Effects Of Radiation And Rotation On An Accelerated Vertical Plate With Uniform Mass Diffusion

Effects Of Radiation And Rotation On An Accelerated Vertical Plate With Uniform Mass Diffusion

A.R. Vijayalakshmi1, A. Florence Kamalam 2

1 Department of Applied Mathematics , Sri Venkateswara College of Engineering, Sriperumbudur- 602 105,INDIA

2 Panimalar Engineering College, Nazarethpet , INDIA

An analysis on effects of thermal radiation and rotation on an accelerated vertical plate with uniform mass is presented here. An exact solution is obtained for the axial and transverse components of the velocity by defining a complex velocity. The effects of velocity, temperature and concentration for different parameters like radiation parameter, rotation parameter, Schmidt number, thermal Grashof number, mass Grashof number, Prandtl number and time on the plate are discussed.

Key words: Accelerated vertical plate, gray, heat and mass transfer, thermal Radiation rotation.

1. Introduction

   The effect of coriolis force has wide applications in science and technology especially in Earth Science and in Ocenography. Radiative heat and mass transfer play an important role in manufacturing industries for the design of reliable equipment. Nuclear power plants, gas turbines and various propulsion devices for aircraft, missiles, satellites and space vehicles are examples of such engineering applications.

   Arpaci [2] studied the interaction between thermal radiation and laminar convection of heated vertical plate in a stagnant radiating gas. England and Emery [3] have studied the thermal radiation effects of an optically thin gray gas bounded by a stationary vertical plate. . Mass transfer effects on flow past a uniformly accelerated vertical plate was analyzed by Soundalgekar [4]. Singh and Singh [5] studied the mass transfer effects on flow past an accelerated vertical plate with uniform heat flux . Singh [6] studied the effects of coriolis as well as magnetic force on the flow field of an electrically conducting fluid past an impulsively started infinite vertical plate

   Bestman and Adjepong [7] studied the magnetohydrodynamic free convection flow, with radiative heat transfer, past an infinite moving plate in rotating incompressible, viscous and optically transparent medium. Das et al. [8] have analyzed radiation effects on flow past an impulsively started infinite isothermal vertical plate. Raptis and Perdikis [9] considered the effects of thermal radiation and free convection flow past a moving vertical plate. Again, Raptis and Perdikis [10] investigated free convection and mass transfer effects on optically thin gray gas past an infinite moving vertical plate. The governing equations were solved analytically.

   However, heat and mass transfer effects on an accelerated isothermal vertical plate in a rotating fluid in the presence of thermal radiation is not studied in the literature. It is proposed to study thermal radiation effects on flow past an accelerated infinite isothermal vertical plate with uniform mass diffusion, in a rotating fluid. The dimensionless governing equations are solved by Laplace transform technique.

2. Basic Equations

   Consider the three dimensional flow of a viscous incompressible fluid induced by uniformly accelerated motion of an infinite vertical isothermal plate with uniform mass diffusion in a rotating fluid [4,5]. On this plate, the x-axis is taken along the plate in the vertically upward direction and the y- axis is taken normal to x-axis in the plane of the plate and z-axis is normal to it. Both the fluid and the plate are in a state of rigid rotation with uniform angular velocity ' about the z-axis. The fluid considered here is a gray, absorbing-emitting radiation but a non-scattering medium. Initially, the plate and fluid are at rest

   with the temperature T

   and concentration C

   every where. At time t 0, the plate starts moving with

   a velocity ct' in its own plane in the vertical direction against gravitational field, in the presence of thermal radiation. At the same time the plate temperature is raised to Tw and the concentration to Cw , which are there after maintained constant. Since the plate occupying the plane z = 0 is of infinite extent, all

   the physical quantities depend only on z and t.

   flow is governed by the following equations:

   Then by usual Boussinesq's approximation, the unsteady

   u 2v g

   t

   (T

   T )

   g *

   (C '

   C ' )

   2 u

   z2

   (1)

   2

   2

   v 2u v

   (2)

   t

   C p

   T

   t

   z2

   2T

   k z2

   • qr

   z

   (3)

   C

   t

   2C

   D z2

   (4)

   The term

   qr

   z

   represents the change in the radiative flux with distance normal to the plate with the

   following initial and boundary conditions:
   t 0 : u 0,	T T ,	C C	for all z
   t 0 : u ct ' ,	T T ,	C C	at z 0
   u 0,	T T ,	C C	as z .

   following initial and boundary conditions:
   t 0 : u 0,	T T ,	C C	for all z
   t 0 : u ct ' ,	T T ,	C C	at z 0
   u 0,	T T ,	C C	as z .

   w w

   (5)

   By Rosseland approximation [6,7], radiative heat flux of an optically thin gray gas is expressed by

   qr

   z

   4a* (T4 T 4 )

   (6)

   It is assume that the temperature differences within the flow are sufficiently small such that

   T 4

   may be

   expressed as a linear function of the temperature. This is accomplished by expanding T 4

   in a Taylor series

   about T

   and neglecting higher-order terms, thus

   T 4 4T 3T 3T 4

   (7)

   By using equations (6) and (7), equation (3) reduces to

   T

   2T

   * 3

   (8)

   Cp t

   k

   y2

   16a

   T (T T )

   On introducing the following dimensionless quantities:

   1 1

   (u,v)

   (u,v) ,

   c 2 3

   t t ,

   z z

   c 3

   ,

   T T ,

   1

   2

   T T

   c3 w

   g (T T )

   C C

   g * (C C )

   (9)

   Gr w , C , Gc w ,

   c Cw C c

   Pr Cp

   1

   3

   16a

   1

   T3 3

   *

   *

   , , R

   k c 2

   ..

   k c 2

   and the complex velocity q = u + iv, i

   reduces to

   1 in equations (1) to (5), the equations relevant to the problem

   q 2i Gr

   t

   GcC

   2 q

   ,

   z 2

   (10)

   1 2 R

   t Pr z 2 Pr

   (11)

   C 1 2C

   t Sc z 2

   (12)

   The initial and boundary conditions in non-dimensional form are

   t 0 :

   q 0, q t, q 0,

   0,

   1,

   p> 0,

   C 0, C 1, C 0,

   for all

   at as

   z 0 & t 0

   z 0

   z .

   (13)

   All the physical variables are defined in the nomenclature. The solutions are obtained for the equations (10) to (12), subject to the boundary conditions (13), by Laplace-transform technique and the solutions are derived as follows:

   C erfc

   Sc

   (14)

   1 exp(2 2

   Rt ) erfc (

   Pr

   at ) exp(2

   Rt ) erfc (

   Pr

   at )

   (15)

   q 1 t Gr

   • Gc

     exp(2

     m t )erfc (d1) exp(2

     m t )erfc (d 2)

     2 b(1 Pr) c(1 Sc)

     t exp(2

     mt ) erfc (d 2) exp(2

     mt )erfc (d1)

     2 m

     • Gr exp(bt) exp(2 2b(1 Pr)

     • Gc exp(ct ) exp(2 2c(1 Sc)

       (b m)t )erfc (d 3) exp(2

       (c m)t )erfc (d 5) exp(2

       (b m)t )erfc (d 4)

       (c m)t )erfc (d 6)

       (16)

       • Gr exp(bt) exp(2 2b(1 Pr)

       Pr(b a)t)erfc (d 9) exp(2

       Pr(b a)t)erfc (d10)

     • Gr exp(2 2b(1 Pr)

       Rt )erfc (d 7) exp(2

       Rt )erfc (d8) Gc erfc

       c(1 Sc)

       Sc

       where

       • Gc exp(ct ) exp(2 2c(1 Sc)

       Sc c t )erfc (d11) exp(2

       Sc c t )erfc (d12)

       d1, d 2

       mt ,

       d 3, d 4 (b m)t ,

       d 5, d 6 (c m)t ,

       d 7, d8 Pr

       at ,

       d 9, d10 Pr

       (a b)t ,

       d11, d12 Sc

       ct ,

       z ,

       2 t

       a R ,

       Pr

       b R m , c

       1 Pr

       m Sc 1

       and

       m 2i.

       In equation(16), the argument of the complementary error function and error function is complex. Hence in order to obtain the u and v components of the velocity and skin-friction, we have used the following formula due to Abramowitz and stegun [1]:

       erf (a

   • ib)

     erf (a)

     exp(a 2 )

     1

     1

     2a

     cos(2ab)

   • i sin(2ab)

   where

   2 exp(a 2 )

   exp(n2 / 4)

   [ f

   n2 4a2 n

   (a,b) ign

   (a,b)]

   (a,b)

   fn 2a 2a cosh(nb) cos(2ab) n sinh(nb)sin(2ab)

   gn 2a cosh(nb) sin(2ab) n sinh(nb) cos(2ab)

3. Discussion of Results

   (a, b) 1016

   erf (a ib) .

   Using the above formula, expressions for u, v are obtained but they are omitted here to save the space. In order to get a physical view of the problem, these expressions are used to obtain the numerical values of u, v, for different values of the various parameter like rotation, radiation, Schmidt number, thermal Grashof number and mass Grashof number.

   Fig. 1, depicts the concentration profiles for different values of the Schmidt number (Sc = 0.16, 0.24, 0.6, 0.78) at time t = 0.2. The wall concentration increases with decreasing Schmidt number. It is observed that there is a fall in concentration due to increasing the values of the Schmidt number.

   Curve R t

   1 0.2 0.4

   2 2 0.4

   3 2 0.2

   4 5 0.2

   Curve R t

   1 0.2 0.4

   2 2 0.4

   3 2 0.2

   4 5 0.2

   1

   0.75

   The temperature profiles for air (Pr = 0.71) are calculated for different values of thermal radiation parameter from Equation

   (15) and these are shown in Fig. 2. The effect of thermal radiation parameter is important in temperature profiles. It is

   0.5

   0.25

   0

   1

   1

   2

   2

   3

   3

   4

   4

   0 0.5 1 1.5 2 2.5

   Z

   Fig.2.Temperature profiles for different R and t

   observed that the temperature increases with decreasing radiation parameter as well as the time.

   The primary velocity profiles of air for different values of the radiation parameter (R = 0.2, 25.0), Gr = 5, Gc = 5, Sc = 0.6, t = 0.2, Pr = 0.71 and rotation parameter ( = 0.5, 2, 3) are shown in Fig. 3. It is observed that the primary velocity increases with decreasing radiation parameter R as well as the rotation parameter in cooling of the plate. This shows that primary velocity decreases in the presence of high

   thermal radiation and rotation

   .

   The secondary velocity profiles of air for different values of the radiation parameter (R = 0.2, 25.0), Gr = 5, Gc = 5, Sc = 0.6, t = 0.2, Pr = 0.71 and rotation parameter ( = 0.5, 2, 3) are shown in fig. 4, the effect of radiation increases the secondary velocity v. But the effect of rotation on v is just reverse to that of radiation parameter.

   The primary velocity profiles for different thermal Grashof number (Gr = 2, 5), mass Grashof number (Gc = 2, 5), Sc = 0.6 and time t = 0.2 are shown in fig. 5. It is clear that the primary velocity increases with increasing thermal Grashof number or mass Grashof number.

   0.6

   0.5

   0.4

   u

   u

   0.3

   0.2

   0.1

   0

   curve Gr Gc

   1	5	5
   2	2	5
   3	5	2
   4	2	2

   1	5	5
   2	2	5
   3	5	2
   4	2	2

   1

   2

   3

   4

   0.2

   Curve Gr Gc 1 5 5

   2 2 5

   3 5 2

   4 2 2

   4

   3

   2

   1

   Curve Gr Gc 1 5 5

   2 2 5

   3 5 2

   4 2 2

   4

   3

   2

   1

   0.1

   0

   v

   v

   -0.1

   -0.2

   -0.3

   -0.4

   -0.1

   0 0.5 1

   z 1.5 2 2.5

   -0.5

   Fig.5.Primary velocity profiles for different Gr and Gc

   0 0.5 1

   1.5 2 2.5

   Z

   Fig.6.Secondary velocity profiles for different Gr and Gc

4. Conclusions

   Theoretical analysis is performed to study flow past an accelerated infinite isothermal vertical plate with uniform mass diffusion, in the presence of thermal radiation in a rotating fluid. The dimensionless governing equations are solved by Laplace-transform technique. The conclusions of the study are as follows:

   • Concentration falls with the raise in Schmidt number.

   • Temperature is enhanced with the decreasing radiation parameter and increasing time

   • The influence of the radiation or rotation parameter on primary flow has a retarding effect for cooling of the plate.

   • The secondary velocity is enhanced with the raise in thermal radiation and opposite phenomenon occurs with the rotation parameter.

   • Primary velocity is enhanced with the raise of thermal Grashof number or mass Grashof number but the secondary velocity has reverse phenomenon .

5. Appendix: Notation

a* absorption coefficient [m 1 ]

C concentration [ kgm3 ]

1. dimensionless concentration

   Cp specific heat at constant pressure [ Jkg1K 1 ]

2. mass diffusion coefficient [ m2s1 ] g acceleration due to gravity [ms 2 ] Gr thermal Grashof number [-]

Gc mass Grashof number [-]

k thermal conductivity of the fluid [Wm1K 1 ]

Pr Prandtl number [-]

qr radiative heat flux in the y-direction [Wm 2 ]

R radiation parameter [-]

Sc Schmidt number [-]

T

temperature of the fluid far away from the plate [-]

Tw temperature of the plate [ K ]

T temperature of the fluid near the plate [-]

t time [ s]/p>

1. dimensionless time

   u velocity of the fluid in the x -direction [ms1 ]

2. dimensionless velocity

   v velocity of the fluid in the y -direction [ms1 ]

3. dimensionless velocity

y coordinate axis normal to x -axis [-]

z coordinate axis normal to the plate [-]

z dimensionless coordinate axis normal to the plate

Greek Symbols

volumetric coefficient of thermal expansion [ K 1 ]

* volumetric coefficient of expansion with concentration [ K 1 ]

coefficient of viscosity [Pa.s ]

kinematic viscosity [ m2.s1 ]

rotation parameter [ rad ]

dimensionless rotation parameter

density [kg.m3]

dimensionless skin-friction [kgm 1 s 2 ]

Stefan-Boltzman constant [Wm 2 K 4 ]

dimensionless temperature

erfc complementary error function

Subscripts

w conditions on the wall

free stream conditions

References

1. Abramowitz, B.M. and Stegun, I.A., Hand book of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Washington, U.S. Govt. Printing Office, 1964.

2. Arpaci, V.S., Effect of thermal radiation on the laminar free convection from a heated vertical plate,

   Int. J. Heat Mass Transfer, 11 (1968), pp. 871881.

3. England, W.G. and Emery, A.F., Thermal radiation effects on the laminar free convection boundary layer of an absorbing gas, Journal of Heat Transfer, 91, (1969), pp. 3744.

4. Soundalgekar V.M. 1982. Effects of mass transfer on flow past a uniformly accelerated vertical plate.

   Letters in heat and mass transfer, Vol.9, pp.65-72

5. Singh A.K. and Singh J. 1983. Mass transfer effects on the flow past an accelerated vertical plate with constant heat flux. Astrophysics and Space science Vol.97, pp.57-61

6. Singh, A.K., Hydromagnetic free convection flow past an impulsively started vertical infinite plate in a rotating fluid, Int. Comm. Heat Mass Transfer, 11, (1984), pp.399406.

7. Bestman, A.R. and Adjepong, S.K., Unsteady hydromagnetic free-convection flow with radiative heat transfer in a rotating fluid, Astrophysics and Space Science, 143, (1988), pp. 7380.

8. Das, U.N., Deka, R.K. and Soundalgekar, V.M., Radiation effects on flow past an impulsively started vertical infinite plate, J. Theoretical Mechanics, 1 (1996), pp. 111115.

9. Raptis, A. and Perdikis, C., Radiation and free convection flow past a moving plate, Applied Mechanics and Engineering, 4 (1999), pp. 817821.

10. Raptis, A. and Perdikis, C., Thermal Radiation of an optically thin gray gas, Int. J. of Applied Mechanics and Engineering, 8 (2003), pp. 131134.

Affiliation:

A.R.Vijayalakshmi (corresponding author) Department of Applied Mathematics, SriVenkateswara College of Engineering, Sriperumbudur 602 105,

India.

e-mail : avijaya@svce.ac.in