International Journal of Mathematical Archive-6(2), 2015, 73-78
Available online through www.ijma.info ISSN 2229 – 5046
MHD BOUNDARY LAYER FLOW AND HEAT TRANSFER
OF AN UPPER-CONVECTED MAXWELL FLUID
Dr. Anuj Kumar Jhankal1 and Manoj Kumar*2
1Army
2Arya
cadet College wing, Indian Military Academy, Dehradun, India.
Institute of Engineering Technology & Mgt., Omaxe City, Jaipur-302026, India.
(Received On: 09-02-15; Revised & Accepted On: 28-02-15)
ABSTRACT
An analysis is made to study boundary layer flow and heat transfer of an upper-convected Maxwell fluid in presence of
transverse magnetic field. The upper-convected Maxwell model is used to characterize the non-Newtonian fluid
behavior. Using similarity transformation, the governing boundary layer equations are transformed into self-similar
nonlinear ordinary differential equations, which are then solved numerically using a very efficient RKF45 (RungeKutta-Fehlberg forth-fifth) method. The effects of various parameters like magnetic parameter, Deborah number and
Prandtl number on the velocity and temperature profiles as well as on the local Skin-friction coefficient and the local
Nusselt number are presented and discussed through graphs and tables.
Mathematics Subject Classification [MSC]: 76A05, 76D10, 76M20, 76W05, 80A20.
Keywords: MHD, boundary layer flow, heat transfer, upper-convected Maxwell fluid, numerical study.
NOMENCLATURE
B0
Cp
f
M
Pr
T
U
u, v
x, y
Constant applied magnetic field
Specific heat at constant pressure
Dimensionless stream function
Magnetic field parameter
Prandtl number
Temperature of the fluid
Free stream velocity
Velocity component of the fluid along the x and y directions, respectively
Cartesian coordinates along the surface and normal to it, respectively
Greek symbols
β Deborah number
η Dimensionless similarity variable
ρ Density of the fluid
μ Viscosity of the fluid
ν Kinematic viscosity
Ψ Stream function
λ Retardation time
σe Electrical conductivity
κ Thermal conductivity
θ Dimensionless temperature
Superscript
′
Derivative with respect to η
Subscripts
w Properties at the plane
∞ Free stream condition
1Army
Corresponding Author: Manoj Kumar*2
cadet College wing, Indian Military Academy, Dehradun, India.
International Journal of Mathematical Archive- 6(2), Feb. – 2015
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Anuj Kumar Jhankal and Manoj Kumar* /
MHD Boundary Layer Flow and Heat Transfer of an Upper-Convected Maxwell Fluid / IJMA- 6(2), Feb.-2015.
INTRODUCTION
The subject of MHD is largely perceived to have been initiated by Swedish electrical engineer Hannes Alfv𝑒𝑒́ n [1] in
1942. If an electrically conducting fluid is placed in a constant magnetic field, the motion of the fluid induces currents
which create forces on the fluid. The production of these currents has led to the design of among other devices the
MHD generators for electricity production. The equations which describe MHD flow are a combination of continuity
equation and Navier-Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. The governing
equations are differential equations that have to be solved either analytically or numerically.
In recent years, the studies of boundary layer flows of non-Newtonian fluid have received considerable attention due to
their numerous industrial and engineering applications. The non-Newtonian fluids in view of their diverse rheological
properties cannot be examined through one constitutive relationship between shear stress and rate of strain. Many
models of non-Newtonian fluids exist. Maxwell model has become the most popular as it can predict stress relaxation
and also excludes the complicating effects of shear dependent viscosity Mukhopadhyay and Gorla [7] (2012). In view
of its simplicity, this fluid model has acquired special status amongst model has acquired special status amongst the
recent workers in the field. This kind of fluid has significant application in viscoelastic problems which have small
dimensionless relaxation time. Sarpakaya [12] (1961) was the first researcher to study the MHD flow a of nonNewtonian fluid. Prandtl’s boundary layer theory proved to be of great use in Newtonian fluids as Navier-Stokes
equations can be converted into much simplified boundary layer equation which is easier to handle. Wang and Tan [14]
(2011) discussed the flow of Maxwell fluid in a porous medium. Zierep and Fetecau [16] (2007) studied RayleighStokes problem using Maxwell fluid and find exact solution. Sadeghy et al.[10] (2006) did a comparative study for
Sakiadis flow of an upper-convected Maxwell fluid on a rigid plate. In view of the Maxwell model, several researchers
have analyzed the MHD flow of a Maxwell fluid under various aspects of rotation, thermal radiation, heat and mass
transfer, chemical reaction, suction/injection, thermophoresis and heat source/sink (Bataller [4] (2011); Zheng et al.[15]
(2013); Noor [8] 2012; Vajravelu [13] (2012); Eosboee et al.[5] (2010)).
In this study we aim at investigating boundary layer flow and heat transfer of an upper-convected Maxwell fluid in
presence of transverse magnetic field. The governing partial differential equation is transformed into ordinary
differential equation by means of similarity transformations. This equation is solved numerically by RKF45 (RungeKutta-Fehlberg forth-fifth) method using symbolic algebra software Maple. The effects of various parameters are
presented with the aid of graphs & tables and discussed.
FORMULATION OF THE PROBLEM
Let us consider the steady and incompressible MHD boundary layer flow an electrically conducting fluid obeying
UCM model. The x- and y-axes are taken along and perpendicular to the flow, respectively. A uniform magnetic field
of strength B0 is imposed along the y-axis. It is assumed that external field is zero, the electric field owing to
polarization of charges and Hall effect are neglected. Under the usual boundary layer approximations, the governing
equation of continuity, momentum and energy [Pai [9] (1956), Schlichting [11] (1964), Bansal [2] (1977)] under the
influence of externally imposed transverse magnetic field [Jeffery [7] (1966), Bansal [3] (1994)] are:
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
+ =0
(1)
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝑢𝑢
𝜕𝜕𝜕𝜕
+ 𝑣𝑣
𝜌𝜌𝐶𝐶𝑝𝑝 �𝑢𝑢
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
+ 𝜆𝜆 �𝑢𝑢2
+ 𝑣𝑣
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕 2 𝑢𝑢
+ 2𝑢𝑢𝑢𝑢
𝜕𝜕𝑥𝑥 2
𝜕𝜕 2 𝑇𝑇
� = 𝜅𝜅
𝜕𝜕𝑦𝑦 2
𝜕𝜕 2 𝑢𝑢
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
+ 𝑣𝑣 2
𝜕𝜕 2 𝑢𝑢
𝜕𝜕𝑦𝑦 2
� = 𝜈𝜈
𝜕𝜕 2 𝑢𝑢
𝜕𝜕𝑥𝑥 2
−
𝜎𝜎𝑒𝑒 𝐵𝐵02
𝜌𝜌
𝑢𝑢
Along with the boundary conditions for the problem are given by
𝑢𝑢(𝑥𝑥, 0) = 𝑈𝑈, 𝑣𝑣(𝑥𝑥, 0) = 0, 𝑇𝑇(𝑥𝑥, 0) = 𝑇𝑇𝑤𝑤 (𝑥𝑥),
𝑢𝑢(𝑥𝑥, ∞) → 0, 𝑇𝑇(𝑥𝑥, ∞) → 𝑇𝑇∞
The continuity equation (1) is satisfied by introducing a stream function Ψ such that u =
(2)
(3)
(4)
∂Ψ
∂y
and v = −
∂Ψ
∂x
The momentum equation can be transformed into the corresponding ordinary nonlinear differential equation by the
following transformation:
𝑈𝑈
𝜂𝜂 = 𝑦𝑦� ,
𝜈𝜈𝜈𝜈
,
𝜃𝜃(𝜂𝜂) =
𝑇𝑇−𝑇𝑇∞
𝛹𝛹 = √𝜈𝜈𝜈𝜈𝜈𝜈𝑓𝑓(𝜂𝜂)
𝑇𝑇𝑤𝑤 −𝑇𝑇∞
© 2015, IJMA. All Rights Reserved
(5)
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Anuj Kumar Jhankal and Manoj Kumar* /
MHD Boundary Layer Flow and Heat Transfer of an Upper-Convected Maxwell Fluid / IJMA- 6(2), Feb.-2015.
Where η is the independent similarity variable. The transformed nonlinear ordinary equation is
2𝑓𝑓 ′′′ − 𝛽𝛽�2𝑓𝑓𝑓𝑓 ′ 𝑓𝑓 ′′ + 𝑓𝑓 2 𝑓𝑓 ′′′ + 𝜂𝜂𝑓𝑓′2 𝑓𝑓′′� + 𝑓𝑓𝑓𝑓 ′′ − 2𝑀𝑀𝑓𝑓 ′ = 0
𝜃𝜃 ′′ − 𝑃𝑃𝑃𝑃(2𝑓𝑓 ′ 𝜃𝜃 − 𝜃𝜃 ′ 𝑓𝑓) = 0
The transformed boundary conditions are:
𝜂𝜂 = 0: 𝑓𝑓 = 0, 𝑓𝑓 ′ = 1, 𝜃𝜃 = 1
𝜂𝜂 → ∞: 𝑓𝑓 ′ = 0, 𝜃𝜃 = 0
Where prime denotes differentiation with respect to 𝜂𝜂, 𝑀𝑀 =
number and 𝑃𝑃𝑃𝑃 =
𝜇𝜇 𝐶𝐶𝑝𝑝
𝜅𝜅
𝜎𝜎𝑒𝑒 𝐵𝐵02 𝑥𝑥
is the Prandtl number.
𝜌𝜌
𝑈𝑈
(6)
(7)
(8)
is the magnetic parameter, 𝛽𝛽 =
𝜆𝜆𝑈𝑈
2𝑥𝑥
is the Deborah
RESULTS AND DISCUSSIONS
The system of governing equations (6)-(7) together with the boundary condition (8) is non-linear ordinary differential
equations depending on the various values of magnetic parameter M, Deborah number β and Prandtl number Pr. The
non-linear differential equations (6) and (7) subject to the boundary conditions (8) are solved numerically using RungeKutta-Fehlberg Forth-Fifth order method. To solve this equation we adopted symbolic algebra software Maple. Maple
uses the well known Runge-Kutta-Feulberg Forth-Fifth (RKF45) order method to generate the numerical solution of
boundary value problem.
In order to verify the accuracy of our present method, we have compared our results with HPM and BVP Eosboee et al.
[5]. Table 1 compares the values of velocity component 𝑓𝑓(𝜂𝜂) for M=1.0 and β=0.4. The comparisons are found to be in
excellent agreement.
Table-1: Numerical values of f(η) compared with the result obtained by Eosboee et al. [5], when M=1.0 and β=0.4.
Eosboee et al. [5] 𝐟𝐟(𝛈𝛈)
η
Present paper results 𝐟𝐟(𝛈𝛈)
HPM
BVP
0
0
0
0
0.5 0.38662
0.386325
0.386128
1.0 0.60613
0.605299
0.604469
1.5 0.72485
0.724257
0.722237
2.0 0.78562
0.786745
0.782737
2.5 0.81461
0.818662
0.811468
3.0 0.82698
0.834332
0.822121
3.5 0.83119
0.841253
0.832975
4.0 0.83192
0.843124
0.843124
Table 2 represents the Skin friction parameter against magnetic parameter M for various values of Deborah number β.
It is noted that the effect of increasing M is to decrease the skin friction at the surface. The positive values of f ′′ (0)
imply that fluid exerts the drag force on the surface, and the negative values f ′′ (0) imply the opposite meaning. Thus as
M becomes very large, there will be a decrease in skin friction f ′′ (0), and then a decrease in the drag force at the
surface. Results also shows that an increase in the Deborah number β leads to increases the skin friction f ′′ (0) and the
drag force at the surface.
Table-2: Numerical values of Skin friction coefficient
M
0
0.5
1
1.5
2
f''(0)
β=0
-0.49664
-0.83234
-1.08534
-1.29338
-1.4733
β=0.1
-0.48124
-0.82348
-1.07869
-1.28783
-1.46844
β=0.2
-0.46522
-0.81449
-1.07198
-1.28223
-1.46354
β=0.3
-0.44854
-0.80536
-1.06521
-1.27660
-1.45860
β=0.4
-0.43117
-0.79608
-1.05836
-1.27092
-1.45364
β=0.5
-0.41316
-0.78665
-1.05144
-1.26520
-1.44864
β=0.6
-0.39465
-0.77706
-1.04445
-1.2594
-1.44361
β=0.7
-0.37598
-0.76731
-1.03738
-1.25362
-1.43855
Table 3 represents the Nusselt number against magnetic parameter M for various values of Deborah number β. It is
noted that the rate of heat transfer increases with magnetic parameter M. While, the rate of heat transfer decreases with
Deborah number β.
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Anuj Kumar Jhankal and Manoj Kumar* /
MHD Boundary Layer Flow and Heat Transfer of an Upper-Convected Maxwell Fluid / IJMA- 6(2), Feb.-2015.
M
Table-3: Numerical values of Nusselt number when Pr = 1.0
−𝛉𝛉′ (𝟎𝟎)
β=0.1
β=0.2
β=0.3
β=0.4
β=0.5
β=0.6
-1.45406 -1.45818 -1.46249 -1.46699 -1.47166 -1.47642
-1.37189 -1.37384 -1.37582 -1.37784 -1.37989 -1.38198
-1.30973 -1.31089 -1.31206 -1.31325 -1.31445 -1.31567
β=0.7
-1.48113
-1.38411
-1.3169
0
0.5
1
β=0
-1.45012
-1.36997
-1.30857
1.5
-1.25892
-1.2597
-1.26048
-1.26127
-1.26207
-1.26287
-1.26368
-1.26449
2
-1.21729
-1.21786
-1.21842
-1.21899
-1.21956
-1.26287
-1.22072
-1.2213
Figures 1 and 2 show the effect of magnetic parameter M for the given values of Deborah number β. An increase in the
M leads in decrease of both the velocity components f(η) and f ′ (η). Accordingly, the thickness of the momentum
boundary layer decreases. This happen due to the Lorentz force arising from the interaction of magnetic and electric
fields during the motion of the electrically conducting fluid. To reduce momentum boundary layer thickness the
generated Lorentz force enhances the fluid motion in the boundary layer region.
Figure 1: Effects of magnetic parameter M on
f(η), when β=0.5.
Figure 2: Effects of magnetic parameter M on f ′ (η),
when β=0.5
Figures 3 and 4 show the effect of Deborah number β on velocity profiles for the given values of magnetic parameter
M. An increase in Deborah number β is seen to decrease both velocity components f(η) and f ′ (η) at a given point and
thereby decreases the thickness of the boundary layer with an increase in β.
Figure 3: Effects of Deborah number β on f(η), when
M=2.5.
Figure 4: Effects of Deborah number β on f ′ (η),
when M=2.5.
Figures 5 and 6 illustrate the temperature profile for various values of magnetic parameter M and Deborah number β
respectively. We infer from this figure that the temperature increase with an increase in the magnetic parameter M and
Deborah number β.
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Anuj Kumar Jhankal and Manoj Kumar* /
MHD Boundary Layer Flow and Heat Transfer of an Upper-Convected Maxwell Fluid / IJMA- 6(2), Feb.-2015.
Figure 5: Effect of magnetic parameter M on the
temperature profiles, when β=0.5 and Pr=1.0
Figure 6: Effects of Deborah number β on the
temperature profiles, when M=2.5 and Pr=1.0
Figure 7 is plotted for the temperature profiles for different values of Prandtl number (Pr). An increase in the Prandtl
number is seen to decrease the fluid temperature θ(η) above the surface. That is not surprising realizing the fact that the
thermal boundary becomes thinner for larger the Prandtl number. Therefore, with an increase in Prandtl number the rate
of thermal diffusion drops. The reason is that smaller values of Prandtl number are equivalent to increasing the thermal
conductivities, and therefore heat is able to diffuse away from the heated surface more rapidly than for higher values of
Prandtl number.
Figure 7: Temperature profiles for
different values of Prandtl number (Pr),
when M=0 and β=0.
CONCLUSION
In the present study, we have theoretically studied the two-dimensional boundary layer flow and heat transfer of an
upper-convected Maxwell fluid in the presence of transverse magnetic field. The governing partial differential
equations are transformed into ordinary differential equations by means of similarity transformations. The resulting
non-linear ordinary differential equations are solved numerically by RKF45 (Runge-Kutta-Fehlberg forth-fifth)
method. The velocity and temperature profiles are discussed numerically and presented through graphs. The numerical
values of Skin-friction coefficient and Nusselt number are derived, discussed numerically and presented through tables.
From the study, following conclusions can be drawn:
1. The study reveals that due to increase of magnetic parameter M, the momentum boundary layer thickness
reduces and the temperature at a point increases. Thus we conclude that we can control the velocity field and
temperature by introducing magnetic field.
2. The effect of Deborah number β is to decreases the momentum boundary layer thickness and the thermal
boundary layer thickness.
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Anuj Kumar Jhankal and Manoj Kumar* /
MHD Boundary Layer Flow and Heat Transfer of an Upper-Convected Maxwell Fluid / IJMA- 6(2), Feb.-2015.
3.
4.
5.
The temperature as well as the thermal boundary layer thickness decrease with increasing values of Prandtl
number.
The skin friction parameter decrease with magnetic parameter M but increases with the increasing values of
Deborah number β.
The heat transfer is enhanced for increase of magnetic parameter M, this heat transfer is very important in
production engineering to improve the quality of the final product.
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Source of support: Nil, Conflict of interest: None Declared
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