Influences of Thermal Radiation, Heat Generation ......T k qr ¶ ¶ = - 4 * * 3 4s where s* = 5. 67...

@IJAPSA-2016, All rights Reserved Page 139

Influences of Thermal Radiation, Heat Generation/Absorption, Variable Electric

Conductivity, Variable Surface Temperature and Variable Concentration with Variable

Viscosity on Heat and Mass Transfer Flow of Micropolar Fluid past a Stretching Sheet in

a Porous Medium

Mohammed Abdur Rahman1*

1 Assistant Professor, Department of Mathematics, Natural Science Group, National University, Gazipur-1704,

Bangladesh

Abstract In this paper we have examined the effect of thermal radiation, heat generation/absorption, variable

electric conductivity, variable surface temperature and variable concentration with variable viscosity

on heat and mass transfer flow of micropolar fluid past a linear stretching sheet in a porous medium.

In this problem the governing partial differential equations are highly non-linear which have been

converted into ordinary differential equations by using the similarity transformations and then solved

numerically by Shooting technique along with the Runge-Kutta numerical integration with

appropriate boundary conditions. The effects of various pertinent non-dimensional governing

parameters involved in the problem like wF , ξ , M , n , p ,Q , R , ∆ , 1λ have been studied on

velocity, microrotation, temperature and concentratiuon and discussed through graphs. The physical

parameters like local skin-friction coefficient, rate of coupling, surface heat transfer coefficient and

surface deposition flux which are very important for engineering interest are also presented for

distinct non-dimensional parameters in graphic and discussed their physical interpretation. The

results in the paper are found for micropolar liquid fluids. The results of the present paper are

compared with earlier studied work and found a close agreement between the results, hence an

encouragement for the use of the present code for our problem.

Keywords-Thermal radiation; heat generation/absorption; variable electric conductivity; variable

surface temperature; variable concentration; variable viscosity; micropolar fluids

I. Introduction

Heat can be transferred by conduction, convection and radiation. The radioactive heat transfer

flow has now become an important fact due to its wide engineering applications in manufacturing

industries where many tasks are operated through high temperature and hence the effect of radiation is

very significant. Gas turbines, nuclear power plants, high temperature plasmas, nuclear reactors, liquid

metal fluids, power generation, designing of reliable equipments and the various propulsion devices for

aircraft, missiles, satellites and space vehicles are examples of such engineering areas. Heat transfer in

polymer processing industry where the quality of the final product depends on the heat controlling

factors can be controlled significantly by the effects of thermal radiation. The Rosseland approximation

has been used in the energy equation to describe the radiative heat flux. The thermal radiation effect on

* corresponding author

International Journal of Applied and Pure Science and Agriculture (IJAPSA)

Volume 02, Issue 03, [March - 2016] e-ISSN: 2394-5532, p-ISSN: 2394-823X


heat and mass transfer problems has been studied by many authors [1-12] during the last few decades for

its uncountable practical applications.

The study of the flow of a porous medium is of great importance to geophysicists and fluid

dynamicists for the last few decades due to many engineering applications such as in heat exchanger

devices, petroleum reservoirs, filtration and nuclear waste repositories. Free convection and mass

transfer flow through a porous medium bounded by an infinite vertical limiting surface with constant

suction was studied by Raptis et al. [13]. Tamayol et al. [14] presented heat transfer in a porous medium

over a stretching surface with injection or suction and with different thermal boundary conditions.

Convective heat transfer in porous media has been discussed by Pop et al. [15] and Nield et al. [16]. A

mathematical model of induced magnetic field bounded by a porous vertical plate in the presence of

radiation was presented by Ahmed [17]. Recently, mixed convection flow of a micropolar fluid over a

continuously moving vertical surface immersed in a thermally and solutally stratified medium with

chemical reaction was studied by Rashad et al. [18]. Very recently, the effects of chemical reaction and

thermal radiation on unsteady free convection flow of a micropolar fluid past a semi-infinite vertical

plate embedded in a porous medium in the presence of heat absorption with Newtonian heating have

been investigated by Ahmed et al. [19].

Micropolar fluids are the fluids containing micro-constituents that are allowed to undergo rotation

which affect the hydrodynamics of the flow. Micropolar fluids are distinctly non-Newtonian in nature.

The basic theory for this type of fluids was first generated by Eringen [20] depending on which a good

number of various flow situations such as the flow of low concentration suspensions, the flow of

colloidal solutions, liquid crystals, human and animal blood, paints, body fluids, polymers, turbulent

shear flows, fluids with additives and many other situations can be explained. Over the years, the

dynamics of micropolar fluids has been a popular area of research and a significant amount of research

papers dealing with micropolar fluid flow was reported. For instance, the boundary layer flow of a

micropolar fluid past a semi-infinite plate was analyzed by Peddieson and McNitt [21]. Gorla [22]

presented heat transfer to a micropolar fluid flow over a flat plate with forced convection. Hady [23]

investigated heat transfer to micropolar fluid from a non-isothermal stretching sheet with injection. The

effect of surface conditions on the flow of a micropolar fluid driven by a porous stretching surface was

analyzed by Kelson and Desseaux [24]. Abo-Eldahab and El Aziz [25] studied flow and heat transfer in

a micropolar fluid past a stretching surface embedded in a non-Darcian porous medium with uniform

free stream. A numerical study for micropolar flow over a stretching sheet was reported by Aouadi [26].

Effects of temperature dependent viscosity and thermal conductivity on the unsteady flow and heat

transfer of a micropolar fluid over a stretching sheet were studied by Modather et al. [27]. The study of

heat generation and radiation effects on steady magnetohydrodynamic free convection flow of

micropolar fluid past a moving surface was accomplished by Reddy [28]. Recently, free convection flow

of a micropolar fluid along a moving vertical plate in a porous medium with velocity and thermal slip

boundary conditions has been studied by Mutlag et al. [29]. Very recently; Alam et al. [30] presented

unsteady hydromagnetic forced convective heat transfer flow of a micropolar fluid along a porous

wedge with convective surface boundary condition.

The study of MHD boundary layer flow and heat transfer on linearly stretched sheet becomes

industrially important matter due to its various possible engineering and physical applications in modern

metallurgical and metal-working process such as hot rolling, wire drawing, drawings of plastic films,

artificial fibers, glass fiber production, metal and plastic extrusion, continuous casting, crystal growing

and paper production. Sakiadis [31] was the first to present boundary layer flow over a stretched surface

moving with a constant velocity. Erickson et al. [32] extended the work of Sakiadis [31] and solved the

problem of mass transfer. Later many investigators analyzed heat and mass transfer on stretching sheet

of which Tsou et al. [33], Crane [34], Gupta and Gupta [35], Chen and Char [36], Elbashbeshy [37],

Ishak et al. [38] are of worth mentioning.




There exists a significant importance of heat generation or absorption that has attracted the

scientists because temperature distribution might be altered with possible heat generation. Recently,

Gnaneswara Reddy [39] investigated heat generation and thermal radiation effects over a stretching

sheet in a micropolar fluid. Very recently, effect of internal heat generation or absorption on MHD

mixed convection flow in a lid driven cavity was examined by Saha et al. [40].

Therefore the purpose of the present work is to investigate the influences of thermal radiation, heat

generation/absorption, variable electric conductivity, variable surface temperature and variable

concentration with variable viscosity on heat and mass transfer fluid flow past a stretching sheet in a

porous medium. In this paper the fluid considered is micropolar liquid. A numerical solution is obtained

for governing momentum, angular momentum, energy and concentration equations using shooting

technique along with the Runge-Kutta numerical integration with appropriate boundary conditions. The

effects of various governing parameters on the non-Newtonian micropolar fluid velocity, microrotation,

temperature, concentration, local skin-friction coefficient, rate of coupling, surface heat transfer, and

surface heat flux are displayed in figures and further analyzed in detail. The data produced by our code

in the problem presented in this paper is compared with that of published earlier by other author and

found a close agreement which is an encouragement for the use of the present code for our problem. It is

hoped that the results obtained in the present analysis will provide useful information for application

purposes as well as further extended studies.

II. Mathematical formulation of the problem

We consider the steady, two-dimensional MHD laminar convective flow of an incompressible,

viscous electrically conducting micropolar fluid on a linear stretching sheet placed in the (y = 0) of a

cartesian coordinates ),( yx system with the x-axis along the sheet. The fluid occupies the half plane (y

> 0). It is assumed that the velocity, temperature and concentration of the sheet are respectively axu = , p

w AxTT += ∞ and r

w DxCC += ∞ where a , A and D are positive constants, wT > ∞T and wC > ∞C in

which ∞T and ∞C being the uniform temperature and uniform concentration of the fluid respectively.

The magnetic field is assumed to be applied in the positive y-direction normal to the sheet and varies in

strength as a function of x and is defined by ))(,0( xBB =r

. The applied magnetic field strength has the

form xBxB /)( 0= and the electrical conductivity is assumed to have the form u0σσ =′ . Magnetic

Reynolds number of the flow is taken to be small enough so that the induced magnetic field is

negligible. Furthermore, we use the Rosseland approximation to define the radiative heat flux as

y

T

kqr

∂

∂−=

4

*

*

3

4σ where 8* 1067.5 −×=σ 42/ kmW is the Stefan-Boltzmann constant and *

k is the mean

absorption coefficient and then expand 4T into the Taylor series about ∞T , which after neglecting

higher order terms takes the form 434 34 ∞∞ −≅ TTTT . We also assume that the viscosity of fluid is an

inverse linear function of temperature and the component of thermophoretic velocity along the surface

of the sheet is negligible compared to that of its normal to the surface.

Under the above assumptions and usual boundary layer approximation, the governing equations for this

problem are given as follows (Rahman and Sultana [41]):

0=∂

∂+

∂

∂

y

v

x

u , (1)




1

2

1

22

00 )())((K

ub

K

uS

x

uB

y

S

y

uS

yy

uv

x

uu −

+−−

∂

∂+

∂

∂+

∂

∂=

∂

∂+

∂

∂

∞∞∞∞ ρ

µ

ρ

σσ

ρρ

µ, (2)

∂

∂+

∞−

∂

∂

∞=

∂

∂+

∂

∂

y

u

j

S

yj

s

yv

xu σ

ρ

σ

ρ

υσσ2

2

2, (3)

pc∞ρ ( )2

2

*

3*

02

2

3

16)(

y

T

k

TTTQ

y

Tk

y

Tv

x

Tu

∂

∂+−+

∂

∂=

∂

∂+

∂

∂ ∞∞

σ, (4)

))((2

2

∞−∂

∂−

∂

∂=

∂

∂+

∂

∂CCV

yy

CD

y

Cv

x

Cu Tm

, (5)

In the above equations, u and v are the velocity components along the x and y direction, µ is the

coefficient of dynamic viscosity, S is the vortex viscosity, ∞ρ is the mass density of the fluid, σ is the

microrotation component normal to the xy -plane, 0σ and 0B are constant strength of electrical

conductivity and magnetic field respectively, jS

s )2

( += µυ is the spin-gradient viscosity, j is the

micro-inertia density, T is the fluid temperature, pc is the specific heat of the fluid at constant pressure,

k is the thermal conductivity of the fluid, 0Q is the heat generation constant, C is the fluid concentration

in the boundary layer, Dm is the molecular diffusivity of the species concentration and TV is the

thermophoretic velocity.

The boundary conditions which are to be satisfied by the solution of the above equations (Rahman and

Sultana [41] and Alam et al. [42]):

( )

→→→→∞→

+==+==∂

∂−=±===

∞∞

∞∞

,CC ,,0,0 :

,Dx(x)CC ,,),(, :0y

r

w0

TTuy

CAxTxTTy

unxvvaxu p

w

σ

σ (6)

Here )(0 xv is a velocity component at the surface representing the permeability of the porous surface

where )(0 xv+ and )(0 xv− to mean fluid injection (blowing) and fluid suction respectively while

0)(0 =xv corresponds to an impermeable sheet and n represents microrotation parameter for 10 ≤≤ n .

The case n= 0 represents strong concentration while the case n = 0.5 indicates weak concentration of

microelements. The case n=1 is used to denote turbulent boundary layer flows which is beyond of our

present problem. A linear relationship between microrotation component σ and surface shear stress y

u

∂

∂

is developed to investigate the effect of various surface conditions.

III. Similarity analysis and method of the problem

For non-dimensionalisation of the problem let us introduce the following non-dimensional

variables which have been used by many investigators in literature like Rahman and Sultana [41] and

Alam et al. [42]:

∞

=υ

ηa

y , ( )ηυψ xfa ∞= , ( )ηυ

σ gxa

∞

=3

,∞

∞

−

−=

TT

TT

w

θ , ( )∞

∞

−

−=

CC

CC

w

ηϕ ,




,r

r

θµ µ

θ θ∞

=

− y

T

T

kV

r

tT

∂

∂−= ∞υ

,r

wt

T

TTk )( ∞−−=τ (7)

where η is the independent similarity variable, )(ηf the non-dimensionl stream function, )(ηg the

dimensionless micorotation, )(ηθ the dimensionless temperature, )(ηϕ the dimensionless

concentration, ψ is the stream function which is defined in the usual way )(ηψ

faxy

u ′=∂

∂= and

( )ηυψ

fax

v 2

1

)( ∞−=∂

∂−= where prime denotes differentiation with respect to η . Further, viscosity

parameter rr

w

T T

T Tθ ∞

∞

−=

− )(

1

* ∞−−=

TTwγ= constant whose value depends upon the viscosity and rT and

*γ are also constants. In the definition of thermophoretic parameter τ the thermophoretic coefficient is

tk .

By using equations (7) into the equations (2)-(5), we obtain the following non-dimensional ordinary

differential equations in the form:

0Re

1

)(

2

2

2

2

=′−′

∆+

−−

′−′∆+′′′−

+′−′′+′′′

∆+

−

fFf

fMgfffff

s

r

r

x

r

r

r

r

λθθ

θλ

θθθ

θ

θθ

θ

(8)

( ) ( )1

2 02

r

r

g g f f g fgθ

ξ ξθ θ

′′ ′′ ′ ′+ ∆ − ∆ + − − =

− (9)

( ) 01Pr3)1(Pr3)43( =

−+′−′−+′′+ θ

θ

θθθ

θ

θθ

r

v

r

v QRfpfRR (10)

0))(1())(1( =′′+′−−′′−−+′′ ϕθτθ

θϕθτ

θ

θϕ rfScfSc

r

v

r

v (11)

The appropriate transformed boundary conditions are:

1,1,,1,:0 ==′′−==′== φθη fngfFf w (12a)

∞→η : 0,0,0,0 ====′ φθgf (12b)

where ∞

±=υa

xvFw

)(0 is the suction/injection velocity at the surface of the sheet for 0>wf and 0⟨wf

respectively, ∞

=∆µ

S is the vortex viscosity parameter,

∞

=ρ

σ 2

00BM is the magnetic field parameter,

∞

=υ

ξja

is the spin-gradient viscosity parameter, ∞

−= PrPr

θθ

θ

r

rv is the variable Prandtl number in

which k

cp∞

∞ =µ

Pr is the constant Prandtl number, ac

QQ

p∞

=ρ

0 is the heat generation/absorption




parameter, ∞

−= ScSc

r

rv θθ

θ is the variable Schmidt number in which

mDSc ∞

∞ =υ

is the constant

Schmidt number, ∞

=υ

2

Reax

x is the Reynolds number based on the fluid kinematic viscosity at constant

properties, aD

11 =λ is the local Darcy parameter in which

2

1

x

KDa = is the Darcy number,

x

bFs = is the

Forchhimer number and 3*

*

4 ∞

=T

KkR

σ is the radiation parameter.

The equations of chief physical interest for the present problem are the local skin-friction coefficient,

local plate couple stress, rate of heat transfer and surface deposition flux.

The equation defining the surface shear stress of the sheet is

00 )())(( == +∂

∂+= yyw S

y

uS σµτ (13)

The local skin-friction coefficient is defined as

( )

( ) ( )01Re

222

fnax

Cfr

r

x

wx

′′

−∆+

−==

∞θθ

θ

ρ

τ (14)

Thus from equation (14) we see that the local values of the skin friction coefficient xCf is proportional

to )0(f ′′ .

The equation defining the local plate couple stress is

0

02

)(=

=

∂

∂

+=

∂

∂=

y

yswy

jS

yM

σµ

συ (15)

The dimensionless couple stress is defined by

)0(

2

)(

2g

ax

MM

r

r

r

r

a

wx

′

∆+−

∆+

−==

∞

θθ

θ

ξθθ

θ

υρ (16)

Thus the local couple stress xM in the boundary layer is proportional to )0(g′ .

The local surface heat flux is written as follows:

( ) 0

4

*

*

0

)(3

4=

=∂

∂−

∂

∂−= y

y

wy

T

ky

Tkxq

σ (17)

The non-dimensional coefficient of surface heat transfer which is known as Nusselt number is defined as

follows:

( ))(

)(

∞−==

TTk

xxq

k

xxhNu

w

wx )0( )

3

41(Re θ ′+−=

Rx (18)

Thus from equation (18) we see that the local Nusselt number xNu is proportional to )0(θ ′− .

Rate of mass transfer per unit area of the surface is written as follows:

0=

∂

∂−=

y

mwy

CDm (19)




The non-dimensional coefficient of mass transfer which is known as Sherwood number is computed as:

)0(Re)(

ϕ ′−=−

=∞

x

wm

wx

CCD

xmSh (20)

The system of transformed governing equations (12)-(15) with the boundary conditions (16a) and (16b)

is solved numerically using the Nachtsheim-Swigert [43] iteration method along with the Runge-Kutta

numerical integration, procedure in conjunction with shooting technique. A step size of 001.0=η were

found to be satisfactory in obtaining sufficient accuracy of 610−

in all cases of the calculations.

IV. Comparison of the results

The numerical results of the present paper are compared by our code with those of previously

published paper of Abdou et al. [44] as shown below and an agreement between the results are noticed,

which assures us using the present code of the model.

Table-1: Comparision of the present results with those of Abdou et al. [44]

Parameter

values

)0(f ′′ )0(g ′ )0(θ ′−

Abdou

et al.

[44]

Present

results

Abdou

et al.

[44]

Present

results

Abdou

et al.

[44]

Present

results

0.0=∆ 0.56612 0.55602 0.00000 0.00000 0.15822 0.15822

0.1=∆ 0.47533 0.47513 0.00977 0.00977 0.29765 0.29765

0.0=Q 0.44255 0.44129 0.01786 0.01786 2.01209 2.01209

2.0=Q 0.44209 0.44103 0.01746 0.01746 1.67580 1.67580

V. Results and discussions

In this paper we have used the parameters values as 0.1=R , 0.2=r , 5.0=∆ , 0.5Re =x ,

0.11 =λ , 01.0=sF , 0.2=ξ , 5.0=Q 5.0=n , vPr =7, 2.1−=rθ , 0.1=M , 5.0=fw , 8.0=τ , vSc = 1.0,

5.0=p unless otherwise stated. To discuss the results of the problem the numerical calculations have

been presented in the form of non-dimensional velocity, microrotation (angular velocity), temperature

and concentration.

Figure 1(a) shows the velocity profiles for distinct values of suction parameter wF . It is seen from

the figure that velocity profiles decrease monotonically with the increase of suction parameter indicating

the usual fact that suction stabilizes the boundary layer growth. Figure 1(b) shows the microrotation

(angular velocity) profiles for different values of suction parameter. In general, the angular velocity of

the microelements increases with the increase of wF very close to the surface of the sheet. As suction

velocity intensifies the rotation of the microconstituents gets induced near the surface of the sheet. After

a short distance from the surface of the sheet ( 5.0≈η ) where kinematic viscosity is dominant these

profiles overlap and then decrease with the increase of wF within the boundary layer region. Figure 1(c)

shows the effect of suction parameter wF on temperature profiles. It is seen from this figure that with

the increase of wF temperature decreases. This is expected because the fluid particles close to the

heated surface absorb more heat from the sheet and consequently the temperature of the fluid within the

boundary layer increases. But when these fluid particles are sucked through the porous sheet there is

decrease to the temperature profiles. Figure 1(d) indicates that the concentration profiles showing the

effect of suction parameter. It can be seen that the concentration profiles increases while concentration

boundary layer growth decreases with the increase of the suction parameter. Thus suction can be used




for controlling the concentration as well as temperature function, which is required in many engineering

applications like nuclear reactors, generators etc.

In Figure 2, we have varied the spin-gradient viscosity parameter ξ keeping all other parameters

values fixed. From this figure we see that microrotation increases slowly with the increase of the spin-

gradient viscosity parameter ξ .

The effect of the variable concentration parameter r on the dimensionless concentration profiles is

shown in Figure 3. The increasing of the variable concentration parameter decreases the concentration

profiles while increases the boundary layer growth significantly inside the boundary layer. The results

for 0=r correspond to constant concentration of the fluid.

Figures 4(a)-(b) respectively show the velocity and microrotation profiles for different values of

magnetic field parameter M . From Figure 4(a) we see that the velocity profiles decreases with the

increase of magnetic field parameter indicating that magnetic field tends to retard the motion of the

fluid. Figure 4(b) reveals that microrotation profiles increase with the increase of M .

It is observed from Figure 5(a) that velocity profiles decrease with the increase of microrotation

parameter n . Figure 5(b) shows the microrotation profiles for different values of n .From this figure we

see that microrotation increases significantly with the increase of n .From Figure 5(c) we observe that

concentration profiles increase while concentration boundary layer growth decreases with the increase of

n . Figure 5(d) shows that the increasing values of radiation parameter R introduce strong decreasing

effect on the temperature profiles.

Figure 6(a) and 6(b) show respectively the velocity and microrotation profiles for different values

of vortex viscosity parameter ∆ . From these figures we see that velocity profiles increase whereas

microrotation profiles decrease very rapidly as the vortex viscosity parameter increases. It is also

understood that as the vortex viscosity increases the rotation of the micropolar constituents gets induced

in most part of the boundary layer except very close to the sheet where kinematic viscosity dominates

the flow.

Figure 7 shows temperature profiles for different values of heat generation/absorption parameter

Q . Owing to the presence of heat generation it is apparent from this figure that there is an increase in

the thermal state of the fluid as a consequence temperature profiles seem to increase. For heat

absorption, opposite phenomenon is revealed.

Insignificant increase of microrotation profiles is observed in Figure 8 with an increasing value of

viscosity parameter rθ .

Figures 9(a)-(c) respectively show the velocity, microrotation and concentration profiles for different

values of the Darcy parameter 1λ . From Figure 9(a) it is observed that the velocity profiles as well as

momentum boundary layer thickness decreases with the increase of 1λ . The case 0=λ corresponds to a

pure fluid, rather than a porous medium. The Darcy number aD the measurement of the porosity of the

medium increases as the porosity of the medium increases and hence λ decreases. Fluid gets more

space for large porosity of the medium to flow, as a consequence fluid velocity increases. Figure 9(b)

shows the effect of 1λ on the microrotation profiles. The microrotation is found to increase as 1λ

increases. From Figure 9(c) it is found that increasing 1λ increases concentration profiles but decreases

concentration boundary layer growth. Figure 9(d) illustrates the influence of variable surface parameter

p on temperature profiles within the boundary layer. This figure indicates that with the increase of

p temperature profiles decrease significantly. The profile correspond to 0=p is for constant surface

temperature which is higher than any other profile for 0⟩p .




(a) 0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

η

f '

Fw = 0.5, 1.0, 1.5, 2.0

(b) 0 1 2 3 4 5 6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

η

g

Fw = 0.5, 1.0, 1.5, 2.0

(c) 0 1 2 3

0

0.2

0.4

0.6

0.8

1

η

θ

Fw = 0.5, 1.0, 1.5, 2.0

(d) 0 1 2 3 4 5 6

0.4

0.6

0.8

1

η

φ

Fw = 0.5, 1.0, 1.5, 2.0

Figure 1: Variations of dimensionless (a) velocity (b) microrotation (c) temperature and

(d) concentration profiles for distinct values of wF .

0 2 4 6

0

0.2

0.4

0.6

0.8

1

η

g

ξ = 1.0, 2.0, 3.0, 4.0

0 2 4 6

0.2

0.4

0.6

0.8

1

r = 0.0, 2.0, 3.0, 5.0

η

φ

Figure 2: Variations of dimensionless Figure 3: Variations of dimensionless

microrotation profiles for distinct concentration profiles for distinct values

values of ξ . of r .




(a) 0 2 4 6

0

0.2

0.4

0.6

0.8

1

η

f 'M = 0.0, 0.5, 0.8, 1.0

(b) 0 2 4 6

0

0.2

0.4

0.6

0.8

1

η

gM = 0.0, 0.5, 0.8, 1.0

Figure 4: Variations of dimensionless (a) velocity and (b) microrotation profiles

for distinct values of M .

(a)0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

n = 0.0, 0.2, 0.5, 0.8

η

f '

(b) 0 2 4 6 8 10

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

n = 0.0, 0.2, 0.5, 0.8

η

g

(c) 0 2 4 6

0.4

0.6

0.8

1

n = 0.0, 0.2, 0.5, 0.8

η

φ

(d) 0 1 2 3

0

0.2

0.4

0.6

0.8

1

h

θ

R = 1.0, 2.0, 3.0, 4.0

Figure 5: Variations of dimensionless (a) velocity (b) microrotation and (c) concentration

profiles for distinct values of n and (d) temperature profiles for distinct values of R .




(a) 0 2 4 6

0

0.2

0.4

0.6

0.8

1

∆ = 0.0, 0.5, 1.0, 1.5

η

f '

0 2 4 6

0

0.2

0.4

0.6

0.8

1

1.2

∆ = 0.0, 0.5, 1.0, 1.5

g

η

Figure 6: Variations of dimensionless (a) velocity and (b) microrotation profiles for

distinct values of ∆ .

0 1 2 3

0

0.2

0.4

0.6

0.8

1

η

θ

Q = -0.2, 0.0, 0.2, 0.5

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

g

η

θr= -2.5, -2.0, -1.5, -1.0

Figure 7: Variations of dimensionless Figure 8: Variations of dimensionless

temperature for distinct values of values of Q . microrotation for distinct values of rθ .

(a) 0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

λ1 = 0.0, 1.0, 5.0, 10.0

η

f '

0 2 4 6

0

0.2

0.4

0.6

0.8

1

1.2

λ1= 0.0, 1.0, 5.0, 10.0

η

g

(c) 0 1 2 3 4 5 6 7 8

0.4

0.6

0.8

1

λ = 0.0, 1.0, 5.0, 10.0

η

φ

1

(d) 0 1 2 3

0

0.2

0.4

0.6

0.8

1

p = 0.0, 0.5, 1.0, 1.5

η

θ

Figure 9: Variations of dimensionless (a) velocity (b) microrotation (c) concentration profiles for distinct values of 1λ

and (d) temperature profiles for distinct values of variable surface temperature parameter p.




(a) 0 0.1 0.2 0.3 0.4 0.5

-1.94

-1.93

-1.92

-1.91

-1.9

-1.89

-1.88

Q

f''

(0)

R = 1, 2, 3, 4

(b)0 0.1 0.2 0.3 0.4 0.5

-1.22

-1.218

-1.216

-1.214

-1.212

-1.21

-1.208

-1.206

Q

g'(

0)

R = 1, 2, 3, 4

(c) 0 0.1 0.2 0.3 0.4 0.5

2

2.5

3

3.5

4

4.5

5

5.5

Q

θ-

'(0

)

R = 1, 2, 3, 4

(d)0 0.1 0.2 0.3 0.4 0.5

-5

-4

-3

-2

Q'(

0)

R = 1, 2, 3, 4

φ

Figure 10: Variations of R and Q on (a) local skin-friction coefficient (b) rate of

coupling (c) surface heat transfer and (d) surface deposition flux.

(a) 0.2 0.4 0.6 0.8 1

-2.15

-2.1

-2.05

-2

-1.95

-1.9

-1.85

-1.8

λ1

= 0, 1, 5

Scv

f''

(0)

(b) 0.2 0.4 0.6 0.8 1

-1.4

-1.3

-1.2

-1.1

Scv

g'(0

)

λ1 = 0, 1, 5

(c) 0.2 0.4 0.6 0.8 1

2.55

2.6

2.65

2.7

2.75

vSc

λ1 = 0, 1, 5

-'(0

)θ

(d) 0.2 0.4 0.6 0.8 1

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

φ'(0

)

= 0, 1, 5λ1

Scv

Figure 11: Variations of 1λ and Sc v on (a) local skin-friction coefficient (b) rate of

coupling (c) surface heat transfer and (d) surface deposition flux.




0.6 0.8 1 1.2 1.4 1.6 1.8 2-8

-7

-6

-5

-4

-3

-2

-1

r = 0, 1, 2, 3

Fw

φ'(

0)

Figure 12: Variations of r and Fw on surface deposition flux.

Figure 10(a) shows the local skin friction coefficient for different values of R and Q . From here

we see that )0(f ′′ decreases as R increases. For fixed value of R , in the case of heat generation

(Q )0⟩ , )0(f ′′ increases and thus to be understood, for heat absorption )0( ⟨Q , )0(f ′′ decreases. Figure

10(b) shows the rate of coupling for different values of Q and R . This figure reveals that for fixed Q ,

the rate of coupling increases with the increases of R . On the contrary, for fixed value of R the rate of

coupling )0(g ′ decreases with the increase of Q . The effect of radiation on the rate of heat transfer from

the surface of the sheet for different values of Q is illustrated in Figure 10(c). From here we see that for

fixed R the rate of heat transfer from the heated surface decreases with the increase of the heat

generation parameter Q . This is expected since the heat generation mechanism will increase the fluid

temperature near the surface. On the other hand, the presence of heat absorption )0( ⟨Q creates a layer of

cold fluid adjacent to the heated surface and therefore the heat transfer rate from the surface increases.

From this figure it is also clear that the rate of heat transfer increases with the increase of Radiation

parameter R . We see from Figure 10(d) that with the increase of R the surface deposition flux )0(ϕ ′

decreases strongly whereas )0(ϕ ′ increases with an increase of heat generation 0⟩Q .

Figures 11(a)-(c) show that if Darcy parameter 1λ increases then the physical parameters )0(f ′′ ,

)0(g ′ and )0(θ ′− decrease. We observe that these physical parameters decrease very quick for higher

value of )5(1 =λ . Again, for a fixed value of 1λ except higher like )5(1 =λ , the values of )0(f ′′ , )0(g ′

and )0(θ ′− remain almost uniform without varying with increase of vSc . Figure 11(d) displays that

)0(ϕ ′ enhances very slowly with the increase of values of 1λ . But for any fixed value of 1λ , with the

increase of vSc , )0(ϕ ′ decreases strongly.The effect of thermophoretic parameter τ and suction

parameter wF on surface deposition flux )0(φ ′ is shown in Figure 12. In this figure we find that

)0(φ ′ decreases effectively with the increase of τ . It is also clear from this figure that for any fixed

value of τ , )0(φ ′ decreases rapidly with the increase of wF .

VI. CONCLUSIONS

This paper presents a numerical study to examine the influences of thermal radiation, heat

generation/absorption, variable electric conductivity, variable surface temperature and variable

concentration with variable viscosity on heat and mass transfer flow of micropolar fluid of liquid type

past a linear stretching sheet in a porous medium. The governing partial differential equations have been




transformed into non-linear ordinary equations using similarity transformations and then solved with

Nachtsheim-Swigert shooting iteration technique. The following results of the numerical computations

can be drawn as conclusions:

1. The velocity profiles increase with the increase of vortex viscosity parameter ∆ whereas velocity

profiles decreases with increase of suction parameter wF , magnetic field parameter M , microrotation

parameter n and Darcy parameter 1λ .

2. The microrotation profiles increeases with increases of suction parameter wF , spin-gradient viscosity

parameter ξ , magnetic field parameter M , microrotation parameter n and viscosity parameter rθ

whereas microrotaion profiles decreases with increase of vortex viscosity parameter ∆ .

3. The temperature profiles increase with the increase of microrotation parameter n and heat

generation/absorption parameter Q whereas temperature profiles decrease with the increase of suction

parameter wF , variable surface temperature parameter p and radiation parameter R .

4. The concentration boundary layer growth increases with the increase of variable concentration

whereas concentration boundary layer growth decreases with the increase of suction parameter wF ,

microrotation parameter n and Darcy parameter 1λ .

5. The local friction coefficient increases with the increase of heat generation 0⟩Q whereas decreases

with the increase of radiation parameter R .

6. The rate of coupling increases with the increase of radiation parameter R whereas this physical

parameter decreases with the increases of heat generation 0⟩Q .

7. The rate of heat transfer from the surface of the sheet increases with the increases of radiation

parameter R whereas this physical parameter decreases with the increases of heat generation 0⟩Q .

8. The surface deposition flux increases with the increase of heat generation 0⟩Q whereas this surface

deposition flux decreases with the increase of radiation parameter R .

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Influences of Thermal Radiation, Heat Generation ......T k qr ¶ ¶ = - 4 * * 3 4s where s* = 5. 67...

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