A Thermal Shock-Chemical Reactive Problem in Flow of
Viscoelastic Fluid with Thermal Relaxation
Magdy A. Ezzat1*, Wisam Khatan2
1Department of Mathematics, Faculty of Sciences and Letters in Al Bukayriyyah, Al-Qassim University, Al-Qassim, KSA
2Department of Mathematics, Faculty of Education, Damanhuur University, Damanhuur, Egypt Email: *maezzat2000@yahoo.com
Received April 20,2012; revised May 20, 2012; accepted May 27, 2012
ABSTRACT
Effects of thermal and species diffusion with one relaxation time on the boundary layer flow of a viscoelastic fluid bounded by a vertical surface in the presence of transverse magnetic field have been studied. The state space approach developed by Ezzat [1] is adopted for the solution of one-dimensional problem for any set of boundary conditions.The resulting formulation together with the Laplace transform techniques are applied to a thermal shock-chemical reactive problem. The inversion of the Laplace transforms is carried out using a numerical approach. The numerical results of dimensionless temperature, concentration, velocity, and induced magnetic and electric fields distributions are given and illustrated graphically for the problem.
Keywords: MHD; Heat and Mass Transfer; Boundary Layer Theory; Viscoelastic Fluid; State Space Approach;
Lord-Shulman Theory; Numerical Results
1. Introduction
Viscoelastic flows are encountered in numerous areas of petrochemical, biomedical and environmental engineer-ing includengineer-ing polypropylene coalescence sinterengineer-ing [2] and geological flows [3]. A wide range of mathematical models have been developed to simulate the nonlinear stress-strain characteristics of such fluids which exhibit both viscous and elastic properties [4].
In nature and many industrial applications, there are plenty of transport processes where simultaneous heat and mass transfer is a common phenomenon. Its applica-tion is found in many diverse fields but not limited to cleaning operations, curing of plastics, manufacturing of pulp-insulated cables, many chemical processes such as analysis of polymers in chemical engineering, condensa-tion and frosting of heat exchangers [5]. The study of convection reduces to the determination of convective heat and mass transfer coefficients. Convective heat and mass transfer coefficients are important parameters, which are a measure of the resistance to heat and mass transfer between a surface and the fluid flowing over that surface. The convective coefficients depend on the hydrodynamic, thermal and concentration boundary layers. In many of the internal flows, both forced and natural convection play major roles in the heat and mass transfer processes.
Whereas in the entrance section of a duct, forced convec-tion becomes dominant, as the flow moves towards the downstream section, natural convection could dominate over forced convection and finally in the thermally de-veloped region natural convection becomes negligible. Natural convection may be due to a temperature or con-centration gradient or both. If the buoyancy forces are due to temperature and concentration gradients that act in the same direction, both the heat and mass transfer will increase. However, if the temperature and concentration gradients act in the opposite direction, both heat and mass transfer reduce [6].
In recent years, the study of viscoelastic fluid flow is an important type of flow occurring in several engineer-ing processes. Such processes are wire drawengineer-ing, glass fiber and paper production, crystal growing, drawing of plastic sheets, among which we also cite many applica-tions in petroleum in drilling, manufacturing of foods and slurry transporting. The boundary layer concept of such fluids is of special importance due to its applications to many engineering problems among which we cite the possibility of reducing frictional drag on the hulls of ships and submarines.
A great deal of works has been carried out on various aspects of momentum and heat transfer characteristics in a viscoelastic boundary layer fluid flow over a stretching plastic boundary [7] since the pioneering work of Sa-
kiadis [8]. Ezzat and Zakaria [9] studied the effects of free convection currents with one relaxation time on the flow of a viscoelastic fluid through a porous medium. Khan and Sanjayanand [10] studied heat and mass trans-fer in a viscoelastic boundary layer flow over exponen-tially stretching sheet.
In this work, we use a more general model of MHD mixed convection flow of conducting viscoelastic fluid which also includes both the relaxation time in the heat and concentration equation and the electric permeability of the electromagnetic field. The unsteady free convec-tion heat and mass transfer flow of electrically conduct-ing incompressible viscoelastic fluid past an infinite ver-tical plate in the presence of a transverse magnetic field and chemical reaction using the state space approach and Laplace transforms technique. The inversion of the Laplace transform is carried out using a numerical tech-nique [11].
2. Formulation of the Problem
The electro-magnetic quantities satisfy Maxwell’s equa-tions [12]: o curl t E
h J (1)
0 curl t h
E (2)
0
divh , divE0 (3)
o o
B H h , DoE (4)
These equations are supplemented by Ohm’s law
o o
o
J E V H (5)
Consider an unsteady free convection flow of electri-cally conducting incompressible, viscoelastic fluid past an infinite vertical plate. The x-axis is taken in the verti-cal direction along the plate and y-axis normal to it. Let u
be the component of the velocity of the fluid in the x di-rection and a constant magnetic field acts in the y direc-tion of strength . This produces an in-duced magnetic field and an induced elec-tric field as well as a conduction current density
0, , 0 o HoH
h, 0, 0 h
0, 0,E
0,
E
0, J
J . All the considered functions will depend on y and the time t only.
Equation (5) reduces to
o o o
J E H u . (6)
The vector Equations (1) and (2) reduced to the fol-lowing scalar equation
o
h J
y t
o
E
t y h
(8)
Eliminating J between Equations (6) and (7) we obtain
o o o o oh E
E
y t
H u (9)
Eliminating E between Equations (8) and (9) we ob-tain 2 2 2 2 o m o o
h h h u
H t y y t
(10) The Lorentz force has a non-vanishing component in the x-direction, given by:
x x o o
h E
F H
y t
o
J B (11)
Assume that the viscoelastic fluid contains some chemically reactive diffusive species then the equations describing the flow in the boundary layer reduce to:
2 2
3 2
0 2 0
0
T c
u u
g T T g c c
t y
u h E
k
H y t
t y (12) 2 2 2 2 o m o o
h h h H u
t y y t
(13)
2 2
1 o
P
T k
t t C y
T
(14)
o o o o o
h E
E H
y t
u (15)
2 2
1 o
c c
D
t t y
Kc (16) Introduce the non-dimensional quantities.
2 2 2 2 2 2 0 2 2 3 3 1 , , , , , , , , , , , , , , , , o o o oo o o o o
c
o o
p r
o
T o C o
T c
y y t t k k E E
H
u h
u k k h x x
H P P
T T c c
T S c
T T D c c
C
K K J J P
H K
g T T g c c
G G , , o (17) E
In Equation (23) the overbar denotes the Laplace transform and the prime indicates differentiations with respect to y.
With the help of the non-dimensional quantities above Equations (12)-(16) reduced to the non-dimensional equ- ations
2 3
0
2 o 2 T c
u u k u G T G c h
t y t y y
E t
(18)
Equation (23) can be written in constracted form as
d ,
, d
V y s
s V y sy A (24)
2 2
1
2 2
h h h
b
t y
y t
u
(19) The formal solution can be expressed as:
y s, exp
s y
0,sV A V
(25)
221 1 o
r T
t t P y
T (20)
The characteristic equation of the matrix A
s is
221 1 o
c
c c
Kc
t t S y
(21)
8 6 4 2
1 2 3 4 0
k I k I k I k I (26)
where
o o oh E
E
y t u
(22)
2 1 2 1 3 1 4 2 3 2
3 4 1 2
3 1 3 4 2 3 4 1 2 3 1 2 4 1 2
1 2 3 4
2
1 1 2 3 4
2 2
2
4
I m m m m bn
4 I m m m m m m m m m m
m m m bn m bn I m m m m m m m m m
m m m m m bn I m m m m
(27) where m b
, 1 o
o b
.
To simplify the algebra, only problems with zero ini-tial conditions are considered. Taking Laplace transform of Equations (18)-(22) and writing the resulting equa-tions in matrix form results in (23).
where The roots k1, k2, k3 and of Equation (25)
satisfy the relations: 4
k
1 r 1 o ,
m P s s
2 2 2 2
1 2 3 4 1
2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 3 1 4 2 3 2 4 3 4 2
2 2 2 2 2 2 2 2 2 1 2 3 1 2 4 2 3 4 3
2 2 2 2 1 2 3 4 4
k k k k I
k k k k k k k k k k k k I
k k k k k k k k k I
k k k k I
(28)
2 c 1 o ,
m S s s K
4 1
m s b s
2 1 1 o o o o s s n sk
Two of the roots, say k1 and have simple
ex-pression given by 2
k and 2 1 1 2 2 2 , . k m k m
(29a)
3 1 1 o o o o o s s m sk
. The other two roots
3
k
and k4 satisfy the relation
1 2 2 3 0 0 4
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
d
0 0 0 0 0 0 0
d
0 0 0 0 0 0 0
0 0 0 0
1 1
0 0 0 0 0 0
c T T T c c u u h h m
y T m T
c G G c
m n
u sk sk u
h m b h
2 2 2
3 4 3
2 2
3 4 3 4.
k k bn m m k k m m
4 (29b)
The Maclaurin series expansion of expA
s y is given by
0 exp ! n n s y s y n A
A .Using the Cayley-Hamilton theorem, the infinite series can be truncated to the following form
2 30 1 2 3 4
5 6 7
5 6 7
exp
,
s y a I a a a a
a a a
A A A A A
A A A
4
(30)
where I is the unit matrix of order 8 and a0 – a7 are some
parameters depending on s and y.
The characteristic roots k1, k2, k3 and k4 of the matrix A must satisfy the equations.
2 31 0 1 1 2 1 3 1 4 1
5 6 7
5 1 6 1 7 1
exp k y a a k a k a k a k a k a k a k
4 4 4 4
2 32 0 1 2 2 2 3 2 4 2
5 6 7
5 2 6 2 7 2
exp k y a a k a k a k a k a k a k a k
2 33 0 1 3 2 3 3 3 4 3
5 6 7
5 3 6 3 7 3
exp k y a a k a k a k a k a k a k a k
2 34 0 1 4 2 4 3 4 4 4
5 6 7
5 4 6 4 7 4
exp k y a a k a k a k a k a k a k a k
The solution of this system of linear equations is given in Appendix A:
Substituting for the parameters a0 - a7 into Equation
(30) and computing A2, A3, A4, A5, A6 and A7, we get, the elements (ℓij i, j = 1, 2, 3, 4, 5, 6, 7, 8) of the matrix L(y, s)
which listed in Appendix B.
It should be noted here that, we have used Equation (29) in order to write these entries in the simplest possi-ble form. It should also be noted that this is a formal ex-pression for the matrix exponential. In the physical prob-lem , we should suppress the positive expo-nential which are unbounded at infinity. Thus we should 0 y
replace each sinh
ky by 1exp
2 ky
and eachby
cosh ky 1exp
2 ky .It is now possible to solve broad class problems in the Laplace transform domain.
3. Thermal Shock-Chemical Reactive
Problem
Consider the free convection flow of an incompressible
viscoelastic fluid in the presence of magnetic field occu-pying a semi-infinite region y 0 of the space bounded by an infinite vertical plate y = 0 with quiescent initial state. A thermal-concentration shock is applied to the boundary plane y = 0 in the form
0, , 0, o oT t T H t c t c H t
(31)
and the mechanical boundary conditions on the plate is taken as
0, 0, 0, 0 u t h t (32)
where and are constant and H(t) is Heaviside unit step function. o
T co
Now we apply the state space approach described above to this problem.
Since the solution is bounded at infinity, then the ex-pressions for ij 1, 2, ,8 can be obtained by sup-pressing the positive exponential terms in Equation (30) which are not bounded at infinity. Thus for , we should replace each
0 x
sinh ky by exp
ky
2 and each cosh
ky by exp
ky
2.The components of the transformed initial state vector
0,V s are known name
0, , 0, o o T T s s c c s s (33)
0, 0, 0, 0 u s h s (34)
In order to obtain the remaining four components
0,T s , c
0,s , u
0,s and h
0,s
we substitutey = 0 into Equations (31) and (32) to obtain the following linear system of equations:
51 55 62 6671 72 75
76 77 78
81 82 85
86 87 88
0, 0,
0,s 0,s
0, 0,
0,s 0, 0,
T c
0, 0,
s s
0,s 0, 0,
o
o
o o
o o
T
T s T s
s c
c c
s
T c
u s T s
s s
c u s h
h s T s
c u s h
s s (35)
1 2 2 22 4 3 4
2 2 2 2 3 4
3 4 2 2 2 2 2 2
1 3 4 3 4
2 2 2 2
3 3 4 3 4 4 4 4 3 4
3 2 2 2 2 4 2 2 2 2
3 3 4 4 3 4
0, , 0,s
0, 2
o o
i i
i i i i
i i i
i i
i i
k k
T s T c c
s s
k bn k m k k
k k
u s A k k k k k
k k k k
k k bn
k bn k m k k k bn k m k k
k k k k
k k k k k k k k
22 3 4 4
2 2 2 2 3 4
3 4 2 2 2 2 2 2
1 3 4 3 4
2 2
3 3 4 3 4 4 3 4 4 4
3 2 2 2 2 4 2 2 2 2
3 3 4 4 3 4
0, 2
( )
i i
i i i i
i i i
i i
i i
k k k k m
k k
h s b A k k k k k
k k bn k k k k
k k k k m k k k k m
k k k k
k k k k k k k k
(36)Finally substituting the above value into (25), we ob-tain the solution of the problem in the transformed do-main as:
, exp
1o T
T y s k y
s
(37)
, oexp
2c
c y s k y
s
(38)
2
1 1 4 1
2
2 2 4 2
2
3 3 4 3
2
4 4 4 4
, exp
exp
exp
exp
u y s A k m k y
A k m k y
A k m k y
A k m k y
(39)
1 1 1 2 2 2
3 3 3 4 4 4
, exp exp
exp exp
h y s b A k k y A k k y
A k k y A k k y
(40)
where the constants Ai, i = 1, 2, 3, 4 are listed in
Appen-dix C.
The induced electric field and current density take the following forms
1 1 2 2
3 3 4 4
exp exp
exp exp ,
E sb A k y A k y
A k y A k y
(41)
2 2
1 1 1
2 2
2 2 2
2 2
3 3 3
2 2
4 4 4
exp exp exp exp . o o o o
J b A k s k y
A k s k y
A k s k y
A k s k y
(42)
The shearing stress at the wall is given by
u
2
2
1 1 1 4 2 2 2 4
2 2
3 3 3 4 4 4 4 4
0 y
s A k k m A k k m y
A k k m A k k m
(43)
4. Inversion of the Laplace Transforms
above In order to invert the Laplace transform in the equations, we adopt a numerical inversion method based on a Fourier series expansion [11]. In this method, the inverse g(t) of the Laplace transform g s
is approxi-mated by the relation
π 1
1 1 1 1 1 ct
e Re π ,
2
0 2 ,
ik t t k
g t g c e g c ik t
t t t
(44)where N is a sufficiently large integer representing the number of terms in the truncated infinite Fourier series.
N must chosen such that
1
π
Re iN t t ct
e e g c iN t π 1 1,
where 1 is a persecuted small positive number that
corresponds to the degree of accuracy to be achieved. The parameter c is a positive free parameter that must be greater than the real parts of all singularities of g s
. The optimal choice of c was obtained according e criteria described in [11].to th
5. Numerical Results and Discussion
and mass
fields as well as the induced magnetic and electric fields The problem of free convective flow with heat
are obtained by using the state space approach. The tech-nique is applied to a thermal shock-chemical reactive problem without heat sources. The effects of flow pa-rameters such as Grashof number for heat and mass transfer GT, Gc, Prandtl number Pr, Schmidt number
c
S , chemical reaction parameter K, viscoelastic parame-ter ko and laxre ation time o have been studied
ana-ically and presented with the help of Figures 1-8 for
the considered problem.
5.1. Velocity Field (u)
lyt
The velocity of the flow variation of the flow param
field varies vastly with the eters such as Grashof number
o e
[image:6.595.127.470.526.719.2]Parameter (ko)
Figure 1depicts the effect of viscoelastic parameter
eping ot
easing
The values of Grashof number for heat have been nt of for heat and mass transfer GT, Gc, viscoelastic
pa-rameter ko, Prandtl number Pr, Schmidt number Sc,
chemical reaction parameter K ffects of these pa-rameters on the velocity fluid f flow field have be n presented in Figures 1-3.
5.1.1. Effect of Viscoelastic
. The e
o
k
her ith on the velocity profiles of the flow field ke
parameters of the flow field constant. The curve w viscoelastic parameter, ko0 corresponds to
Newto-nian flow and in other two curves the viscoelastic pa-rameter is taken in incr order. The viscoelastic parameter is found to decelerate the velocity of the flow field. The above parameter is in good agreement with the result obtained in cases of Khan and Sanjayanand [10].
5.1.2. Effect of Grashof Number for Heat (GT)
T
chosen as they are interesting from physical poi
G
e
view. The free convection of heat is du to the tem-perature difference ToT and hence GT 0 when
0
o
T T which ph y corresponds t ling of
by free convection currents. Then GT 0
ysicall o coo
the surface
correspond to heating of the surface by free convection currents. In Figure 2, we observe that the effect of
cool-ing and heatcool-ing by free convection currents when
0
T
G and GT0 are in agreement with physical
tions th ling of the surface by free convec-tion currents occurs for positive values of GT while
heating corresponds to negative values of GT It was
also noticed that the velocity increase with the increase of GT.
observa
5.1.3. Effect
ica
5.2. Temper
at coo
of Different P
ld.
ature Field (
.
, Sc
o
, K)
o-re or less
arameters (Gc, P
T)
to change
r
m
Figure 3 present the effect of Grashof number for mass
transfer Gc, Prandtl number Pr, Schmidt number Sc
and chem l reaction paramete K on the velocity pr files of the flow fluid. Comparing the curve (1) and (2) of the figure, it is observed that the Grashof number for mass transfer is to enhance the velocity of the flow field at all points. The effect of both Prandtl number Pr and
chemical reaction parameter K on the velocity field is shown by the curves (1), (3) and (5). It was found that the increasing of Pr and K lead to decelerate the
veloc-ity of the flow fie Curves (1) and (4) describe the ef-fect of Schmidt number Sc on the velocity profiles of
the flow field which reve that the presence of heavier diffusing species has a retarding effect on the velocity of the flow field. The effect of the above parameters has the same behavior as in case of Das et al. [13].
r
al
The temperature field is found
with the variation of the Prandtl number Pr. Figure 4 is
plot of non-dimensional temperature and di ance for five different values of the Prandtl number. The temperature
st
Figure 2. Velocity distribution for different values of GT.
Figure 3. Velocity distribution for different values of Gc, Pr, Sc and K.
[image:7.595.128.468.517.720.2]Figure 5. Concentration distribution for different values of Sc.
Figure 6. Induced magnetic field distribution for different values of GT, Gc.
Figure 8. Current density distribution for different values of GT, Pr, Sc and K.
f the flow field diminishes as the Prandtl number
in-5.3. Concentration Field (c)
ion distribution of the
5.4. Induced Magnetic Field (h)
induced
mag-5.5. Induced Electric Field (E) and Current
The induce nd current density of the flow
field and current density have been presented in Figures
ifferent Parameters(GT, Pr, Sc, K)
Figures 7 and 8present the effect of Grashof number for
t number
flu
skin friction coefficient τ for
K and
corre-ng
e
nature of the governing D viscoelastic flow, few at
o
creases. Higher the Prandtl number, the sharper is the reduction in the temperature of the flow field.
The variation of the concentrat
flow field with the diffusion of the mass is shown in Fig-ure 5. The concentration distribution decreases at all
points of the flow field with the increase of the Schmidt number Sc. This shows that the heavier the diffusing
species h e a greater retarding effect on the concentra-tion distribuconcentra-tion of the flow field. The concentraconcentra-tion pro-files are in good agreement with the results obtained in case of Hsiao [14].
av
Figure 6 concentrate on variations in the
netic field profiles h for cooling GT 0 and heating 0
T
G of the plate due to chang he values of number for mass transfer Gc. It is observed that
for cooling (heating) of the plate the induced magnetic field increases (decreases) rapidly in the vicinity of the plate and decreases (increases) asymptotically for higher values of y. It is also observed that an increase in Grashof number for mass transfer Gc increases (decreases) the
induced magnetic field.
e in t Grashof
Density (J)
d electric field a
field vary vastly with the variation of the flow parameters such as Grashof number GT , Prandtl number Pr ,
Schmidt number Sc and che al reaction parameter .
The effects of th parameters on the induced electric
7 and 8.
Effect of D
mic K
ese
heat transfer GT, Prandtl number Pr, Schmid
c
S and chemical reaction parameter K on the induced electric field E nd current density rofiles of the flow id, respectively. It was found that the increasing of these parameters lead to decelerate the magnitude of both the induced magnetic field and current density but Grashof number for heat transfer is to enhance them.
5.6. Skin Friction (τ)
, a J p
The numerical values of
different values of GT, c r c o
sponding to cooling of the plate GT 0 are entered in Table 1. It is observed th th in icient τ
increases due to increase in GT, Gc ile decrease due
to increase in Pr, Sc, K and ko.
6. Concludi
R marks
G ,at
P,
e sk
S , k
ff friction coe
wh
1) Owing to the complicated equations for the unsteady MH
Table 1. Values of the skin friction τ due to the variation of GT, Gc, Pr, Sc, K nd ko.
T c r c o
a
G G P S K k τ
1 2 0.73 0.22 0.2 0.2 731.23 915
5 2 0.73 0.22 0.2 0.2 1.2373925
1 4 0.73 0.22 0.2 0.2 2.4747825
1 2 1 0.22 0.2 0.2 1.0391526
1 2 0. 73 0.62 0.2 0.2 0.2902698
1 2 0.73 0.22 0.5 0.2 1.1652091
1 2 0.73 0.22 0.2 0.4 1.1499502
omain without any assumed restrictions on the applied
n this work, we use a more general model of equa-tio
d
magnetic field or viscoelastic parameters. The same ap-proach was used quite successfully in dealing with prob-lems in generalized thermoelasticity theory by Ezzat et al.
[15]. 2) I
ns, which includes the relaxation time of heat conduc-tion o and the electric permeability of the
electromag-netic field o. The inclusion of the relaxation time and
electric permeability modifies the governing thermal, concentration and electromagnetic equations, changing them from parabolic to hyperbolic type, and thereby eliminating the unrealistic result that thermal and chemi-cal reactive disturbance is realized instantaneously eve-rywhere within a fluid.
3) The work considered here in reflects the effects tra
over the boundaries has many ap
agriculture, engineering, petroleum industries, and heat
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doi:10.1016/j.ijheatmasstransfer.2004.10.032
[11] G. Honig and U. Hirdes, “A method for the Numerical Inversion of Laplace Transforms,” Computational a Applied Mathematics, Vol. 10, No. 1, 1984, pp. 113-132. nd [12] M. A. Ezzat, “Free Convection Effects on Perfectly
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(00)00095-1
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Appendix A: The Solution of the Linear System
2 2 2 2 2 2 2 2 2 2 2 2
0 2 3 4 1 1 3 4 2 1 2 4 3 1 2 3 4
a F k k k C k k k C k k k C k k k C ,
2 2 2 2 2 2 2 2 2 2 2 2
1 2 3 4 1 1 3 4 2 1 2 4 3 1 2 3 4
a F k k k S k k k S k k k S k k k S
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
2 2 3 2 4 3 4 1 3 4 3 1 1 4 2 4 1 4 2 1 2 3 1 2 1 3 2 3
a F k k k k k k C k k k k k k C k k k k k k C k k k k k k C4
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
3 2 3 2 4 3 4 1 3 4 3 1 1 4 2 4 1 4 2 1 2 3 1 2 1 3 2 3 4
a F k k k k k k S k k k k k k S k k k k k k S k k k k k k S
2 2 2
2 2 2
2 2 2
2 2 2
4 2 3 4 1 1 3 4 2 1 2 4 3 1 2 3 4
a F k k k C k k k C k k k C k k k C
2 2 2
2 2 2
2 2 2
2 2 2
5 2 3 4 1 1 3 4 2 1 2 4 3 1 2 3
a F k k k S k k k S k k k S k k k S4 4
6 1 2 3
a F C C C C4 , a7 F S
1S2S3S
where
2 2
2 2
2 2
2 2
2 2
2 2
1 2 1 3 1 4 2 3 2 4 3 4
1 F
k k k k k k k k k k k k
2 2
2 2
2 2
1 2 3 3 4 4 2 cosh 1
C k k k k k k k y ,
2 2
2 2
2 2
2 3 4 4 1 3 1 cosh 2
C k k k k k k k y
2 2
2 2
2 2
3 4 1 1 2 2 4 cosh 3
C k k k k k k k y ,
2 2
2 2
2 2
4 1 2 2 3 1 3 cosh 4
C k k k k k k k y
2 2
2 2
2 2
1 2 3 3 4 4 2 1
1 1
sinh
S k k k k k k k y
k
,
2 2
2 2
2 2
2 3 4 4 1 3 1 2
2 1
sinh
S k k k k k k k y
k
2 2
2 2
2 2
3 4 1 1 2 2 4 3
3
1 sinh
S k k k k k k k y
k
,
2 2
2 2
2 2
4 1 2 2 3 1 3 4
4
1 sinh
S k k k k k k k y
k
Appendix B:
Elements of the Matrix
L
(
s
,
y
)
2 2
2 2
2 2
11 F k1 k2 k1 k3 k1 k C4 1
, 12 0, 130, 140
2 2
2 2
2 2
15 F k1 k2 k1 k3 k1 k S4 1
, 160, 17 0, 18 0, 210
2 2
2 2
2 2
22 F k2 k1 k2k3 k2k C4 2
, 230, 240,25 0
2 2
2 2
2 2
26 F k2 k1 k2k3 k2k S4 2
, 270, 280
2 2
2
2 2
2
2 2
2
31 2 1 1 4 1 2 3 3 4 3 2 4 4 4 4
0 1
T
G F
k k k m C k k k m C k k k m C
sk
2 2
2
2 2
2
2 2
2
32 1 2 2 4 2 1 3 3 4 3 1 4 4 4 4
0 1
c
G F
k k k m C k k k m C k k k m C
sk
2 2 2 2 2 2 2 2 2 2 2 2
33 4 2 3 1 3 4 3 3 3 2 4 1 4 4 4 4
4
F k k k k k m k C k k k k k m k C
m
2 2 2 2 2 2 2 2 2
34 n m F k4 1 k3 k3 k S2 3 k1 k4 k4 k S2 4
2 2
2
2 2
2
2 2
2
35 2 1 1 4 1 2 3 3 4 3 2 4 4 4 4
0 1
T
G F
k k k m S k k k m S k k k m S
sk
2 2
2
2 2
2
2 2
2
36 1 2 2 4 2 1 3 3 4 3 1 4 4 4 4
0 1
c
G F
k k k m S k k k m S k k k m S
sk
2 2
2 2
2
2 2
2 2
2
2 2 2 2 2 2 2 2 2
38n F k 3 k1 k3 k C2 3 k4k1 k4k C2 4
,
2 2 2 2 2 2 2 2 2
2
41 1 1 2 1 3 3 2 3 4 4 4
0 1
T
G bF
k k k S k k k S k k k S sk
2 2 2 2 2 2 2 2 2
42 2 2 1 2 3 3 1 3 4 4 1 4
0 1
c
G bF
k k k S k k k S k k k S sk
2 2
2 2
2 2
2 2
43 bm F k3 3 k1 k3 k S2 3 k4k1 k4k S2 4
2 2
2 2
2
2 2
2 2
2
44 F k 1 k3 k2k3 k4m C4 3 k1 k4 k2 k4 4 k3m C4
2 2
2 2
2 2
45 2 1 1 2 3 3 2 4 4
0 1
T
G bF k k C k k C k k C
sk
,
2 2
2 2
2 2
46 1 2 2 1 3 3 1 4 4
0 1
c
G bF
k k C k k C k k C
sk
2 2
2 2
2 2
2 2
47 bF k 1 k3 k2k C3 3 k1 k4 k2k C4 4
2 2 2 2 2 2 2 2 2 2 2 2
48 3 3 1 3 2 4 4 3 4 4 1 4 2 4 3 4
4
F k k k k k m k S k k k k k m k S m
2 2 2 2 2 2 2
51 k F k1 1 k2 k1 k3 k1 k S4 1
, 520, 530, 54 0
2 2
2 2
2 2
55 F k1 k2 k1 k3 k1 k C4 1
, 56 0, 57 0, 580, 610
2 2 2 2 2 2 2
62 Fk k2 2 k1 k2k3 k2k S4 2
, 630, 640, 650
2 2
2 2
2 2
66 F k2 k1 k2 k3 k2k C4 2
, 67 0, 680
2
3 2 4
k
2 2 2 2 2 2 2 2 2 2 2
71 1 1 2 1 4 1 3 3 3 4 4 2 4 4 4
0 1
T
G F
k k k k m S k k k m S k k k k m S
sk
2 2 2 2 2 2 2 2 2 2 2 2
72 2 2 1 2 4 2 3 3 1 3 4 3 4 4 1 4 4 4
0 1
c
G F
k k k k m S k k k k m S k k K k m S
sk
2 2
2 2
2
2 2
2 2
2
73 m F k3 3k1 k3 k2 k3 m S4 3 k4 k1 k4k2 k4m S4 4
2 2 2 2 2 2 2 2 2
74 n m F k4 3k1 k3k C2 3 k4k1 k4 k C2 4
2 2
2
2 2
2
2 2
2
75 1 2 1 4 1 3 2 3 4 3 4 2 4 4 4
0 1
T
G F
k k k m C k k k m C k k k m C
sk
2 2
2
2 2
2
2 2
2
76 2 1 2 4 2 3 1 3 4 3 4 1 4 4 4
0 1
c
G F
k k k m C k k k m C k k k m C
sk
2 2
2 2
2
2 2
2 2
2
77 F k 3k1 k3k2 k3 m C4 3 k4k1 k4k2 k4m C4 4
2 2 2 2 2 2 2 2 2 2
78 n F k k 3 3 k1 k3 k S2 3 k4k1 k4 k S2 4
2 2 2 2 2 2 2 2 2
81 1 1 2 1 3 3 2 3 4 4 2 4
0 1
T
G bF
k k k C k k k C k k k C sk
2 2 2 2 2 2 2 2 2
82 2 2 1 2 3 3 1 3 4 4 1 4
0 1
c
G bF
k k k C k k k C k k k C sk
2 2
2 2
2 2
2 2
83m bF k3 3 k1 k3k C2 3 k4k1 k4k C2 4
2 2 2 2 2 2 2 2 2 2 2 2
84F k k 3 3 k1 k3 k2 k4m S4 3k k4 4k1 k4k2 k3m S4 4
2 2 2 2 2 2 2 2 2
85 1 1 2 1 3 3 2 3 4 4 2 4
0 1
T
G bF
k k k S k k k S k k k S sk
2 2 2 2 2 2 2 2 2
86 2 2 1 2 3 3 1 3 4 4 1 4
0 1
c
G bF
k k k S k k k S k k k S sk
2 2 2 2 2 2 2 2 2 2
87 bF k k 3 3 k1 k3 k S2 3k k4 4k1 k4k S2 4
2 2 2 2 2 2 2 2 2 2 2 2
88 3 3 1 3 2 4 4 3 4 4 1 4 2 3 4 4
4
F k k k k k k m C k k k k k k m C
m
Appendix C: The Constants for the Problem
1 2 2 2 2
3 1 1 4
1 o
A G TT o
s sk k k k k
, 2
2
3
1 o
A
2 2 2
2 2 4
c o
G c s sk k k
k k
2
3 4 4 2 2
3 2 2 4
1
4 3 4 3 4
i i i
i k k m
A A k k m
m k k bn k k
4 k k4 m4
2
3 4 4 2 2
4 2 2 3
1
4 3 4 3 4
i i i
i k k m
A A k k m
m k k bn k k
4 k k3 m4Nomenclature
.
P
C k
specific heat at constant pressure viscoelastic parameter
thermal Grashof number
density o
GT
G t time
x y z, ,
space coordinates solute Grashof number torc
P Prandtl number
V velocity vec r
S Schmidt number
H magnetic field intensity vector
ctor
vector
c
H constant component of magnetic field
B magnetic induction ve o
electrical conductivity
D electric induction
magnetic permeability
E induced electric field vector h induced magnetic field vector
J conduction electric density vector
x-direction
on parameter
u velocity of the fluid along the
T temperature
c concentration
D mass diffusivity of chemically reactive species
K chemical reacti
g acceleration
T
c
temperature of the fluid away from the surface
concentration of the fluid away from the
sur-face
thermal conductivity dynamic viscosity
, kinematics viscosity
m
1
o o
, magnetic diffusivity
o
thermal relaxation time
2
oHo
, Alfven velocity
