Convective conditions on magnetohydrodynamic flow over stretched cylinder with time and space dependent heat source or sink
B. Madhusudhana Rao1*, V. Nagendramma2, C.S.K. Raju3, A. Leelaratnam4, P. Prakash5
1Lecturer in Mathematics, Higher College of Technology, Muscat, Oman-105
2,4Department of Mathematics, S.P.M.V.V, Tirupati, A.P.
3Department of Mathematics, GITAM University, Bangalore Campus, K.A.
5Research Scholar, Dept. of Mechanical Engineering, NIT Warangal, Warangal (Telangana), India
ABSTRACT:
The present study emphases steady boundary layer flow and heat
transfer of a hyperbolic tangent fluid flowing over a vertical exponentially
stretching cylinder in its axial directionwith non-uniform heat source/sink. Proposed
mathematical model has a tendency to characterize the effect of the non-uniform
heat source/sink. The non-linear ordinary differential equations are solved
using the Runge-Kutta Feldberg (RKF) integration method. The characteristics of
velocity and temperature boundary layers in the presence of Weissennberg number are presented for different physical
parameters such as heat source/ sink parameter, Reynolds number
, the Prandtl number
, the Weissennberg number
and the natural convection parameter
, magnetic field parameter and porosity
parameter
. Moreover,
the friction factor coefficients, Nusselt number are also estimated and
discussed for aforesaid physical parameters. In addition, the rate of heat
transfer rate is higher in case of
compared
to
with
.
Key words: Weissennberg number, stretching cylinder, non-uniform heat source/sink, non-Newtonian fluid.
INTRODUCTION:
Non-Newtonian fluids are imperative in boundary layer flows because of their engineering and technology related applications. Paints, ketchup, polymeric liquids, apple sauce, jellies, tomato sauce, glues, soaps, blood, cosmetic products are examples of non-Newtonian fluids. As there are diversity of non-Newtonian fluids so it is reasonably complicated to form an equation expressing the viscous and elastic properties of these fluids. In comparison to viscous fluids, mathematical modeling of non-Newtonian fluids is considerably complex and challenging.
The Navier-Stokes expressions are proved inadequate for examining rheological characteristics of non-Newtonian materials. Generally the differential, integral and rate types classification have been made in this direction. Further, the phenomenon of stratification is very important in many engineering and manufacturing processes involving non-Newtonian materials. It arises in the flow fields due to change in concentration differences and temperature or fluids with different densities. Double stratification occurs through simultaneous effects of heat and mass transfer. The relevant examples include stratification of lakes and oceans, salinity stratification in rivers, ground water reservoirs, industrial, heterogeneous mixtures in atmosphere, manufacturing food processes and several others. Density differences through gravity are useful for dynamics and involvement of heterogeneous fluid. For instance in lakes, thermal stratification diminishes the blending of oxygen to the base water to develop anoxic from the effort of biological methods. The biomedical aspects are pathology, ophthalmological, public health and nursing etc. The Stratification has a key role in ponds and lakes since it influences the difference between concentration and temperature of oxygen and hydrogen.
Many researchers are contributed on these fluids highlighting the different aspects Ahmad et al. [1] and Ramzan et al. [2]. Crane [3] is the first to introduce the boundary layer flow of viscous fluid through a stretching sheet. Further, Cortell [4] proposed the effects of suction or blowing and heat generation or absorption through porous a stretching surface. Recently, various aspects of such problem have been investigated by many authors such as [5-9]. Wang [10] proposed non-Newtonian fluids for mixed convection heat transfer from a vertical plate. Xu et al. [11] reported the series solutions of unsteady boundary layer flows of non-Newtonian fluids near a forward stagnation point. An important branch of the non-Newtonian fluid models is the hyperbolic tangent fluid model. The hyperbolic tangent fluid is used extensively for different laboratory experiments. Friedman et al. [12] have used the hyperbolic tangent fluid model for large-scale magneto-rheological fluid damper coils. Nadeem and Akram [13] studied the peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel. In another paper Nadeem and Akram [14] have presented the effects of partial slip on the peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel. Only few researchers [15-19] have worked on different non-Newtonian fluid models. Ishak et al. [20] discussed the uniform suction/blowing effect on flow and heat transfer due to a stretching cylinder. Wang [21] examined the natural convection on a vertical stretching cylinder. Naseer et al. [22] have explained the boundary layer flow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder.
In view of these facts the present study focuses on the numerical investigation of two dimensional boundary layerflow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder with non-uniform heat source/sink. The boundary layer equations given as a set of partial differential equations (PDEs) are first changed into non-linear ordinary differential equations (ODEs) ahead being solved numerically via Runge-Kutta-Fehlberg integration method. The effects of the governing flow parameters on the velocity and temperature profiles are discussed and presented in table and graphs.
FLUID MODEL:
For the hyperbolic tangent fluid the continuity and momentum equations are given as
Where is
the density,
is the
velocity vector,
is
the Cauchy stress tensor, b represents the specific body force vector and
represents
the material time derivative. The constitutive equations for hyperbolic tangent
fluid model is given by [22]
We consider the case with and
. Therefore,
the component of extra stress tensor can be written as
FORMULATION:
Consider the problem of natural convection boundary
layer flow of a hyperbolic tangent fluid flowing over a vertical circular
cylinder of radius a. The cylinder is assumed to be stretched exponentially
along the axial direction with velocity . The
temperature at the surface of the cylinder is assumed to be
and the
uniform ambient temperature is taken as
. Under
these assumptions the boundary layer equations of motion and heat transfer are
(1)
(2)
(3)
Where the velocity components along the (r,z) axes are
(u,w), is density
,
is the
kinematic viscosity, p is pressure, g is gravitational acceleration along the
Z–direction ,
is the
coefficient of thermal expansion ,T is the temperature ,
is the
infinite shear rate viscosity,
is
the zero shear rate viscosity ,
is
the time constant , n is the power law index, and
is the
thermal diffusivity. The corresponding boundary conditions for the problem are
(
4)
(5)
The space dependent and
temperature heat generation/absorption (non-uniform heat source/sink) is defined by
Where and
are
parameters of the space and temperature dependent internal heat
generation/absorption. The positive and negative values of and represents heat
generation and absorption respectively.
(
is
dimensional constant)is the field velocity at the surface of the cylinder.
SOLUTION OF THE PROBLEM:
Introduce the following similarity transformations:
(6)
(7)
Wehere the characteristic
temperature difference is calculated from the relations With the help
of transformations (6) and (7) ,Eqs. (1)-(3) take the form
(8)
(9)
In which is the natural
convection parameter,
is the
Prandtal number
is the
Weissennberg number and
is the
Reynolds number. The boundary conditions in nondimentsional form become
The important physical
quantities such as the shear stress at the surface , the skin
friction coefficient
, the heat
flux at the surface of the cylinder
and the local
Nusselt number
are
,
The solution of the present problem is obtained numerically by using the Runge-Kutta-Fehlberg method.
RESULTS AND DISCUSSION:
In this paper an analysis is carried out
for natural convection boundary layer flow of a hyperbolic tangent fluid over
an exponentially stretched cylinder with non-uniform heat source/sink
parameter. It is assumed that the cylinder is stretching exponentially along it
axial direction. Expression is the assumed exponential stretching
velocity at the surface of the cylinder. For the solution of the problem
Runge-Kutta-Felhberg method is used. The impact of the different parameters
such as the Reynolds number
, the Prandtl number
, the Weissennberg number
and the natural convection parameter
, magnetic field parameter and porosity
parameter
over the non-dimensional velocity and temperature
profiles are presented graphically and in the form of table in the presence and
absence of Weissennberg number
.
Fig. 1 depicts that the effects of Reynolds
number decelerates the velocity when
. Fig.2 shows the influence of Reynolds
number
on the temperature profile when
. The temperature profile increases by
increasing the values of
.Figs. 3 and 4 display the impact of space and temperature dependent
heat source/sink parameter
and
on
dimensional temperature field. It is evident that from the rising values of
and
enhanced the
velocity and temperature fields (Figs.3 and 4) for both
and
cases. It is
well known that the positive values of A* and B*
indicate the heat generation while negative values represent the heat
absorption of the system. Figs. 5 and 6 display that the influence of the
natural convection parameter
on
velocity and temperature profiles
.
From the graph we observed that the increase in natural convection parameter
increases in velocity and decreases in temperature profiles.
Figs. 7-8 are plotted to examine the influence of
magnetic field parameter on
the velocity and temperature profiles. It can be found that increasing values
of
decelerates
the velocity profile (Figs. 7). From these we observed that the effects of
transverse magnetic field on electrically, conducting fluid give rise to
resistive-type force called Lorentz force. This force has tendency to slow down
the motion of the fluid in the boundary layer. These results quantitatively
agree with the expectations, since magnetic field
exerts
retarding force on natural convection flow, while the magnetic field parameter
M increases in temperature field as shown in Fig.8.
In Fig.9, the influence of porosity parameteron velocity
distribution
with fixed
values of other parameters. It is discovered that
falls as
increase.
Also it is noticed that the
diminish
as
evolves. This
development can be explained as, the increasing value of porosity parameter
increases the
viscosity of the fluid and so the friction in between the surface and the fluid
increases.
Table 1 shows the deviation in skin friction
coefficient, local Nusselt number with various values of Reynolds number, the Prandtl number
, the Weissennberg number
and the natural convection parameter
, magnetic field parameter and porosity parameter
. It is found
that
decreases
the heat transfer rate and increases skin friction coefficient. The magnetic
field parameter and porosity parameter are shown the reverse behavior on heat
transfer. It is observed that both friction factor coefficients increase with
an increase magnetic field parameter and porosity parameter. The velocity and
temperature fields are decelerates with increase in Reynolds number.
Fig.1 Velocity field for different values of
Fig.2Temperature
field for different values of
Fig.2Temperature field for different values of
Fig.3Temperature field for different values of
Fig.4Temperature field for different values of
Fig.5 Velocity field for different values of
Fig.6Temperature field for different values of
Fig.7Velocity field for different values of
Fig.8Temperature field for different values of
Fig.9Velocity field for different values of
Table-1:Variation of friction factor coefficient and
Nusselt number for different non-dimensional parameters for and
with
.
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|
0.1 |
|
|
|
|
|
-0.773316 |
-0.540847 |
4.190680 |
4.172795 |
0.3 |
|
|
|
|
|
-0.819420 |
-0.564065 |
3.875090 |
3.852582 |
0.5 |
|
|
|
|
|
-0.861830 |
-0.584200 |
3.534092 |
3.506239 |
0.7 |
|
|
|
|
|
-0.901068 |
-0.601765 |
3.160831 |
3.127352 |
|
0.1 |
|
|
|
|
-0.797050 |
-0.552960 |
4.262200 |
4.249505 |
|
0.3 |
|
|
|
|
-0.796689 |
-0.552797 |
3.809157 |
3.781549 |
|
0.5 |
|
|
|
|
-0.796327 |
-0.552634 |
3.355368 |
3.312786 |
|
0.7 |
|
|
|
|
-0.795964 |
-0.552470 |
2.900825 |
2.843213 |
|
|
0.1 |
|
|
|
-0.796957 |
-0.552918 |
4.176282 |
4.157549 |
|
|
0.3 |
|
|
|
-0.796772 |
-0.552835 |
3.884988 |
3.863177 |
|
|
0.5 |
|
|
|
-0.796537 |
-0.552729 |
3.543129 |
3.516869 |
|
|
0.7 |
|
|
|
-0.796219 |
-0.552588 |
3.121382 |
3.088045 |
|
|
|
1 |
|
|
-0.763437 |
-0.535244 |
4.017669 |
3.995662 |
|
|
|
3 |
|
|
-0.687931 |
-0.492568 |
3.975452 |
3.949078 |
|
|
|
5 |
|
|
-0.610588 |
-0.444866 |
3.930161 |
3.899026 |
|
|
|
7 |
|
|
-0.531165 |
-0.391800 |
3.881312 |
3.844897 |
|
|
|
|
1 |
|
-0.911470 |
-0.606031 |
4.111608 |
4.098827 |
|
|
|
|
3 |
|
-1.257040 |
-0.706399 |
4.266848 |
4.268861 |
|
|
|
|
5 |
|
-1.517532 |
-0.704174 |
4.341019 |
4.351089 |
|
|
|
|
7 |
|
-1.735585 |
-0.722665 |
4.387449 |
4.403533 |
|
|
|
|
|
1 |
-0.911470 |
-0.606031 |
4.111608 |
4.098827 |
|
|
|
|
|
3 |
-1.257040 |
-0.706399 |
4.266848 |
4.268861 |
|
|
|
|
|
5 |
-1.517532 |
-0.704174 |
4.341019 |
4.351089 |
|
|
|
|
|
7 |
-1.735585 |
-0.622665 |
4.387449 |
4.403533 |
CLOSING REMARKS:
The present computational results characterize the effects ofMagnetic field parameter on steady boundary layer flow and heat transfer of a hyperbolic tangent fluid flowing over a vertical exponentially stretching cylinder in its axial direction with non-uniform heat source/sink. The resultant non-linear governing partial differential equations (PDEs) are solved using the robust RKF integration method. Based on the present computational investigation the following observations are made:
· The Reynoldsnumber decelerates in friction factor coefficientand heat transfer rate.
· A rise in porosity parameter enhances the rate of heat transfer.
· Increase in the values of natural convection parameter increases in friction factor coefficient while decreases in heat transfer rate.
·
The dimensional
temperature dependent heat source/sink parameter and
enhances in
skin friction while decreases in heat transfer rate.
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Received on 18.09.2017 Modified on 30.10.2017
Accepted on 07.12.2017 ©A&V Publications All right reserved
Research J. Science and Tech. 2017; 9(4): 569-575.
DOI: 10.5958/2349-2988.2017.00096.1