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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 18 (2018) pp. 13705-13726
© Research India Publications. http://www.ripublication.com
13705
Heat and Mass transfer on MHD flow of Nanofluid with thermal slip effects
B.V.Swarnalathamma
Department of Science and Humanities, JB institute of Engineering & Technology, Moinabad, Hyderabad, Telangana-500075, India.
Abstract:
In this paper, we have discussed on heat and mass transfer of
magnetohydrodynamic (MHD) flow of Eyring-Powel
Nanofluid through an Isothermal Sphere with thermal slip
effects. The governing partial differential equations are
reduced to a system of non-linear ordinary differential
equations, using appropriate non-similarity transformations,
and then solved numerically by using a Keller box method.
The influence of the leading parameters i.e. Eyring-Powel
fluid parameter, Buoyancy ratio parameter, Brownian motion
parameter, Hartmann number and Prandtl number on velocity,
temperature, and Nano-particle concentration distributions
illustrated graphically.
Keywords: Nano Fluid, Species diffusion, Thermal slip,
MHD, Eyring-Powell fluid model, Keller-Box Method.
Nomenclature
a Radius of the sphere
C dimensional concentration
cp specific heat at constant pressure
DB Brownian diffusion coefficient
DT thermophoretic diffusion coefficient
f non-dimensional stream function
g acceleration due to gravity
T temperature
u, v non-dimensional velocity components along the x-
and y- directions, respectively
x stream wise coordinate
y transverse coordinate
Greek symbols
thermal diffusivity
local non-Newtonian parameter
dimensionless transverse coordinate
ν kinematic viscosity
non-dimensional temperature
density of nanofluid
electrical conductivity of nanofluid
dimensionless steam wise coordinate
kinematic viscosity
dynamic viscosity
dimensionless stream function
Subscripts:
w conditions on the wall
free stream conditions
INTRODUCTION
Boundary layer fluid flow problems in different dimensions
through a stretching sheet with heat transfer and MHD effects
have plentiful and inclusive applications in several
engineering and industrial sectors. They include glass blowing
melt spinning, heat exchanger design, fiber and wire coating,
production of glass fibers, industrialization of rubber and
plastic sheets, etc. In addition, the action of thermal radiation
is vital to calculating heat transmission in the polymer treating
industry. In investigations of all these applications, many
investigators deliberate the flow of different fluid models over
a stretching sheet. Sakiadis [1] studied boundary layer flow
over a flat surface. Crane [2] obtained the closed-form
solution for the flow instigated by the stretching of a flexible
parallel sheet moving periodically. Gupta and Gupta [3]
extended this work by considering suction/blowing at the
surface of the sheet. The dissemination of chemically reactive
species over a moving continuous sheet was studied by
Anderson et al. [4]. Pop [5] studied time-dependent flow over
a stretched surface. The impact of heat transmission on
second-grade fluid over a stretching sheet was explored by
Cortell et al. [6]. Areal [7] studied an asymmetric viscoelastic
fluid flow past a stretching sheet for different purposes in the
fluid field. Rashdi et al. [8,9] studied entropy generation in
magneto hydrodynamic Eyring–Powell fluid and Carreau
nanofluid through a permeable stretching surface. Hayat et al.
[10–13] studied boundary layer flow using different
phenomena. There are no solitary constitutive equations for
non-Newtonian fluid that clarify all the distinctive aspects of
compound rheological fluids. The Eyring–Powell model [14],
an important subclass of these, models from the kinetic theory
of liquids instead of experimental relations. Recently, Prasad
[15] studied heat transfer and momentum in Eyring–Powell
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fluid over a nonisothermal stretching sheet. Noreen et al. [16]
examined the peristaltic flow of MHD Eyring–Powell fluid in
a channel. Ellahi [17] recently completed a numerical study of
MHD generalized Couette flow of Eyring–Powell fluid with
heat transfer and the slip condition. Ellahi et al. [18] examined
the shape effects of spherical and nonspherical nanoparticles
in mixed convection flow over a vertical stretching permeable
sheet. Other related studies concerning Eyring–Powell fluid
can be seen in [19–25]. Thermal radiation is the procedure in
which energy is released in the form of electromagnetic
radiation by a surface in all directions. Thermal radiation has
numerous uses in the areas of engineering and heat transfer
analysis. In the case of conduction and convection, energy
transmission amongst objects depends almost entirely on the
temperature. For natural free convection, or when variable
property effects are included, the power of the temperature
difference may be slightly larger than one, and can reach two.
Tawade et al. [26] investigated a thin liquid flow through a
stretching surface with the influence of thermal radiation and
a magnetic field. A brief discussion was given on physical
parameters in his work. Ellahi et al. [27] examined the
boundary layer magnetic flow of nano-ferroliquid under the
influence of low oscillation over a stretchable rotating disk.
Zeeshan et al. [28] studied the effect of a magnetic dipole on
viscous ferrofluid past a stretching surface with thermal
radiation. The Hall effect on Falkner–Skan boundary layer
fluid flow over a stretching sheet was examined by Maqbool
et al. [29]. The enhancement of heat transfer and heat
exchange effectiveness in a double-pipe heat exchanger filled
with porous media was examined by Shirvan et al. [30].
Ramesh et al. [31] studied the Casson fluid flow near the
stagnation point over a stretching sheet with variable
thickness and radiation. Other related studies concerning
stretching sheets can be seen in [32–34]. Bakier and Moradi et
al. [35,36] studied the influence of thermal radiation on
assorted convective flows on an upright surface in a
permeable medium. Chaudhary et al. [37] investigated the
thermal radiation effects of fluid on an exponentially
extending surface.
Recently, Krishna and Gangadhar Reddy [38] discussed the
unsteady MHD free convection in a boundary layer flow of an
electrically conducting fluid through porous medium subject
to uniform transverse magnetic field over a moving infinite
vertical plate in the presence of heat source and chemical
reaction. Krishna and Subba Reddy [39] have investigated the
simulation on the MHD forced convective flow through
stumpy permeable porous medium (oil sands, sand) using
Lattice Boltzmann method. Krishna and Jyothi [40] discussed
the Hall effects on MHD Rotating flow of a visco-elastic fluid
through a porous medium over an infinite oscillating porous
plate with heat source and chemical reaction. Reddy et al.[41]
investigated MHD flow of viscous incompressible nano-fluid
through a saturating porous medium. Recently, Krishna et al.
[42-45] discussed the MHD flows of an incompressible and
electrically conducting fluid in planar channel. Veera Krishna
et al. [46] discussed heat and mass transfer on unsteady MHD
oscillatory flow of blood through porous arteriole. The effects
of radiation and Hall current on an unsteady MHD free
convective flow in a vertical channel filled with a porous
medium have been studied by Veera Krishna et al. [47]. The
heat generation/absorption and thermo-diffusion on an
unsteady free convective MHD flow of radiating and
chemically reactive second grade fluid near an infinite vertical
plate through a porous medium and taking the Hall current
into account have been studied by Veera Krishna and
Chamkha [48]. Veera Krishna et al. [49] discussed the heat
and mass transfer on unsteady, MHD oscillatory flow of
second-grade fluid through a porous medium between two
vertical plates under the influence of fluctuating heat
source/sink, and chemical reaction. Veera Krishna et al. [50]
investigated the heat and mass transfer on MHD free
convective flow over an infinite non-conducting vertical flat
porous plate. Veera Krishna and Jyothi [51] discussed the
effect of heat and mass transfer on free convective rotating
flow of a visco-elastic incompressible electrically conducting
fluid past a vertical porous plate with time dependent
oscillatory permeability and suction in presence of a uniform
transverse magnetic field and heat source. Veera Krishna and
Subba Reddy [52] investigated the transient MHD flow of a
reactive second grade fluid through a porous medium between
two infinitely long horizontal parallel plates. Veera Krishna et
al. [53] discussed heat and mass-transfer effects on an
unsteady flow of a chemically reacting micropolar fluid over
an infinite vertical porous plate in the presence of an inclined
magnetic field, Hall current effect, and thermal radiation taken
into account. Veera Krishna et al.[54] discussed Hall effects
on steady hydromagnetic flow of a couple stress fluid through
a composite medium in a rotating parallelplate channel with
porous bed on the lower half.
The aim of the current research is to investigate the heat and
mass transfer of MHD flow of Eyring-Powel Nanofluid
through an Isothermal Sphere with thermal slip effects.
FORMULATION AND SOLUTION OF THE PROBLEM
In the present study a subclass of non-Newtonian fluids
known as the Eyring–Powell fluid is employed owing toits
simplicity. Mathematically, the Eyring–Powell model isgiven
as
A PI (1)
where is the extra stress tensor is defined as
1
1 1
1 1A Sinh AC
(2)
Here μ is dynamic viscosity, β and C are the
rheologicalEyring–Powell fluid model [37] parameters.
Consideringthe second-order approximation of the 1Sinh
function as
3
1
1
1 1 1 1 1, 1
6Sinh A
C C C C
(3)
Therefore, Eq. (2) takes the form
2
1 1
3
1 1( )
6A A
C C
(4)
https://www.sciencedirect.com/science/article/pii/S2214785317323003
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Where 21
1
2trA and the kinematical tensor A1 is
1 ( )TA V V
The introduction of the appropriate terms into the flowmodel
is considered next. The resulting boundary valueproblem is
found to be well-posed and permits an excellentmechanism
for the assessment of rheological characteristics-on the flow
behavior.
Steady, double-diffusive, laminar, incompressible,buoyancy-
driven convection flow, heat and mass transfer ofan Eyring–
Powellnanofluid from an isothermal sphere with the effects of
MHD and radiation is illustrated in Fig. 1. The x-axis taken
along the isothermal sphere surface measured from the origin
and the y-axis are measured normal to the surface,
sin xr a a with ‘a’ denoting the radius of the sphere. The gravitational acceleration, g acts downwards. Both the sphere
and the fluid are maintained initially at the same temperature.
Instantaneously they are raised to a temperature T Tw i.e. the ambient temperature of the fluid which remains
unchanged. The Boussineq approximation holds, i.e. the
density variation is only experienced in the buoyancy term in
the momentum equation. Introducing the boundary layer
approximations, the equations of continuity, momentum,
energy and species can be written as follows:
Figure 1: Physical Configuration of the Problem
The two-dimensional momentum, energy and concentration
species boundary layer equations governing the flow in an
(x,y) coordinate system may be shown to take the form:
( ) ( )0
ru rvx y
(5)
2
2
2 22
0
3 2
1
1( )sin
2
u u uu vx y c y
Bu u xg T T uc y y a
(6)
22
2
TB
DT T T C T Tu v Dx y y y y T y
(7)
2 2
2 2
TB
DC C C Tu v Dx y y T y
(8)
The boundary conditions imposed at the sphere surface and in
the free stream are:
0, 0, ( ), 0
0, 0, ,
w w wTu v K h T T C C at yy
u v T T C C as y
(9)
Here u and v are the velocity components in the x- and y- directions, respectively, ν- the kinematic viscosity of the electrically-conducting Nano fluids. The stream function is
defined by the Cauchy-Riemann equations, ru y and rv x , and therefore, the continuity equation is automatically satisfied. In order to write the governing
equations and the boundary conditions in dimensionless form,
the following non-dimensional quantities are introduced.
1/4, , ( , ) ,
1/4
( , ) , ( , )
yx Gr fa x GrT T C C
T T C Cw w
(10)
The transformed boundary layer equations for momentum,
energy and concentration emerge as:
2 2 2(1 ) (1 cot )
sin(11)
f ff f f f
f fMf f f
21 cotPr
b tff N N f
(12)
1
1 cot b
t
N ff fLe Le N
(13)
The corresponding transformed dimensionless boundary
conditions are:
0, 0, 1 (0), 1 0
0, 0, 0
f f S atTf as
(14)
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Here, primes indicate the differentiation with respect to
and
14ahwS GrT k
is the thermal slip parameter.
Where,
Pr / is the Prandtl number;
/Le DB is the Lewis number;
/Nt c D T T c Twp fT is the
thermophoresis parameter;
β ( ) / β ( )N T T C Cw wT C is the concentration of thermal buoyancy ratio parameter.
/Nb c D C C cwp fB is the
Brownian parameter
and 1
2 20 /M B a Gr is the Magnetic parameter.
The wall thermal boundary condition in (14) relates to
convective cooling. The skin friction coefficient and Nusselt
number (heat transfer rate) can be defined using the
transformations depicted in the above with the following
expressions.
3
334 (1 ) ( ,0) ( ,0)3
xGr C f ff
(15)
14 ,0xGr Nu
(16)
14 ,0xGr Sh
(17)
The strongly coupled, nonlinear conservation equations do not
admit analytical (closed-form) solutions. An elegant, implicit
finite difference numerical method developed by Keller
(1970) is therefore adopted to solve the general flow model
defined by equations (11)-(13) with boundary conditions (14).
This method is especially appropriate for boundary layer flow
equations which are parabolic in nature. It remains one of the most widely applied computational methods in viscous fluid
dynamics. Recent problems which have used Keller’s method
include radiative magnetic forced convection flow (2006),
stretching sheet hydromagnetic flow (2013),
magnetohydrodynamic Falkner-Skan “wedge” flows (2014),
magneto-rheological flow from an extending cylinder (2015),
Hall magneto-gas dynamic generator slip flows (2016) and
radiative-convective Casson slip boundary layer flows (2016).
Keller’s method provides unconditional stability and rapid
convergence for strongly non-linear flows. It involves four
key stages, summarized below.
1) Reduction of the Nth order partial differential equation system to Nfirst order equations
2) Finite difference discretization of reduced equations
3) Quasilinearization of non-linear Keller algebraic equations
4) Block-tridiagonal elimination of linearized Keller algebraic equations
Stage 1: Reduction of the Nthorder partial
differential equation system to N first order
equations
Equations (11) – (13) and (14) subject to the
boundary conditions are first written as a system of
first-order equations. For this purpose, we reset Equations (9) – (10) as a set of simultaneous
equations by introducing the new variables
f u (18) (18)
'u v (19) (19)
s (20) (20)
s t (21) (21)
g p (22) (22)
2 2sin
1 1 cotu fv fv v v s Mu u v
(23)
21 cotPr
b tt s fft N pt N t u t
(24)
1
1 cot b
t
Np g ffp t u pLe Le N
(25)
where primes denote differentiation with respect to . In
terms of the dependent variables, the boundary conditions become:
0 : 0, 0, 1, 1
: 0, 0, 0, 0
At u f s gAs u v s g
(26)
Stage 2: Finite difference discretization of reduced
boundary layer equations
A two-dimensional computational grid (mesh) is imposed on
the -η plane as sketched in Fig.2. The stepping process is defined by:
0 10, , 1,2,..., ,j j j Jh j J (27)
0 10, , 1,2,...,n n nk n N
(28)
Where, kn and hj denote the step distances in the ξ (stream wise) and η (span wise) directions respectively
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Fig 2: Keller Box element and boundary layer mesh
If njg denotes the value of any variable at , nj , then the variables and derivatives of Eqns. (18) – (25) at 1/21/2 , nj are
replaced by:
1/2 1 11/2 1 11
4
n n n n nj j j j jg g g g g
(29)
1/2
1 1
1 1
1/2
1
2
nn n n nj j j j
jj
g g g g gh
(30)
1/2
1 1
1 1
1/2
1
2
nn n n nj j j jn
j
g g g g gk
(31)
The finite-difference approximation of equations (18) – (25) for the mid-point 1/2 , nj , below:
1 1 1/2n n nj j j jh f f u
(32)
1 1 1/2n n nj j j jh u u v
(33)
1 1 1/2n n nj j j jh g g p
(34)
1 1 1/2n n nj j j jh t (35)
1 1 1 1
2 221 1 1 1
11 11/2 1 1/2 1 1 1/2
1 1 cot4 2
14 2
2 2
j jj j j j j j j j
j jj j j j j j j j
nj jn nj j j j j j j
h hv v f f v v M u u
h hu u A s s v v v v
h hv f f f v v R
(36)
1 1 1 1 1 1 1
2 1 11 1/2 1 1/2 1
1 11/2 1 1/2 1 2 1/2
11 cot
Pr 4 4 4
4 2 2
2 2
j j jj j j j j j j j j j j j j j
j j jn nj j j j j j j j
nj jn nj j j j j j j
h h ht t f f t t u u s s Nb t t p p
h h hNt t t s u u u s s
h hf t t t f f R
1
(37)
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1 1 1 1 1 1
11 1 1 11/2 1 1/2 1 1/2 1 1/2 1 2 1/2
11 cot
4 4
2 2 2 2
j jj j j j j j j j j j j j
nj j j jn n n nj j j j j j j j j j j j j
h hBp p f f p p t t u u g gLe Le
h h h hs u u u g g f p p p f f R
(38)
Where the following notation applies:
1/21/21/2
sin, ,
nn
nn
NtA Bk Nb
(39)
1
1/2 1/2
1
1 1/22 2 12
1/2 1/2 1/2 1/2
1 1 cot
1
j jj j
jnjj
j jj j j j
j
v vf v
hR h
v vu As v Mu
h
(40)
21 1
2 1/2 1/2 1/2 1/2 1/2 1/2 1/21/2
11 cot
Pr
n j jj j j j j j j jj
j
t tR h f t u s Nb p t Nt t
h
(41)
1 1 13 1/2 1/2 1/2 1/21/21
1 cotn j j j j
j j j j jjj j
p p t tBR h f p u gLe h Le h
(42)
The boundary conditions are
0 0 0 00, 1, 0, 0, 0, 1, 0n n n n n n n n
J J J Jf u u v (43)
Stage 3: Quasilinearization of non-linear Keller algebraic
equations
If we assume 1 1 1 1 1 1
1 1 1 1 1 1, , , , , ,n n n n n nj j j j j jf u v p s t to be known for
, Equations (33) – (39) comprise a system of
6J+6 equations for the solution of 6J+6 unknowns
, , , , , ,n n n n n nj j j j j jf u v p s t , j = 0, 1, 2 …, J. This non-linear system of algebraic equations is linearized by means of Newton’s method as elaborated by Keller (1970).
Stage 4: Block-tridiagonal elimination of linear Keller
algebraic equations
The linearized version of eqns. (33) – (39) can now be solved
by the block-elimination method, since they possess a block-tridiagonal structure since it consists of block matrices. The complete linearized system is formulated as a block matrix system, where each element in the coefficient matrix is a matrix itself. Then, this system is solved using the efficient
Keller-box method. The numerical results are affected by the
number of mesh points in both directions. After some trials in
the η-direction (radial coordinate) a larger number of mesh points are selected whereas in the ξ direction (tangential coordinate) significantly less mesh points are utilized. ηmax has
been set at 10 and this defines an adequately large value at
which the prescribed boundary conditions are satisfied. ξmax is set at 3.0 for this flow domain. Mesh independence testing is also performed to ensure that the converged solutions are
correct. The computer program of the algorithm is executed in
MATLAB running on a PC.
The present Keller box method (KBM) algorithm has been
tested rigorously and benchmarked in numerous studies by the
authors. However to further increase confidence in the present
solutions, we have validated the general model with an
alternative finite difference procedure due to Nakamura (1994). The Nakamura tridiagonal method (NTM) generally
achieves fast convergence for nonlinear viscous flows which
may be described by either parabolic (boundary layer) or
elliptic (Navier-Stokes) equations.The coupled 7th order
system of nonlinear, multi-degree, ordinary differential
equations defined by (11)–(13) with boundary conditions (14)
is solved using the NANONAK code in double precision
arithmetic in Fortran 90, as elaborated by Bég (2013).
Computations are performed on an SGI Octane Desk
workstation with dual processors and take seconds for
compilation. As with other difference schemes, a reduction in
the higher order differential equations, is also fundamental to
Nakamura’s method. The method has been employed
successfully to simulate many sophisticated nonlinear
transport phenomena problems e.g. magnetized bio-polymer
enrobing coating flows (Béget al. (2014)). Intrinsic to this method is the discretization of the flow regime using an equi-
spaced finite difference mesh in the transformed coordinate
() and the central difference scheme is applied on the -
0 j J
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variable. A backward difference scheme is applied on the -
variable. Two iteration loops are used and once the solution
for has converged, the code progresses to the next station.
The partial derivatives for f,,with respect to are as explained evaluated by central difference approximations. An
iteration loop based on the method of successive substitution is utilized to advance the solution i.e. march along. The finite
difference discretized equations are solved in a step-by-step
fashion on the -domain in the inner loop and thereafter on the -domain in the outer loop. For the energy and nano-
particle species conservation Eqns. (12)-(13) which are
second order multi-degree ordinary differential equations, only a direct substitution is needed. However a reduction is required for the third order momentum (velocity) boundary layer eqn. (11). We apply the following substitutions:
P = f (44)
Q = (45)
R = (46)
The eqns. (11)-(13) then retract to:
Nakamura momentum equation:
(47)
Nakamura energy equation:
(48)
Nakamura nano-particle species equation:
(49)
Here Ai=1,2,3, Bi=1,2,3, Ci=1,2,3, are the Nakamura matrix
coefficients, Ti=1,2,3, are the Nakamura source terms
containing a mixture of variables and derivatives associated
with the respective lead variable (P, Q, R). The Nakamura
Eqns. (30)–(33) are transformed to finite difference equations
and these are orchestrated to form a tridiagonal system which
due to the high nonlinearity of the numerous coupled, multi-
degree terms in the momentum, energy, nano-particle species
and motile micro-organism density conservation equations, is
solved iteratively. Householder’s technique is ideal for this
iteration. The boundary conditions (14) are also easily
transformed. Further details of the NTM approach are
provided in Nakamura (1994). Comparisons are documented
in Table 1 for skin friction and very good correlation is
attained. Table 1 further indicates that increase in Eyring-
Powel fluid parameter ( ) induces a strong retardation in the flow i.e. suppresses skin friction magnitudes. In both cases
however positive magnitudes indicate flow reversal is not
generated.
Table 1. Numerical values of skin-friction coefficient
33(1 ) ( ,0) ( ,0)
3f f of with
Pr 7.0, 1.0, 5.0, 0.1, 0.02,
1.0, 1.0, 0.3, 0.1
T
M Le N Nt NbS
33(1 ) ( ,0) ( ,0)3
f f
(KBM)
33(1 ) ( ,0) ( ,0)
3f f
(NTM)
0.0 0.3226 0.3221
0.5 0.3691 0.3694
1.0 0.4054 0.4052
1.5 0.4355 0.4352
2.0 0.4620 0.4622
RESULTS AND DISCUSSION
The impact of Eyring-Powell fluid parameter ε, on velocity,
temperature and concentration as shown in the Figs.3(a)- (c)
and observed that the increase in Eyring-Powell fluid
parameter, the boundary layer flow is accelerated with
increasing Eyring-Powell fluid parameter and velocity,
temperature and concentration profiles are diminished
throughout the boundary layer regime.
Figs.4 (a) - (c)depicts the velocity , temperature and
concentration distributions with increasing local non-
Newtonian fluid parameter .Very little tangible effect is observed inFig. 4a, although there is a very slight increase in
velocity with increase in , but there is only a very slight depression in temperature magnitudes in Fig. 4b with arise in
and Fig.4(c) indicates there is a substantial enhancement in nano-particle concentration (and species boundary layer
thickness) with enhance values. Figs. 5(a) – (c) illustrates the effect of the thermophoresis parameter (Nt) on the velocity, temperatureand concentration distributions,
respectively. Thermophoretic migration of nano-particles
results in exacerbated transfer of heat from the nanofluid
regime to the sphere surface. This de-energizes the boundary
layer and inhibits simultaneously the diffusion of momentum,
manifesting in a reduction in velocity i.e. retardation in the
boundary layer flow and increasing momentum
(hydrodynamic) boundary layer thickness, as computed in fig.
5a. Temperature is similarly decreased with greater
thermophoresis parameter (fig.5b). Fig.5(c) indicates there is a
substantial enhancement in nano-particle concentration (and
species boundary layer thickness) with greater Nt values.
Figs. 6(a) – (c) depict the response in velocity, temperature
and concentration functions to a variation in the Brownian
motion parameter (Nb). Increasing Brownian motion parameter physically correlates with smaller nanoparticle diameters. Smaller values of Nb corresponding to larger nanoparticles, and imply that surface area is reduced which in
turn decreases thermal conduction heat transfer to the cylinder
surface. This coupled with enhanced macro-convection within
the nanofluid energizes the boundary layer and accelerates the
11
/
1
//
1 TPCPBPA
22
/
2
//
2 TQCQBQA
33
/
3
//
3 TRCRBRA
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flow as observed in fig. 6a. Similarly the energization of the
boundary layer elevates thermal energy which increases
temperature in the viscoelastic Nano fluid observed in Fig. 6b.
Fig 6c however indicates that the contrary response is
computed in the nano-particle concentration field. With
greater Brownian motion number species diffusion is
suppressed. Effectively therefore momentum and nanoparticle
concentration boundary layer thickness is decreased whereas
thermal boundary layer thickness is increased with higher
Brownian motion parameter values. Figs. 7 (a)-(c) exhibit the
profiles for velocity, temperature and concentration,
respectively with increasing buoyancy ratio parameter, N. In
general, increases in the value of N have the prevalent to
cause more induced flow along the Sphere surface. This
behaviour in the flow velocity increases in the fluid
temperature and volume fraction species as well as slight
decreased in the thermal and species boundary layers
thickness as N increases. Figs. 8(a) – (c) illustrate the
evolution of velocity, temperature and concentration functions
with a variation in the Lewis number, is depicted. Lewis
number is the ratio of thermal diffusivity to mass (nano-
particle) species diffusivity. In fig 8a, a consistently weak
decrease in velocity accompanies an increase in Lewis
number. Momentum boundary layer thickness is therefore
increased with greater Lewis number. This is sustained
throughout the boundary layer. Fig. 8b shows that increasing
Lewis number also incresses the temperature magnitudes and
therefore increases thermal boundary layer thickness. Fig 8c
demonstrates that a more dramatic depression in nano-particle
concentration results from an increase in Lewis number.
Figs. 9 (a)-(c) present the evolution in velocity, temperature
and concentration functions with a variation in magnetic body
force parameter (M). The radial magnetic field generates a
transverse retarding body force. This decelerates the boundary
layer flow and velocities are therefore reduced as observed in
fig. 9a. The momentum development in the viscoelastic
coating can therefore be controlled using a radial magnetic
field. The effect is prominent throughout the boundary layer
from the Sphere surface to the free stream. Momentum
(hydrodynamic) boundary layer thickness is therefore
decreased with greater magnetic field. Fig. 9b and 9c shows
that both temperatures and nano-particle concentrations are
strongly enhanced with greater magnetic parameter. The
excess work expended in dragging the polymer against the
action of the magnetic field is dissipated as thermal energy
(heat). This energizes the boundary layer and increases
thermal boundary layer thickness. Again the influence of
magnetic field is sustained throughout the entire boundary
layer domain. This energizes the boundary layer since the
kinetic energy is dissipated as thermal energy, and this further
serves to agitate improved species diffusion. As a result, both
thermal and nano-particle (species) concentration boundary
layer thicknesses are increased.
Figs. 10(a)-(c) depict the evolution in velocity, temperature
and nanoparticle concentration characteristics with transverse
coordinate i.e. normal to the Sphere surface for various
Prandtl numbers, Pr. Relatively high values of Pr are
considered. Prandtl number embodies the ratio of momentum
diffusivity to thermal diffusivity in the boundary layer regime.
It also represents the ratio of the product of specific heat
capacity and dynamic viscosity, to the fluid thermal
conductivity. For polymers momentum diffusion rate greatly
exceeds thermal diffusion rate. The low values of thermal
conductivity in most polymers also result in a high Prandtl
number. With increasing Pr from 1 to 50 there is evidently a
substantial deceleration in boundary layer flow i.e. a
thickening in the momentum boundary layer (fig. 10a). The
effect is most prominent close to the Sphere surface. Also fig.
10b shows that with greater Prandtl number the temperature
values are strongly decreased throughout the boundary layer
transverse to the Sphere surface. Thermal boundary layer
thickness is therefore significantly reduced. Inspection of fig.
10c reveals that increasing Prandtl number strongly elevates
the nano-particle concentration magnitudes. In fact, a
concentration overshoot is induced near the Sphere surface.
Therefore, while thermal transport is reduced with greater
Prandtl number, species diffusion is encouraged and nano-
particle concentration boundary layer thickness grows. The
asymptotically smooth profiles in the free stream
(highvalues) confirm that an adequately large infinity boundary condition has been imposed in the Keller box
numerical code. Figs. 11(a) – (c) illustrate the variation of
velocity, temperature and nano-particle concentration with
transverse coordinate (), for different values of thermal slip parameter (ST). Thermal slip is imposed in the augmented wall boundary condition in eqn. (10). With increasing thermal slip
less heat is transmitted to the fluid and this de-energizes the
boundary layer. This also leads to a general deceleration as
observed in fig. 11a and also to a more pronounced depletion
in temperatures in fig.11b, in particular near the wall. The
effect of thermal slip is progressively reduced with further
distance from the wall (curved surface) into the boundary
layer and vanishes some distance before the free stream. It is
also apparent from fig. 11c that nanoparticle concentration is
reduced with greater thermal slip effect. Momentum boundary
layer thickness is therefore increased whereas thermal and
species boundary layer thickness are depressed. Evidently the
non-trivial responses computed in figs. 11a-c further
emphasize the need to incorporate thermal slip effects in
realistic nanofluid enrobing flows. Figs. 12(a) – (c) presents
the in fluence of Eyring–Powell fluid parameter, ε, on
dimensionless skin friction coefficient, Nusselt number and
Sherwood number at the sphere surface. It is observed that the
dimensionless skin friction is enhanced with the increase in ε,
i.e. the boundary layer flow is accelerated with decreasing
viscosity effects in the non-Newtonian regime. Conversely,
Nusselt number and Sherwood number is substantially
decreased with increasing ε values. Decreasing viscosity ofthe
fluid (induced by increasing the ε value) reduces thermal
diffusion as compared with momentum diffusion. A decrease
in heat transfer rate and mass transfer rate at the wall implies
less heat is convected from the fluid regime to the sphere,
thereby heating the boundary layer and enhancing
temperatures and concentrations.
Figs. 13(a)-(c) illustrate the skin friction, Nusselt number and
Sherwood number distributions with various values of thermal
slip effect (ST). Both skin friction and Nusselt number are
strongly reduced and Sherwood number is enhanced with an
increase thermal slip (ST). The boundary layer is therefore
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13713
decelerated and heated with stronger thermal slip. With
thermal slip absent therefore the skin friction is maximized at
the Sphere surface. The inclusion of thermal slip, which is
encountered in various slippy polymer flows, is therefore
important in more physically realistic simulations.
(a)
(b)
(c)
Figure 3.Influence of on (a)velocity profiles (b) temperature profiles (c) concentration profiles
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(a)
(b)
(c)
Figure 4. Influence of on (a)velocity profiles (b) temperature profiles(c) concentration profiles
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(a)
(b)
(c)
Figure 5. Influence of Nt on (a)velocity profiles (b) temperature profiles(c) concentration profiles.
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(a)
(b)
(c)
Figure 6. Influence of Nb on (a)velocity profiles (b) temperature profiles (c) concentration profiles
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(a)
(b)
(c)
Figure 7. Influence of N on (a)velocity profiles (b) temperature profiles(c) concentration profiles.
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(a)
(b)
(c)
Figure 8. Influence of Le on (a) velocity profiles (b) temperature profiles(c) concentration profiles.
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(a)
(b)
(c)
Figure 9. Influence of M on (a) velocity profiles (b) temperature profiles(c) concentration profiles.
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(a)
(b)
(c)
Figure 10. Influence of Pr on (a) velocity profiles (b) temperature profiles(c) concentration profiles.
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(a)
(b)
(c)
Figure 11. Influence of TS on (a) velocity profiles (b) temperature profiles(c) concentration profiles.
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(a)
(b)
(c)
Figure 12. Effect of on (a) Skin friction profiles (b) Nusselt number profiles (c) Shear wood number profiles
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(a)
(b)
(c)
Figure 13. Effect of TS on (a) Skin friction profiles (b) Nusselt number profiles (c) Shear wood number profiles
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CONCLUSIONS
1. Increasing viscoplastic Eyring-Powell fluid parameter decelerates the flow and also decreases
thermal and nano-particle concentration boundary
layer thickness.
2. Increasing Lewis number, the velocity and concentration profiles decelerates the flow whereas it
enhances temperatures.
3. Increasing Brownian motion accelerates the flow and enhances temperatures whereas it reduces
nanoparticle concentration boundary layer thickness.
4. Increasing thermal slip strongly reduces velocities, temperatures and nano-particle concentrations.
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