Heat and Mass transfer on MHD flow of Nanofluid with thermal … · 2018-10-04 · Department of...

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



13706

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



13707

Where 2

1

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 2 sin1 1 cot

u 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 , n

j , then the variables and derivatives of Eqns. (18) – (25) at 1/2

1/2 , nj

are

replaced by:

1/2 1 1

1/2 1 1

1

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:

11 1/2

n n nj j j jh f f u

(32)

11 1/2

n n nj j j jh u u v

(33)

11 1/2

n n nj j j jh g g p

(34)

11 1/2

n 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:

1/21/2

1/2

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 1

3 1/2 1/2 1/2 1/21/2

11 cot

n j j j jj j j j jj

j 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 nJ 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



13711

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



13712

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



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



13714

(a)

(b)

(c)

Figure 4. Influence of on (a)velocity profiles (b) temperature profiles(c) concentration profiles



13715

(a)

(b)

(c)

Figure 5. Influence of Nt on (a)velocity profiles (b) temperature profiles(c) concentration profiles.



13716

(a)

(b)

(c)

Figure 6. Influence of Nb on (a)velocity profiles (b) temperature profiles (c) concentration profiles



13717

(a)

(b)

(c)

Figure 7. Influence of N on (a)velocity profiles (b) temperature profiles(c) concentration profiles.



13718

(a)

(b)

(c)

Figure 8. Influence of Le on (a) velocity profiles (b) temperature profiles(c) concentration profiles.



13719

(a)

(b)

(c)

Figure 9. Influence of M on (a) velocity profiles (b) temperature profiles(c) concentration profiles.



13720

(a)

(b)

(c)

Figure 10. Influence of Pr on (a) velocity profiles (b) temperature profiles(c) concentration profiles.



13721

(a)

(b)

(c)

Figure 11. Influence of TS on (a) velocity profiles (b) temperature profiles(c) concentration profiles.



13722

(a)

(b)

(c)

Figure 12. Effect of on (a) Skin friction profiles (b) Nusselt number profiles (c) Shear wood number profiles



13723

(a)

(b)

(c)

Figure 13. Effect of TS on (a) Skin friction profiles (b) Nusselt number profiles (c) Shear wood number profiles



13724

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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