Effects Of Radiation And Heat Source On Micropolar Flow With Heat And Mass Transfer Over A Vertical Porous Moving Plate

Effects Of Radiation And Heat Source On Micropolar Flow With Heat And Mass Transfer Over A Vertical Porous Moving Plate

*I. J. Uwanta and **A. Hussaini

*Department of Mathematics, Usmanu Danfodiyo University Sokoto,Nigeria

**Teachers Service Board Sokoto, Nigeria

ABSTRACT

We present an analytical study for the unsteady, coupled, hydrodynamic, heat and mass transfer with radiation and heat source of an incompressible micropolar fluid flowing past a semi- infinite vertical moving porous plate embedded in a porous medium. The governing differential equations are solved analytically using twoterm harmonic and nonharmonic functions. The dimensionless translational velocity, microrotation, temperature and mass distribution function are computed for the different thermophysical parameters controlling the flow regime. All results are shown graphically. Additionally skin friction, Nusselt number and Sherwood number, which provide an estimate of the surface shear stress, rate of cooling of the surface and rate of mass transfer coefficient, respectively, are also computed.

Key words: Micropolar fluid, heat transfer, radiation, heat source, boundary layer.

INTRODUCTION

In numerous industrial transportation processes, convective heat and mass transfer take place simultaneously. Phenomena involving stretching sheets feature widely in for example, aerospace component production metal casting Dieter [1]. Unsteady free convective flows in a porous medium have received much attention in recent time due to its wide application in geothermal and oil reservour engineering as well as other geophysical and astrophysical studies. Moreover, considerable interest has been shown in radiation interaction with convection for heat and mass transfer in fluids. This is due to the significant role of thermal radiation in the surface heat transfer when convection heat is small, espercially in free convection problems involving absorbing emitting fluids. The unsteady fluid flow past a moving plate in the presence of free convection and radiation were studied by Monsour [2], Cogley et al. [3], Raptis and Perdikis [4]; Das et al. [5], Grief et al. [6], Ganeasan and Loganathem [7], Mbeledogo et al. [8], Makinde [9] and Satter and Kalim [10]. All these studies have been confined to unsteady fluid flow, we observe that little work were done in a non- porous medium. The effect of radiation on MHD flow and heat treansfer must be considered when high temperatures are reached. El Hakiem [11], studied the unsteady MHD oscillatory flow on free convection radiation through a porous medium with a vertical infinite surface that absorbs the fluid with a constant velocity. Raptis et al. [12], studied the effect of radiation on two dimensional steady MHD optically thin gray gas

flow along an infinite vertical plates taking into account the induced magnetic field. Cookey et al. [13], researched the influence of viscous dissipation and radiation on unsteady MHD free convection flow past on infinite heated vertical plate in a porous medium with time dependent suction. Singh and Dikshit [14], investigated the hydromagnetic flow past a continously moving semiinfinite plate at large suction. Kim [15], studied unsteady MHD convective heat transfer past a semiinfinite vertical porous moving plate.

The study of heat generation or absorption effects in moving fluid is important in view of several physical problems, such as fluids undergoing exothermic or endothermic chemical reactions. Possible heat generation effects may alter the temperature distribution and consequently, the particle deposition rate in nuclear reactors, electric chips and semiconductors wafers. Seddeek [16], studied the effects of chemical reaction, thermophoresis and variable viscosity on steady hydromagnetic flow with heat and mass transfer over a flat plate in the presence of heat generation/absorption. Patil and Kulkarmi [17], studied the effects of chemical reaction on free convection flow of a polar fluid through porous medium in the presence of internal heat generation. Radiation effects on unsteady MHD convection heat and mass transfer flow past a semiinfinite vertical permeable moving plate embedded in a porous medium was studied by Ramachandroprasad et al. [18]. Recently, Pal et al. [19], studied perturbation analysis of unsteady magnetohydrodynamic convection heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction.

All the above investigations were restricted to Newtonian or non- Newtonian fluids. In many enviromental and industrial flows the classical theory of Newtonian fluids is unable to explain the microfluid mechanical characteristics observed. Micropolar fluid are fluids with microstructure belonging to a class of complex fliuds with nonsymmerical stress tensor refered to as micromorphic fluids. Physically they represent many industrially important liquids consisting of randomly oriented particles suspended in viscous medium. The theory of micropolar fluids and its extension, the thermomicropolar fluid constitute suitable non Newtonian hydrodynamic and thermo hydrodynamic models which can stimulate the flow dynamics of colloidal fluids, liquid crystals, polymeric suspension, haemotological fluids etc. Hassannien and Gorla (1990), investigated the heat transfer to a micropolar fluid from nonisothermal stretching sheet with suction and blowing. Flow over a porous stretching sheet with strong suction or injection was investigated by Kelson and Farell, (2001).

The purpose of the present paper is to study the effects of radiation and heat source on micropolar flow with heat and mass transfer over a vertical porous moving plate. It is assumed that the plate is embedded in a uniform porous medium and moves with a constant velocity in the flow direction in the presence of a transverse magnetic field. The governing system of conservation equations, which are solved by using a regular perturbation method. Graphs are plotted for dimensionless translational velocity, micro rotational (angular velocity), temperature and mass transfer function for various values of the flow parameters. We also

compute dimensionless wall shear stress function, heat transfer rate and mass transfer coefficient which have been discussed.

FORMULATION OF THE PROBLEM

Two dimensional unsteady, laminar, incompressible,viscous, electrically conducting and heat/absorption micropolar fluid flowing past a semi infinite vertical moving porous plate embedded in a uniform porous medium and subjected to a uniform transverse magnetic field in the presence of a pressure gradient. According to the coordinate system the x axis is taken along the porous plate in the upward direction and y axis normal to it. The simplified two dimensional equations governing the flow in the boundary of unsteady, laminar and incompressible micropolar fluid are:

Continuity equation:

V *

y* 0

(1)

Momentum equation

u*

u*

1 p*

2u*

u* 2 *

*

t*

v y* x* vr y*2 g f T T k* 0 u

• 2vr

y*

1 3u *

K0 t*y*2 g

C C

(2)

Angular Momentum equation

j

* *

t*

*

*

y

2*

y*2

(3)

Energy equation

T

• T

  2T Q

  p p

  1 q

  t*

  v y*

  y*2

  C T T

  r

  p>C y

  (4)

  Concentration equation

  C

• C

2C

*

t*

v D K C

y* y*2

C

(5)

The third and the eighth terms on the right hand side of equation (2) denote the thermal and concentration buoyancy effects respectively. Furthermore, the last term on the right hand of the

energy equation (3) represents the radiative heat flux term. where

x , y and t are the

dimensional distances along and perpendicular to the plate and dimensional time, respectively.

u and v are the components of dimensional velocities along x and y direction, is the

fluid density, is the fluid electrical conductivity, 0

is the magnetic induction, j the micro-

inertia density of the component of the angular velocity vector normal to the xy plane , the

spine gradient viscosity, v is the fluid kinematic viscosity, vr

is the fluid kinematic rotational

viscosity, g is the acceleration due to gravity,

f and

are the coefficients of volume

k

0

expansions for temperature and concentration, k the permeability of the porous medium, 1 is

the elastic parameter, K is the chemical reaction, is the component of angular velocity, T is

r

the temperature, C is the concentration , is the fluid thermal diffusivity, and q is the local

radiative heat flux. The term Q T T

is assumed to be amount of heat generated or absorbed

per unit volume, Q is a constant, which may take on either positive or negative values. When the

wall temperature T exceeds the free stream temperature T

, the source term Q is greater than

zero and heat sink when Q is less than zero.

The boundary conditions for the velocity, temperature and concentration fields are:

u u , v v 1 ent ,T T T T ent ,

p 0

2

u , C C C C ent at y 0

y

u u

y2

u

1 ent ,T T

, C C ,

(6)

0

0

as y

Where u is the velocity of the moving plate, T and C are the temperature and the

p

concentration respectively, U

is the free stream velocity, and

U and n are constants. From

0

equation (2), it is obvious that the suction velocity at the plate is either a constant or a function of time. Hence the suction velocity normal to the plate is assumed in the form

v V

1 Aent

(7)

0

Where A is a real positive constant,

and A

is small values less than unity, and V0

is the

scale of suction velocity which is non-zero positive constant. The negative sign indicates that the suction is towards the plate.

Outside the boundary layer, equation (2) gives

1 p*

dU

* 2

x*

dt*

p

k* U 0 U

(8)

Eliminating

x

between equation (2) and equation (8), we obtain

u*

t*

u*

*

v y*

dU *

dt*

v U *

k

2U * v v

2u*

y*2

g f T T

0 r

u* 2 *

*

1 3u

k

v *1 0

0 u

• 2vr

y*

• K0

  • g

    t2

    C C

    (9)

    Where v

    is the coefficient of the kinematic viscosity. The fifth term on the RHS of the

    momentum equation (8) denote body force due to nonuniform temperature, the last term denote body force due to nonuniform concentration.

    The governing boundary layer equation with temperature dependent heat generation/absorption, constant magnetic field and radiation is given by equation (4), by using Rosseland approximation the radiation heat flux is given by

    4 T 4

    qr 3k y

    (10)

    Where

    and k respectively, the stephan Boltzmann constant and mean absorption

    coefficient. We assume that the temperature difference within the flow is such that T 4 may be

    expand in a Taylor series. Hence, expanding T 4

    get

    T 4 4T 3T 3T 4

    about T and neglecting higher order terms we

    (11)

    Now using equations (10) and (11) , equation (4) becomes

    T T

    2T Q

    16 T 2T

    V

    • T

    T

    (12)

    t*

    y*

    y*2

    Cp

    3k

    y2

    To write the governing equations and the boundary conditions in dimensionless form, the following non dimensional quantities are introduced

    u v

    V y U u v

    tV 2

    u , v , v 0 ,U ,U p , v , v 0 ,

    U V v U p U U V V

    0 0 0 0 0 0 0

    T T C C T T nv

    k 1V 2

    V 2 j

    , C , , n , k1 0 0 , j 0 ,

    T T C C T T V 2 0 V 2

    V 2

    0 0 0

    (13)

    n v

    2v

    , M 0 , Gr

    v f

    g T

    T

    , Gc

    v g T

    T

    , N M 1

    V 2

    U V 2

    U V 2 k

    0

    j

    0 0 0 0 0

    V

    2 K v

    2 , Kr 2

    Then substituting from equation (13) into equations (9), (3) , (12) and (5) and taking into account equation (7) we obtain

    u

    t

    1 Aent

    u

    y

    dU dt

    2

    • u

    1 y2 Gr N U u

    2 K

    y

    3u

    ty2

    • GcC

      (14)

      nt

      1 2

      t

      (15)

      1 Ae

      y

      y2

      nt 2

      t 1 Ae y z y2 S

      (16)

      C

      nt C

      1 2C

      t 1 Ae

      y

      Sc y2

• KrC

(17)

Where

Gr

v f

g T

T

is the thermal Grashof number,

Gc

v g C

C

is the solutal

U V

U V

2 2

0 0 0 0

2v vQ

Grashof number,

M 0

0

V 2

is the magnetic field parameter,

vC

S is the dimensionless

0 p

V 2C

4 T 3

heat generation/absorption coefficient,

Pr

p is the Prandtl number,

R

k

is the

thermal radiation parameter,

kv

V

Kr 2

0

is the chemical reaction parameter,

Gr v

D

is the

Schmidt number and

N M 1 .

k

The dimensionless form of the boundary conditions (6) becomes

p

u U

, v 1 Aent , 1 Aent , C 1 Aent ,

y

2u

y2

at y 0

(18)

u U , 0, 0, C 0,

as y

ANALYTICAL SOLUTIONS

In order to reduce the above system of partial differential equations to a system of ordinary equations in dimentionless form, we may represent the velocity, angular velocity, temperature and concentration as

u u

y entu y 0 2 ,

(19)

0 1

y ent y 0 2 ,

(20)

0 1

0 1

y ent y 0 2 ,

(21)

C C

y entC y 0 2 ,

(22)

0 1

By substituting the above equations (19)(22) into equations (14)(17) and equating the

harmonic and nonharmonic terms and neglecting the higherorder terms of the following pairs of equations for u0 ,0 ,0 ,C0 and u1,1,1,C1 .

1 u0 u0 Nu0 N Gr0 GcC0 20

Eu1 u1 N nu1 N n Au0 Gr1 GcC1 21

0 0 0

11 n1 A0

Z11 n S 1 A0

0 2 . We obtain

(23)

(24)

(25)

(26)

(27)

Z11 n S 1 A0

C0 ScC0 kr ScC0 0

C1 ScC1 n kr ScC1 AScC0

Where primes denotes ordinary differentiation with respect to y

The corresponding boundary conditions are

u0 up ,u1 0,0 u ,1 u1,0 1,1 1,C0 1,C1 1 at y 0

(28)

(29)

(30)

u 1,u 1, 0, 0, 0, 0, C 0, C 0 as y

(31)

0 1 0 1 0 1 0 1

The analytical solutions of equations (23)(30) stisfying boundary conditions (31) are given by

4 6 7 8

u y,t 1 B em2 y B ePry B em1 y B e y

1 B em7 y B e Pry B em5 y B em6 y

ent

15 18 21 23

(32)

B em4 y B em2 y B em1 y B e y

25 27 28 29

1 9 10

y,t B e y ent B em4 y B e y

(33)

2 11 12

y,t B em1 y ent B em5 y B em1 y

3 13 14

C y,t B em2 y ent B em6 y B em2 y

(34)

(35)

Where

m 1 1

1

1 2Z

1 4SV ,

m 1

2 2

Sc

S 2c 4ScK ,

r

1

,

m3 21

1 4N 1 ,

m4

1

4n

1

m5

1 1

1 4Z n S ,

m6

Sc

1

1 4n Kr

,

2 2Z

2 Sc

m 1 1 1 4N n E , B U ,

B 1 ,

B 1,

B Gr ,

7 2E

1 p 2 5

6 1 P2r Pr N

B Gc ,

B 2B1 ,

B U

1 B B

B ,

1 1

7 1 m2 m N

8 1 2 N 4 p

6 7 8

B AB1 ,

B u B ,

B Am1 ,

B 1 B ,

10 n

9 1 10

12 Zm2 m n S

11 12

1 1

2 2 r

B Am2 Sc ,

B 1 B , B 1,

B AB4m2 ,

2 2

14 m2 m Sc n K Sc

13 14 16

17 Em2 m n S

B18

AB6 Pr

,

EP2r Pr N n

B19

AB7 m1

1 1

,

Em2 m N n

B20

AB8

,

E 2 N n

B GcB11 ,

B GrB12 ,

B GcB13 ,

6 6

5 5

1 1

21 Em2 m N n

22 Em2 m N n

23 Em2 m N n

1. GcB14 ,

   B 2B9 m4 ,

   B 2B10 ,

   2 2

   4 4

   24 Em2 m N n

   25 Em2 m N n

   26 E 2 N n

   B15 1 B17 B18 B19 B20 B21 B22 B23 B24 B25 B26 ,

   B27 B17 B24 ,

   B28 B19 B22 , B29 B20 B26 .

   Skin friction

   Using equation (32), the skin friction at the wall is given by

2. m B Pr B m B B ent m B Pr B m B m B

f 2 4 6 1 7 8 7 15 18 5 21 6 23

m4 B25 m2 B27 m1B28 B29

(36)

Nusselt number

Similarly, Nusselt number from (34) is given by

u

N m B

ent m B

• m B

  (37)

  1 2 5 11 1 12

  Sherwood number

  Also the sherwood number from (35) gives

  h

  S m B

  ent m B

• m B

(38)

2 3 6 13 2 14

RESULTS AND DISCUSSION

The problem of effects of radiation and heat source on micropolar flow with heat and mass transfer over a vertical porous moving plate in the presence of variable suction has been formulated, analysed and solved by using multi parameter perturbation technique. Approximate solutions has been derived for the mean velocity, angular velocity, mean temperature and mean concentration. The effects of the flow parameters such as Hartmann number (M ) , Schmidt number (Sc) , Prandtl number (Pr) , Grashof number for heat and mass

transfer (Gr,Gc) , heat source (S ) , porosity parameter (K0 ) , chemical reaction parameter (Kr ) ,

and radiation parameter (R)

on the mean velocity, angular velocity, mean temperature and mean

concentration profiles of the flow field are presented with help of mean velocity profiles (Figures 112), angular velocity profiles (figures 1315), mean temperature profiles (Figures 1618) and concentration profiles (Figures 1920).

Figures 13 represent the mean velocity profiles due to variations in M (magnetic parameter), Sc (Schmidt number) and Pr (Prandtl number) . It is observed that the mean velocity decreases with increase of Magnetic parameter. It is also observed that the increases in Schmidt number and Prandtl number causes the decrease in mean velocity.

Figures 49 represent the mean velocity profiles due to variations in Gr (Thermal Grashoff

number), Gc (solutal Grashoff number), S (heat source parameter), K0 (permeability

parameter), Kr (chemical reaction parameter) and R (radiation parameter). It is obsreved that

the mean velocity increases with increase of thermal Grashoff number, solutal Grashoff number, heat source parameter and permeability parameter whereas it decreases with increase in chemical reaction parameter and radiation parameter.

Figures 1015 reveals the mean velocity profiles due to variations in U p

(velocity of the moving

porous plate), (epsilon) and t (time). It is noticed that whenever the velocity of the moving porous plate, epsilon and time increases causes the increase in mean velocity.

Figures 1315 are graphed to see the influence of

U p ,

and t

on micro rotation velocity

(angular velocity) . It is noticed that whenever U p ,

micro rotation velocity.

and t

increases causes the decrease in

Figures 1618 reveals the mean temperature profiles due to variations in Pr (Prandtl number), R (radiation parameter) and S (heat source parameter). It is noticed that whenever Prandtl number and radiation parameter increases the mean temperature decreases whereas the mean temperature increases with increase in heat source parameter.

Figures 19 and 20 elucidate the effects of Sc (Schmidt number) and Kr

(chemical reaction

parameter) on the mean concentration profiles of the flow field. It is noticed that the increases in Schmidt number and chemical reaction parameter cause the decrease in mean concentration.

Table 1: The effects of the Schmidt number Sc , Prandtl number Pr , and radiation parameter R ,

on the skinfriction coefficients

C f , Nusselt number Nu

and the Sherwood number Sh are

given in table (2). It is seen from the table that as Sc increases the skinfriction coefficients decreases and Sherwood number increases whereas the Nusselt number remains unchanged. However, as the Prandtl number effects increase, the skinfriction coefficients, Nusselt number and Sherwood number increases. Also, increases in the heat generation effects result in increase in skinfriction coefficient, Nusselt number and Shrewood number.

Table 2: The effects of heat source parameter S , chemical reaction parameter Kr

and

permeability parameter

K0 , on the skinfriction coefficient

C f , Nusselt number Nu

and the

Sherwood number Sh , are given in table 2. It is observed from this table that as S increases, the skinfriction coefficients, the Nusselt number and the Sherwood number decrease. However, as the chemical reaction parameter effects increase the skinfriction coefficient decreases and the Sherwood number increases whereas the Nusselt number remains unchanged. Increases in the permeability effects increase the skinfriction coefficient whereas the Nusselt number and the Shrwood number remains unaltered.

Table 3: The numerical values of the skinfriction coefficients

C f , Nusselt number Nu

and the

Sherwood number Sh for different values of the thermal Grashof number Gr , solutal Grashof number Gc and magnetic field parameter M are shown in table 3. It can be noticed from this table that an increasing in Gr and Gc lead to an increasing in the value of the skinfriction

coefficients, whereas the Nusselt number and Sherwood number remains unchanged. However, as M increases, the skinfriction coefficient decreases whereas the Nusselt number and Sherwood number remains unchanged.

Table 1: Shows the effects of Schmidt number Sc , Prandtl number Pr and

radiationparameter R on skin – friction coefficient C f

number Sh

, Nusselt number Nu

and Sherwood

Sc	Pr	R	Cf	Nu	Sh
0.2	0.71	0.1	0.2891	0.8139	0.8017
0.3	0.71	0.1	0.1362	0.8139	0.8049
0.4	0.71	0.1	0.0136	0.8139	0.8079
0.2	0.81	0.1	0.3996	0.9282	0.9111
0.2	0.91	0.1	0.5199	1.0431	1.0198
0.2	1.01	0.1	0.6402	1.1484	1.1173
0.2	0.71	0.2	0.3240	0.9093	0.8931
0.2	0.71	0.3	0.3619	1.0051	0.9841

0.2

0.71

0.4

0.4041

1.1019

1.0747

Table 2: Shows the effects of heat source parameter S , chemical reaction parameter Kr

and permeability parameter R on skin – -friction coefficient C f

Sherwood number Sh

, Nusselt number Nu

and

S	Kr	K0	C f	Nu	Sh
0.02	0.10	0.50	0.2891	0.8139	0.8017
0.03	0.10	0.50	0.2867	0.8071	0.7950
0.04	0.10	0.50	0.2843	0.8002	0.7881
0.02	0.20	0.50	0.2161	0.8139	0.8021
0.02	0.30	0.50	0.1567	0.8139	0.8025
0.02	0.40	0.50	0.1054	0.8139	0.8028
0.02	0.10	0.60	0.3418	0.8139	0.8017
0.02	0.10	0.70	0.3877	0.8139	0.8017
0.02	0.10	0.80	0.4280	0.8139	0.8017

Table 3: Shows the effects of thermal Grashof number Gr , solutal Grashof number Gc

and magnetic field parameter M on skin – friction coefficient C f

and Sherwood number Sh

, Nusselt number Nu

Gr	Gc	M	C f	Nu	Sh
2.0	1.0	1.0	0.2891	0.8139	0.8017
3.0	1.0	1.0	0.4269	0.8139	0.8017
4.0	1.0	1.0	0.5647	0.8139	0.8017
2.0	2.0	1.0	0.4519	0.8139	0.8017
2.0	3.0	1.0	0.6147	0.8139	0.8017
2.0	4.0	1.0	0.7775	0.8139	0.8017
2.0	1.0	2.0	0.1808	0.8139	0.8017
2.0	1.0	3.0	0.1141	0.8139	0.8017
2.0	1.0	4.0	0.0686	0.8139	0.8017

Fig 1: Effects of M on velocity profile

Fig 2: Effects of Sc on velocity profile

Fig 3: Effects of Pr on velocity profile

Fig 4: Effects of Gr on velocity profile

Fig 5: Effects of Gc on velocity profile

Fig 6: Effects of S on velocity profile

Fig 7: Effects of

K0 on velocity profile

Fig 8: Effects of

Kr on velocity profile

Fig 9: Effects of R on velocity profile

Fig 10: Effects of U p on velocity profile

Fig 11: Effects of on velocity profile

Fig 12: Effects of t on velocity profile

Fig 13: Effects of on angular velocity profile

Fig 14: Effects of U p on angular velocity profile

Fig 15: Effects of t onangular velocity profile

Fig 16: Effects of Pr ontemperature profile

Fig 17: Effects of R on temperature profile

Fig 18: Effects of S on temperature profile

Fig 19: Effects of Sc on concentration profile

Fig 20: Effects of

Kr on concentration profile

CONCLUSION: We have examined the governing equations for effects of radiation and heat source on micropolar flow with heat and mass transfer over a vertical porous moving plate. Analytical solution for the governing momentum, microrotation, energy and diffusion equations was obtained which allows the computation of the flow and heat transfer characteristics and their dependence on the material parameters. It is observed that the velocity decreases with an increasing magnetic field parameter, Schmidt number, Prandtl number, chemical reaction parameter and radiation parameter, however, an increase in the thermal Grashof number, solutal Grashof number, heat source parameter and permeability parameter results to an increase in velocity. The micro-rotation velocity decreases with an increasing epsilon, velocity of the moving porous plate and time respectively. The temperature field decreases with an increasing Prandtl number and radiation parameter whreas it increases with an increasing heat source parameter. Finally, the concentration profile decreases with an incresing Schmidt number and chemical reaction parameter respectively.

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