Chapter 4 MHD Non-Darcy Flow with Thermal Radiation ...shodhganga.inflibnet.ac.in/bitstream/10603/23993/11/11_chapter4.pdf · Highly non-linear momentum and heat transfer equations

Chapter 4

MHD Non-Darcy Flow with

Thermal Radiation, Viscous and

Ohmic Dissipation ∗

4.1 Introduction

Viscous mechanical dissipation effects are important in geophysical flows and also in cer-

tain industrial operations and are usually characterized by the Eckert number. Mahajan

and Gebhart (1989) reported the influence of viscous heating dissipation effects in natu-

ral convection flows, showing that heat transfer rates are reduced by an increase in the

dissipation parameter. With this awareness, the effect of Ohmic heating on the MHD

free convection heat transfer has been examined for a Newtonian fluid by Hossain (1992).

Chamka and Khanafer (1999) analyzed the nonsimilar combined convection flow over a

vertical surface embedded in a variable porosity medium. Abo-Eldhab (2000) studied the

radiation effect in heat transfer in an electrically conducting fluid at stretching surface.

The influence of viscous dissipation and radiation on unsteady MHD free-convection flow

past an infinite heated vertical plate in a porous medium with time dependent suction

was studied by Israel-Cookey et al (2003). Abo-Eldahab EM and El-Aziz MA (2004)

studied the blowing/suction effect on hydromagnetic heat transfer by mixed convection

from an inclined stretching surface with internal heat generation/absorption. Heat and

mass transfer effects on moving plate in the presence of thermal radiation have been stud-

∗Heat transfer part published in Communications in Nonlinear Science and Numerical Simulation

(Elsevier Journal) 15 (2010) 1197-1209; Impact Factor: 2.697

73

CHAPTER 4. VISCOUS AND OHMIC DISSIPATION EFFECTS 74

ied by Muthukumarswamy (2004) using Laplace technique. For the problem of coupled

heat and mass transfer in MHD free convection, the effect of both viscous dissipation and

Ohmic heating are not studied in the above investigations. However, it is more realistic

to include these two effects to explore the impact of the magnetic field on the thermal

transport in the boundary layer. Abo-Eldahab and Abd El Aziz (2004) studied the effect

of Ohmic heating on mixed convection boundary layer flow of a micropolar fluid from a

rotating cone with power-law variation in surface temperature. Chen (2004) studied the

problem of combined heat and mass transfer of an electrically conducting fluid in MHD

natural convection, adjacent to a vertical surface with Ohmic heating. Tak and Lodha

(2005) studied the flow and heat transfer due to a stretching porous surface in the pres-

ence of transverse magnetic field to include heat due to viscous dissipation. Chaudhary

et al. (2006) analyzed the radiation effect with simultaneous thermal and mass diffusion

in MHD mixed convection flow from a vertical surface with Ohmic heating. Osalusi et al.

(2007) analyzed the effects of Ohmic heating and viscous dissipation on unsteady MHD

and slip flow over a porous rotating disk with variable properties in the presence of Hall

and ion-slip currents. Abel et al. (2008) studied momentum and heat transfer character-

istics in an incompressible electrically conducting viscoelastic boundary layer flow over a

linear stretching sheet in the presence of viscous and Ohmic dissipations. Mahantesh et

al. (2010) investigated the flow and heat transfer characteristics of a viscoelastic fluid in

a porous medium over an impermeable stretching sheet with viscous dissipation.

In view of the above investigations, the effects of thermal radiation, viscous dissipation

and Ohmic heating on MHD non-Darcy heat transfer and mass diffusion of species over a

continuous stretching sheet with electric and magnetic fields are analyzed in this chapter.

The flow is subjected to a transverse magnetic field normal to the plate. Highly non-

linear momentum and heat transfer equations are solved numerically using fifth-order

Runge-Kutta Fehlberg method with shooting technique.

4.2 Mathematical Formulations

4.2.1 Flow Analysis

We consider two-dimensional steady incompressible electrically conducting fluid flow over

a continuous stretching sheet embedded in a porous medium. The flow region is exposed

under uniform transverse magnetic fields−→B = (0, B0, 0) and uniform electric field

−→E =


(0, 0,−E0). Since such imposition of electric and magnetic fields stabilizes the boundary

layer flow (Dandapat and Mukhopadhyay, (2003)). It is assumed that the flow is generated

by stretching of an elastic boundary sheet from a slit by imposing two equal and opposite

forces in such a way that velocity of the boundary sheet is of linear order of the flow

direction. (see Fig.4.1). We know from Maxwell’s equation that ∇.−→B = 0 and ∇×−→E = 0.

When magnetic field is not so strong then electric field and magnetic field obey Ohm’s law−→J = σ(

−→E+−→q ×−→

B ), where−→J is the Joule current, σ is the magnetic permeability and−→q is

the fluid velocity. We assume that magnetic Reynolds number of the fluid is small so that

induced magnetic field and Hall effect may be neglected. We take into account of magnetic

field effect as well as electric field in momentum and thermal boundary layer equations.

Under the above stated physical situation, the governing boundary layer equations for

momentum and energy for mixed convection under Boussinesq’s approximation are

∂u

∂x+∂v

∂y= 0 (4.1)

u∂u

∂x+ v

∂u

∂y= ν

∂2u

∂y2+σ

ρ(E0B0 −B2

0u)−ν

ku− Fu2 + gβT (T − T∞) (4.2)

where u and v are the velocity components in the x and y directions respectively; ν is

the kinematic viscosity; g is the acceleration due to gravity; ρ is the density of the fluid;

βT is the co-efficient of thermal expansion; T is the temperature of the fluid inside the

thermal boundary layer, Tw is the plate temperature and T∞ is the fluid temperature in

the free stream. k is the permeability of the porous medium; qr is the radiative heat flux

in the y-direction; F is the emperical constant (Forchheimer number) in the second-order

resistance and setting F = 0 in Eq. (4.2), the equation is then reduced to the Darcy’s

law. The third and fourth terms on the right hand side of Eq. (4.2) stand for the first-

order (Darcy) resistance and second-order porous inertia resistance, respectively. It is

assumed that the normal stress is of the same order of magnitude as that of the shear

stress in addition to usual boundary layer approximations for deriving the momentum

boundary layer Eq. (4.2). The following appropriate boundary conditions on velocity are

appropriate in order to employ the effect of stretching of the boundary surface causing

flow in x-direction as

u = Uw(x) = bx, v = 0 at y = 0 (4.3)

u = 0 as y → ∞. (4.4)

To solve the governing boundary layer Eq. (4.2), the following similarity transformations


are introduced

u = bxf ′(η), v = −√bνf(η), η =

√b

νy. (4.5)

Here, f(η) is the dimensionless stream function and η is the similarity variable. Substitu-

tion of Eq. (4.5) in the Eq. (4.2) results in a third-order non-linear ordinary differential

equation of the following form

f′2 − ff

′′= f

′′′+Ha2(E1 − f

′)− k1f

′ − F ∗f′2+ λθ (4.6)

where k1 = νKb

is the porous parameter, Ha =√

σρbB0 is Hartmann number, E1 = E0

B0bx

is the local electric parameter, F ∗ = Fx is the local inertia-coefficient, λ = GrxRe2x

is the

buoyancy or mixed convection parameter, Grx =gβT (Tw−T∞)x3

ν2is the local Grashof number

and Rex =Uxν

is the local Reynolds number. In view of the transformations, the boundary

conditions (4.3)-(4.4) take the following non-dimensional form on stream function f :

f(0) = 0, f′(0) = 1, f

′(∞) = 0. (4.7)

The physical quantities of interest are the skin-friction coefficient Cf , which is defined as

Cf =τw

ρU2/2, (4.8)

where wall sharing stress τw is given by

τw = µ(∂u

∂y

)y=0

. (4.9)

Using the non-dimensional variables (4.5), we get from Eqs. (4.8) and (4.9) as

1

2CfRe

1/2x = f ′′(0). (4.10)

We now consider the heat transfer in the flow using appropriate boundary conditions in

the next section.

4.2.2 Similarity Solution of the Heat Transfer Equation

The governing boundary layer heat transfer with thermal radiation, vi scous and Ohmic

dissipations is given by

u∂T

∂x+ v

∂T

∂y=

κ

ρCp

∂2T

∂y2+

µ

ρCp

(∂u

∂y

)2

+σ

ρCp(uB0 − E0)

2 − 1

ρCp

∂qr∂y

(4.11)

where Cp is the specific heat at constant pressure and κ is the thermal conductivity. Ther-

mal boundary layer Eq. (4.11) takes into account the Joule heating or Ohmic dissipation


due to the magnetic as well as electric fields. To solve the thermal boundary layer Eq.

(4.11), we consider non-isothermal temperature boundary condition as follows:

T = Tw = T∞ + A0

(x

l

)2

at y = 0 (4.12)

T → T∞ as y → ∞ (4.13)

where A0 is the parameters of temperature distribution on the stretching surface, Tw

stands for stretching sheet temperature and T∞ is the temperature far away from the

stretching sheet.

The thermal radiation heat flux qr is employed according to Rosseland approximation

such that

qr = −4σ∗

3K

∂T 4

∂y(4.14)

where σ∗ and K are the Stefan-Boltzmann constant and the mean absorption coefficient,

respectively. As done by Raptis (1998), the fluid-phase temperature differences within the

flow are assumed to be sufficient small so that T 4 may be expressed as a linear function

of temperature. This is done by expanding T 4 in a Taylor series about the free stream

temperature T∞ and neglecting higher order terms to yield

T 4 = 4T 3∞T − 3T 4

∞. (4.15)

We introduce a dimensionless temperature variable θ(η) of the form:

θ =T − T∞Tw − T∞

, (4.16)

where expression for Tw − T∞ is given by Eqs. (4.12)-(4.13). Making use of the Eqs.

(4.14)- (4.15) in Eq. (4.11) we obtain non-dimensional thermal boundary layer equation

as1 +Nr

Prθ′′ + (fθ

′ − 2f′θ) = −Ecf ′′2 − EcHa

2(f ′ − E1)2 (4.17)

where Pr = ναis the Prandtl number, Ec =

b2l2

A0Cpis the Eckert number and Nr = 16σ∗T 3

∞3Kκ

is the thermal radiation parameter. Temperature boundary conditions of the Eq. (4.11)

take the following non-dimensional forms

θ(0) = 1, θ(∞) = 0. (4.18)

We solve Eqs. (4.6) and (4.17) using boundary conditions (4.7) and (4.18) numerically

by applying the method of fifth-order Runge-Kutta-Fehlberg method with shooting tech-

nique. The local Nusselt number which are defined as

Nux =xqw

κ(Tw − T∞), (4.19)


where qw is the heat transfer from the sheet is given by

qw = −κ(∂T

∂y

)y=0

, (4.20)

Using the non-dimensional variables (4.5) and (4.16), we get from Eqs. (4.19) and (4.20)

as

Nux/Re1/2x = −θ′(0). (4.21)

4.2.3 Similarity Solution of the Concentration Equation

To study the effects of various physical parameters on concentration profiles as well as on

local Sherwood number which are presented graphically. Conservation of species:

u∂C

∂x+ v

∂C

∂y= D

∂2C

∂y2. (4.22)

The boundary conditions for Eq. (4.22) as

u = Uw(x) = bx, v = 0, T = Tw = T∞ + A0

(x

l

)2

, (4.23)

C = Cw = C∞ + A1

(x

l

)2

at y = 0, (4.24)

u = 0, T → T∞, C → C∞, as y → ∞. (4.25)

where u and v are the velocity components in the x and y- directions respectively; ν is the

kinematic viscosity; C∞ is the concentration of the species in the free stream, Tw stands

for stretching sheet temperature and T∞ is the temperature far away from the stretching

sheet, Cw stands for concentration at the wall and C∞ is the concentration far away from

the stretching sheet. To solve the governing boundary layer Eq. (4.22), the following

similarity transformations are introduced

u = bxf ′(η), v = −√bνf(η), η =

√b

νy. (4.26)

θ =T − T∞Tw − T∞

, ϕ(η) =C − C∞

Cw − C∞. (4.27)

Substitution of Eqs. (4.26)-(4.27) into the governing Eq. (4.22) and using the above

relations we finally obtain a system of non-linear ordinary differential equations with

appropriate boundary conditions

ϕ′′ + Sc(fϕ′ − 2f ′ϕ) = 0. (4.28)


The boundary conditions (4.26)-(4.27) becomes

f(0) = 0, f′(0) = 1, θ(0) = 1, ϕ(0) = 1 at η → 0 (4.29)

f′(∞) = 0, θ(∞) = 0, ϕ(∞) = 0 as η → ∞, (4.30)

where, Sc = νD

is the Schmidt number. The physical quantity of interest is the local

Sherwood number which are defined as

Shx =xqm

D(Cw − C∞)(4.31)

where qm is the mass transfer which is defined by

qm = −D(∂C

∂y

)y=0

. (4.32)

Using the non-dimensional variables (4.26) and (4.27), we get from Eq. (4.31) as

Shx/Re1/2x = −ϕ′(0). (4.33)

4.3 Results and Discussion

Numerical solutions for effects of thermal radiation on non-Darcy mixed convection heat

transfer over a stretching sheet in the presence of magnetic field and thermal radiation

are reported. The results are presented graphically in Figs. 4.2-4.21 and conclusions are

drawn for flow field and other physical quantities of interest that have significant effects.

Comparisons of the present results with previously works are performed and excellent

agreement has been obtained. Non-linear ordinary differential equations are integrated

by Runge-Kutta Fehlberg method with shooting technique. Comparison of our results of

−θ′(0) with those obtained by Ishak et al. (2008), Chen (1998) and Grubka and Bobba

(1985)( see Table 4.1) in absence of buoyancy force and magnetic field show a very good

agreement. The values of the skin-friction co-efficient and the local Nusselt number in

terms of f ′′(0) and −θ′(0), respectively are presented in Table 4.2 for various values of

Hartmann number Ha. From Table 4.2, it is understood that the skin-friction and rate

of heat decrease with increase in Hartmann number. Table 4.3 gives the values of wall

temperature gradient −θ′(0) for different values of Hartmann number (Ha), Eckert num-

ber (Ec), local electric parameter (E1) and Prandtl number (Pr). Analysis of the tabular

data shows that magnetic field enhance the rate of heat transfer across the stretching

sheet to the fluid. However, the effect of Prandtl number (Pr), in absence of local electric


field parameter (E1), is to reduce the rate of heat transfer from boundary stretching sheet

to the fluid whereas in the presence of local electric parameter (E1), the effect of Prandtl

number (Pr) is to increase the rate of heat transfer. Thus application of electric field may

change the limitation of heat transfer.

Figs 4.2 and 4.3 show the effect of Hartmann number (Ha) on velocity and temperature

profiles, respectively by keeping other physical parameter fixed. Fig. 4.2 depicts that the

effect of Hartmann number is to reduce the velocity distribution in the boundary layer

which results in thinning of the boundary layer thickness. Temperature in the boundary

layer increases with increase in the value of Ha, as shown in Fig. 4.3. From this plot

it is observed that the transverse magnetic field contributes to the thickening of thermal

boundary layer. This is evident from the fact that the applied transverse magnetic field

produces a body force such as Lorentz force, which opposes the motion. Hence the

resistance offered by this body force to the flow is the cause of enhancing the temperature.

Fig. 4.4 is the plot of velocity profile for various values of electric field parameter E1.

It is clearly observed from this figure that the effect of electric parameter E1 on velocity

is to increase its value throughout the boundary layer but more significantly little away

from the stretching sheet. Analysis of the graph reveals that the effect of local electric

field parameter E1 is to shift the streamlines away from the stretching boundary. This

shifting of streamlines is significant little away from the stretching sheet. This is because

Lorentz force arising due to electric field acts as an accelerating force in reducing the

frictional resistance. Fig. 4.5 indicates the variation of temperature profile with electric

field parameter E1 when Hartmann number is fixed to 0.1. From this graph we observe

that the temperature decreases with the increases of electric field parameter E1, because

the Lorentz force arising due to electric field reduces frictional resistance which help is

reducing temperature distribution in the boundary layer for a fixed value of Hartmann

number.

Figs. 4.6 and 4.7 represent the variations of velocity and temperature distribution in

the boundary layer for various values of porous permeability parameter k1. It as observed

from these figures that the velocity distribution decreases with increasing the porous per-

meability parameter k1 whereas reverse trend is seen on temperature distribution because

the presence of porous medium is to increase the resistance to the flow, which causes

the fluid velocity to decrease. Thus rise in the temperature is observed by increasing the

porous permeability parameter k1. Fig. 4.8 depicts the graph of non-dimensional temper-

ature profile θ(η) for different values of Eckert number Ec. By analyzing the graph it is


revealed that the effect of Eckert number Ec is to increase the temperature distribution

in flow region. This is due to the fact that heat energy is stored in the liquid due to the

frictional heating. Thus the effect of increasing Ec, is to enhance the temperature at any

point.

Fig. 4.9 represents the graph of temperature profile for different values of Prandtl

number Pr. It is seen that the effect of Prandtl number Pr is to decrease, temperature

throughout the boundary layer which results in decrease of the thermal boundary layer

thickness with the increase of values of Prandtl number Pr. The increase of Prandtl

number means slow rate of thermal diffusion. The effect of radiation parameter Nr on

temperature profile is shown in Fig. 4.10. The effect of Nr is prominently seen through-

out the boundary layer. It is interesting to observe that the effect of Nr is to increase

the temperature distribution in the thermal boundary layer. This is due to the fact that

increase in the value of Nr implies increasing of radiation in the thermal boundary layer,

and hence increases the value of temperature profile in the thermal boundary layer. Fig.

4.11 illustrates the effect of drag (inertia) co-efficient of porous medium F ∗ in the mo-

mentum boundary layer. From this figure it is observed that the effect of drag co-efficient

is to decrease the velocity profile in the thermal boundary layer.

Figs. 4.12 and 4.13 represents the variations of velocity and temperature distribution

in the boundary layer for various values of mixed convection parameter or buoyancy

parameter λ. It is observed from these figures that the velocity distribution increases

with increasing the buoyancy parameter λ whereas reverse trend is seen on temperature

distribution. From this figure it is observed that the effect of buoyancy parameter λ

decrease the temperature profile in the boundary layer.

Fig 4.14. depicts the concentration profiles for various values of porous parameter

k1 in the solutal boundary layer. It is observed from this figure that the concentration

profiles increases with increase in the value of the porous parameter because of the fact

that increase in the value of k1 increases the permeability of the porous medium which

results is increase in fluid velocity and hence increases the concentration of the species in

the solutal boundary layer. Further, it is observed that there is also increase in the solutal

boundary layer with increase in the porous parameter k1. The variation of concentrations

profiles for various values of mixed convection parameter λ along η are seen from Fig.

4.15. It is found that the increase in the value of λ decreases the concentration in the

solutal boundary layer. Fig. 4.16 is the plot of variation of concentration profiles for

various values of Eckert number Ec along η. From this figure it is seen that there is not


much applicable change in the concentration profiles but if a close look is given on this

figure, it is noticed that that the effect of Eckert number Ec is to decrease the value of

the concentration in the solutal boundary layer.

Fig. 4.17 shows the effect of thermal radiation parameter Nr on the concentration

profiles along η. The effect of radiation is noticeably observed from this figure for 2 ≤ η ≤12, i.e. the effect of increasing the value of Nr is to reduce the concentration of species

due to reduction in the solutal boundary layer. Fig. 4.18 is the plot of concentration

of species in the solutal boundary layer for various values of Sc. It is observed that the

effect of increasing the value of Schmidt number is to decrease the concentration of the

diffusive species. The effects of porous parameter k1, electric parameter E1 and inertial

parameter F ∗ on Sherwood number with variations in Schmidt number are shown in Figs.

4.19-4.21. From these figures it is seen that the effect of increasing the porous parameter

k1 and inertial parameter F ∗ is to increase the value of the Sherwood number (see Figs.

4.19 and 4.20) whereas reverse trend is seen on increasing the strength of the electric field

E1 (see Fig.4.21).

4.4 Conclusions

Mathematical analysis has been carried out to study the MHD non-Darcy boundary layer

flow and heat transfer characteristics in an incompressible electrically conducting fluid

over a linear stretching sheet in presence of radiation and viscous dissipation. Highly

non-linear third-order momentum boundary layer equation is converted into a ordinary

differential equation using similarity transformations. Fifth-order Runge-Kutta-Fehlberg

method with shooting is used to solve momentum and heat transfer equations numerically.

The effects of various physical parameters like Prandtl number, Eckert number, Hartmann

number and local electric parameter on velocity and temperature profiles are obtained.

The following main conclusions can be drawn from the present study:

(i) Boundary layer flow attains minimum velocity for higher values of Hartmann number

(Ha).

(ii)The effect of increasing the values of Prandtl number (Pr) is to increase temperature

largely near the stretching sheet and the thermal boundary layer thickness decreases with

Prandtl number.

(iii) The effect of Eckert number and thermal radiation is to increase the thermal bound-

ary layer thickness.


(iv) Inertia effect is to decrease the velocity distribution in the momentum boundary layer.

(v) The effect of the local electric field is to increase velocity distribution and decrease

temperature in the boundary layer more significant little away from the stretching sheet.

(vi) The effect of porous permeability parameter is to decrease velocity and increase tem-

perature profile through out the boundary layer.

(vii) Boundary layer flow attains minimum concentration for higher values of Hartmann

number (Ha).

(viii)The effect of increasing the values of Schmidt number (Sc) is to increase concentra-

tion largely near the stretching sheet and the solutal boundary layer thickness decreases

with Prandtl number.

(ix) The effect of Eckert number and thermal radiation is to increase the solutal boundary

layer thickness.

(x) Inertia effect is to decrease the concentration distribution in the solutal boundary

layer.

(xi) The effect of the local electric field is to increase concentration distribution and de-

crease concentration in the boundary layer more significant little away from the stretching

sheet.

(xii) The effect of porous permeability parameter is to increase concentration profile

throughout the boundary layer.


Table 4.1: Comparison of Local Nusselt number −θ′(0) for Ha = 0, λ = 0 and various

values of Pr with Ishak et al.(2008), Chen (1998) and Grubka and Bobba (1985)

Pr Ishak et al. (2008) Chen (1998) Grubka and Bobba (1985) Present Results

1.0 1.3333 1.33334 1.3333 1.333333

2.0 −− −− −− 1.999996

3.0 2.5097 2.50997 2.5097 2.509715

4.0 −− −− −− 2.938782

5.0 −− −− −− 3.316479

6.0 −− −− −− 3.657769

7.0 −− −− −− 3.971509

8.0 −− −− −− 4.263457

9.0 −− −− −− 4.537609

10.0 4.7969 4.79686 4.7969 4.796871

Table 4.2: Analysis for Skin friction −f ′′(0), and local Nusselt number −θ′

(0) for various

values of Ha when Pr = 1, λ = 1 in absence of E1, Ec, Nr, F∗, k1

Ha −f ′′(0) −θ′

(0)

0.0 0.615066 1.412357

0.6 0.788658 1.377062

1.0 1.065770 1.335962

1.5 1.467078 1.252360

2.0 1.949595 1.238607


Table 4.3: Values of wall temperature gradient −θ′(0) for different values of Hartmann

number Ha, Eckert number Ec, local electric parameter E1 and Prandtl number Pr for

Nr = 0, λ = 0, F ∗ = 0, k1 = 0.

Ha Ec E1 Pr −θ′(0)

0.0 0.0 0.0 3.0 2.509715

0.0 0.0 0.0 5.0 3.316479

0.0 1.0 1.0 3.0 1.745111

0.0 1.0 1.0 5.0 2.219381

1.0 1.0 1.0 3.0 2.227830

1.0 1.0 1.0 5.0 2.916217

1.0 1.0 0.0 3.0 0.459953

1.0 1.0 0.0 5.0 0.366367

Figure 4.1: Boundary layer over stretching sheet


0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Pr=3.0, Ec=1.0, Nr=0.1, k

1=0.2,

F*=0.1, E1=0.0, λ=0.1

f'(η)

η

Ha=0.0 Ha=0.5 Ha=1.0 Ha=2.0

Figure 4.2: Influence of the Hartmann number Ha, on the dimensionless velocity profile

f ′(η).

0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

Ha = 0.0 Ha = 0.5 Ha = 1.0 Ha = 2.0

Pr=3.0,Ec=1.0,E

1=0.0,

k1=0.2,Nr=0.1,F*=0.1,=0.1

( )

Figure 4.3: Influence of the Hartmann number Ha, on the dimensionless temperature

profile f ′(η).


0 1 2 3 4 5 6 7 8 90.0

0.2

0.4

0.6

0.8

1.0

Pr=1.0, Ec=0.1, Ha=0.1, Nr=0.1, k

1=0.1,

F*=0.1, =0.1

f'( ) E

1=0.1

E1=1.0

E1=1.5

E1=2.5

E1=3.0

Figure 4.4: Variation of velocity profile for different values of electric parameter E1.

0 1 2 3 4 5 6 7 8 90.0

0.2

0.4

0.6

0.8

1.0

Pr=1.0, Ec=0.1, Ha=0.1, Nr=0.1, k

1=0.1

F*=0.1, =0.1( )

E1=0.1

E1=1.0

E1=1.5

E1=2.5

E1=3.0

Figure 4.5: Variation of temperature profile for different values of electric parameter E1.


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.0

0.2

0.4

0.6

0.8

1.0

Nr=0.1, Ha=0.1, Pr=3.0, E1=0.1, E

c=0.1

F*=0.1, =0.1f'( )

k1=0.0

k1=0.5

k1=1.0

k1=2.0

Figure 4.6: Variation of velocity profile for different values of k1.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.0

0.2

0.4

0.6

0.8

1.0

Nr=0.1, Ha=0.1, Pr=3.0, E1=0.1, E

c=0.1,

F*=0.1, =0.1

( ) k

1=0.0

k1=0.5

k1=1.0

k1=2.0

Figure 4.7: Variation of temperature profile for different values of k1.


0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

Pr=1.0, Ha=0.1, Nr=0.1, k1=0.1

F*=0.1, E1=0.1, =0.1

( )

Ec=0.1

Ec=0.5

Ec=1.0

Figure 4.8: Variation of temperature profile for different values of Eckert number Ec.

0 1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

Nr=0.1, Ha=0.1, E1=0.1, E

c=0.1,

k1=0.1, F*=0.1, λ=0.1

θ(η)

η

Pr=0.7 Pr=1.0 Pr=2.0 Pr=3.0 Pr=4.0

Figure 4.9: Variation of temperature profile for different values of Prandtl number Pr.


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.50.0

0.2

0.4

0.6

0.8

1.0

Ha=0.1, Pr=2.0, E1=0.1, E

c=0.1, k

1=0.1,

F*=0.1, =0.1( )

Nr=0.0 Nr=0.5 Nr=1.0 Nr=2.0

Figure 4.10: Effects of Nr on the temperature profile in the boundary layer.

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

Nr=0.1, Ha=0.1, Pr=2.0, E1=0.1, E

c=0.1,

k1=0.1, =0.1

f'( ) F*=0.0 F*=0.5 F*=1.5 F*=2.0

Figure 4.11: Effects of inertia co-efficient of porous medium F ∗ on the velocity profile in

the boundary layer.


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.0

0.2

0.4

0.6

0.8

1.0

Pr=2.0, Ec=0.1, E

1=0.1, Ha=0.1,

Nr=0.1, k1=0.1, F*=0.1

f'( )

=-0.5 =-0.2 = 0.2 = 0.5

Figure 4.12: Effects of buoyancy or mixed convection parameter λ on the velocity profile

in the boundary layer.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.0

0.2

0.4

0.6

0.8

1.0

Pr=2.0, Ec=0.1, E

1=0.1, Ha=0.1,

Nr=0.1, k1=0.1, F*=0.1( )

=-0.5 =-0.2 = 0.2 = 0.5

Figure 4.13: Effects of buoyancy or mixed convection parameter λ on the temperature

profile in the boundary layer.


0 5 10 15 20 25 30 350.0

0.2

0.4

0.6

0.8

1.0

Ha=0.1, E1=0.1, F*=0.1, λ=0.1,

Nr=0.1, Pr=3.0, Ec=0.1, Sc=0.22

φ(η)

η

k1=0.0

k1=0.5

k1=1.0

k1=2.0

Figure 4.14: Concentration profile for different values of porous permeability parameter

k1.

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

Pr=2.0, Ec=0.1, Ha=0.1, Nr=0.1,

k1=0.1, F*=0.1, Sc=2.0

φ(η)

η

λ= - 0.5 λ= - 0.2 λ= 0.2 λ= 0.5

Figure 4.15: Concentration profile for different values of mixed convection parameter λ.


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.0

0.2

0.4

0.6

0.8

1.0

Pr=4.0, Ha=0.5, E1=0.2, k

1=0.5,

F*=0.2, λ=0.5, Nr=2.5, Sc=0.22

φ(η)

η

Ec=0.2

Ec=0.8

Ec=1.5

Ec=2.0

Ec=3.0

Figure 4.16: Concentration profile for different values of Eckert number EC .

0 2 4 6 8 10 12 14 16 18 200.0

0.2

0.4

0.6

0.8

1.0

Ha=0.1, E1=0.1, k

1=0.1, F*=0.1,

λ=0.1, Pr=2.0, Sc=0.22φ(η)

η

Nr=0.0 Nr=0.5 Nr=1.0 Nr=2.0

Figure 4.17: Concentration profile for different values of thermal radiation parameter Nr.


0 1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

Pr=4.0, Ha=0.5, E1=0.2, k

1=0.2,

F*=0.2, λ=0.2, Nr=0.5, Ec=0.2

φ(η)

η

Sc=0.5 Sc=1.0 Sc=1.5 Sc=2.0

Figure 4.18: Concentration profile for different values of Schmidt number Sc.

1 2 3 4 5 6-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

Pr=4.0, Ha=0.5, E1=0.2, F*=0.2,

λ=0.5, Nr=2.5, Ec=0.2

- φ'(0)

Sc

k1=0.5

k1=2.0

k1=3.0

k1=4.0

Figure 4.19: Effect of Sc on Sherwood number for various values of k1.


1 2 3 4 5 6-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

Pr=4.0, Ha=0.5, E1=0.2, k

1=0.5,

λ=0.5,Nr=2.5, Ec=0.2

- φ' (0)

Sc

F* = 0.5 F* = 2.0 F* = 3.0 F* = 4.0

Figure 4.20: Effect of Sc on Sherwood number for various values of F ∗.

1 2 3 4 5 6-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

Pr=4.0, Ha=0.5, k1=0.5,

F*=0.2, λ=0.5, Nr=2.5,

- φ'(0)

Sc

E1=0.2

E1=1.5

E1=3.0

E1=4.0

Figure 4.21: Effect of Sc on Sherwood number for various values of E1.

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Chapter 4 MHD Non-Darcy Flow with Thermal Radiation ...shodhganga.inflibnet.ac.in/bitstream/10603/23993/11/11_chapter4.pdf · Highly non-linear momentum and heat transfer equations