Unsteady Flow Past an Exponentially Accelerated Vertical Plate With Ramped Plate Heat Flux

Unsteady Flow Past an Exponentially Accelerated Vertical Plate With Ramped Plate Heat Flux

Bhaskar Chandra Sarkar

Department of Mathematics, Ramananda College, Bishnupur 722122, India

Abstract: An exact solution of an unsteady flow past an exponentially accelerated vertical plate with ramped plate heat flux has been studied. The governing coupled differential equations describing the flow are solved analytically by using Laplace transform technique. The influences of the various type of parameters on the velocity field, temperature distribution, shear stress and rate of heat transfer at the moving plate have been analyzed either graphically or in tabular form. It is found that both the velocity as well as the temperature of the fluid decrease with an increase in Prandtl number whereas they increase as time progresses. Prandtl number reduce the shear stress but enhance the rate of heat transfer at the moving plate.

Keywords: Exponentially Accelerated Plate, Ramped Wall Heat Flux, Prandtl Number and Grashof Number.

1. INTRODUCTION

Heat transfer is the area that deals with the mechanism responsible for transferring energy from one place to another when a temperature difference exists. Natural convection is one of the most economical and practical methods of cooling and heating. Natural convection is caused by temperature or concentration induced density gradient within the fluid. Natural convection flow occurs as a result of influence of gravity forces on fluids in which density gradients have been thermally established. With the growing sophistication in technology and with the increasing concern with energy and the environment, the study of heat transfer has, over the past several years, been related to a very wide variety of problems, each with its own demands of precision and elaboration in the understanding of the particular processes of interest. Areas of study range from atmospheric, geophysical and environmental problems to those in heat rejection, space research and manufacturing systems. In a wide class of natural convection processes, heat transfer occurs from a heated vertical surface placed in a quiescent medium at a uniform temperature. If the plate surface temperature is greater than the ambient temperature, the fluid adjacent to the vertical surface gets heated, becomes light and rises. Heavier fluid from the neighboring areas rushes into take the place of the rising fluid. Similarly the flow for a cooled surface is downwards. Gupta et al. [1] have studied free convective effects on the flow past an accelerated vertical plate in an incompressible vertical plate. Isachenko [2] has reviewed the problems of heat transfer. Singh and Naveen [3] have investigated free convection flow past an exponentially accelerated vertical plate. Hossain and

accelerated boundary motion. Barletta [7] has presented an analysis on the heat transfer by fully developed flow and viscous heating in a vertical channel with prescribed wall heat fluxes. The transient free convection flow past an infinite vertical plate with periodic temperature variation has been discussed by Das et al. [8]. Narahari et al.[9] have considered the transient free convection flow between infinitely long vertical parallel plates with constant heat flux at one boundary. Chandran et al.[10] have studied the natural convection near a vertical plate with ramped wall temperature. The developing flow near a semi-infinite vertical wall with ramped temperature has been investigated by Singh et al. [11]. Muthucumaraswamy et al.[12] have studied the heat transfer effects on the flow past an exponentially accelerated vertical plate with variable temperature. Singh and Singh [13] have presented the transient HHD free convective near a semi infinite vertical wall having ramped temperature. The effects of heat transfer and viscous dissipation on MHD free convection flow past an exponentially accelerated vertical plate with variable temperature have been investigated by Kishore et al.[14]. Chandrakala [15] has studied thermal radiation effects on moving infinite vertical plate with uniform Heat flux. Chandrakala and Bhaskar [16] have considered the effects of heat transfer on flow past an exponentially accelerated vertical plate with uniform Heat flux. Asogwa et al.[17] have investigated the flow past an exponentially accelerated infinite vertical plate and temperature with variable mass diffusion. Chandrakala [18] has studied effects of radiation on flow past an impulsively started infinite vertical plate with uniform Heat and mass flux. Das et al.[19] have investigated an unsteady free convection flow past a vertical plate with heat and mass fluxes in the presence of thermal radiation. Unsteady slip flow past an infinite vertical plate with ramped plate temperature and concentration in the presence of thermal radiation and buoyancy has been studied by Maiti and Mandal [20]. Reddy et al.[21] have presented the radiation and heat absorption effects on an unsteady MHD boundary layer flow along an accelerated infinite vertical plate with ramped plate temperature. Recently, an unsteady flow past an accelerated vertical plate with variable temperature has been investigated by Kalita et al.[22].

The motivation of our present investigation is to study the unsteady free convection flow of a viscous incompressible fluid past an exponentially accelerated vertical plate with ramped wall

Shayo [4] have studied analytically the skin friction in the

heat flux. Initially, at time

t 0 , the plate and fluid are at the

unsteady free convection flow past an accelerated plate. The

mass transfer effects on the flow past an exponentially

same constant temperature T

in a stationary condition. At

accelerated vertical plate with constant heat flux have been

time

t > 0 , the plate starts to move with exponential accelerated

considered by Jha et al.[5]. Chandran et al.[6] have studied the

velocity

0

u e t , where

0

u and are constants. The heat flux

unsteady hydromagnetic free convective flow with heat flux and

at the plate changes rampedly with time. The flow related

dimensionless governing equations have been solved analytically with the help of Laplace transform technique. The effects of pertinent flow parameters on the fluid velocity,

u = 0, T = T

for

y and

t 0,

temperature, shear stress and heat transfer rate at the moving

q t

for 0 < t t0

plate have been discussed with the help of graphs and table.

u = u e t ,

T =

k t0

at y = 0,

(3)

0 y q

2 FORMULATION OF THE PROBLEM AND ITS SOLUTIONS

k

for

t > t0

Consider an unsteady viscous incompressible flow of a fluid past

u 0, T T

as y

for

t > 0,

an exponentially accelerated infinite vertical plate with ramped wall heat flux. A graphic view of the flow model and physical coordinate system is shown in Fig.1. Choose a Cartesian

where q is the constant heat flux.

We introduce the non-dimensional variables

co-ordinates system in such a way that x -axis is taken along the yu

1. u

   u k(T T )

   0

   0

   (4)

   plate in a vertically upward direction and y -axis is is assumed

   = , =

   t

   , t0 = 2 , u1 =

   u u

   , = ,

   q

   to be normal to the plate. At time

   t 0 , both the fluid and plate

   0 0 0

   are at rest with constant temperature T . At time

   t > 0 , the

   On the use of (4), equations (1) and (2) become

   plate starts to move in its own plane with exponential accelerated

   u 2 u

   (5)

   1 1

   velocity

   0 e

2. t

(where u0

is the mean velocity of the plate and

=

2

• Gr ,

is the accelerating parameter) and the heat flux at the plate changes rampedly with time. Since the plate is infinitely long in

2

(6)

the x – direction, all the physical variables are the functions of

y and t only.

Pr = ,

2

where the non-dimensional parameters, specified based on the

c

properties of the pure fluid are taken as

g 2 q

Pr = p the Prandtl

k

number and

Gr =

0

ku4

the Grashof number.

The initial and boundary conditions given by equation

(3) become

u1 = 0, = 0 for

and 0,

for 0 < 1

u1 = e

, =

1 for > 1

at = 0,

(7)

Fig.1 : Geometry of the problem.

u1 0, 0 as

for > 0,

The basic flow in the medium is entirely due to buoyancy force

caused by temperature difference between the plate and the fluid. Under Boussinesq approximation, the the fully developed fluid

where

=

u

2

is the non-dimensional accelerated

0

flow be governed by the following system of equations

parameter.

u =

t

2 u

y2

g (T T ),

(1)

On the use of Laplace transformation technique, equations (5) and (6) become

T k 2T

= ,

(2)

d 2 u

su = 1

1 d 2

• Gr ,

  (8)

  t cp

  y2

  d 2

  Prs = ,

  d 2

  (9)

  where u is the velocity in the x -direction, T the

  temperature of the fluid, g the acceleration due to gravity,

  where

  the coefficient of thermal expansion, the kinematic

  s

  s

  (10)

  coefficient of viscosity, the fluid density, k the thermal

  u1 (, s) = 0 u1 (, s)e d

  and

  (, s) = 0 (, s)e d

  conductivity and cp

  the specific heat at constant pressure.

  The velocity and temperature boundary conditions are

  and s is the Laplace transform variable.

  The corresponding boundary conditions for u1

  and become

  3

  d

  • e s

2

1 2 1 1 2 2

e 2 3 erfc ,

(19)

u1 =

1

s

, = 1

d s2

at = 0,

(11)

3

u1 0, 0 as .

A ( , ) = 1 e e2 erfc(

) e2 erfc(

) ,

(20)

1 2

The solution of the equations (8) and (9) subject to the boundary conditions (11) can easily be obtained and are given by

B1 (, ) = G2 (, ) G1 (, ),

= ,

(21)

(22)

s 2

1 s Gr (1 e ) s s Pr

e e e

for Pr 1,

and erfc (x)

is the complementary error function and

H ( 1)

s 7

u1 (, s) =

1 s

e

s

(1 e s )

s2 Pr (Pr 1)

s

Gr (1 e ) e s

2s3

(12)

for Pr = 1,

(13)

the unit step function.

3 RESULTS AND DISCUSSION

In the following subsections, we highlights the effects of various thermophysical parameters such as Prandtl number Pr , Grashof number Gr , accelerating parameter and time

(, s) =

s2

s Pr

e s Pr ,

on the velocity field, temperature distribution, shear stress and the rate of heat transfer at the exponentially accelerated moving plate =0 with the help of graphs and table. The value of the

The inverse Laplace transform of the equations (12) and (13)

Prandtl number

Pr = 0.71

is chosen to represent air at

200 C

gives the solution of velocity field and temperature distributions in terms of exponential and complimentary error function as

Gr

temperature and 1 atmospheric pressure. The Grashof number or

buoyancy parameter Gr represents the effect of free

convection current. Only positive values of the buoyancy parameter ( Gr > 0 ) is considered (which corresponds to the

A1 (, )

B1 (, ) H ( 1)B1 (, 1)

for Pr 1,

cooling problem) here.

u (, ) =

Pr (Pr 1)

1

(14)

A1 (, )

Gr

2 G3 (, ) H ( 1)G3 (, 1)

for Pr = 1,

1. EFFECTS OF PARAMETERS ON THE VELOCITY PROFILES

   (, ) = F1 (, ) H ( 1)F1 (, 1),

   where

   (15)

   The fluid velocity profiles are shown in Figs. 2-5. These figures

   show that the velocity field is maximum near the moving plate and gradually decreases away from the plate and finally tends to

   zero. It is observed from Fig.2 that the fluid velocity u1

   F1 (, ) =

   1 Pr 2 e Pr

   3 4 2 2

   Pr 3 3

   Pr 3 2Pr 2 erfc

   Pr ,

   decreases with an increase in Prandtl number Pr . This is

   (16) consistent with the physical point of view that the fluids with high Prandtl number have greater viscosity, which makes the

   fluid thick and hence move slowly. Fig.3 reveals that an increase

   in Grashof number Gr leads to rise in the fluid velocity

   u1 .

   5

   2 1

   2 2 4

   Pr 2

   This is due to the contribution from the buoyancy force near the plate. The Grashof number Gr signifies the relative effect of

   15

   G1 ( , ) =

   16 36Pr

   • 8Pr e

     the thermal buoyancy force to the viscous hydrodynamic force in the boundary layer. The fluid velocity increases due to the enhancement of the thermal buoyancy force. It is seen from Fig.4

     2

     Pr (15 20Pr 2 4Pr2 4 ) erfc(

     Pr ) , (17)

     that the fluid velocity u1

     increases with an increase in

     5

     2 1

     2 4 2

     2 4

     accelerating parameter . It means that the increase of acceleration parameter increase the motion of exponentially accelerating moving plate at =0 which in turn increase the

     15

     G2 (, ) =

     16 36

     8 e

     2 (15 20

     4 ) erfc() ,

     fluid motion also. It is observed from Fig.5 that the fluid velocity

     u1 increases with time progresses. That means enhancement

     5

     2

     1 2

     2 1

     of time accelerates the fluid motion.

     G3 ( , ) = 2

     erfc( ) 2 e

     erfc( )

     2

     (18)

     Fig.5: Velocity profiles for when

     Pr = 0.71 ,

     Fig.2: Velocity profiles for Pr when

     Gr = 5 ,

     Gr = 5 and

     = 0.5

     = 0.5

     and = 0.2

2. EFFECTS OF PARAMETERS ON THE TEMPERATURE DISTRIBUTION

   The effects of Prandtl number Pr and time on the

   temperature distribution have been shown in Figs.6 and 7. The temperature is highest near the plate surface and decreases asymptotically to the free stream zero value far away from the plate. Fig.6 displays that the temperature decreases with an increase in Prandtl number Pr . It may be noted that an increase of Prandtl number causes the decrease of thermal boundary layer thickness that is why the temperature distribution across the thermal boundary layer decreases. It is observed from Fig.7 that the temperature increases with an increase in time .

   Fig.3: Velocity profiles for Gr when

   P = 0.71 ,

   = 0.5

   and = 0.2

   Fig.6: Temperature profiles for Pr when and = 0.2

   = 0.5

   Fig.4: Velocity profiles for when

   Gr = 5 and = 0.2

   Pr = 0.71 ,

   Generally, increase of shear stress is a disadvantage in the different technical applications. Numerical values of the shear

   stress x

   at the plate =0 due to the flow are presented in

   Table 1 for several values of Prandtl number Pr , Grashof number Gr , accelerating parameter and time . Table 1

   shows that the magnitude of the shear stress x

   at the plate

   =0 decreases with an increase in Prandtl number Pr while

   it increases with an increase in either Grashof number Gr or accelerating parameter for fixed values of time as it expected since the fluid velocity decreases with an increase in Prandtl number Pr and it increases with an increase in either Grashof number Gr or accelerating parameter . Further, it is seen that for fixed values of Prandtl number Pr , Grashof number Gr and accelerating parameter , the magnitude of

   x

   Fig.7: Temperature profiles for when

   decreases as time progresses.

   Pr = 0.71

   and

   = 0.5

3. EFFECTS OF PARAMETERS ON THE SHEAR STRESS AT THE PLATE

   The non-dimensional shear stress x

   due to the flow is given by

   at the moving plate =0

   1 1

   e erf (

   )

   Gr 2 2

   ( 1)

   for Pr 1,

   x

   =

   Pr (

   Pr 1)

   1 1

   e erf (

   )

   3/2 3/2

   3Gr

   ( 1)

   for Pr = 1,

   (23)

   2

   Table 1. Shear stress

   x

   at the moving plate

   =0

   for

   = 0.5 ,

   Pr = 0.71 ,

   Gr = 5

   	Pr			Gr
   	0.71	2	3	1	3	5	0.2	0.5	1.0
   0.1	4.54490	3.14018	2.81390	2.48386	3.51438	4.54490	4.43275	4.54490	4.74201
   0.2	3.46362	2.41007	2.16537	1.91784	2.69073	3.46362	3.29745	3.46362	3.77141
   0.3	2.66007	1.95771	1.79457	1.62955	2.14481	2.66007	2.44689	2.66007	3.07606
   0.4	1.94457	1.59339	1.51182	1.42931	1.68694	1.94457	1.68673	1.94457	2.47436

4. EFFECTS OF PARAMETERS ON THE HEAT TRANSFER RATE AT THE PLATE

The rate of Heat transfer at the plate =0 is given by

• It is seen that the temperature distribution decreases with an increase in Prandtl number where as it increases with an increase in time.

• The magnitude of shear stress at the moving plate due to the flow decreases with an increase in Prandtl number while it

'

=

=0

= F2 ( ) H ( 1)F2 ( 1),

(24)

increases with an increase in either Grashof number or accelerating parameter.

• The rate of heat transfer at the moving plate increases with

  where

  F ( ) = 4

  (25)

  the enhancement of either Prandtl number or time.

  2 3 Pr

  The numerical values of the rate of heat transfer at the plate

  =0 are depicted in Fig.8 for different values of Prandtl number Pr and time . It is seen from Fig.8 that the rate of

  5 REFERENCES

  [1] A. S. Gupta, I. Pop and V. M. Sounldalgekar, Free convective effects on the flow past an accelerated vertical

  plate in an incompressible vertical plate, Rev. Roum. Science and applied tech., vol.24, 1979, pp.561-568.

  [2] V.P. Isachenko, Heat Transfer, Mir Publishers, USSR, 3rd

  edition 1980.

  [3] A. K. Singh and K. Naveen, Free convection flow past an

  heat transfer at the moving plate

  =0

  increases with an

  exponentially accelerated vertical plate, Astrophysics and

  increase in either Prandtl number Pr or time . The negative

  Space, vol.98, 1984, pp.245-258.

  [4] M.A. Hossain and L.K. Shayo, The skin friction in the

  values of

  ' (0)

  physically clarify that there is heat flow from

  unsteady free convection flow past an accelerated plate,

  the plate to the fluid.

  Fig.8: Rate of Heat transfer

  for Pr and

  Astrophysics and Space Science, vol.125, 1986, pp.315-324. [5] B.K. Jha, R. Prasad and S. Rai, Mass transfer effects on the

  flow past an exponentially accelerated vertical plate with constant heat flux, Astrophysics and Space Science, vol.181, 1991, pp.125-134.

  [6] P. Chandran, N.C. Sacheti and A.K. Singh, Unsteady hydromagnetic free convection flow with heat flux and accelerated boundary motion, J. Phys. Soc. Jpn., vol.67, 1998, pp.124-129.
  [7] A. Barletta, Heat transfer by fully developed flow and viscous heating in a vertical channel with prescribed wall heat fluxes, Int. Heat and Mass Transfer, vol.42, 1999, pp.3873-3885.
  [8] U.N. Das, R.K. Deka and V.M. Soundalgekar, Transient free convection flow past an infinite vertical plate with periodic temperature variation, J. Heat Transfer, vol.121, 1999, pp.1091-1094.
  [9] M. Narahari, S. Sreenadh and V. M. Soundalgekar, Transient free convection flow between long vertical parallel plates with constant heat flux at one boundary, J. Thermophysics and Aeromechanics, vol.9, No.2, 2002, pp.2287-293.
  [10] P. Chandran, N.C. Sacheti and A. K. Singh, Natural Convection near a vertical plate with ramped wall temperature, Heat Mass Tranasfer, vol.41, 2005, pp.459-464.
  [11] A.K. Singh, P. Chandran and N.C. Sacheti, Developing flow near a semi-infinite vertical wall with ramped temperature, Int. J. Appl.Math. Stat., vol.13, 2008, pp.34-45.
  [12] R. Muthucumaraswamy, K.E. Sathappan and R. Natarajan,

  4 CONCLUSION

  Effects of heat transfer on an unsteady flow past an exponentially accelerated vertical plate with ramped plate heat flux have been investigated theoretically. The flow related governing equations are solved analytically with the help of Laplace transform technique. Some conclusions of the study are as below:

• The velocity field and temperature distribution are maximum near the moving plate and gradually decrease asymptotically away from the plate and finally tends to zero.

• Increase of Prandtl number reduce the fluid velocity field. On the other hand, enhancement of either buoyancy parameter or accelerating parameter or time increases the fluid velocity field.

Heat transfer effects on flow past an exponentially accelerated vertical plate with variable temperature, Theo. Appl. Mech., vol.35, No.4, 2008, pp.323-331.

[13] R.K. Singh and A.K. Singh, Transient HHD free convective near a semi infinite vertical wall having ramped temperature, Int. J. Appl. Math. Mech., vol.6, no.5, 2010, pp.69-79.

[14] P. M. Kishore, V. Rajesh and S. V. Verma, Effects of heat transfer and viscous dissipation on mhd free convection flow past an exponentially accelerated vertical plate with variable temperature, J. Naval Architecture and Marine Engineering, vol.7, 2010, pp.101-110.

[15] P. Chandrakala, Thermal radiation effects on moving nfinite vertical plate with uniform Heat flux, Int. J. Dynamics Flu., vol.1, 2010, pp.49-55.

[16] P. Chandrakala and P. Bhaskar, Effects of Heat transfer on flow past an exponentially accelerated vertical plate with uniform Heat flux, Int. J. Dynamics Flu., vol.7, no.1, 2011, pp.9-16.

[17] K.K. Asogwa, I. J. Uwanta and A. A. Aliero, Flow past an exponentially accelerated infinite vertical plate and temperature with variable mass diffusion, Int. J. Comp. Appl., vol.45, no.2, 2012, pp.1-7.

[18] P. Chandrakala, Effects of radiation on flow past an impulssively started infinite vertical plate with uniform Heat and mass flux, Int. J Fluids Eng., vol.6, no.1, 2014,

pp.73-85.

[19] S. Das, R. N. Jana and A. J. Chamkha, Unsteady free convection flow past a vertical plate with heat and mass fluxes in the presence of thermal radiation, J. Appl. Flu. Mech., vol.8, no.4, 2015, pp.845-854.

[20] D. K. Maiti and H. Mandal, Unsteady slip flow past an infinite vertical plate with ramped plate temperature and concentration in the presence of thermal radiation and buoyancy, J. Eng. Thermophysics, vol.28, 2019, pp.431-452.

[21] Y. D. Reddy, B. Shankar Goud and M. Anil Kumar, Radiation and heat absorption effects on an unsteady MHD boundary layer flow along an accelerated infinite vertical plate with ramped plate temperature in the existence of slip condition, Partial Diff. Equ. Appl. Math., vol.4, 2021, pp.1-10.

[22] N. Kalita, R.K. Deka and R. S. Nath, Unsteady flow past an accelerated vertical plate with variable temperature in presence of thermal stratification and chemical reaction, East European J. Physics, vol.3, 2023, pp.441-450.