

ρβ φ ρβ φ ρβ

ρ φ ρ φ ρ

φ

φ

= − +

=−

=

= − +

= − +

+ − −=

+ + −

(5)

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

V

( )dV 0.δ σ − − =∫ ψ Φψ Φψ Φψ Φ (6)

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

, ( 1,2,3, )i i ia i fρ σ+ ∇ ⋅ = =J& L (7)

are satisfied. In Equation (6), δ is the variational symbol, σ is the entropy production, ψψψψ and ΦΦΦΦ are

dissipation potentials and V is the total volume of the thermodynamic system. In Equation (7), ρ is

the mass density and ia& , 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





1

0f

i

i iσ=

= ⋅ ≥∑J X (8)

where Ji and Xi are fluxes and forces respectively. According to Onsager’s linear theory [19, 20]

the fluxes are linear functions of forces, that is

1

, ( 1,2,3, )f

i ik k

k

L i f=

= =∑J X L (9)

or alternatively

1

, ( 1, 2,3, )f

i ik k

k

R i f=

= =∑X J L (10)

The constants Lik and Rik are conductivities and resistances respectively satisfying the reciprocal

relations [19, 20]

Lik = Lki and Rik = Rki, (i, k = 1,2,3,…f ) (11)

The matrices of Lik and Rik are mutually reciprocals and they are symmetric, that is

1 1

, ( , 1,2,3, )f f

im mk mk im ik

m m

L R L R i k fδ= =

= = =∑ ∑ L (12)

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

, 1

( , ) (1/ 2) 0f

ik i k

i k

L=

= ⋅ ≥∑X X X Xψψψψ (13)

, 1

( , ) (1/ 2) 0f

ik i k

i k

R=

= ⋅ ≥∑J J J JΦΦΦΦ (14)

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

1 , 1 , 1V

(1/ 2) (1/ 2) V 0f f f

i i ik i k ik i k

i i k i k

L R dδ= = =

⋅∇Γ − ∇Γ ⋅∇Γ − ⋅ =

∑ ∑ ∑∫ J J J (16)

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

0, ( )u v∇ ⋅ = = +V V i j (17)





2

0( ) P = ( B )[ ( )] ( ) ( )nf nf

U g T Tρ κ ρβ∞ ∞⋅∇ + ∇ ⋅ − ⋅ + −V V i V i (18)

2 2

0 0( ) ( ) = ( ) ( B )( )[ ( )] ( )p nf q nf yC T u U Q T Tρ µ κ ∞ ∞⋅∇ + ∇ ⋅ − ⋅ − ⋅ + −V J i V i V (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 P vspδ= + (20)

where p is the hydrostatic pressure, δ is the unit tensor and

o

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,

12( / ) ( / )q

T J lnT y P u yσ = − ∂ ∂ − ∂ ∂ (21)

where Jq is the heat flux and P12 is the only component of momentum flux

o

P vs , satisfy the

constitutive relations connecting the independent fluxes and forces as

12( / ) and ( / )q s

J L lnT y P L u yλ= − ∂ ∂ = − ∂ ∂ (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

2 2(1/ 2) ( ( / )s

T L lnT / y) L u yλ = ∂ ∂ + ∂ ∂ ψψψψ (23)

2 2

12(1/ 2)q s

T R J R Pλ = + ΦΦΦΦ (24)

where 1 1ands s

L R L Rλ λ− −= = .

Using Equations (21-24), Gyarmati’s variational principle (6) is formulated in the following form

2

12

2 2 20 0 12

( ) ( ) ( 2)( )0

( 2)( ) ( 2) ( 2)

lq

s q s

J lnT y P u y L lnT ydydx

L u y R J R P

λ

λ

δ∞ − ∂ ∂ − ∂ ∂ − ∂ ∂

=− ∂ ∂ − −

∫ ∫ , (25)

in which l is the representative length of the surface.

4. METHOD OF SOLUTION

It is assumed that the trial functions for velocity and temperature fields inside the respective

boundary layers are as follows





3 3 4 4

1 1 1 1

1

0

3 3 4 4

2 2 2 2

2

2 2 ( )

( )

( ) ( )

1 2 2 ( )

( )

u U y d y d y d y d

u U y d

T T T T

y d y d y d y d

T T y d

θ

∞

∞

∞ ∞

∞

= − + <

= ≥

− − =

= − + − <

= ≥

(26)

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 = 0v , T = T0, ∂ 2T/∂ y

2 = 0

y = d1 ; u = U∞ = constant, ∂ u /∂ y = 0 (smooth fit), ∂ 2u /∂ y

2 = 0 (27)

y = d2 ;T = T∞, ∂ T/∂ y = 0 (smooth fit), ∂ 2T/∂ y

2 = 0

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

5 5 4 4 2 2 '

1 1 1 1 0(4 / 5 3 / 2 / )y dv y d y d d vU∞ − + += , (28)

where 0v 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,

2 9 9 8 8 7 7 6 6 5 5

12 1 1 1 1 1 1

3 3 4 4 3 3

1 0 1 1 1

2 5 4 4 3 2

0 1 1 1 1 1

5 4 4 3 2

0 2 2 2

/ ( / )( 4 45 2 5 3 7 11 15 7 5

2 3 101/1800) ( )( 2 2 7/10)

( )( 5 2 7 30)

( )/ ( 5 2

s nf

nf

nf

nf nf

P L U d y d y d y d y d y d

y d vU y d y d y d

BU y d y d y d y d U d

gT T y d y d y d y

υ

υ

κ µ

β υ

∞

∞

∞ ∞

∞

′− = − + − − +

− + + − + −

+ − + − + +

− − − + − + + 5 4 4 3

1 2 1 2

2

1 2 1 1

30 10

3 2) (0 )

d d d d

d d d y d

−

+ − ≤ ≤

(29)





9 4 5 8 3 5 8 4 4 7 3 4

0 2 1 2 1 2 1 2 1 2

6 5 6 4 2 5 4 5 3 2 3 2 5 5

1 2 1 2 1 2 1 2 1 2 1 2

4 4 2 2 9 5 4 8 5 3

1 2 1 2 1 1 2 1 2

8 4 4

1 2

( ) (4 9 3 4 12 7

4 3 3 12 5 4 5 4 3 45

9 140 2 15 3 /10) ( 16 45 3 5

3 4 9

q nfJ L U T T d y d d y d d y d d y d d

y d d y d d y d d y d d y d d d d

d d d d d y d d y d d

y d d

λ α∞ ∞ ′ − = − − − +

+ + − − + +

′− + − + − +

+ − 7 4 3 6 2 4 6 5 5 4

1 2 1 2 1 2 1 2

5 2 3 3 2 4 4 3 3

1 2 1 2 1 2 1 2 1 2

4 4 3 3 4 4 3 3

0 2 2 2 1 2 1 2 1 2

2 7 8 6 7 5 6 4 5

1 1 1 1

3

1

7 2 3 4 15 3 5

6 5 2 3 49 180 18 35 3 )

( )( 2 2 2 2 )

( ( ) )( 16 7 8 36 5 4

8

nf nf p nf

y d d y d d y d d y d d

y d d y d d d d d d d d

v U y d y d y d d d d d d d

U C y d y d y d y d

y d

µ α ρ

∞

∞

− − +

+ − + − +

+ − + − + − +

+ − + − −

+ 4 2 2 2 9 8

1 1 0 1

8 7 7 6 6 5 5 4 4 3 3 2

1 1 1 1 1 1

2 5 4 4 3 2

1 1 0 0 2 2 2

2 1

4 52 35 ) ( ( ) )( 9

2 4 7 2 3 9 5 2 4 3

37 315) ( ( ) ( ) )( 5 2

3 10) (0 ); ( 1)

nf p nf

nf p nf

y d d B U C y d

y d y d y d y d y d y d

y d d Q T T C y d y d y d

y d y d Pr

κ α ρ

α ρ

∞

∞

− + + −

+ − − + − −

+ − + − − +

− + ≤ ≤ ≤

(30)

5 5 4 4 2 2

0 2 2 2 2

5 4 4 3 2

0 0 2 2 2 2

1 2

( ) (4 5 3 2 3/10)

( ( ) ( ) )( 5 2 3 10)

( ); ( 1)

q nf

nf p nf

J L U T T d y d y d y d

Q T T C y d y d y d y d

d y d Pr

λ α

α ρ

∞ ∞

∞

′− = − − + −

+ − − + − +

≤ ≤ ≤

(31)

9 4 5 8 3 5 8 4 4 7 3 4

0 2 1 2 1 2 1 2 1 2

6 5 6 4 2 5 4 5 3 2 3 2

1 2 1 2 1 2 1 2 1 2

4 4 3 3 9 5 4 8 5 3

2 1 2 1 2 1 1 1 2 1 2

8 4 4 7 4

1 2 1 2

( ) (4 9 3 4 12 7

4 3 3 12 5 4 5 4 3

36 3 35 4 15 ) ( 16 45 3 5

3 4 9 7

q nfJ L U T T d y d d y d d y d d y d d

y d d y d d y d d y d d y d d

d d d d d d d y d d y d d

y d d y d d

λ α∞ ∞ ′ − = − − − +

+ + − − +

′− + − + − +

+ − 3 6 2 4 6 5 5 4

1 2 1 2 1 2

5 2 3 3 2 5 5 4 4 2 2

1 2 1 2 2 1 2 1 2 1

4 4 3 3 2

0 2 2 2

7 8 6 7 5 6 4 5 3 4 2

1 1 1 1 1 1

7 8 6

2 1 2

2 3 4 15 3 5

6 5 2 3 45 9 140 2 15 )

( )( 2 2 1) ( ( ) )

(16 7 8 36 5 4 8 4 (32)

16 7 8

nf nf p nf

y d d y d d y d d

y d d y d d d d d d d d

v U y d y d y d U C

y d y d y d y d y d y d

d d d d

µ α ρ∞ ∞

− − +

+ − + − +

+ − + − + −

− + + − +

− + 7 5 6 4 5 3 4

1 2 1 2 1 2 1

2 2 2 9 8 8 7 7 6

2 1 0 1 1 1

6 5 5 4 4 3 3 2 2 9 8

1 1 1 1 1 2 1

8 7 7 6 6 5 5 4 4 3 3 2

2 1 2 1 2 1 2 1 2 1 2 1

2

2 1 0 0

36 5 4 8

4 ) ( ( ) )( 9 2 4 7

2 3 9 5 2 4 3 9

2 4 7 2 3 9 5 2 4 3

) ( ( ) (

nf p nf

nf p

d d d d d d

d d B U C y d y d y d

y d y d y d y d y d d d

d d d d d d d d d d d d

d d Q T T C

κ α ρ

α ρ

∞

∞

− − +

− + − + −

− + − − + +

− + + − + +

− + − 5 4 4 3 2

2 2 2 2

2

) )( 5 2 3 10)

(0 ); ( 1)

nfy d y d y d y d

y d Pr

− + − +

≤ ≤ ≥

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,

1 1 2 1 2

0

[ , , , ] 0, ( 1)

l

L d d d d dx Prδ ′ ′ = ≤∫ (33)





2 1 2 1 2

0

[ , , , ] 0, ( 1)

l

L d d d d dx Prδ ′ ′ = ≥∫ (34)

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

1,2 1 1,2 1( ) ( )( ) 0L d d dx L d ′∂ ∂ − ∂ ∂ = (35)

1,2 2 1,2 2( ) ( )( ) 0, ( 1, 1)L d d dx L d Pr Pr′∂ ∂ − ∂ ∂ = ≤ ≥ (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 *

1d , *

2d and are given by

* *

1 1 2 2/ and /f fd d x U d d x Uυ υ∞ ∞= = (37)

These variational principles (33) and (34) are subject to the transformations (37). The

resulting Euler-Lagrange equations are obtained as simple polynomial equations,

*

1,2 1 0L d∂ ∂ = (38)

*

1,2 2 0, ( 1, 1)L d Pr Pr∂ ∂ = ≤ ≥ (39)

The coefficients of these Equations (38) and (39) dependent on the independent parameters

Pr, ξ, Q, Ri, H and Ec where f fPr υ α= (Prandtl number), 2

0B x Uξ κ ρ ∞= (magnetic parameter),

0 / ( )p fQ Q x U Cρ∞= (heat generation/ absorption parameter) Ri=Gr/Re2 (Richardson number),

3 2

0( )f fGr g T T xβ υ∞= − (Grashof number), fRe U x υ∞= (Reynolds number), 0 fH v x Uυ ∞=

(suction/injection parameter) and 2

0( )pEc U C T 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 *

1d and *

2d are as

the only one set of positive real roots. After obtaining the values of *

1d and *

2d 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,

fy U xη υ∞= (40)

3

12 0( )w f s yx U P Lτ υ ∞ == − (41)





and 2

0 0( ) ( )l f q y

Nu x U T T J Lλυ ∞ ∞ == − . (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 * *

1 2 ( 1)d d Pr≤ ≤ and * *

1 2 ( 1)d d Pr≥ ≥ . 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-1

K-1

) k (Wm-1

K-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.

Pr Present Results

Nul

Chamkha et al. [21]

Nul

Aydin & Kaya [16]

Nul

Rana & Bhargava [7]

Nul

0.01 0.054742313 0.051830 0.051437 0.0596

0.1 0.147754551 0.142003 0.148123 0.1579

1 0.334277544 0.332173 0.332000 0.3319

10 0.738452128 0.728310 0.727801 0.7278

100 1.599967934 1.572180 1.573141 1.5721

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.





Fig. 1: Velocity profile for different values of Fig. 2: Temperature profile for different

Ri with Pr = 6.2 and Q = 0.05 when ξ = H= values of Ri with Pr = 6.2 and Q = 0.05

Ec = 0. 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 Fig. 4: Skin friction values as a function of

function of ξ for different values of Ri with ξ for different values of Ri with Pr = 6.2

Pr = 6.2 and Q = 0.05 when H = Ec = 0. and Q = 0.05 when H = Ec = 0.





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 nano- Fig. 6: Temperature profile for different

particles with φ = 0.04, Pr = 6.2, Ri = 1 and nanoparticles with φ = 0.04, Pr = 6.2,

Q = 0.05 when ξ = H = Ec = 0. 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.





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.





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 Fig. 16: Skin friction values as a function of

as a function of ξ for different values of heat ξ for different values of heat source/sink

source/sink parameter Q with Pr = 6.2 and parameter Q with Pr = 6.2 and Ri = 1

Ri = 1 when H = Ec = 0. 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.





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.

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