Research Journal of Mathematics and Statistics 4(3): 63-69, 2012
ISSN: 2040-7505
© Maxwell Scientific Organization, 2012
Submitted: April 25, 2012
Accepted: June 08, 2012
Published: August 25, 2012
Viscoelastic Fluid Flow in a Fixed Plane with Heat and Mass Transfer
I.J. Uwanta and E. Omokhuale
Department of Mathematics, Usmanu Danfodiyo University, P.M.B. 2346, Sokoto, Nigeria
Abstract: The aim of the present study is to investigate viscoelastic fluid in a fixed plane, taking the effect of heat
and mass transfer into account. The resulting equations have been solved analytically using perturbation technique.
Solutions are obtained for velocity, temperature and concentration distributions, as well as the Skin friction, Nussetl
number and Sherwood number. The influence of various parameters such as the thermal Grashof number, Hartmann
number, mass Grashof number, thermal Radiation, Schmidt number, Prandtl number, permeability parameter,
frequency parameter and chemical reaction parameter on the flow field is examined. It is observed that the velocity
increases with the increase in Gc, Gr and R and it decreases with the increase in  , M, K, k, Sc, Pr and  .
Temperature decreases with increase in Pr and concentration decreases with the increase in Sc and k.
Keywords: Heat transfer, mass transfer, viscoelastic
The study of viscoelastic fluid has become of
increasing importance during recent times. This is
mainly due to its many applications in petroleum
drilling, manufacturing of foods and study and many
other similar activities (Ambethkar and singh, 2011).
Elastic fluid flow of magneto hydro dynamic free
convection through a porous medium was studied by
Ezzat et al. (1996). Later, magneto hydro dynamic flow
and heat transfer in a rectangular duct with temperature
dependent viscosity and Hall effect was investigated by
Sayed-Ahmed and Attia (2001). Atul Kumar et al.
(2003) studied heat and mass transfer in MHD flow of a
viscous fluid past a vertical plate under oscillating
suction velocity. Muthucumaraswamy and Senthil
(2004) have studied heat and mass transfer effects on
moving vertical plate in the presence of thermal
radiation.
INTRODUCTION
The study of viscoelastic fluid has become
important in the last few years. Qualitative analysis of
these studies has significant bearing on several
industrial applications such as polymer sheet extrusion
from a dye, drawing of plastic firms etc. When
manufacturing processes at high temperature need
cooling, the flow may need viscoelastic fluid to produce
a good effect or reduce the temperature (Kai-Long,
2010). It is a well-known fact in the study of nonNewtonian fluid flow by Hartnett (1992). Raptis (1989)
studied heat transfer of a viscoelastic fluid, Rafael
(2007a) studied effect of viscous dissipation and
radiation on the thermal boundary layer over a nonlinearly stretching sheet. Sanjayanand and Khan (2006)
studied heat and mass transfer in a viscoelastic
boundary layer flow over an exponentially stretching
sheet, Rafael (2007b) and Seddeek (2007) studied
MHD flow and mass transfer of an electrically
conducting fluid of second grade in a porous medium
over a stretching sheet with chemically reactive species
and the heat and mass transfer problems about the
viscoelastic boundary layer flow over a stretching sheet
with magnetic effect respectively, but did not
considered the mixed convection with radiation effect.
Kai-Long (2010) studied heat and mass transfer for
viscous flow with radiation effect past a non-linear
stretching sheet. Unsteady two dimensional flow and
heat transfer through an elastic-viscous liquid along an
infinite hot vertical porous medium was studied by
Sharma and Sharma (2005). Noushima et al. (2009)
studied unsteady MHD memory flow and heat transfer
over a moving continuous porous horizontal surface.
MATHEMATICAL FORMULATION
Consider the flow of incompressible memory fluid
in a fixed plane with heat and mass transfer, under the
influence of a uniform transverse magnetic field and
constant suction. The x-axis is taken along the plane in
the upward direction and a straight line perpendicular to
that of the y-axis. All fluid properties are assumed
constant. The magnetic field of small intensity H0 is
introduced in the y-direction. Since the fluid is slightly
conducting, the magnetic Reynolds number is much
less than unity and hence the induced magnetic field is
neglected.
The governing equations for the momentum,
continuity, energy and concentration are as follows:
Corresponding Author: I.J. Uwanta, Department of Mathematics, Usmanu Danfodiyo University, P.M.B. 2346, Sokoto, Nigeria
63
Res. J. Math. Stat., 4(3): 63-69, 2012



g  T0  Td 
Kv02
 e 2 H 02
1v02 
 Cp
Gr 
, Pr 
,K  2 ,M 
, Rm  2 

 v02
 
v03
KT
3

g  2  C0  Cd 
4  T

 4R  1

Gc 
, Sc , R 
, N  1 

 03
D
KaR
3
Pr



y
g
v
 0
y
(2)
T
T
K T  2T
1 qr
v


 c py 2  c p y
t
y
(3)
C
C
 C
v
D
 K * (C  C d )
t
y
y 2
2
where,
u and v :
g
β and β2
β1
ρ
KT
Cp
:
:
:
:
:
:
:
:
σ
μe
D
T0
Td
:
:
:
:
:
C0
Cd
:
:


(1)
,t 
qr 
(4)
tv02
T  Td
C  Cd
u
, u  , 
,C 
4
0
T0  Td
C0  C d
 4 *  T 4
3a R y
(8)
We assume that the temperature difference within
the flow is such that T4 may be expressed as a linear
function of the temperature T. This is accomplished by
expanding T4 in Taylor series about Td and neglecting
higher order terms:
T 4  4Td3T  3Td4
(9)
Substituting the dimensionless variable in (7) into
(1), (3), (4) and also using (8) and (9). We get
(dropping the bars):
  3u
u  u
 2u
 3u 

 Gr  2  Rm 
 3   Ku  Mu  GcC
2
4 t y
y
y 
 4t y


N  2


0
y 2
y 4t
1  2C C
C


 kC  0
y
4t
Sc y 2
The mathematical formulation is an extension of
Noushima et al. (2010). The rate of change of
radioactive heat flux with respect to distance and mass
transfer were added to the existing equations
Integrating (2) gives:
(10)
(11)
(12)
With boundary conditions:
y  0 : U  0,   1   e i t , C  1   e i t 

y  1: U  0,   0, C  0

(5)
where,
Gr : The thermal Grashof number
M : The Hartmann number
Gc : The mass Grashof number
R : The thermal radiation
Sc : The Schmidt number
Pr : The Prandtl number
Rm : The Magnetic Reynold number
K : The permeability Parameter
K : The chemical reaction parameter
where, v0>0 and the negative sign indicate that the
suction is towards the plane.
The boundary conditions of the problem are:
y  0 : U  0, T  Td   (T0  Td )e i t , C  C d   (C 0  C d ) e i t 

y  1: U  0,   0, C  0


(7)
The term ∂qr/∂y represents the change in
radioactive heat flux with distance. The radioactive heat
flux term by using Rosseland approximation, is given
as:
The components of velocity in the x and y
direction respectively
The acceleration due to gravity
The coefficient of volume expansion
The kinematic viscoelasticity
The density
The viscosity
The kinematic viscosity
The thermal conductivity
The specific heat in the fluid at constant
pressure
The electrical conductivity of the fluid
The magnetic permeability
The molecular diffusivity
The temperature of the plane
The temperature of the fluid far away
from plane
The concentration of the plane
The concentration of the fluid far away
from the plane
v  v0
yv0
(6)
Introducing the following non-dimensionless quantities:
64 (13)
Res. J. Math. Stat., 4(3): 63-69, 2012
C1  ScC1  D2C1  0
METHOD OF SOLUTIONS
To solve (10), (11) and (12), we take solutions of
the form:
U ( y, t)  U 0 ( y)  U 1  y  e
 ( y , t )   0 ( y )   1  y   e i t
(15)
C ( y , t )  C 0 ( y )  C 1  y   e i t
(16)
where U 0  y  ,  0  y  , U1  y  , 1  y  and
With boundary conditions:
y  0 : U 0  0,  0  1, C0  1 

y  1: U 0  0,  0  0, C0  0 
(14)
i t
C1  y  are
determined.
Substituting (14), (15), (16) into (10), (11), (12)
respectively, Comparing harmonic and non harmonic
terms, we obtain:
 0 
U 0  U 00  RmU 01 

U 1  U 11  RmU 12 
(17)
1
 0  0
N
(19)
and boundary conditions:
y  0 : U 0  0,0  1, C0  1 

y  1: U 0  0,0  0, C0  0
(20)
Also,
1
i
1 
1  0
N
4N
  U 00
  L1U 00   Gr 0  GcC 0
U 00
(26)
U 11  U 11  D1U 11   Gr1  GcC1
(27)
  U 01
  L1U 01  U 00

U 01
(28)
  U 12
  D1U 12  U 11  iU 11
U 12
(29)
The corresponding boundary conditions are:
RmU1  i RmU1  U1  U1  D1U1  Gr1  GcC1 (21)
1 
(25)
Substituting (25) into (17) and (21) with their
boundary conditions, we obtain the following:
(18)
C 0  ScC 0  K ScC 0  0
(24)
where, the primes denotes differentiation with respect
to y.
The Eq. (17) and (21) are third order differential
equation due to the presence of viscoelasticity. Since
the viscoelasticity coefficient β1 is very small in Rm
therefore U0 and U1 is expanded in term of Rm to first
order (Beard and Walters, 1964):
to be
RmU 0  U 0  U 0  LU
1 0  Gr 0  GcC0
(23)
y  0 : U 00  U 01  U11  U12  0

y  1: U 00  U 01  U11  U12  0 
(22)
(30)
Solving (26) to (29) under the boundary condition (30) and substituting the obtained solution into (14).
The velocity field can be expressed as:
U  y, t   [( A1em1 y  A2e m2 y  A3  A4e

1
y
N
 A5er1 y  A6e r2 y )
 Rm( B1em1 y  B2e m2 y  B3 yem1 y  B4 ye  m2 y  B5e

1
y
N
 B6er1 y  B7 e r2 y )]
  e i t {( D 0 e my  D 20 e  ny  D 3 e gy  D 4 e  fy  D 5 e ry  D 6 e  sy )
+ Rm( E1e my  E2 e  ny  E15 ye my  E16 ye  ny  E17 e gy  E18 e  fy  E19 e ry  E20 e  sy )}
(31)
Similarly, solving (18) and (22) subject to the boundary conditions (20) and (24). The temperature field is:
  y , t   [ L30  L31 e

1
N
(32)
]   e i t [ L 60 e gy  L 61 e  fy ]
Also, solving (19) and (23) subject to the boundary conditions (20) and (24). The concentration field is:
65 Res. J. Math. Stat., 4(3): 63-69, 2012
C  y , t   [ L32 e r1 y  L33 e  r2 y ]   e i t [ L62 e ry  L63 e  sy ]
(33)
Skin friction is obtained when (31) is differentiated and evaluated at y  0 :
U ( y , t )
y
 [( m1 A1  m2 A2 
y 0
1
1
A4  r1 A5  r2 A6 )  Rm ( m1 B1  m2 B2  B3  B4  B5  r1 B6  r2 B7 )]
N
N
(34)
 e i t {( mD0  nD20  gD3  fD4  rD5  sD6 )  Rm ( mE1  nE2  E15  E16  gE17  fE18  rE19  sE20 )}
Nusselt number is obtained when (32) is differentiated and evaluated at y = 0:
  y , t 
y
 [
y0
(35)
1
L31 ]   e i t [ gL60  fL61 ]
N
Sherwood number is obtained when (33) is differentiated and evaluated at y = 0:
C
 y,t 
y
(36)
 [ r1 L 3 2  r2 L 3 3 ]   e i t [ r L 6 2  s L 6 3 ]
y0
1.01
RESULTS AND DISCUSSION
Pr = 0.025, 0.71, 1, 3
0.99
0.98
0.97
0.96
0.95
0
1
2
3
4
6
5
t
7
8
9
10
Fig. 1: Temperature profiles plotted against time for different
values of Pr
1.0
Pr = 0.71, 1, 3, 7, 10
Temperature
0.8
0.6
0.4
0.2
0
0
0.1 0.2
0.3 0.4
0.5 0.6
Y
0.7 0.8
0.9 1.0
Fig. 2: Temperature profiles for different values of Pr
0.65
0.60
Concentration
Viscoelastic fluid flow in a fixed point with heat
and mass transfer has been formulated and solved
analytically. In order to understand the flow of the
fluid, computations are performed for different
parameters such as Gr, M, Gc, R, Sc, Pr, K, k,  and 
.
Figure 1 and 2 represent the temperature profiles,
Fig. 3-5 are the concentration profiles and Fig. 6-16
represent the velocity profiles with varying parameters
respectively.
The effect of temperature for different values of (Pr
= 0.025, 0.71, 1, 3) when y = 0.5 and 0  t  10 is
presented in Fig. 1. In Fig. 2, the effect of temperature
for different values of (Pr = 0.71, 1, 3, 7, 10) is shown.
The graphs show that temperature decreases with
increase in Pr.
The effect of concentration for different values of (Sc =
0.22, 0.6, 0.8, 2.01) when y = 0.5 and 0  t  10 is given
in Fig. 3. Also, the effect of concentration for different
values of (k = 2, 4, 6, 8) is shown
in
Fig. 4.
Figure 5 denote the effect of concentration for (Sc =
0.22, 0.6, 0.8, 2.01). The graphs show that
concentration decreases with the increase in Sc and k.
The effect of velocity for different values of (Gc =
2, 3, 5, 10) is presented in Fig. 6, the effect of velocity
for different values of (Gr = 2, 3, 4, 5) is given in Fig. 7
and 8 denote the effect of velocity for (R = 1.0, 1.5, 2.0,
2.5). The graphs show that velocity increases with the
increase in Gc, Gr and R.
The effect of velocity for different values (  = 1, 5, 8,
10) is presented in Fig. 9, the effect of velocity for
different values of (M = 2, 2.5, 5, 7) is shown in
Temperature
1.00
Sc = 0.22, 0.6 , 0.8, 2.01
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0
1
2
3
4
5
t
6
7
8
9
10
Fig. 3: Concentration profiles plotted against time for different
values of Sc
66 Res. J. Math. Stat., 4(3): 63-69, 2012
2.5
1.4
2.0
K = 2, 4, 6, 8
1.0
1.5
Velocity
Concentration
1.2
0.8
0.6
1.0
0.4
0
0
0.1
0.2 0.3 0.4 0.5
Y
0.6 0.7
0
0.8 0.9 1.0
0.16
0.14
0.6
0.4
0.10
0.08
0.06
0
0.1
0.2 0.3 0.4 0.5
Y
0.6 0.7
0.02
0.00
0.8 0.9 1.0
0
Fig. 5: Concentration profiles for different values of Sc
0.1
0.2 0.3 0.4 0.5
Y
0.6
0.7
Fig. 9: Velocity profiles for different values of
0.35
1.6
1.4
0.25
1.2
0.20
1.0
Velocity
0.30
0.15
0.8
0.9 1.0

M = 2, 2.5, 5, 7
0.8
0.6
0.10
0.05
0.4
Gc = 2, 3, 5, 10
0
1.0
2.0 3.0
4.0 5.0
Y
0.2
6.0 7.0
0.0
8.0 9.0 1.0
0
Fig. 6: Velocity profiles for different values of Gc
0.1
0.2 0.3
0.4 0.5
Y
0.6 0.7
0.8 0.9 1.0
Fig. 10: Velocity profiles for different values of M
1.8
K = 0.1, 0.5, 1, 2
0.6
0.5
1.2
0.4
Velocity
1.6
1.4
Velocity
 = 1, 5, 8, 10
0.04
0.2
0.00
0.8 0.9 1.0
0.12
Sc = 0.2, 0.6, 0.8, 2.01
0.8
0
0.2 0.3 0.4 0.5 0.6 0.7
Y
Fig. 8: Velocity profiles for different values of R
1.2
1.0
0.1
0
1.4
Velocity
Conce ntration
Fig. 4: Concentration profiles for different values of k
Velocity
R = 1.0, 1.5, 2.0, 2.5
0.5
0.2
1.0
0.8
0.6
0.3
0.2
0.4
0.1
Gr = 2, 3, 4, 5
0.2
0
0.0
0
0.1
0.2 0.3 0.4 0.5
Y
0.6
0.7
0
0.8 0.9 1.0
0.1
0.2 0.3 0.4 0.5
Y
0.6 0.7
0.8
0.9 1.0
Fig. 11: Velocity profiles for different values of K
Fig. 7: Velocity profiles for different values of Gr
67 Res. J. Math. Stat., 4(3): 63-69, 2012
K = 1, 5, 10, 20
0.5
0.155
0.4
0.150
0.3
0.145
0.2
0.140
0.1
0.135
0.130
0.0
0
0.1
0.2 0.3 0.4 0.5
Y
0.6 0.7
0.8 0.9
3
2
4
5
t
0.7
Sc = 0.22, 0.6, 2.01, 2.65
0.7
6
7
9
8
10
Fig. 15: Velocity profiles plotted against time for different
values of Sc
0.8
 = 0.2, 2, 5, 7
0.6
0.6
0.5
Velocity
Velocity
1
0
1.0
Fig. 12: Velocity profiles for different values of k
0.5
0.4
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0
0.1
0.2 0.3 0.4 0.5 0.6
Y
0.7
0
0.8 0.9 1.0
Fig. 13: Velocity profiles for different values of Sc
Pr = 0.71, 1, 3, 7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
0.1
0.2 0.3 0.4 0.5 0.6 0.7
Y
0.8 0.9 1.0
Fig. 14: Velocity profiles for different values of Pr
Table 3: Skin friction τ
ε
ω
0.2
1
0.2
1
0.3
1
0.2
2
0.2
1
0.2
1
0.2
1
0.2
1
0.2
1
0.2
1
0.2
1
Pr
0.71
0.71
0.71
0.71
1.00
0.71
0.71
0.71
0.71
0.71
0.71
Sc
0.3
0.6
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
R
1
1
1
1
1
3
1
1
1
1
1
0.1
0.2 0.3 0.4 0.5
Y
0.8 0.9 1.0
0.6 0.7
Fig. 16: Velocity profiles for different values of
0.7
Velocity
Sc = 0.22, 0.5, 0.8, 2.01
0.160
Velocity
Velocity
0.6
Rm
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
Table 1: Sherwood number
ε
ω
0.2
1
0.2
1
0.2
1
0.2
2
0.3
1
Sc
0.3
0.6
0.3
0.3
0.3
K
1
1
2
1
1
Sh
-1.2505
-1.5109
-1.3448
-1.2457
-1.2481
Table 2: Nusselt number
ε
ω
0.2
1
0.2
1
0.2
2
0.3
1
0.2
1
Sc
0.71
1.00
0.71
0.71
0.71
K
1
1
1
1
2
Sh
-1.1548
-1.2224
-1.1497
-1.1523
-1.0967
M
2
2
2
2
2
2
2
2
2
2
4
τ
0.3212
0.3068
0.1819
0.3193
0.3179
0.4202
0.3042
0.3140
0.5245
0.6360
0.2860
K
1
1
1
1
1
1
4
1
1
1
1
68 
K
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.3
0.2
0.2
0.2
Gc
4
4
4
4
4
4
4
4
8
4
4
Gr
2
2
2
2
2
2
2
2
2
4
2
Res. J. Math. Stat., 4(3): 63-69, 2012
Beard, D.W. and K. Walters, 1964. Camb. Phil. Soc.,
60: 667.
Ezzat, M., M. Zakaria, O. Shaker and F. Barakat, 1996.
Elastic fluid flow of magneto hydrodynamics free
convection through a porous medium. Acta
mechanical, 119(5): 147-164.
Hartnett, J.P., 1992. Viscoelastic fluids a new challenge
in heat and mass transfer. Trans. ATME, pp:
296-303.
Kai-Long, H., 2010. Heat and mass transfer for a
viscous flow with radiation effect past a non-linear
stretching sheet. World Acad. Sci. Eng. Techn.,
62(3): 326-330.
Muthucumaraswamy, R. and G.K. Senthil, 2004. Heat
and mass effects on moving vertical plate in the
presence of thermal radiation. Int. J. Appl. Theoret.
Mech., 17(20): 801-820.
Noushima, H.G., M.V. Ramana Murthy, Rafiudinn and
R.M. Chennakrishna, 2009. Unsteady MHD
memory flow and heat transfer over a moving
continuous porous horizontal surface. Bull. Cal.
Math. Soc., 101(3): 281-290.
Noushima,
H.G.,
M.V.
Ramana
murthy,
R.M. Chennakrishna, A. Ramu and Rajender,
2010. Hydromagnetic free convective RevlinEricson flow through a porous medium with
variable permeability. Int. J. Comput. Appl. Math.,
5(3): 267-275.
Rafael, C., 2007a. MHD flow and mass transfer of an
electrically conducting fluid of second grade in a
porous medium over a stretching sheet with
chemically reactive species. Chem. Eng. Proc.,
46(8): 721-728.
Rafael, C., 2007b. Effect of viscous dissipation and
radiation on the thermal boundary layer over a nonlinearly stretching sheet. Appl. Math. Comput.,
184(2): 864-873.
Raptis, A.A., 1989. Heat transfer in a elastico viscous
fluid. Int. J. Heat Mass Transfer, 16: 193-197.
Sanjayanand, E. and S. Khan, 2006. Heat and mass
transfer in a viscoelastic boundary layer flow over
an exponentially stretching sheet. Int. J. Thermal
Sci., 45(8): 819-828.
Sayed-Ahmed, E.M. and H.A. Attia, 2000. MHD flow
and heat transfer in a rectangular duct with
temperature dependent viscosity and Hall effects.
Int. J. Heat Mass Transfer, 27(8): 1177-1187.
Seddeek, M.A., 2007. Heat and mass transfer on a
stretching sheet with a magnetic field in a
viscoelastic fluid flow in a porous medium with
heat source. Comput. Mater. Sci., 46(8): 721-728.
Sharma, P.R. and S. Sharma, 2005. Unsteady two
dimensional flow and heat transfer of through an
elastic-viscous liquid along an infinite hot vertical
porous medium. Bull. Cal. Math. Soc., 98(3):
477-488. Fig. 10, the effect of velocity for different values of (K
= 0.1, 0.5, 1, 2) is given in Fig. 11. Figure 12 depict the
effect of velocity for (k = 1, 5, 10, 20).
The effect of velocity for different values (Sc =
0.22, 0.6, 2.01, 2.65) is presented in Fig. 13, the effect
of velocity for different values (Pr = 0.71, 1.0, 3.0, 7.0)
is shown in Fig. 14, the effect of velocity for different
values of (Sc = 0.22, 0.6, 0.8, 2.01) when y = 0.5 and
0  t  10 is presented in Fig. 15and16 depict the effect
of velocity for (  = 0.2, 2, 5, 7). The graphs show that
velocity decreases with increase in  , M, K, k, Sc, Pr
and  .
Table 1 to 3 represent the Skin friction, Nusselt
number and Sherwood number respectively.
Table 1 depicts that the Sherwood number
decreases with increase in Sc, k and increases with
increase in  and  .
Table 2 represents that the rate of heat transfer
decreases with increase in Pr and increases with
increase in  ,  and R.
Table 3 shows that Shear stress increases with
increase in R, Gc and Gr and decreases with the
increase in Sc,  , Pr, R, k, K,  and M.
SUMMARY AND CONCLUSION
We have examined and solved the governing
equations for the viscoelastic fluid flow in fixed plane
with heat and mass transfer analytically using
perturbation technique, in order to point out the effect
of physical parameters namely; thermal Grashof
number, Hartmann number, mass Grashof number,
thermal Radiation, Schmidt number, Prandtl number,
permeability Parameter, frequency parameter and
chemical reaction parameter on the flow field. We
observe that, the velocity increases with the increase in
Gc, Gr and R and it decreases with increase in  , M,
K, k, Sc, Pr and  . Temperature decreases with
increase in Pr and concentration decreases with the
increase in Sc and k.
REFERENCES
Ambethkar, V. and P.K. Singh, 2011. Effect of
magnetic field on an oscillatory flow of a viscous
fluid with thermal radiation. Appl. Math. Sci.,
5(19): 935-946.
Atul Kumar, S., K. Ajay and N.P. Singh, 2003. Heat
and mass Transfer in MHD flow of a vioscous fluid
over a vertical plate under oscillatory suction
velocity. Indian J. Pure Appl. Math., 34(3):
429-442.
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