Effect of viscous dissipation on mhd flow of a free convection power law fluid with a pressure

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME
302
EFFECT OF VISCOUS DISSIPATION ON MHD FLOW OF A FREE
CONVECTION POWER-LAW FLUID WITH A PRESSURE GRADIENT
1
M N Raja Shekar
Department of Mathematics,
JNTUH College of Engineering,Nachupally, Karimnagar,
2
Shaik Magbul Hussain
Professor,Dept. of Mechanical engineering.
Royal Institute of technology and Science.Chevella.India.
ABSTRACT
The paper deals with the study of steady two-dimensional flow of a electrically conducting
power-law fluid past a flat plate in the presence of transverse magnetic field under the influence of a
pressure gradient by considering viscous dissipation effects is studied. The resulting governing partial
differential equations are transformed into set of non linear ordinary differential equations using
appropriate transformation. The set of non linear ordinary differential equations are first linearized by
using Quasi-linearization technique and then solved numerically by using implicit finite difference
scheme. The system of algebraic equations is solved by using Gauss-Seidal iterative method. The
energy equation for a special case for which similarity solution exist is also considered. The special
interest is the effects of the power-law index, magnetic parameter, viscous dissipation and generalized
prandtl number on the velocity and temperature profiles. Numerical results are tabulated for skin
friction co-efficient. Velocity and Temperature profiles are drawn for different controlling parameters
which reveal the tendency of the solution.
Key words: Non-Newtonian fluids, Magnetic field effects, Prandtl number, Quasi-linearization, finite
difference method and viscous dissipation.
INTRODUCTION
A non-Newtonian fluid is a fluid in which the viscosity changes with applied strain rate. As a
result non-Newtonian fluids may not have well defined viscosity. In modern technology and in
industrial applications, non-Newtonian fluids play an important role. Many processes in modern
technology use non-Newtonian fluids as working fluids in heat exchangers. Heat transfer
characteristics of these fluids have been studied widely during the past decades due to the growing use
of these non-Newtonian substances in various manufacturing and processing industries. In the recent
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN
ENGINEERING AND TECHNOLOGY (IJARET)
ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 4, Issue 3, April 2013, pp. 302-307
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303
years, the non-Newtonian fluids find an increasing applications in industries such as the flow of
nuclear fuel slurries, liquid metal and alloys, plasma and mercury, lubrication with heavy oils and
greases, coating of papers, polymer extrusion, continuous stretching of plastic films and artificial
fibres and many others. During the past three decades there have been extensive research works on
various aspects of non-Newtonian power-law fluids over bodies of different shapes which are
documented in books by Skelland [1], Irvine and Karni [2] and others have presented an excellent
review of non-Newtonian fluids. Many flow problems, both internal and external flows, have been
investigated. The important concept of boundary layer was applied to power-law fluids by Schowalter
[3].
Similarity solutions were obtained by Kapur and Srivastava [4], Lee and Ames [5], Acrivos
et.al [6], Berkoveskii [7], Hansen and Na [8] and others. Rao [9] investigated momentum and heat
transfer phenomena on a continuous moving surface in power-law fluid. Pop et.al [10] considered the
steady laminar forced convection boundary layer of power-law non-Newtonian fluids on a continually
moving cylinder with the surface maintained at a uniform temperature or uniform heat flux. An
analysis of steady laminar forced convection heat transfer from a moving or stationary slender
cylinder to a quiescent or flowing non-Newtonian fluid is presented by Tian-Yih Wang [11]. Agarwal
et.al [12] presented the laminar momentum and thermal boundary layers of power-law fluids over a
slender cylinder.Thomson and Snuder [13-14] studied the effect of injection on the flow of power-law
fluid over a flat plate. Liu [15] presented a class of asymptotic solutions for the flow of power-law
fluids over a flat plate with suction.
In recent years, the non-Newtonian fluids in the presence of a magnetic field find increasing
applications in many areas such as chemical engineering, electromagnetic propulsion, nuclear reactors
etc. Sarpakaya [16] has given many possible applications of non-newtonian fluids in various fields.
The flow of non-Newtonian power-law fluids in the presence of a magnetic field over two-
dimensional bodies was investigated by Sarpakaya[16], Sapunkov [17], Vujanovic et.al [18] and
Djukic [19], [20]. Anderson et.al [21] have studied the MHD flow of a non-Newtonian power-law
fluid over a stretching sheet in an ambient fluid. The effect of MHD heat transfer to non-Newtonian
power-law fluids flowing over a wedge in the presence of magnetic field taking into consideration
viscous dissipation is studied by Kishan and Amrutha [22].
Recently the steady two-dimensional incompressible flow of a conducting power-law fluid past a flat
plate in the presence of a transverse magnetic field and under the influence of a pressure gradient was
studied by T. C. Chiam [23].
In this paper we have investigated the effect of viscous dissipation on the steady two-
dimensional flow of a conducting power-law fluid past a flat plate in the presence of transverse
magnetic field under the influence of a pressure gradient. The similarity transformations were applied
to partial differential equations governing the flow and heat transfer under boundary layer
approximations to transform the non-Newtonian two-dimensional steady boundary layer equations
into non-linear ordinary differential equations system. Numerical solutions of out coming non-linear
ordinary differential equations are found by using an implicit finite difference method.
FORMULATION OF THE PROBLEM:
As in Sapunkov[17], we consider a steady two-dimensional incompressible conductive
power-law fluid flow past a semi-infinite flat plate under the influence of a pressure gradient and in
the presence of a transverse magnetic field with magnetic field intensity H. The continuity and
momentum equations are
,0=
∂
∂
+
∂
∂
y
v
x
u
----- (1)
,)( 2
21
Huu
y
u
y
u
y
k
dx
du
u
y
u
v
x
u
u
n
−+








∂
∂
∂
∂
∂
∂
+=
∂
∂
+
∂
∂
∞
−
∞
∞
ρ
σµ
ρ
----- (2)

304
where x and y are cartesian coordinates along and normal to the body surface respectively, and u, v
are the corresponding velocity components, k is the fluid consistency index, n the flow behavior
index, ρ the fluid density, µ is the magnetic permeability and σ is the electrical conductivity of the
fluid. The external velocity distribution u∞ is given by
u∞= cxm
, ----- (3)
Where c and m are constants. The magnetic Reynolds number is assumed small.
Sapunkov[17] has shown that similarity solutions exist if H = H0 x(m-1)/2
.
METHOD OF SOLUTION
Solution for Momentum equation:
We shall transform equation (2) into a ordinary differential equation amenable to a numerical
solution by introducing a similarity variable η and a stream function ψ as follows:
,2
1
α
αη yx= And )()1(
1
η
α
ψ fx
c nr +
= ----- (4)
Where r = (2n-1)m+1,
1
1
2
1
)1(
1)12( +−






+
+−
=
nn
nn
nm
k
cρ
α ,
1
1)2(
2
+
−−
=
n
nm
α
Where the dimensionless stream function f satisfies the continuity equation with
y
u
∂
∂
=
ψ
and
x
v
∂
∂
−=
ψ
.
Under the transformation (4) the differential equations (2) is reduce to
0)1()1( 21
=′−+′−+′′+′′′′′
−
fMfffff
n
β ----- (5)
Where β=m(1+n)/r represents the flow behavior and
M = σµ2
H0
2
(1+n)/cρr is the magnetic field parameter.
Subject to the boundary conditions
1)(0)0(,0)0( =∞′=′= fandff . ----- (6)
Equation (5) has been solved numerically. As this system of equation is highly non-linear. We have
applied Quasi-Linearization technique to linearize this system. This method converts the non-linear
two-point boundary value problem into an iterative scheme of solution. This method is discussed in
detail by Bellman and Kalaba [24]. This technique has been used successfully by my authors for the
solution of the Falkner-Skan type equations. Applying this technique to equation (5) we obtain
0)1())(21(][}{ 2111
=′−+′+′′−+′′−′′+′′+′′′′′−′′′′′+′′′′′
−−−
fMFfFFFFffFFFFffF
nnn
β
----- (7)
Where F is assumed to be a known function and the above equation can be rewritten as

305
1
254310 ][][][][][][
−
′′−=+′+′′+′′′
n
fiAiAfiAfiAfiAfiA ----- (8)
Where
1
0 ][
−
′′=
n
FiA , FiA =][1 , FiA ′′′=][2 ,
,2][3 MFiA −′−= β FiA ′′=][4 , MFFFFFiA n
−′−−′′+′′′′′= − 21
5 )(][][ ββ ,
Using implicit finite difference formulae, the equations (8)is transformed to
C0 [i] f [i+2] + C1[i] f [i+1] + C2[i] f [i] + C3 [i] f [i-1] = C4[i] -----(9)
Where C0[i] = 2A0[i], C1[i] = -6A0[i]+2hA1[i]+h2
A3[i],
C2[i] = -6A0[i]-4hA1[i]+2h3
A3[i], C3[i] = -2A0[i]+2hA1[i]-h2
A3[i],
And ( )1
25
3
4 ][][2][
−
′′−=
n
FiAiAhiC
Here ‘h’ represents the mesh size in η direction. The transformed equation (9) is solved
under the boundary conditions (6) by Gauss-Seidel iteration method and computations were carried
out by using C programming. The numerical solutions of ƒ are considered as (n+1)th
order iterative
solutions and F are the nth
order iterative solutions. After each cycle of iteration the convergence
check is performed, and the process is terminated when 6
10fF −
<− .
Solution of the energy equation:
By considering the viscous dissipation effects into account, the energy equation takes the
simplified form as :
,
1
2
2 +
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
n
y
u
y
T
K
y
T
v
x
T
u µ ----- (10)
Using
∞−
−
=
TT
TT
w
w
θ
This equation can be transformed into the form
,
1
2
2 +
∞ ∂
∂
−
+
∂
∂
=
∂
∂
+
∂
∂
n
w y
u
TTy
K
y
v
x
u
µθθθ
----- (11)
Where T∞ is the uniform temperature of the free stream and Tw is the temperature at the wall. We
assume that the wall is isothermal (i.e Tw = constant). Lee and Ames [5] have shown that equation
(11) possesses similarity solutions only for β = 0.5, which is the flow past a right-angled wedge when
there is no imposed magnetic field. In this case, energy equation reduces to the ordinary differential
equation:
0
2
3
2
3
Pr
1 11
1
1
1
=′′





+′





+′′
++
−
+
−
nn
n
n
n
fEcfθθ -----(12)
Where the Prandtl number and Eckert are defined as
)!(
)1(3
1
2
1
Pr +
−
+






= n
n
n
c
nk
K ρ
And
)TT(kn
xc
Ec
w
322
∞−
ρµ
=
And f is the solution of equation (05).The boundary conditions for equation (12) are
.0)(1)0( =∞= θθ and ----- (13)

306
Now, equation (12) can be expressed in the simplified form as
0][][][ 210 =+′+′′ iBiBiB θθ ----- (14)
Where
Pr
1
][0 =iB , fiB
n
n
+
−






=
1
1
1
2
3
][ and
11
1
2
2
3
][
++
−
′′





=
nn
n
fEciB
To solve the equation (10), apply the implicit finite difference scheme to the transformed equation
(14) for obtaining
0][]1[][][][]1[][ 3210 =+−+++ iDiiDiiDiiD θθθ -----(15)
Where D0[i] = 2B0[i] + hB1[i], D1[i] = -4B0[i],
D2[i] = 2B0[i] – hB1[i] and D3[i] = 2h2
B2[i]
Here ‘h’ represents the mesh size in η direction. Equation (14) is solved under the boundary
conditions (13) by Gauss-Seidel iteration method and computations were carried out by using C
programming.
RESULTS AND DISCUSSIONS
The parametric study is performed to explore the effect of magnetic field parameter M on the
velocity profiles for different values of the flow behavior parameter β for both the pseudo plastic and
dilatant fluids. And the effect of prandtl number Pr and Eckert number Ec on the temperature profiles
for various values of magnetic field parameter M is studied.
The values of the skin friction coefficient )0(f ′′ for various values of magnetic field
parameter M and flow behavior parameter β are tabulated in tables 1 and 2 for pseudo plastic and
dilatant fluids respectively. It noticed that the skin friction coefficient )0(f ′′ increases with the
increase in the magnetic field parameter M for fixed value of β. For a constant magnetic field
parameter M the skin friction coefficient )0(f ′′ increases with the increase of β for both pseudo
plastic (n=0.5) and dilatant fluids (n=1.5). It can also be noticed that the skin friction coefficient
)0(f ′′ increases as the power-law index n increases for a constant β value when there is no magnetic
field, while in the presence of a magnetic field it decreases with the increase in power-law index n for
a constant β value.
The effect of magnetic field parameter M on the velocity profiles of pseudo plastic fluids
(n=0.5) for β=0 (flat plate flow), β=0.5 and β=1 (stagnation point flow) are shown in figure 1. It is
evident from these figures that the velocity profiles f ′increases with the increases with the increase of
magnetic field parameter M. For dilatant fluids (n=1.5) the effect of magnetic field parameter M
accelerates the velocity profiles f ′in all cases i.e (a) for flat plate flow (β=0.0), (b) for flow with
β=0.5 and (c) for stagnation point flow (β=1.0) which is shown in figure 2.
The set of temperature profiles are presented in the figures 3 and 4. It can be seen from figure
3 that for a given n and β, the magnetic field decreases the thickness of thermal boundary layer for
different values of Pr. The effect of magnetic field parameter M on the temperature profiles is more in
case of pseudo plastic fluid (n=0.5) than that of dilatant fluid (n=1.5). It is also noticed that the
temperature profiles decreases with the increase of Prandtl number Pr.
The effect of viscous dissipation on the temperature profiles is shown in the figure 4. The viscous
dissipation effects is to accelerates the temperature profiles rapidly for flow with β=0.5 in both the
cases of pseudo plastic (n=0.5) and dilatant fluids (n=1.5). The viscous dissipation effect is more in
case of dilatant fluid when compared to pseudo plastic fluids.

307
REFERENCES
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