INTRODUCTION:
Over recent
years, researchers are allured by the characteristics and performance of
non-Newtonian fluids. Such motivation is due to complex physical behavior of
non-Newtonian fluids. A non-Newtonian fluid finds numerous applications in
pharmaceuticals, food products, crystal growth, fiber technology, chemical and
material engineering. Hence several constitutive models for non-Newtonian
fluids are available in the existing literature [1-4]. Non- Newtonian Casson
fluid is a shear thinning fluid which exhibit yield shear stress. If suppose
the yield stress is lesser than the applied shear stress than the liquid would
start moving. For example tomato sauce, honey, human blood and fruit juices.
Casson
fluid finds several applications in cancer homeo-therapy, fibrinogen, protein
cells and red blood cells. Because of these applications researchers Raju et
al.[5] studied heat and mass transfer of MHD Casson fluid on an exponential
permeable stretching surface. Further Raju et al. [6] took up a comparative
study of non-Newtonian Casson and Carreau fluid over a stretching surface and
found that heat and mass transfer in Casson fluid is considerable high compared
with Carreau fluid.
Phenomenon of suspended
particles finds significant role in advanced processes of engineering and
scientific field concerned with polymer technology, metallurgy, powder
technology, power generation, atmospheric fallout, nuclear reactor cooling,
combustion, waste water treatment, electrostatic precipitation of dust
,fluidization etc. Fundamental studies of suspended particles in the flow were
initiated by Saffman [7]. Recently Krupa Lakshmi et al.[8] took up heat and
mass transfer of a dusty liquid past a stretching sheet by considering thermal
radiation in the energy equation. Gireesha et
al. [8] studied influence of viscous dissipation on flow of dusty fluid over
stretching sheet. Krishnamurthy et al. [9] took up theoretical study of
suspended particle on slip flow and nanofluid in a porous Medium with the
presence of nonlinear thermal radiation.
Extensive research has
been undertaken by the researchers to study flow induced by a stretching
surface in view of its large range of applications in chemical processing
equipment, polymer extrusion, wire and fiber coating, glass fiber production,
water pipes, irrigation channels etc. Due to its vast applications, initially,
Sakiadis [11] presented the concept of two-dimensional boundary layer flows
past a continuously stretching sheet with constant speed. Later on, several
authors [12-22] extended the research over a stretching sheet for various fluid
flow models over several geometries such as cone, Cylinder, Plate etc.
Heat and mass
transportation from varied geometries embedded with porous media has several
engineering and geophysical applications in geothermal reservoirs, thermal
insulation, hydrology, agriculture, mineral processing, etc. The classical
Darcy’s theory which has been used in most of the flow problems is suitable for
lower velocity and smaller porosity. When inertial and boundary effects take
place with higher flow rate Darcy’s theory is inadequate. Keeping this in view
researchers [23-25] studied heat and mass transfer of fluid by considering Darcy-
Forchheimer porous medium.
By studying the
above literature, in this paper we proposed a mathematical model of boundary
layer flow of Casson fluid with suspended uniform sized dust particles towards
a stretching sheet in the presence of magnetic field, Darcy-Forchheimer porous
medium, exponentially decaying space dependent internal heat source and
convective boundary condition. The governing nonlinear partial differential
equations are reduced into a set of non-linear ordinary differential equations by
using local similarity transformation. These nonlinear ordinary differential
equations then solved numerically by Runge-Kutta based shooting process for
different values of parameters of interest. Influence of governing parameters
on velocity, temperature, Skin friction coefficient and local Nusselt number
are analyzed and discussed in detail with help of plotted graphs and tables.
Fig.1 Schematic
representation of the flow diagram
2. MODELING:
A steady
two-dimensional, incompressible, magnetohydrodynamic boundary layer flow of Casson
fluid with suspended uniform size dust particles over a stretching sheet characterizing
Darcy-Forchheimerporous medium is developed. The stretching sheet is aligned
with the 
axis and
the flow is confined to
.
The surface is stretched with the velocity 
in 
direction
with rate
.Magnetic
field of strength 
is
imposed normal to the fluid flow as shown in Figure 1 By assuming
electrical conductivity of the fluid to be small we have neglected the induced
magnetic field. Convective surface temperature is characterized by 
and heat
transfer coefficient by 
.The
temperature 
at the
surface of the sheet is considered to be more than the ambient fluid
temperature
. The
associated governing boundary layer equations are:
(1)
(2)
(3)
(4)
(5)
(6)
With
boundary conditions
(7)
Here 
- velocity
components along direction
of the fluid and dust particle phase, 
- velocity
components along 
direction
of the fluid and dust particle phase,
are
density of the fluid, density of the dust particle phase, mass of the dust
particles, number density of the dust particle, kinematic viscosity, thermal
conductivity, electric conductivity, uniform magnetic field, specific heat of
the fluid and dust phase.
-permeability
of porous medium, A is power law index, 
denotes
stokes drag coefficient. 
-radius
of the dust particle.
is
the non-uniform inertia coefficient of porous medium, where 
is the
drag coefficient [24] .
represents
temperature of the fluid, temperature of the dust particle phase, ambient fluid
temperature, convective fluid temperature, and convective heat transfer
coefficient. 
and
represents
relaxation time of dust particles and thermal equilibrium time.
Coefficient
of the dimensionless space-dependent internal heat generation [26].
Following
similarity transformation were used to convert PDEs into set of ODEs.
(8)
Continuity equations
(1) and (3) are identically satisfied. Equations (2), (4)-(7) are transformed
as follows.
(9)
(10)
(11)
(12)
The boundary conditions defined in equation (7)
will be transformed to:
(13)
in the above expressions, represents magnetic
parameter, mass concentration of dust particles, fluid-particle interaction parameter,Casson
fluid parameter, porosity parameter, inertia coefficient, Prandtl number, heat
source parameter, Eckert number, specific heat ratio, fluid particle
interaction parameters,biot number.
(14)
Distinctive
measures of practical interest are skin friction coefficient 
and local
Nusselt number 
Which
are defined as ;
Where
is the local
Reynold’snumber.
Fig.2 The effect
of 
on
velocity field Fig.3 The
effect of on
temperature field
Fig.4 The effect
o
on
temperature field Fig.5 The
effect o
on
velocity field
Fig.6 The effect
o
on
velocity field Fig.7 The effect o
on
velocity field
Fig.8 The effect
oon
temperature field Fig.9 The effect of on
temperature field
RESULTS AND DISCUSSION:
For analyzing
the approximate solutions of velocity 
as well as
temperature 
fields,
the non-linear ordinary differential Eqs. (9)-(12) with reference to the
boundary conditions (13) are solved numerically with the assistance of
Runge-Kutta and Newton’s methods. For numerical solutions we have considered
the values of non- dimensional parameters as
these
values are taken constant in this study besides the varied parameters as
mentioned in the respective figure. In Figure 2and 3 the influence of
magnetic field parameter 
on
velocity
and
temperature profiles 
of
fluid and dust phases can be observed. As 
accelerates,
The Lorentz force which opposes the flow regime also improves hence there is
decrement in the and
Lorentz force generates friction on the flow, which results more heat energy
causing increase in the temperature distribution of the flow.
Figure 4
demonstrates the influence of convective heating parameter
on the
temperature distribution of fluid and dust phase. It is noticed that increasing
values of 
improves
thermal boundary layer.
Physically convective heat exchange at the surface of the sheet helps to
improve the thermal boundary layer thickness. Figure 5 presents larger
values of non –uniform inertia coefficient 
declines
the velocity profiles of both fluid and dust phase. Figure 6 and 7
depicts the variation in velocity and temperature profiles for larger values of
porosity parameter
.
It is observed that increment in 
reduces
velocity field and improves temperature field for both fluid and dust phase. In
Figure 8 and 9 it is noticed that increment in Casson fluid parameter 
decreases
velocity boundary layer and improves thermal boundary layer for both fluid and
dust phase. Figure 10 demonstrates the influence of exponentially
decaying internal heat generation parameter 
over
temperature profiles of fluid and dust phase. It is observed that increase in enhances
heat generation which causes the temperature profiles of fluid and dust phase
to enhance.
Table 1 Shows
the comparative study of thepresent result with the published results. Present
results agree with the published results.
Table 2
display the effect of non-dimensional parameters
on skin
friction coefficient 
and
local Nusselt number
.
It is noticed that rise in magnetic field parameter minimize the friction
coefficient along with Nusselt number. Increment in Biot number decrease
Nusselt number. But no variation is observed in friction factor. Rising values
of inertia coefficient, porosity parameter and Casson fluid parameter decreases
skin friction and Nusselt number. Increment in exponentially decaying internal
heat generation parameter enhances heat transfer rate and friction factor
remains same.
4. CONCLUSION:
Darcy
–Forchheimer two- dimensional flow of non-Newtonian Casson fluid with suspended
uniform size dust particles over a stretching sheet with exponentially decaying
heat generation and convective boundary condition has been discussed. The major
findings of present study are as follows.
·
The
magnetic interaction parameter motivates friction factor coefficients and heat
transfer rate.
·
Increment
in Biot number improves the thermal boundary layer of both phases. But retards
heat transfer rate.
·
Heat
source parameter declines thermal boundary layer but improves heat transfer
rate.
·
Non-uniform
inertia coefficient, Porosity parameter, Casson fluid parameter reduces
friction factor and heat transfer rate but improves thermal boundary layer in
both the fluid and dust phase
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