rakesh ijamm 2
TRANSCRIPT
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
UNSTEADY MHD FLOW OF RADIATING AND REACTING FLUID
PAST A VERTICAL POROUS PLATE WITH COSINUSOIDALLY
FLUCTUATING TEMPERATURE
R. Kumar1 and K. D. Singh
2
1Department of Mathematics, Govt. College for Girls (RKMV), Shimla 171 001, India
2Department of Mathematics (ICDEOL), H.P. University, Shimla 171 005, India
Email: [email protected]
Received 11 March 2010; accepted 10 February 2011
ABSTRACT
An analysis of an unsteady MHD free convective flow of a viscous, incompressible,
electrically – conducting and radiating fluid past an infinite hot vertical porous plate with
chemical reaction of first order has been carried out by taking into account the effect of
viscous dissipation. The temperature of the plate is assumed to be spanwise cosinusoidally
fluctuating with time. The governing equations are solved by perturbation technique.
Numerical evaluation of the analytical results is performed. Graphical results for transient
velocity and transient temperature profiles and tabulated results for skin-friction coefficient
and Nusselt number are presented and discussed. It is found that velocity and temperature
decrease for destructive chemical reactions and increase for generative chemical reactions.
Keywords: Spanwise Cosinusoidally Fluctuating Temperature, Chemical Reaction, Thermal
Radiation, Unsteady Flow
1 INTRODUCTION
Process involving coupled heat and mass transfer occur frequently in nature. It occurs not
only due to temperature difference, but also due to concentration difference or the
combination of these two. Combined heat and mass transfer problems with chemical reaction
are of importance in many processes and have, therefore, received a considerable amount of
attention in recent years. In processes such as drying, evaporation at the surface of a water
body, energy transfer in a wet cooling tower and the flow in a desert cooler, heat and mass
transfer occur simultaneously. Possible applications of this type of flow can be found in many
industries. Representative applications of interest include: solidification of binary alloy and
crystal growth, dispersion of dissolved materials or particulate water in flows, drying and
dehydration operations in chemical and food processing plants and combustion of atomized
liquid fuels. Muthucumaraswamy and Ganesan (2001) pointed out that chemically reacting
flows are classified as heterogeneous or homogeneous depending on whether they occur at an
interface or as a single phase volume. Furthermore, the presence of a foreign mass in air or
water causes some kind of chemical reaction. During a chemical reaction between two species
heat is also generated. The effect of mass transfer on flow past an impulsively started infinite
vertical plate with constant heat flux and chemical reaction were studied by Das et al. (1994).
R. Kumar and K. D. Singh
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
20
Anderson et al. (1994) studied the flow and mass diffusion of a chemical species with first
order and higher order reactions over a linearly stretching surface. Fan et al. (1998) studied
the mixed convective heat and mass transfer over a horizontal moving plate with a chemical
reaction effect. El-Kabeir et al. (2007) investigated heat and mass transfer on MHD flow over
a vertical isothermal cone surface with heat sources and chemical reactions.
Muthucumaraswamy and Vijayalakshmi (2008) investigated the effects of heat and mass
transfer on flow past an oscillating vertical plate with variable temperature. Ahmed (2009)
studied free and forced convective three dimensional flows with heat and mass transfer.
Mansour et al. (2008) studied the effects of chemical reaction and viscous dissipation on
MHD natural convection flows saturated in porous media. Moreover chemical reaction effects
on heat and mass transfer laminar boundary layer flow have been studied by many scholars
e.g. Chamkha (2003), Kandasamy et al. (2005), Afify (2004), Takhar et al. (2000), Raptis and
Perdikis (2006) and Anjalidevi and Kandasamy (2000) etc.
Radiative convective flows are encountered in countless industrial and environment processes
e.g. heating and cooling chambers, fossil fuel combustion energy processes, evaporation from
large open water reservoirs, astrophysical flows, solar power technology and space vehicle re-
entry. 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 air craft, missiles, satellites and space vehicles are examples of such engineering
applications. Radiation effect on mixed convection along an isothermal vertical plate was
studied by Hossain and Takhar (1996). Raptis and Perdikis (2003) investigated the effects of
thermal radiation on a moving vertical plate in the presence of mass diffusion. Prasad and
Reddy (2008) studied the radiation effects on unsteady MHD flow with viscous dissipation.
Cookey et al. (2003) have investigated unsteady two-dimensional flow of a radiating and
chemically reacting MHD fluid with time dependent suction and Ogulu et al. (2008) extended
it for micro-polar fluid. Alam et al. (2009) investigated the effects of variable reaction,
thermophoresis and radiation on MHD free convection flows. Loganathan et al. (2008)
investigated first order chemical reaction past a semi-infinite vertical plate for optically thin
gray gas neglecting viscous dissipation.
The objective of the present paper is to investigate the effects of radiation and first order
chemical reaction on an unsteady MHD free convective heat and mass transfer flow past a hot
vertical non-conducting porous plate with constant suction normal to the plate when the plate
temperature is spanwise cosinusoidally fluctuating with time. Solution of the flow problem is
obtained by using perturbation technique by assuming the Eckert number as the perturbation
parameter.
2 MATHEMATICAL FORMULATION
We consider the flow past an infinite, hot porous plate lying vertically on ** zx plane. The
*x axis is oriented in the direction of the buoyancy force and *y axis is taken
perpendicular to the plane of the plate. A magnetic field of uniform strength 0H is introduced
normal to the plane of the plate. Let )w,v,u( *** be the components of velocity in the
)z,y,x( *** directions respectively. The plate is being considered infinite in *x direction;
hence all physical quantities will be independent of *x . Further, since the plate is subjected to
Unsteady MHD Flow Of Radiating And Reacting Fluid Past A Vertical Porous Plate
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
21
a constant suction velocity, i.e. Vv* , thus, following Singh and Chand (2000), *w is
independent of *z and so we assume 0*w throughout. The temperature of the plate is
considered to vary spanwise cosinusoidally fluctuating with time and assumed to be of the
form
tz
cosTTTt,zTw
00, (1)
where ** T,T 0 and *
wT are the mean, ambient temperature and wall temperature of the plate
respectively, * is the frequency, *t is the time, is the wave length and is a small
parameter i.e., 1 .
We further assume that (i) the magnetic Reynolds number is small so that the induced
magnetic field is negligible in comparison to the applied magnetic field, (ii) the fluid is
considered to be gray; absorbing – emitting radiation but a non-scattering medium, (iii) the
effects of Joule heating is negligible as small velocity usually encountered in the free
convection flows, (iv) no external electric field is applied and effect of polarization of ionized
fluid is negligible, therefore electric field is assumed to be zero, (v) there exists a first order
chemical reaction between the fluid and species concentration, (vi) the level of species
concentration is very low so that the heat generated during chemical reaction can be
neglected.
Using the Boussinesq and boundary–layer approximation, the governing equations for this
problem can be written as follows:
00
V,Vv
y
v *
*
*
, (2)
**
*
*
*
*
*
**
*
*
TTgz
u
y
u
y
uv
t
u
2
2
2
2
*e**
c uH
CCg
2
0
2
, (3)
*
*
*
*
*
*
*
*
*
*
*
**
*
*
Py
q
z
u
y
u
z
T
y
Tk
y
Tv
t
TC
22
2
2
2
2
, (4)
**
*
*
*
*
*
**
*
*
CCKz
C
y
CD
y
Cv
t
C
12
2
2
2
, (5)
where *
cPe
** q,,,,,C,k,,D,H,,C,T,g 0 and
1K are acceleration due to gravity,
fluid temperature, species concentration, magnetic permeability, magnetic field, chemical
molecular diffusivity, electrical conductivity, thermal conductivity, specific heat at constant
pressure, kinematic viscosity, density, coefficient of volume expansion for heat transfer,
volumetric coefficient of expansion with species concentration, radiative heat flux and
chemical reaction parameter.
R. Kumar and K. D. Singh
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
22
The boundary conditions of the problem are:
yasCC,TT,u
yatCC,tz
cos)TT(TT,u
*****
*****
*****
0
00 000
. (6)
For the case of an optically thin gray gas the local radiant absorption is expressed by
44
4 ****
*
*
TTay
q
, (7)
where *a is the mean absorption coefficient and * is Stefan – Boltzmann constant.
We assume that the temperature differences with in the flow are sufficiently small such that 4*T may be expressed as a linear function of the temperature. This is accomplished by
expanding 4*T in a Taylor series about *T and neglecting higher-order terms, thus
434
34 *** TTTT
(8)
By using equations (6) and (7), equation (4) reduces to
22
2
2
2
2
*
*
*
*
*
*
*
*
*
**
*
*
Pz
u
y
u
z
T
y
Tk
y
Tv
t
TC ***** TTTa
3
16 (9)
Introducing the following non-dimensional parameters
,,,***
V
uu
zz
yy
,**
0
**
TT
TT (Dimensionless temperature)
,**
0
**
CC
CCC (Dimensionless concentration)
,**tt (Dimensionless time)
**
P TTC
VEc
0
2
(Eckert number),
k
CPr
p (Prandtl number),
Unsteady MHD Flow Of Radiating And Reacting Fluid Past A Vertical Porous Plate
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
23
VRe (Reynolds number),
3
0
V
TTgGr
**
(Grashoff number),
3
0
V
)CC(gGm
**
c
(Modified Grashoff number),
2*
, (Dimensionless frequency of oscillation)
DSc
(Schmidt number),
22
0
2
2 HM e (Hartmann number),
1
2 K (Chemical reaction parameter),
2
2 3
16
Vk
TaR
**
(Radiation parameter)
into the equations (3), (5) and (9), we get
uMGmCGrRez
u
y
u
y
uRe
t
u 22
2
2
2
2
, (10)
Pr
RRe
z
u
y
uEc
zyPryRe
t
222
2
2
2
21
, (11)
Cz
C
y
C
Scy
CRe
t
C
2
2
2
21
. (12)
The boundary conditions (6) reduce to
yasC,,u
yatC,tzcos,u
000
0110
(13)
R. Kumar and K. D. Singh
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
24
3 SOLUTION OF THE PROBLEM
Assume
tzie)y(f)y(f)t,z,y(f
10 , (14)
where f stands for u, and C . Then substituting equation (14) into equations (10) to (12) and
equating the like powers of, we get
Zeroth-order equations
00
2
0
20
2
0
2
GmCGrReuMyd
udRe
yd
ud , (15a)
0
2
0
0
20
2
0
2
yd
duPrEcRRe
yd
dPrRe
yd
d
, (15b)
00
0
2
0
2
ScCyd
dCScRe
yd
Cd . (15c)
The corresponding boundary conditions reduce to
yasC,,u
yatC,,u
000
0110
000
000
(16)
First-order equations
11
2
1
221
2
1
2
GmCGrReuiMyd
udRe
yd
ud , (17a)
yd
du
yd
duEcPriRRe
yd
dPrRe
yd
d 10
1
221
2
1
2
2
, (17b)
01
21
2
1
2
CSciScyd
dCScRe
yd
Cd . (17c)
The corresponding boundary conditions reduce to
yasC,,u
yatC,,u
000
0010
111
111
(18)
Unsteady MHD Flow Of Radiating And Reacting Fluid Past A Vertical Porous Plate
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
25
The equations (15c) and (17c) are ordinary second order differential equations and solved
under the boundary conditions given in (16) and (18), respectively. Hence the expressions of
)y(C0 and )y(C1 are given by
y
e)y(C 1
0
, (19a)
01 )y(C . (19b)
Since equations (15a), (15b), (17a) and (17b) are coupled differential equations, therefore
approximate solution is obtained by perturbation technique for small values of Ec as the
Eckert number is small for incompressible fluid flows. Hence assuming
2
11101
2
01000
EcOEcFFF
EcOEcFFF, (20)
where F stands for u and .
Using (20) in (15a), (15b), (17a) and (17b) and equating like powers of Ec, we obtain:
Zeroth-order equations
000
2
00
2
0000 GmCGrReuMuReu , (21a)
10
2
10
22
1010 GrReuiMuReu , (21b)
000
2
0000 RRePrRe , (21c)
010
22
1010 PriRRePrRe . (21d)
The corresponding boundary conditions are
yas,u,,u
yat,u,,u
0000
01010
10100000
10100000
(22)
First - order equations
01
2
01
2
0101 GrReuMuReu , (23a)
11
2
11
22
1111 GrReuiMuReu , (23b)
2
0001
2
0101 uPrRRePrRe , (23c)
100011
22
1111 2 uu)PriR(RePrRe , (23d)
R. Kumar and K. D. Singh
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
26
where prime denotes differentiation with respect to y.
The corresponding boundary conditions are
yas,u,,u
yat,u,,u
0000
00000
11110101
11110101
(24)
Solving (21a) to (21d) and (23a) to (23d) under the boundary conditions (22) and (24)
respectively, we get
y
e 2
00
, (25)
y
e 3
10
, (26)
yyyyeeAeeAu 1424
2100
, (27)
yyeeAu 35
310
, (28)
2 1 2 1 4 2 1 42
01 1 4 7 5 6 8 9
y y y y y y y yB A e A e e A e A e A e A e e , (29)
3 52 1 4 2 1 4
11 2 10 11 12 13 14 15
y yy y y y y yB A e A e A e e A e A e A e e , (30)
yyyy
yyyy
eAeeAeAA
eeAeAeABu
1212
4124
2
18201716
222119301
, (31)
yyyy
yyyy
eeAeAeAA
eeAeAeABu
3412
5412
26252423
292827411
. (32)
The solutions obtained in equations (25) to (32) are in complex variable notations and only
the real part of it will have the physical significance. From equation (14) we get the
expressions for velocity and temperature profiles as
)tz(sinf)tz(cosf)y(f)t,z,y(f ir 0 , (33)
where
ir fiff 1 .
Hence we can obtain the expressions for the transient velocity and transient temperature
profiles from (33) for 0z and 2
t as
Unsteady MHD Flow Of Radiating And Reacting Fluid Past A Vertical Porous Plate
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
27
if)y(f,,yf
0
20
. (34)
Skin-friction: Knowing the velocity field, the expression for the skin-friction coefficient at
the plate in the *x direction is given by:
0
yy
u . (35)
Nusselt number: From the temperature field, the rate of heat transfer coefficient in terms of
the Nusselt number Nu at the plate is given by:
0
yy
Nu
. (36)
4 PARTICULAR CASES
(1) Our results are similar to the results of Singh and Chand (2000) in the absence of mass
transfer, chemical reaction, viscous dissipation and radiation effect.
(2) In the absence of viscous dissipation, chemical reaction and radiation effect for mixed
convective MHD flow with constant wall temperature past an accelerated infinite
vertical porous plate our results reduce to the results of Reddy et al. (2009).
(3) In the absence of radiation effects and in the presence of heat generation/absorption
for an ordinary medium our results are found in good agreement with Singh and
Kumar (2010).
5 RESULTS AND DISCUSSION
The numerical values of the transient velocity, transient temperature, coefficient of skin-
friction and Nusselt number are computed for different parameters like modified Grashoff
number, Grashoff number, Hartmann number, Reynolds number, Prandtl number, Eckert
number, Schmidt number, frequency parameter, chemical reaction parameter and radiation
parameter. To be realistic, the values of Prandtl number Pr , are chosen 710.Pr and
007.Pr to represent air and water respectively. The values of Schmidt number are taken for
Oxygen ( 600.Sc ) and Ammonia ( 780.Sc ).
R. Kumar and K. D. Singh
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
28
Figure 1: Transient velocity for 2
20
t,. , 20. and 0z .
The transient velocity profiles and transient temperature profiles are shown in Figure 1 and
Figure 2 respectively. It is observed from these figures that the velocity and temperature
profiles decrease with the increase of Hartmann number, Reynolds number, Prandtl number,
Schmidt number and radiation parameter, however, modified Grashoff number, Grashoff
number, Eckert number and frequency parameter have opposite effect on velocity and
temperature profiles.
Figure 2: Transient temperature for 2
20
t,. 20. and 0z .
Gm Gr M Re Pr Ec Sc R
5 5 1 2 0.71 0.01 0.60 5 2 I
10 5 1 2 0.71 0.01 0.60 5 2 II
5 10 1 2 0.71 0.01 0.60 5 2 III
5 5 2 2 0.71 0.01 0.60 5 2 IV
5 5 1 4 0.71 0.01 0.60 5 2 V
5 5 1 2 7.00 0.01 0.60 5 2 VI
5 5 1 2 0.71 0.02 0.60 5 2 VII
5 5 1 2 0.71 0.01 0.78 5 2 VIII
5 5 1 2 0.71 0.01 0.60 10 2 IX
5 5 1 2 0.71 0.01 0.60 5 5 X
y
t,z,yu
Gm Gr M Re Pr Ec Sc R
5 5 1 2 0.71 0.01 0.60 5 2 I
10 5 1 2 0.71 0.01 0.60 5 2 II
5 10 1 2 0.71 0.01 0.60 5 2 III
5 5 2 2 0.71 0.01 0.60 5 2 IV
5 5 1 4 0.71 0.01 0.60 5 2 V
5 5 1 2 7.00 0.01 0.60 5 2 VI
5 5 1 2 0.71 0.02 0.60 5 2 VII
5 5 1 2 0.71 0.01 0.78 5 2 VIII
5 5 1 2 0.71 0.01 0.60 10 2 IX
5 5 1 2 0.71 0.01 0.60 5 5 X
t,z,y
y
Unsteady MHD Flow Of Radiating And Reacting Fluid Past A Vertical Porous Plate
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
29
Figure 3: Effect of chemical reactions on transient velocity for ,.202
t and 0z .
Figure 3 and Figure 4 show the effect of chemical reaction on velocity and temperature
profiles respectively. It is clear from these Figures that velocity and temperature decrease for
destructive chemical reactions ( 0 ) and increase for generative chemical reactions ( 0 ).
Figure 4: Effect of chemical reactions on transient temperature for ,.202
t and 0z .
The values of skin-friction coefficient and Nusselt number at the plate are shown in Table 1. It
is clear from Table 1 that both coefficient of skin-friction increases and Nusselt number
Curve I 20.
Curve II 0
Curve III 20.
y
t,z,yu
25
600010710
2155
R,
,.Sc,.Ec,.Pr
,Re,M,Gr,Gm
Curve I 20.
Curve II 0
Curve III 20.
y
t,z,y
25
600010710
2155
R,
,.Sc,.Ec,.Pr
,Re,M,Gr,Gm
R. Kumar and K. D. Singh
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
30
decreases with the increase of modified Grashoff number, Grashoff number, Eckert number
and frequency parameter.
Table 1: Values of Skin-friction coefficient and Nusselt number Nu at the plate when
220
t,. and 0z .
Gm Gr M Re Pr Ec Sc R Nu
5 5 1 2 0.71 0.01 0.60 5 0.2 2 16.853 3.3736
10 5 1 2 0.71 0.01 0.60 5 0.2 2 28.812 3.0267
5 10 1 2 0.71 0.01 0.60 5 0.2 2 22.096 3.2740
5 5 2 2 0.71 0.01 0.60 5 0.2 2 12.116 3.4678
5 5 1 4 0.71 0.01 0.60 5 0.2 2 41.087 6.6543
5 5 1 2 7.00 0.01 0.60 5 0.2 2 13.382 12.901
5 5 1 2 0.71 0.02 0.60 5 0.2 2 16.964 3.2035
5 5 1 2 0.71 0.01 0.78 5 0.2 2 14.773 3.4207
5 5 1 2 0.71 0.01 0.60 10 0.2 2 16.909 3.2919
5 5 1 2 0.71 0.01 0.60 5 0 2 17.540 3.3562
5 5 1 2 0.71 0.01 0.60 5 -0.2 2 18.464 3.3312
5 5 1 2 0.71 0.01 0.60 5 0.2 5 15.358 5.0514
However, Hartmann number, Prandtl number, Schmidt number and radiation parameter have
opposite effect on coefficient of skin-friction and Nusselt number. It is interesting to note that
both coefficient of skin-friction increases and Nusselt number increase with the increase of
Reynolds number. It is also observed from Table 1 that Nusselt number increases and skin-
friction coefficient decreases for destructive chemical reactions ( 0 ) and Nusselt number
decreases and skin-friction coefficient increases for generative chemical reactions ( 0 ).
6 SUMMARY
The governing equations for unsteady MHD free convective heat and mass transfer flow of
thermally radiating and chemically reacting fluid past a vertical porous plate with viscous
dissipation was formulated. The plate temperature is taken spanwise cosinusoidally
fluctuating with time. The resulting partial differential equations were transformed into a set
of ordinary differential equations using two-term series and solved in closed-form. The
conclusions of the study are as follows:
1) The velocity, temperature, skin-friction coefficient decrease and Nusselt number increases
with increasing radiation parameter.
2) The temperature decreases tremendously with increasing Prandtl number as compared to
Hartmann number.
3) The velocity increases tremendously with increasing modified Grashoff number as
compared to Grashoff number.
Unsteady MHD Flow Of Radiating And Reacting Fluid Past A Vertical Porous Plate
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
31
4) Velocity, temperature, skin-friction coefficient increase and Nusselt number decreases
with increasing values of Grashoff number or modified Grashoff number.
5) Velocity, temperature, skin-friction coefficient increase and Nusselt number decreases for
generative chemical reactions ( 0 ). However, destructive chemical reactions ( 0 )
have opposite effect on velocity, temperature, skin-friction coefficient and Nusselt
number.
ACKNOWLEDGEMENT
The authors are thankful to the learned referees for their valuable suggestions towards the
definite improvement of the paper.
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Unsteady MHD Flow Of Radiating And Reacting Fluid Past A Vertical Porous Plate
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APPENDIX
PrRe1 ,
RRe 2
2 ,
iM 22
3 ,
PriRRe 22
4 ,
2
422
1
ScScReScRe
,
2
4 2
2
11
2
,
2
4 4
2
11
3
,
2
4 22
4
MReRe ,
2
4 3
2
5
ReRe,
2
2
2
2
2
1MRe
GrReA
,
2
1
2
1
2
2MRe
GmReA
,
33
2
3
2
3
Re
GrReA ,
221
2
2
2
2
2
14
24
APrA ,
R. Kumar and K. D. Singh
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
34
211
2
1
2
1
2
25
24
APrA ,
241
2
4
2
4
2
216
24
AAPrA ,
2211
2
21
21217
2
AAPrA ,
2421
2
42
422118
2
AAAPrA ,
2411
2
41
412129
2
AAAPrA ,
4321
2
32
3213
10
2
AAA ,
4311
2
31
3123
11
2
AAA ,
4431
2
43
43213
12
2
AAAA ,
4521
2
52
5213
13
2
AAA ,
4511
2
51
5123
14
2
AAA ,
4541
2
54
54213
15
2
AAAA ,
2
2
2
2
1
2
16MRe
BGrReA
,
2
2
2
2
4
2
1724 MRe
AGrReA
,
2
1
2
1
5
2
1824 MRe
AGrReA
,
Unsteady MHD Flow Of Radiating And Reacting Fluid Past A Vertical Porous Plate
Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.
35
2
4
2
4
6
2
1924 MRe
AGrReA
,
2
21
2
21
7
2
20MRe
AGrReA
,
2
42
2
42
8
2
21MRe
AGrReA
,
2
41
2
41
9
2
22MRe
AGrReA
,
33
2
3
2
2
23
Re
BGrReA ,
33232
10
2
24
Re
AGrReA ,
33131
11
2
25
Re
AGrReA ,
34343
12
2
26
Re
AGrReA ,
35252
13
2
27
Re
AGrReA ,
35151
14
2
28
Re
AGrReA ,
35454
15
2
29
Re
AGrReA ,
9876541 AAAAAAB ,
1514131211102 AAAAAAB ,
222120191817163 AAAAAAAB ,
292827262524234 AAAAAAAB .