ABSTRACT: Theoretical solution of unsteady flow past an infinite uniformly accelerated plate has been presented in presence of a variable plate temperature and uniform mass diffusion. The plate temperature is raised linearly with time. The dimensionless governing equations are solved using Laplace-transform technique. The velocity profile, the concentration, Skin friction and the rate of heat transfer in terms of Nusselt Number are studied for different physical parameters like thermal Grashof number, mass Grashof number, Schmidt number and time.

1. International
OPEN ACCESS Journal
Of Modern Engineering Research (IJMER)
| IJMER | ISSN: 2249–6645 www.ijmer.com | Vol. 7 | Iss. 3 | Mar. 2017 | 33 |
Free Convective FlowFromAn Accelerated Infinite Vertical
Platewith Varying PlateTemperatureand Uniform Mass Diffusion
D. Barman
Department Of Mathematics, Pandu College, Guwahati, Assam, India
I. INTRODUCTION
Heat and mass transfer plays an important role in manufacturing industries for design of fins, steel
rolling, nuclear power plants, gas turbines and various propulsion devices for aircrafts, missiles, spacecraft
design, solar energy collectors, design of chemical processing equipments, satellites and space vehicles are
examples of such engineering applications. Free convective flow past a linearly accelerated plate in presence of
viscous dissipative heat was studied by Gupta et al. [1] using perturbation technique. Kafousias and Raptis
[2]extended the above problem and included mass transfer effects subjected to variable suction or injection. Free
convection effects on flow past an accelerated plate with variable suction and uniform heat flux in the presence
of magnetic field was studied byRaptis et al. [3]. Mass transfer effects on flow past a uniformly accelerated
vertical plate was studied by Soundalgekar [4]. Again, mass transfer effects on flow past an accelerated vertical
plate with uniform heat flux was analyzed by Singh and Singh [5]. Basant Kumar Jha and RavindraPrasad [6]
analyzed mass transfer effects on the flow past an accelerated infinite vertical plate with heat sources. R.
Muthucumaraswamy et al. [7] investigated the effects of a flow past a uniformlyaccelerated infinite vertical
plate in the presence of variable temperature and uniform mass diffusion.
This paper deals with the study of magnetic and massdiffusion effects on the free convection flow from a
uniformly accelerated infinite vertical plate with variable temperature. The solutions are in terms of exponential
and complementary error function.
II. FORMULATION OF THE PROBLEM
We consider unsteady, free convective, two dimensional MHD flow of an incompressible and
electrically conductingviscous fluid along an infinite vertical plate. The x-axis is taken along the plate vertically
upward direction and y-axis is taken normal to the plate. A magnetic field of uniform strength Bo is applied in
the direction of the flow and induced magnetic field is neglected. Initially, the plate and the fluid are at the same
temperature in a stationary condition with concentration level at all points. At time >0 the plate starts
moving with a velocity . Its temperature is raised to and the concentration of the plate
is raised to . Using the Boussinesq approximation, the governing equations for the flow are given by
.
With the following initial and boundary conditions

T 
C t
tu 0   tATTT   
C
     
22
* 0
2
1
B uu u
g T T g C C
t y

  
 
  
       
 
2
2
(2)
p
T T
t C y


 
 
 
 
2
2
3
C C
D
t y
  
 
 
ABSTRACT: Theoretical solution of unsteady flow past an infinite uniformly accelerated plate has been presented in
presence of a variable plate temperature and uniform mass diffusion. The plate temperature is raised linearly with time.
The dimensionless governing equations are solved using Laplace-transform technique. The velocity profile, the
concentration, Skin friction and the rate of heat transfer in terms of Nusselt Number are studied for different physical
parameters like thermal Grashof number, mass Grashof number, Schmidt number and time.
Keywords: Accelerated,heattransfer, MHD,massdiffusion,vertical plate

2. Free Convective Flow From An Accelerated Infinite Vertical Platewith Varying ….
| IJMER | ISSN: 2249–6645 www.ijmer.com | Vol. 7 | Iss. 3 | Mar. 2017 | 34 |
Where
We introduce following non-dimensional quantities
The equations (1), (2), (3) become:
.
Under boundary conditions:
III. METHOD OF SOLUTION
We solve the governing equations in an exact form by using Laplace-transformations.The Laplace
Transformation of the equations (6), (7), (8) and the boundary conditions (9) are given by:
.
Under the boundary conditions
Solving equations (10), (11), and (12) under the boundary conditions (13), we get
   4
,,0
0,,:0
0,,,0
0 














yasCCTTu
yatCCtATTTTtuut
tyallforCCTTu

3/12
0








u
A
 
 
 
 5
,,,,
,,,,
3/2
0
3/12
0
0
*
0
3/1
2
0
3/12
0
3/1
0









































u
B
M
D
S
C
P
u
CCg
G
CC
CC
C
u
TTg
G
TT
TTu
yY
u
tt
u
u
U
c
p
rc
r











 
 
 8
1
7
1
6
2
2
2
2
2
2


















Y
C
St
C
YPt
MU
Y
U
CGG
t
U
c
r
cr


 9
0;0;0
01,,:0
0,0,0,0












YasCU
YatCttUt
tYforCU



   102
2
 CGGuMS
dY
ud
cr 
 1102
2
 

sP
dY
d
r
 1202
2
 CsS
dY
Cd
c
 13
00,0,0
0
1
,
1
,
1
:0
0,0,0,0
22














YasCU
Yat
s
C
ss
Utwhen
tYwhenCU




3. Free Convective Flow From An Accelerated Infinite Vertical Platewith Varying ….
| IJMER | ISSN: 2249–6645 www.ijmer.com | Vol. 7 | Iss. 3 | Mar. 2017 | 35 |
Inverting equations (14), (15) and (16), we get
For
And for
SKIN-FRICTION
We know study skin-friction from velocity field. It is given by
This reduces to
Now for
 
  
 
     YsPYMS
P
M
sPs
G
s
YMS
sYu r
r
r
r











 expexp
1
1
exp
,
2
2
 
       14expexp
1
1









 YsSYMS
S
M
sSs
G
c
c
c
c
 
   
 
   16
exp
,
15
exp
, 2






s
YsS
sYC
s
YsP
sY
c
r

1,1  cr SP
     


















 Mterfc
Mt
Mterfc
Mt
t
tYu 



211
2
,      
   
2 24
2 1 2 exp
2
1 1 2
r r r r
r
P erfc P P P
G t
M
erfc Mt erfc Mt
Mt Mt

  

 
 

  

 
                 
 
2
1
exp exp 2
2 1 1 1
r r r r
r r r
G P MPt MPtMt
erfc
M P P P
 
       
                 
exp 2
1 1 1 1
r r
r r r r
MPt MPtMt Mt
erfc erfc erfc
P P P P
   
           
                                      
















































111
2exp
1
exp
2 cc
c
c
c
c
c
S
Mt
erfc
S
tMS
erfc
S
tMS
S
Mt
M
G

  c
c
cc
c
c
c
Serfc
M
G
S
Mt
erfc
S
tMS
erfc
S
tMS
 2
2111
2exp 










































         exp 2 exp 2 17Mt erfc Mt Mt erfc Mt        
         2 22
, 1 2 exp 18r
r r r
P
Y t t P erfc P P

   

 
     
  
     , 19cC Y t erfc S 
1 cr SP
     , 1 exp 2
2
t
u Y t Mt erfc Mt
Mt

 
 
    
 
   1 exp 2 Mt erfc Mt
Mt

 
 
    
 
         2 24
2 1 2 exp 1 exp 2rG
erfc Mt erfc Mt
M Mt
 
    

  
       
 
   
       
1 exp 2
2 exp 2 20
2
c
Mt erfc Mt
Mt
G
erfc Mt erfc Mt
M

 
  
 
     
  
   

         2 22
, 1 2 exp 21Y t t erfc

   

 
     
 
     , 22C Y t erfc  
0



y
y
u

0









YY
u

1 cr SP
   
1 1
exp
2
t M erf Mt Mt
t M t


  
      
  

4. Free Convective Flow From An Accelerated Infinite Vertical Platewith Varying ….
| IJMER | ISSN: 2249–6645 www.ijmer.com | Vol. 7 | Iss. 3 | Mar. 2017 | 36 |
And when
Nusselt Number
In non-dimensional form the rate of heat transfer is given by
IV. RESULTS AND DISCUSSION
In order to point out the effects of various parameters on the flow characteristics, the following
discussion is set out. The values of the Prandtl number are chosen such that they represent air (Pr=0.71) and
water (Pr=7). The values of Schmidt number are chosen to represent the presence of species of hydrogen
(Sc=0.22), water vapour (Sc=0.6) and ammonia (Sc=0.78). The numerical values of the velocity, temperature and
concentration are computed for different physical parameters like Prandtl number, Schmidt number, thermal
Grashof number, mass Grashof number and time and are represented graphically.The velocity profiles for
variation of time t, Schmidt number Sc, Grashof numbers Gr and Gc, and the Hartmann number M are studied
and presented in fig.1 to fig.4Fig. 1 displays the effect of time t on the velocity field for the fluid when Pr=0.71,
Sc=0.6 and Gr=Gc=5. Here we see that in case of cooling of the plate (as Gr>0), the velocity near the plate
increases owing to the presence of water vapour in the flow field.In Fig. 2 we notice that although there is a rise
in the velocity due to the presence of water vapour and ammonia, but it is not so high in the presence of
Hydrogen. Fig. 3 reveals the velocity variations with Gr, Gc and Pr. It is observed that the greater cooling of the
surface (increase in Gr) and increase in Gc results in an increase in the velocity for air (Pr=0.71). It is due to the
fact that an increase in the
Fig. 1 Velocity profile for different values of t
 1 1
2
2
r rG t P
M erf Mt
M t t t M 
  
     
  
 
 
 
 
2
2
1 1
1
1
exp
1 1 1 1
exp
1 1 1 1
r r r
c c
r r r r
r r r r
c c c
c c c c
G P P
M erf Mt
M t t
G S
M erf Mt
M t t
G P MP MPtMt Mt
erf erf
M P P P P
G MS MS tMt Mt
erf erf
M S S S S
 
 
 
   
 
 
   
 
      
                   
     
                   
 23

1 cr SP
   
 
   
2 1
exp
4 1
1 1
exp( ) 24
2
r c
r c r
G G
M erf Mt Mt
M t
G G G
erf Mt
M t M Mt
t M erf Mt Mt
t M t




   
     
  
 
 
 
  
      
  
0









YY
Nu

 2 25rP t
Nu

 
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2
t=0.2
t=0.4
t=0.6
t=0.8
t=1

5. Free Convective Flow From An Accelerated Infinite Vertical Platewith Varying ….
| IJMER | ISSN: 2249–6645 www.ijmer.com | Vol. 7 | Iss. 3 | Mar. 2017 | 37 |
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3
Gr=5, Gc=5, Pr=0.71
Gr=2, Gc=5, Pr=0.71
Gr=2, Gc=2, Pr=0.71
Fig. 3: Velocity profiles for different values of Gr, Gc (t=0.4, M=2.0 Sc=0.6)
η
u
Values of Gr and Gc has the tendency to increase the thermal and mass buoyancy effect.
Fig. 2 Velocity profile for different values of Sc
We also observe that the velocity near the plate is greater than at the plate. The maximum velocity
attains near the plate and is in the neighbourhood of the point . After > 0.2, the velocity decreases and
tend to zero as .
Fig 3: Velocity profiles for different values of Gr, Gc (t= 0.4, M=2.0, Sc=0.6)
It is also found in Fig.4 that the velocity decreases with the increase in the magnetic parameter M. It is
because of the fact that when a transverse magnetic field is applied, a resistive type of force (known as Lorentz
force) get produced which tends to resist the fluid motion and thus reducing the velocity
Fig. 4: Velocity profiles for variation of M (t=0.4, Pr=0.71, Sc=0.6, Gr=5, Gc=5)
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1 2 3 4
Sc=0.16
Sc=0.22
Sc=0.6
Sc=0.78
2.0 

0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.5 1 1.5 2 2.5
M=1
M=2
M=3
M=4

6. Free Convective Flow From An Accelerated Infinite Vertical Platewith Varying ….
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Fig. 5: Temperature profiles for variation of t (Pr=0.71, Sc=0.6)
Fig. 5 and Fig. 6 depict the temperature profiles against for variation of time and the Prandtl number. In fig.
5, we notice that the magnitude of temperature is the maximum at the plate and decays to zero asymptotically.
Fig.6: Temperature profiles for variation of Pr (t=0.4, Sc=0.6)
Fig. 7 demonstrates the effect of Schmidt number (Sc) on the concentration. Like temperature, the
concentration is maximum at the surface and falls exponentially. The concentration decreases with an increase
in Sc. Further it is seen that the concentration falls slowly and steadily for Hydrogen (Sc=0.22) as compared to
other gases.
Fig. 7: Concentration profiles for variation of Sc (Pr=0.71)
Fig. 8 depicts the nature of skin-friction against time t for different values of the parameters Gr, Gc, M,
Sc and Pr. Here we notice that Skin-friction increases with an increase of Sc. Further we observe that it increases
with M. This is due to the fact that Lorentz force gets enhanced with increasing M, which imports additional
momentum in the boundary layer. On the other hand we see that skin-friction decreases with the increase of Gr
and Gc.
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3
t=.1
t=0.3
t=0.6
t=0.8
t=1

-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.5 1 1.5 2
Pr=0.7
Pr=7
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4
Sc=0.22
Sc=0.6
Sc=0.78

7. Free Convective Flow From An Accelerated Infinite Vertical Platewith Varying ….
| IJMER | ISSN: 2249–6645 www.ijmer.com | Vol. 7 | Iss. 3 | Mar. 2017 | 39 |
Fig. 8: Skin-friction
Fig. 9 exhibits the behavioure of co-efficient of rate of heat transfer (Nusselt number) against time.
Here we observe that Nusselt number decreases with time. Also we notice that it is higher for Pr=7 than that for
Pr=0.71. This is due to the fact that Pr is small means thermal conductivity is more, which results more rapid
diffusion of heat from the plate and hence rate of heat transfer get reduced.
Fig.9:Nusselt number
V. CONCLUSION
From the above discussions, the following conclusions are set out:
1) The velocity decreases with an increase in magnetic parameter, Schmidt number and Prandtl number. On the
other hand, it increases with an increase in the values of thermal Grashof number, mass Grashof number and
time.
2) The concentration decreases with an increase in Schmidt number.
3) The skin-friction increases with the increase in magnetic parameter, Schmidt number and Prandtl number,
while it decreases with an increase of thermal Grashof number, mass Grashof number and time.
4) Nusselt number increases with the increase in Prandtl number.
APPENDIX
A. Nomenclature
A constant
Species concentration in the fluid
Dimensionless concentration
Specific heat at constant pressure
Mass diffusion coefficient
Mass Grashof number
Thermal Grashof number
Acceleration due to gravity
Thermal conductivity
Prandtl number
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1 1.2
10 5 2 0.22 0.71
5 5 2 0.22 0.71
5 5 2 0.6 0.71
5 5 5 0.22 0.71
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2 2.5
Pr=0.71
Pr=7
C
C
pC
D
cG
rG
g

rP

8. Free Convective Flow From An Accelerated Infinite Vertical Platewith Varying ….
| IJMER | ISSN: 2249–6645 www.ijmer.com | Vol. 7 | Iss. 3 | Mar. 2017 | 40 |
Schmidt number
T Temperature of the fluid near the fluid
Time
Dimensionless time
Velocity of the fluid in x-direction
Velocity of the plate
Dimensionless velocity
Spatial co-ordinate along the plate
Co-ordinate axis normal to the plate
Dimensionless co-ordinate axis normal to the plate
B.Greek Symbols
Volumetric coefficient of thermal expansion
Volumetric coefficient of thermal expansion with
Concentration
Coefficient of viscosity
Kinematic viscosity
Density of the fluid
Dimensionless skin-friction
Dimensionless temperature
Similarity parameter
Error function
Complementary error function
C. Subscripts
Conditions at the plate
Conditions in the free stream
REFERENCES
[1]. Gupta AS,andSoudalgekar VM: Free convection effects on the flow past an accelerated vertical plate in an
incompressible dissipative fluid, Rev. Roum. Sci. Techn.-Mec. Apl., 24, pp. 561-568, 1979
[2]. Kafousias NG and Raptis AA: Mass transfer and free convection effects on the flow past an accelerated vertical
infinite plate with variable suction or injection, Rev. Roum. Sci. Techn.-Mec. Apl., 26, pp. 11-22, 1981
[3]. Raptis AA, Tzivanidis GJ, and Peridikis CP: Hydromagnetic free convection flow past an accelerated vertical
infinite plate with variable suction and heat flux, Letters in heat and mass transfer, 8, pp. 137-143, 1981
[4]. Soundalgekar VM: Effects of Mass transfer on the flow past an accelerated vertical plate, Letters in heat and mass
transfer, 9, pp. 65-72, 1982
[5]. Singh A. K. and Singh J. Mass transfer effects on the flow past an accelerated vertical plate with constant heat
flux, Astrophysics and space science, 97, pp. 57-61, 1983.
[6]. Basanth Kumar Jha and Ravindra Prasad: Free convection and mass transfer effects on the flow past an accelerated
vertical plate with heat source, Mechanics Research Communications, 17, pp. 143-148, 1990.
[7]. Muthucumaraswamy R. Sundar Raj M and Subramanian VSA: Unsteady flow past an accelerated infinite vertical
plate with variable temperature and uniform mass diffusion. Int. J. of Appl. Math. and Mech., 5(6), pp.51-56,
2009.
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