Temperature Dependent Suction/Injection and Variable Properties on Non-Newtonian Casson

17 Pages·2017·0.74 MB·English

Columbia International Publishing
American Journal of Heat and Mass Transfer
(2017) Vol. 4 No. 2 pp. 104-120
doi:10.7726/ajhmt.2017.1007
Research Article

Temperature Dependent Suction/Injection and Variable
Properties on Non-Newtonian Casson Mixed Convective
MHD Laminar Fluid Flow with Viscous Dissipation and
Thermal Radiation

Kh. Abdul Maleque1*

Received: 15 December 2016; Returned for revision: 12 January 2017; Received in revised form: 12 January
2017; Accepted: 18 January 2017; Published online: 4 February 2017

© Columbia International Publishing 2017.Published at www.uscip.us

Abstract

Similarity solution of an unsteady MHD laminar combined convective boundary layer incompressible non-
Newtonian casson fluid flow for a vertical porous plate with the effects of viscous dissipation, variable fluid
 
properties ( viscosity ( ) and thermal conductivity ( )), thermal radiation and time and strongly
temperature dependent suction/injection velocity, has been studied under the action of transverse applied
magnetic field. Assuming the suction/injection is a nonlinear function of time and temperature as
B
T 
v(t,T ) w
  . The boundary layer equations have been transformed into dimensionless nonlinear
w T
 

ordinary differential equations by similarity transformations. The nonlinear ordinary differential equations
are then solved numerically by using Range-Kutta six order integration scheme and Nachtsheim-Swigert
iteration technique. The obtained numerical results are presented graphically in the form of velocity and
()
temperature. Finally the effects of the relative temperature difference parameter on the skin friction and
heat transfer coefficients are also examined.

Keywords: Variable Suction; Variable Properties; MHD Casson Fluid; Viscous Dissipation; Thermal Radiation

1. Introduction

It is known that the physical properties (density (), viscosity () and thermal conductivity ())
______________________________________________________________________________________________________________________________
*Corresponding e-mail: [email protected]
1. Professor of Mathematics, American International University-Bangladesh, House-23, 17,Kamal Ataturk
avenue, Banani,Dhaka-1213, Bangladesh. Email: [email protected] and [email protected]

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Kh. Abdul Maleque/ American Journal of Heat and Mass Transfer
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may change significantly with temperature of the flow. To predict the flow behavior accurately, it
may be necessary to take into account these variable properties. In light of this concept, Hodnett [1]
studied the low Reynolds number flow of a variable property gas past an infinite heated circular
cylinder. Bhat and Bose [2] examined the fluid flow with variable properties in a two dimensional
channel. Zakerullah and Ackroyd [3] investigated the free convection flow above a horizontal
circular disc for variable fluid properties. Correlations have been developed by Gokoglu and Rosner
[4] for deposition rates in force convection systems with variable properties. Herwing [5] analyzed
the influence of variable properties on laminar fully developed pipe flow with constant heat flux
across the wall. It was shown how the exponent in the property ratio method depends on the fluid
properties. The influence of temperature dependent fluid properties on laminar boundary layers
was further discussed by Herwig and Wickern [6] for wedge flows. In case of fully developed
laminar flow in concentric annuli, the effect of the variable properties has been studied by Herwig
and Klemp [7]. Herwing [8] examined an asymptotic analysis of laminar film boiling on a vertical
plate including variable property effects. Maleque and Sattar [9] investigated the effects of Hall
current and variable viscosity on an unsteady MHD laminar convective flow due to rotating disc.
Maleque and Sattar [10, 11] presented the variable properties on steady laminar flow and heat
transfer for a viscous fluid due to an impulsively started rotating porous infinite disk. A theoretical
study of the effect of variable fluid properties on the classical Blasius and Sakiadis flow taking into
account the variation of the physical properties with temperature studied recently by recently
Asterios Pantokratoras [12]. More recently Maleque [13] presented the effects of Combined
Temperature- and Depth-Dependent Viscosity and Hall Current on an Unsteady MHD Laminar
Convective Flow Due to a Rotating Disk.

Very recently Maleque [14] investigated temperature dependent variable properties on
mixed convective unsteady MHD laminar incompressible fluid flow with heat transfer and viscous
dissipation.

Earlier in 1959 for the flow of viscoelastic fluid Casson[15] has first established the casson fluid
model. This model is cast off by fuel engineers in the description of adhesive slurry and is improved
for forecasting high shear-rate viscosities when only low and transitional shear-rate data are
accessible. Examples of Casson fluid include jelly, tomato sauce, honey, soup and concentrated fruit
juices, ect. Human blood can also be treated as Casson fluid. Many authors have published their
research papers on non Newtonian Casson fluid flow. Some of them Eldabe et al [16], Dash et. al
[17], Mostafa et al [18], Mukhopadhyay et al [19]. Nadeem et al [20] considered
magnetohydrodynamic Casson fluid flow in two lateral directions past a porous linear stretching
sheet. Recently Maleque [21] investigated the unsteady MHD non-Newtonain casson fluid flow due
to a porous rotating disk with a uniform angular velocity in the presence of an axial uniform
magnetic field and uniform electric field is examined. Very recently Maleque [22] investigated
exothermic-endothermic binary chemical reaction on unsteady non-Newtonian Casson fluid flow
with heat and mass transfer. The effect of the Casson parameter on the velocity profiles for cooling
and heating plate and the effects of chemical reaction rate and Arrhenius activation energy on the
concentration are also studied.

Considering the importance of MHD combined convective non-Newtonian casson fluid flow, in
present study we are to investigate the effects of temperature dependent suction/injection, variable
fluid properties (viscosity () and thermal conductivity ()), viscous dissipation and thermal
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radiation on an unsteady MHD laminar incompressible fluid flow for a vertical porous plate in the
presence of transverse magnetic field. The governing partial differential equations of the MHD
combined convective boundary layer flow are reduced to nonlinear ordinary differential equations
by similarity transformations. The dimensionless nonlinear differential equations are then solved
numerically by using Range-Kutta six order integration scheme and Nachtsheim-Swigert iteration
technique. The obtained numerical results are presented graphically in the form of velocity and
temperature profiles. Finally the effects of the relative temperature difference parameter ()on
the skin friction and heat transfer coefficients are also examined.

2. Governing Equations

Let us consider an unsteady combined forced and free convective MHD laminar boundary layer
y0
flow of an electrically conducting viscous fluid past an infinite vertical porous plate . The flow
is assumed to be moving in the upward direction with a constant velocity U0. We take x-axis along

the plate in upward direction and y-axis normal to it. A uniform magnetic field B parallel to the y-
axis is applied to the plate which is electrically non-conducting. The magnetic field is of the form
 
B(0,B ,0) . The equation of conservation of electric charge is J 0 , where
0

J (J ,J ,J )
x y z , the direction of propagation is considered only along the y-axis and does not
J
y
have any variation along the y-axis. The derivative of J with respect to y is namely 0,
y
y
which gives result J = constant. Since the plate is electrically non-conducting, therefore this
y
constant is zero and hence J = 0 everywhere in the flow. It is assumed that the plate is infinite in
y
extent and hence all the physical quantities depend on y and t.

()
We assume that the dependency of the fluid properties, viscosity coefficient and thermal
conductivity coefficient () are function of temperature alone and obey the following laws (see
Sattar and Maleque[10])

A C
  T    T 
  ,   , (1)
 T  T
     

 
where A and C are arbitrary exponents,  is the uniform viscosity of the fluid and  is a
uniform thermal conductivity of heat. For the present analysis the fluid is considered to be flue gas.
For flue gases the values of the exponents A, and C are taken as A0.7, and C0.83 (see
Jayaraj[23]).

Thus accordance with the above assumptions and Boussinesq’s approximation, the basic equations
relevant to the problem are
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v
0, (2)
y
u u 1 2u u
t vy(1)(y2 y y) g(TT)B02(U0u), (3)

T T 2T T  u2 q
cp t vyy2  y y y B02(U0u) yr (4)

The boundary conditions are
B 
 T 
u(0)0, v(0)v (t,T)  , 
 
w T   (5)


T(0)T ,u()U , T()T ,

w 0 

,
where u and v are velocity components along x-axis and y-axis respectively; the density of the
fluid; Bo, the applied constant magnetic field; g, the acceleration due to gravity; , the coefficient
of volume expansion for temperature; T, and T the temperature of the fluid inside the thermal
,
boundary layer and the fluid temperature in the free stream respectively; Tw,the plate
temperature; , the electrical conductivity; and v(t), the normal velocity at the plate having a

positive value for suction and negative value for injection and , the casson parameter.
q
The radiative heat flux r is described by Roseland approximation such that

4 T4
q  1
(6)
r 3k y
1
 k
where 1 and 1 are the Stefan Boltzmann constant and mean absorption coefficient
respectively. We assume that the temperature differences within the flow are sufficiently small so
n n
that the T can be expressed as a linear function. By using Taylor’s series, we expand T about
T
the free stream temperature  and neglecting higher order terms. This result of the following
approximation:

n n n1
T (1n)T nT T (7)

Using equation (7) we have

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2 4 3 2
q 4  T 16T   T
r  1   1 (8)
y 3k1 y2 3k1 y2
n
 T  T
and   (1n)n . (9)
T T
  

x

T
w C ,T

C  
w

B

y

Fig. 1. The flow configuration and the coordinate system.

3. Mathematical Formulations

In order to solve the governing equations (2)–(4) under the boundary conditions (5), we adopt the
well-defined similarity technique to obtain the similarity solutions. For this purpose the following
non-dimensional variables are now introduced:
y u
 ,  f(), TT T. (10)
0
(t) U
0
From the continuity equation (2) we have
B
v   T 
v 0  
  , (11)
(t) T
 
 y0
T
where  is the relative temperature difference parameter which is positive for a heated
T

plate, negative for a cooled plate and zero for uniform properties, v0 is the dimensionless
suction/injection velocity. v0 0 corresponds to suction and v0 0 corresponds to injection and
B B0 v
is a unit less constant exponent . represents at the wall is the linear/nonlinear
B0 v
function of temperature and corresponds respectively does not depends on temperature.

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Introducing the dimensionless quantities from equation (10) and v from equation (11) in
equations (3)—(4) we finally obtain the nonlinear ordinary differential equations as

1   
(1 )(fA(1)1 f) 1A  fG 1A
 
  r
 

 
M(1 f)v (1)B f 0
, (12)
0

C112Pr[E 1ACf2 ME 1C1 f2
c c

 
1C   v (1)B1C]/(1N)0
  (13)
 0
 


where,
2gT 2B2
G  (Grashof Number), M 0 (Magnetic parameter),
r
U  
0  
  c U2
Pr   p (Prandtl Number) E  0 (Eckert Number)
c
 Tc
 p
3
16T
1 
and N (Radiation parameter)
3k
1

The equations (12)—(13) are similar except for the term , where time t appears explicitly.



Thus the similar condition requires that must be a constant quantity. Hence following


Maleque [14] one can try a class of solutions of the equations (12)—(13) by assuming that

K
(constant) (14)


From equation (14) we have

 2K t L
 (15)
where the constant of integration L is determined through the condition that L when t 0.
Here K0 implies that L represents the length scale for steady flow and C0that is, 
represents the length scale for unsteady flow (Maleque [23]).

K
It thus appear from equation (14) that by making a realistic choice of to be equal to 2 in equation
(t) 2  t L
(14), the length scale becomes which exactly corresponds to the usual

scaling factor considered for viscous unsteady boundary layer flows (Schlichting [25]. Since  is a
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K
scale factor as well as similarity parameter, any value of in equation (15) would not change the
K2
nature of solutions except that the scale would be different. Finally introducing (Maleque
[26]) in equation (14) , we have

2 (16)


Using equation (12), the equations (8)—(9) yield

1
(1 )(fA(1)1 f)


M(1 f)v (1)B f(1)A 2fG1A 0
(17)
0 r

C112 Pr[Ec1AC f2 MPrEc1C1 f2
2Pr1C v Pr(1)B1C]/(1N)0, (18)
0

The corresponding boundary conditions are obtained from equation (5) as,

f(0)0, (0)1, f()1, ()0
. (19)


In the above equations prime denotes the ordinary differentiation with respect to .
The skin friction coefficient and the rate of heat transfer to the plate, which are of chief physical
interest, are also calculated out. The equation defining the wall shearing stress () is
1  u 1 1 1
(1 )   (1 )U2 1AR2 f(0)
 y 2  0 e
y0
 
Hence the skin friction C is
f
1 1
1AR2 C (1 )f(0) (20)
e f 
The rate of heat transfer (q) from the plate to the fluid is computed by the application of Fourier’s
law as given in the following
 T  T1C
q    (0).
 y 
y0

Hence the Nusselt number (Nu) is obtained as
1

R 21C Nu(0), (21)
e
1
U 
where Re2  0 is the square root of local Reynolds number.

  
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In equations (20)-(21), the gradient values of f and  at the plate are evaluated when the
corresponding differential equations are solved satisfying the convergence criteria.

4. Numerical Solutions

The set of coupled, nonlinear ordinary differential equations (17)-(18) are solved numerically using
a standard initial value solver called the shooting method. For this purpose Nachtsheim and Swigert
[27] iteration technique has been employed. Thus adopting this numerical technique, a computer
program was set up for the solutions of the basic non-linear differential equations of our problem
where the integration technique was adopted as a six ordered Range-Kutta method of integration.
Various groups of parameters Gr,M,,Ec,N,and v0 were considered in different phases.
0.01
In all computations the step size was selected that satisfied a convergence criterion of
106in almost all phases mentioned above. The value of  was found to each iteration loop by
 
setting  . Thus  max, to each group of parameters, has been determined when the
value of unknown boundary conditions at 0(that arise in the Shooting method) does not
change to successful loop with error less than 106.

5. Results and Discussion

As a result of the numerical calculation, the velocity and temperature distributions for the flow are
G ,M, ,
obtained from equations (17)-(18) and are displayed in Figs.2-10 for different values r
 
Ec, v0, , N and B. In the present analysis the fluid considered is flue gas Pr0.64 .

G
Fig.2 presents the effects of r(Grashof number) on the velocity and temperature profiles. It
G 0
appears from Fig.2 that when r that is for forced convection the temperature profile shows its
(G )
usual trend of gradual decay. As Grashof number r becomes larger the profiles overshoot the
uniform temperature close to the boundary. Fig.2 also shows that the velocity profiles increase as
G
r increases.

 Gr0,M0.5,
The effects of on the velocity and temperature profiles are shown in Fig.3 for
E 0.5,v 0.5,N0.5,0.5, B1
. In this figure comparison is made between constant
c o

property and variable property solutions. In Fig.3, it is observed that an increase values of leads
to the decrease in the values of the velocity and the increase in the values of temperature. It is also
observed that the velocity profile increases and temperature profile decreases for negatively

increasing values of relative temperature different parameter .

Imposition of a magnetic field to an electrically conducting fluid creates a drag like force called
Lorentz force. The force has the tendency to slow down the flow around the plate at the expense of
increasing its temperature. This is depicted by decreases in velocity profiles and increases in the
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temperature profiles as M increases as shown in Fig.4. In addition, the increases in the temperature
profiles as M increases are accompanied by increases in the thermal boundary layer.
1.5
1

Gr=10
Gr=1
0.5
Gr=0
f
0
0 1 2 3


G
Fig. 2. Effects of r on velocity and temperature profiles.
1


 
0.5


f

0
0  1 2 3


Fig. 3. Effects of on velocity and temperature profiles for Gr = 0.
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1.5
f
1

M=0
M=2
0.5 M=5
M=5
M=0
0
0  1 2 3

M
Fig. 4. Effects of on velocity and temperature profiles.

1.5
f
1
 Ec=1.0
Ec=0.5
0.5 Ec=0.0
0
0  1 2 3

Fig. 5. Effects of Ec on velocity and temperature profiles

113