Effects of Variable Viscosity and Thermal Conductivity on MHD Flow over a Radially Stretching Disk

Effects of Variable Viscosity and Thermal Conductivity on MHD Flow over a Radially Stretching Disk

G. C. Hazarika

and Joydeep Borah

1

Department of Mathematics,

Dibrugarh University, Dibrugarh, Assam- 786004, INDIA.

email:

[email protected],

[email protected]

(Received on: September 15, 2018) ABSTRACT

A numerical model is presented to study a steady magnetohydrodynamic (MHD) flow past a radially stretching or shrinking disk. In the analysis, effects of variable viscosity and variable thermal conductivity with viscous dissipation and radiation are taken into account. The governing non linear partial differential equations are made dimensionless using suitable similarity transformations and they are solved using the fourth order Runge-Kutta shooting method. The effects of viscosity parameter, thermal conductivity parameter, magnetic field parameter, Prandtl number, Schimdt number, radiation parameter, Eckert number etc. on velocity, temperature and species concentration fields are shown in graphs and interpretations are made accordingly. The values of skin friction coefficient, Nusselt number and Sherwood number for values of various parameters are shown in tabular form.

Keywords: Variable viscosity, variable thermal conductivity, MHD flow, stretching disk, viscous dissipation.

1. INTRODUCTION

The study of fluid flow through permeable wall with suction and injection is a very important branch of fluid mechanics. MHD is the study of the magnetic properties of electrically conducting fluids. There are vast applications of MHD fluid flow including the transpiration cooling of rocket engines and gas turbines, modelling of cross flow filtration systems, blood transportation, gastrointestinal system, kidneys and lungs where mass transfer between air, blood and tissue occurs at the walls.

Stretching or shrinking plates are commonly and naturally created. Researches on both

stretching and shrinking plates have been initiated since last decade. Fluid flow past a

stretching plate was first studied by Crane

where an exact analytical solution to the Navier- Stokes equations was reported. Meanwhile, Wang

pioneered the boundary layer flow over a shrinking plate, where a specific unsteady shrinking film solution was discussed. The industrial applications are arising in manufacturing of plastic and rubber sheet, polymer industry, continuous casting and spinning of fibres, glass glowing

etc. Stretching plate can be considered as typical plate; however the shrinking plate has a unique character which featured with a reverse flow in the boundary layer that would create a complexity. There are two conditions where the solutions of the fluid flow towards the shrinking case is possible to exist, having an adequate suction imposed on the plate or creating a stagnation flow or both

. Amen et. al.

studied the MHD stagnation point flow towards a stretching- shrinking sheet with slip effects.

Turkyilmazoglu

investigated the MHD fluid flow and heat transfer due to a stretching rotating disk. Freidoonimehr et al.

analyzed the MHD flow over a permeable rotating disk in the presence of Soret and Dufour effects. MHD fluid flow and heat transfer over a radially stretching/shrinking disk has been studied by Soid et al.

.

From the above studies, it is seen that most of the researchers restricted their investigations by assuming viscosity and thermal conductivity of the fluid as constant. But, it was known from the work of Herwig and Wicken

that these properties may change with temperature. Ling and Dybbs

had been made a numerical investigation of the effects of temperature dependent viscosity on forced convection over a flat plate in porous medium. So, a more accurate prediction can be achieved for the flow, heat transfer and mass transfer by taking the variation of such properties with temperature.

Motivated by the above mentioned investigations and applications, the aim of this study is to investigate the effects of temperature dependent fluid viscosity and thermal conductivity on MHD boundary layer flow due to a stretching disk with radiation and viscous dissipation. Shooting technique with fourth order Runge-Kutta method is used to obtain the solution of the boundary value problem.

2. MATHEMATICAL FORMULATION

Let us consider a steady flow of an electrically conducting fluid over a radially stretching disk as shown in Fig.1. The flow is generated due to a stretching disk. It is assumed that the fluid is extent infinitely in the positive z direction. The temperature at the free stream T is taken as constant. An external magnetic field B 0, 0, B

is applied perpendicularly to the surface of the disk under the following assumption.

At z 0 the disk is stretched with equal velocity u

cr , which is proportional to the distance from the centre of the disk in the radial direction. Here c is a constant.

The surface is permeable with the mass flux velocity w

, where w

0 is for suction and

0

0

w is for injection.

The fluid properties are considered as constant except the viscosity and thermal conductivity.

The induced magnetic field is ignored as the magnetic Reynolds number is assumed to be so small.

Fig.1: Flow configuration

On the basis of the above assumption, the governing equations are as follows:

Equation of continuity:

u u w 0

r r z (1)

Equation of conservation of momentum:

2 2

0 2

1 B u

u u u u

u w

r z z z z ⁽²⁾

Equation of conservation of energy:

2 2 2 3

2 2

0

2 2

16 1

p p p p

3

B u T

T T T T u T

u w

r z C z C z z C z C a C z

(3) Equation of concentration:

2 2

2 2

C C 3 C C D D C C D C

u w D D

r z r r r r r z z z ⁽⁴⁾

The boundary conditions are:

0 : ,

, ,

: 0, ,

w w w

z u u w w T T C C

z u T T C C (5)

Here, u , 0, w is the velocity of the fluid, T is the fluid temperature, C is the

concentration of the fluid, is the dynamic viscosity of the fluid, is the kinematic

viscosity, is the electrical conductivity of the fluid, is the thermal conductivity, is the

fluid density, C

is the specific heat at constant pressure, D is the mass diffusivity and a is

the mean absorption coefficient. is taken as the constant stretching parameter. It is positive for stretching and is negative for shrinking.

The last term of the equation (3) refers to the radiation parameter. Here, Rosseland diffusion model has been considered, where the radiation heat flux q

is expressed for radiation heat transfer as

4

3 q T

a z . Using the Taylor’s series, T

may be expressed as a linear function of T and expanding it about T and neglecting the second and higher order terms, we get T

4 T T

3 T

. T is the free stream temperature.

In order to obtain ordinary differential equations, the following similarity transformations are introduced:

, 2 , , ,

w w

T T C C c

u crf w c f z

T T C C ⁽⁶⁾

It is assumed that the temperature varies with r as T

T T r

.

Following Lai and Kulacki

, the viscosity and thermal conductivity of the fluid are assumed to be inverse linear function of temperature as

1 1

1 T T

(7)

1 1

1 T T

(8)

We define two parameters as

r w

T T

T T

, called viscosity parameter and

k w

T T

T T

, called thermal conductivity parameter.

Using these two parameters in (7) and (8), we have the viscosity and thermal conductivity respectively as:

r

r

,

k

(9)

Using equations (6) and (9) we get Dimensionless equation of momentum:

2

2

2 0

r r

r r

f f f f Mf (10)

Dimensionless equation of energy:

2 2 2

2

2 . . 0

k k r

k k r

Kr Pr f f EcPr f M Ec Prf (11)

Dimensionless equation of concentration:

2 2 0

r r

r r

Scf

(12)

The corresponding boundary conditions are:

0 : , , 1, 1

: 0, 0, 0

f f S

f

(13)

where,

2

B0

M c

is the magnetic field parameter,

16

3 Kr T

a is the radiation parameter, C

Pr = is the Prandtl number,

0 p

Ec c

C T

is the Eckert number, Sc

D is the

Schmidt number and

2 S w

c

is the mass transfer parameter.

2.1. Skin friction coefficient:

The skin friction coefficient C

is defined by

2

2

f

w

C u ^{, where}

u w

z r is the wall shear stress.

2.2 Nusselt Number:

The Nusselt number Nu is defined as

w w

Nu rq

T T ^{, where}

q T

z is the heat transfer from the surface of the disk.

2.3 Sherwood Number:

The Sherwood number Sh is defined as

w

Sh rm

D C C ^{, where}

m D C

z ^{is the} mass flux at the centre and D is the diffusion constant at the free stream.

Using the similarity transformations we get

2 0

1

r f

r

C f (14)

12

0

1

k k

Nu Re (15)

12

0

1

r r

Sh Re (16)

3. RESULTS AND DISCUSSION

The system of differential equations (10), (11) and (12) together with the boundary conditions (13) is solved numerically using fourth order Runge-Kutta shooting method.

Numerical values are computed by developing suitable codes in MATLAB. The present study has been done to obtain the effects of the parameters

,

, M Pr Kr Sc , , , and Ec on velocity, temperature and concentration profiles. The values of the parameters are taken as

r

0.1

K , Pr = 0.71 , M 1, Ec 0.01, Sc = 0.22,

5 and

5 . These values are applied unless otherwise stated.

With the increasing of the viscosity parameter

the velocity of the fluid flow decreases (Fig.2). Physically as

increases, the resistance to the relative motion of different layers of the fluid increases due to the increase of viscous force and as a result velocity of the fluid decreases. The concentration of the fluid decreases for increasing value of

(Fig.3).

Fig.2: Velocity distribution for various values Fig.3: Species concentration distribution for various values of _r

for various values of _r

In Fig.4, it is observed that the fluid temperature is reduced with an increase in the

thermal conductivity parameter

. This is due to the fact that thermal conductivity is an

inverse linear function of temperature.

Fig.4: Temperature distribution for various Fig.5: Velocity distribution for various values of _k

values of M

Figures 5-7 illustrate the effect of magnetic field on velocity, temperature and concentration profiles. As the magnetic field parameter M increases, velocity decreases. The presence of transverse magnetic field sets a resistive force, called Lorentz force, which opposes the velocity field. An increase in M increases the temperature as well as concentration profiles.

This is due to the resistive Lorentz force, which leads to increase the friction between the adjacent fluid layers, as a result heat is produced and hence temperature and concentration increases.

Fig.6: Temperature distribution for various Fig.7: Species concentration distribution for

values of M various values of M

In Fig.8 it is seen that temperature increases with the increase of radiation parameter

Kr . The temperature of the fluid is maximum at the surface of the disk and decreases to zero

as η increases.

Fig.9 depicts the variations of temperature profiles for different values of Eckert number Ec. We have noticed that an increase of Ec leads to increase the temperature distributions due to the fact that the heat energy is stored in the fluid because of frictional heating.

Fig.8: Temperature distribution for various Fig.9: Temperature distribution for various values of Kr values of Ec

An enhancement of Prandtl number ( Pr ) decreases the velocity (Fig.10). With the increase of Pr , viscosity increases, as a result velocity decreases. Fig.11 depicts the temperature profile for various values of Pr. It is observed that temperature of the fluid decreases with the increase of Pr . For higher Prandtl number the fluid has a relatively high thermal conductivity which decreases the temperature.

Fig.10: Velocity distribution for various values Fig.11: Temperature distribution for various

values of Pr values of Pr

Fig.12 displays the variation of concentration profile for various values of Sc. The

species concentration decreases with the increasing value of Sc. With the increase of Sc, the

concentration boundary layer becomes thinner due to which the concentration gradient

increases. As a result, the species concentration decreases.

Fig.12:Species concentration distribution for various values of Sc

Table 1: Effects of _r

,

, M Pr , and Sc

C

r k

M Pr Sc

C

3 4 5 6

5 1 0.71 0.22 -0.647664 -0.569157 -0.529703 -0.506051

2.414078 2.409765 2.406951 2.405011

0.089868 0.021292 0.009051 0.005140 5 3

4 5 6

1 0.71 0.22 -0.530413 -0.529949 -0.529706 -0.529565

3.018865 2.647720 2.406951 2.288119

0.009190 0.009098 0.009051 0.009023

5 5 0

1 2 3

0.71 0.22 -1.393263 -1.529703 -1.693447 -1.897558

2.623090 2.406951 2.181395 1.950759

0.020185 0.013094 0.009051 0.006598 5 5 1 0.5

0.6 0.7 0.8

0.22 -0.528679 -0.529222 -0.529664 -0.530027

2.084820 2.238732 2.391702 2.543833

0.008832 0.008947 0.009043 0.009122 5 5 1 0.71 0

0.1 0.2 0.3

-0.529708 -0.529701 -0.529703 -0.529707

2.406951 2.406951 2.406951 2.406951

0.000337 0.018775 0.543732 3.063866

4. CONCLUSION

From the above analysis the following conclusions can be made.

(i) With the increase of viscosity parameter velocity and species concentration decreases.

(ii) Temperature decreases for increasing value of thermal conductivity parameter and

Prandtl number.

(iii) Both temperature and species concentration increases while velocity decreases for increasing of magnetic field parameter.

(iv) With the increase of radiation parameter and Eckert number, temperature increases.

(v) Increasing of Prandtl number decreases the velocity.

(vi) Species concentration decreases with the increase of Schmidt number.

(vii) Skin friction coefficient increases with the increase of magnetic field parameter and Prandtl number and decreases with the increase of viscosity and thermal conductivity parameter.

(viii) Nusselt number increases with the increase of Prandtl number and decreases with the increase of viscosity parameter, thermal conductivity parameter and Magnetic field parameter.

(ix) Sherwood number increases with the increase of Prandtl number but decreases with the increase of viscosity parameter, thermal conductivity parameter, magnetic field parameter and Schmidt number.

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