MHD Stagnation Point Flow of Viscoelastic Nanofluid Past a …ijens.org/Vol_19_I_06/192506-7373-IJMME-IJENS.pdf · 2020. 1. 16. · stagnation point chemically reactive MHD flow past

International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:06 145

192506-7373-IJMME-IJENS © December 2019 IJENS I J E N S

MHD Stagnation Point Flow of Viscoelastic

Nanofluid Past a Circular Cylinder with non-Fourier

heat Flux and Variable Fluid Properties Adigun, J. A1*., 2Adeniyan, A., 3Sobamowo, M. G.

1Bells University of Technology, Ota 2University of Lagos, Akoka 3University of Lagos, Akoka

*[email protected]

https://orcid.org/0000-0003-3834-1225

Abstract-- Stagnation point MHD flow of a viscoelastic

nanofluid over a stretching cylinder with temperature-

dependent dynamic viscosity is studied. Heat transfer is

characterized by Cattaneo-Christov heat flux and temperature

dependent thermal conductivity. The nonlinear governing

equations with velocity slip and thermal slip boundary

conditions are redacted and transmuted into dimensionless

forms by similarity transformations to obtain a set of coupled

non-linear ordinary differential equations which are

accordingly solved via the Spectral Quasi-linearization method

(SQLM). The usual heat source parameter is used and a new one

emerges from the heat generation/absorption body force

through mathematical derivation from first principle. The

upshots of the engrafted flow parameters on the dimensionless

velocity and temperature profiles, as well as on the skin friction

coefficient are analysed. Obtained outcome shows that

increasing values of the curvature parameter and viscoelastic

parameter enhances fluid velocity but nanoparticle volume

fraction slows it down. Temperature quashes for larger values

of thermal relaxation parameter, indicating that the non-

Fourier heat flux model produces lower temperature within the

flow system than that of the Fourier law. However, the nanofluid

temperature is raised by increasingly varying values of heat

generation parameter. The skin-friction coefficient exhibits an

increasing behaviour against both curvature and viscoelastic

parameters.

Index Term-- Catteneo-Christov heat flux; Cylinder;

Nanofluid; Quasi-linearization Method; Second grade fluid

1. INTRODUCTION

Applications of nanotechnology has been all-encompassing. Due to its higher thermal conductivity and

convective heat transfer rates, nanofluids are used in a wide

range of areas. Its applications to biotechnology include like

delivering drugs, heat, light or other substances to specific

types of cells in cancer therapy while in engineering and

industry at large, it's used in microelectronics and thermal

generation and as cooling agents in nuclear reactors and heat-

transfer devices in air planes, cars and microreactors among

others. However, the traditional fluids which would be

referred to as base fluids in this context, such as water,

mineral oils, ethylene glycol have very minimal appreciable

level of thermal conductivity. Since the main property of an ideal cooling fluid is its heat absorption (due to its high

thermal conductivity), nanofluids became most appropriate

for the aforementioned processes. Firstly, introduced by Choi

[1], a nanofluid is referred to as a two-phase mixture of a base

fluid with suspended particle solid phase which consists of

ultrafine particles. Nanoparticles could be metallic, non-

metallic or polymeric, could portray in rod-like, tabular and

spherical shapes and are less than 100nanometers in shape,

hence, behave more like fluids than a mixture. Usually, Al,

Cu, TiO2, Al2O3 and many others are used as nanoparticles.

Xuan and Roetzel [2], proposed a homogeneous flow model where the convective transport equations of pure fluids are

directly extended to nanofluids. This implied that all the

conventional heat transfer correlations could be used for

nanofluids provided the properties of pure fluids are replaced

by those of nanofluids involving the volume fraction of the

nanoparticles. A conflicting report to this homogeneous flow

model ensued from the experimental observations of Maliga

et al. [3] who considered the case of forced convection, as

they underpredict the heat transfer coefficients of nanofluids.

The basic flow in a nanofluid involves the effects of gravity,

Brownian force and the friction force between the fluid and

the ultrafine particles, the phenomena of Brownian diffusion, sedimentation and dispersion. Thus, although the

nanoparticles are ultrafine, the slip between the fluid and the

particles may not be zero. Later, Buongiorno [4] developed a

mathematical model which exhibits the thermophoretic

properties and Brownian motion of nanoparticles. Eastman et

al. [5] investigated the enhanced thermal conductivity

through the development of nanofluids and ascertained that

with 5% volume of CuO nanoparticles in water, there would

be an increase in thermal conductivity of the resulting

nanofluid by approximately 60%. This is ascribed to the

increase in surface area due to the suspension of nanoparticles. For example, copper (Cu) has a thermal

conductivity 700 times greater than water and 3000 times

greater than engine oil. Masuda [6] also had a similar

conclusion. Tiwari and Das [7] brought a new dimension to

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the study of nanofluids by studying the numerical modelling

of mixed convection in two-sided lid-driven differentially

heated square cavity filled with nanofluid. They developed

the model to analyse the behaviour of nanofluids by

considering the solid volume fraction and found out that

nanoparticles were able to change the flow pattern of a fluid from natural convection to forced convection regime. Das

and Jana [8] took an investigation of the hydromagnetic

boundary layer flow past a moving vertical plate in nanofluids

in the presence of a uniform transverse magnetic field and

thermal radiation. They established that fluid temperature

increases as volume fraction parameter enlarged. Also, the

thermal boundary layer for Cu–water was greater than pure

water.

Stagnation point flow has gained much attention

lately as a result of its diverse applications in industrial

processes and engineering procedures. These include

polymer extrusion, transpiration, oil recovery, nuclear reactors and production. The works of Massoudi and Rameza

[9] introduced a new dimension to its study when they

analysed the heat transfer characteristics of a boundary layer

flow of viscoelastic fluid towards a stagnation point.

Adeniyan and Adigun [10] analysed the convective plane

stagnation point chemically reactive MHD flow past a

vertical porous plate with a convective boundary condition in

the presence of a uniform magnetic field. Bachok et al. [11]

studied the stagnation point flow of a nanofluid over a

stretching or shrinking plate and observed in their research

that the skin friction and heat transfer coefficients are enhanced with nanofluid. Hayat et al. [12] discussed the

MHD stagnation point flow of Jeffrey nanofluid by a

stretching cylinder using the Buongiorno model. By applying

the Homotopy Analysis Method, they discovered that the

temperature and concentration profiles were reduced by the

Brownian movement but the thermophoretic parameter

enhanced the Nusselt number. Ogunseye et al. [13] studied

the mixed convection flow of a radiative

magnetohydrodynamic Eyring-Powell copper-water

nanofluid over a stretching cylinder. Aman et al. [14] studied

the steady two-dimensional stagnation-point flow over a

linearly stretching/shrinking sheet in a viscous and incompressible fluid in the presence of a magnetic field is

studied. Their results showed that the skin friction coefficient

decreases, but the heat transfer rate at the surface increases

when the effect of slip at the boundary is taken into

consideration. Also, dual solutions were found to exist for the

shrinking sheet, the solution was unique for the stretching

sheet.

Much attention in the past has been given to the law

of heat conduction suggested by Fourier [15]. The

presentation of its energy equation in parabolic form which

shows that the whole system is instantly influenced by the initial disturbance was its limitation. In a bid to undo this, in

1948, Cattaneo [16] modified the classical Fourier’s law

through thermal relaxation time. Subsequently, Christov [17]

proposed a material-invariant version of the Maxwell-

Catteneo law wherein the thermal relaxation time is given by

Oldroyd’s upper-convected derivative. He showed that the

new formulation allows for the elimination of heat flux, thus

yielding a single equation for the temperature field. This

expression is named as Cattaneo–Christov heat flux model.

This mechanism plays a critical role in several processes such as milk pasteurization and thermization, making of

microchips and some other electronic devices. Han et al. [18]

in his letter, presented a study on coupled flow and heat

transfer of an upper-convected viscoelastic fluid above a

stretching plate with velocity boundary slip. Contrary to most

classical works, he used the newly proposed Christov new

heat flux model. Also, Hayat et al. [19] employed the

Cattaneo-Christov flux model for heat transfer in the

stagnation point flow due to a stretching cylinder with

thermal stratification and found out that the temperature for

Cattaneo-Christov heat flux model is less than the Fourier's

expression. Dil and Khan [20] took a non-Fourier approach to model the heat transfer phenomenon in nanofluids having

application to the automotive industries.

Beg et al. [21] numerically studied the free

convection magnetohydrodynamic heat and mass transfer

from a stretching surface to a saturated porous medium with

Soret and Dufour effects and examined the combined effects

of Soret and Dufour diffusion and porous impedance on

laminar magnetohydrodynamic flow. Makinde [22] and

Mukhopadhay [23] also incorporated the influence of

saturated porous medium into the study of MHD flow and

heat transfer past a vertical plate and cylinder respectively. Abel-Waheed and El-Said [24] studied the MHD flow and

heat transfer over a moving cylinder in a nanofluid under

convective boundary conditions and heat generation.

Mabood and Das [25] wrote an article to discuss the analysis

of MHD flow and melting heat transfer of a nanofluid over a

stretching surface wherein they used a second-order slip

model and thermal radiation. Nandeppanavar et al. [26] also

analysed the MHD flow and heat transfer over a stretching

sheet with non-linear Navier boundary condition. They

investigated the effects of the first and second-order slip

parameters on the fluid flow using both the prescribed surface

temperature and prescribed heat flux boundary conditions. Goyal and Bhargava [27] examined the effect of viscoelastic

parameter velocity slip boundary on the flow and heat

transfer of non-Newtonian nanofluid over a stretching sheet

with heat source/sink, under the action of a uniform magnetic

field. They employed variational finite element method

(VFEM) to obtain numerical solutions to the problem and

discovered that the viscoelastic parameter increased the

velocity profiles. Hayat et al. [28] investigated the heat and

mass transfer analysis in mixed convective radiative flow of

Jeffrey fluid over a moving surface with the effects of thermal

and concentration stratification. Makinde et al. [29] investigated the combined

effects of thermal radiation, thermophoresis, Brownian

motion, magnetic field and variable viscosity on boundary

layer flow, heat and mass transfer of an electrically



conducting nanofluid over a radially stretching convectively

heated surface but did not consider the case for a cylinder

where the curvature parameter is not zero. Hayat et al. [30]

took a comparative study for the flow of viscoelastic fluids

with Cattaneo-Christov heat flux but did not consider the case

of variable viscosity and thermal conductivity. Khan et al. [31] explored the stagnation point flow of second-grade fluid

towards an impermeable stretched cylinder with Non-Fourier

heat flux and thermal stratification but neglected the influence

of magnetic field. Hayat et al. [32] addressed double stratified

mixed convection stagnation point flow induced by an

impermeable inclined stretching cylinder with both thermal

and solutal buoyancy forces. Mahat et al. [33] examined the

steady case of two-dimensional convection boundary layer

flow of a viscoelastic nanofluid over a circular cylinder but

both [32, 33] limited their research to the case of the Fourier

law. Hence, the main goal of this research is to do a numerical

investigation of the effects of Cattaneo-Christov non-Fourier heat flux, variable viscosity, variable thermal conductivity,

velocity and thermal partial slip boundary conditions on the

MHD flow of nanofluid past a vertically positioned stretching

cylinder, using the Spectral Quasi-Linearization Method

(SQLM). To the best of our knowledge, this problem is yet to

be considered, haven gone through existing literature.

2. MATHEMATICAL FORMULATION

Consider a steady two-dimensional, laminar and

incompressible hydromagnetic boundary layer flow of a

viscous, electrically conducting and chemically reactive

nanofluid past a vertical circular cylinder of infinite extent embedded in a uniform porous medium. A magnetic field is

of strength 𝐵𝑜 is applied normal to the surface and parallel to

the 𝑟-axis and the induced magnetic field due to the motion

of a conducting nanofluid is neglected. The flow is invoked

by the stretching of the free stream fluid along the axial

cylinder along the axial direction (𝑧) with velocity e zU

due to the application of two but opposite forces localized

along the axial direction of the cylinder and far from the

cylinder.

Fig. 1. Physical flow geometry

The following equations govern the present flow

consideration

2.1 Equation of conservation of mass

0 v . (1)

2.2 Equation of motion:

d

tb

d

vτ , (2)

where d

dt

v is the material derivative and the Cauchy stress

tensor τ for the second-grade fluid, given by Rivlin and

Ericksen [34] is

21 1 2 2 1p τ I + A A A

, (3)

where p I is the spherical stress due to incompressibility

constraint in which p is the fluid pressure and I is the unit

tensor of order two, 1 2, are material constants. 1A and

2A represent first and second Rivlin-Ericksen tensors of

order two which are defined as

1

T

v vA and 12 1 1

Td

dt

v vA

A A A (4)

*T represents transpose.

Besides, for consistency of the model with thermodynamics,

it is assumed that when the fluid is locally at rest i.e. at

equilibrium, the specific Helmholtz free fluid energy is

minimal, and thus

1 2 10, 0, . (5)



In the cylindrical coordinates,

1

1

1

u u v u

r r r z

v v u v

r r r z

w w w

r r z

v = and

1

21

1

r rrrr zr

r z r

zrz zz rz

r r z r

r r z r

r r z r

τ = (6)

Utilizing the velocity field ( , z), 0, ( , z)u r w rV and the

Prandtl boundary-layer approximations for flow theory

within the boundary layer, 2

2, , ,

u u pu

z zz

are taken to be of

order 1 i.e. (1)O and ,r u to be of order , ( )V VO

respectively, where V is the thickness of boundary-layer

near the cylinder surface at 𝑟 = 𝑎. Thus, the governing equations of momentum and continuity reduce to

( ) ( ) 0ru rwr z

(7)

20

3 2 2

3 2 2

12 2

2

1( )

( )

1

nf nf

nf nf

nf

T nfP

ee

e

e

dw ww u

z r dz

wr T B w

r r r

Tw g T T

K

w w w uu w

z rr r r

w w w w w uu w

r z r z r r rr

UU

U

U (8)

and the applicable boundary conditions are

( , ) , 0, ats ww z a u U z u r a , (9)

,( s, a) ew z r z r U . (10)

In the aforementioned expressions, w and u are the

velocity components in the z and r directions respectively,

PK is the permeability of the porous medium, L is the

characteristic length, nf is the density of nanofluid, ( )nf T

is the temperature-dependent dynamic viscosity of the

nanofluid, nf is the kinematic viscosity of the nanofluid,

nf is the electrical conductivity of the nanofluid and a is

the radius of cylinder, 0w

U zU z

L and e

eL

zU z

U are

velocities with which the cylinder and free-stream stretch

linearly. Also, 2

2

w wu A B

s r r

is the nonlinear Navier

(second-order) slip velocity, where

23

4 2 *2

2

*3(12 3 1 21

3 2 4

), ,

n n

LLA B L L

K K

1min ,1

n

LK

, 0,1 is momentum accommodation

coefficient, * 0 is molecular mean free path,

*

nKL

is Knudsen number. It is known empirically that

for any value of nK , the characteristic length of the

geometry flow field 0,1 ,L and consequently constants

0A and 0B always. The thermophysical properties of

the nanofluid showing relationships between various

properties of the nanoparticles and base fluid are given below

[8], Sobamowo [35], Makinde et al. [36], Pandey and Kumar

[37]:

2.5

3 1

1 , 1 ,

2 1

1 ,

2 2, ,

21

( ), 1 (11)

s

nf f s nf f

s s

f f

nf f s

s f f sf nf nf

nf nfnf f s f f s

nf

nf P P Pnf f sP nf

k k k kk

k k k k k

k TC C C

C



Table I

Thermophysical properties of water and nanoparticles Das and Jana [8].

Physical properties Water/base fluid Cu (copper) 𝐴𝑙2𝑂3 (Alumina) 𝑇𝑖𝑂2 (Titanium oxide)

𝜌 (𝑘𝑔/𝑚3) 997.1 8933 3970 4250

𝑐𝑝 (𝐽/𝑘𝑔𝐾) 4179 385 765 686.2

𝑘 (𝑊/𝑚𝐾) 0.613 401 40 8.9538

𝛽 × 105 (𝐾−1) 21 1.67 0.85 0.90

𝜑 0.0 0.05 0.15 0.2

𝜎 (𝑆/𝑚) 5.5 × 10−6 59.6 × 106 35 × 106 2.6 × 106

Here, P nfC is the heat capacitance of nanofluid, m is

shape factor, is nanoparticle volume fraction, subscript s

is nano solid particle, subscript f is base fluid, and subscript

nf is nanofluid. However, the dynamic viscosity of the of the

water base fluid which was proposed by Lings and Dybbs

[38] and seen in Animashaun and Aluko [39], Choudhury and

Hazarika [40], Prasad et al [41], Eldabe et al. [42] is

temperature-dependent and given as 11

( )f

T T

T

,

wherein

is a constant and is the freestream viscosity.

Considering the derivation of the dimensionless viscosity

variation parameter, let b T , and so nf could be

finally expressed as

2.5

1 1nf

b

. (12)

Implementing the similarity transformation, 2 2

2

Wr a

a z

U and the stream function ( )Wa z f U

, such that ur

and 1

vr z

, the continuity equation

is verified identically, and the momentum equation with its

boundary conditions transmutes as

2.5 2.5

32.5

2 21

2

22 2.5

1 11 2 2

(1 ) (1 )

11 2 (1 ) M

1 (1 )

(1 ) M 4 (1 )

(1 ) 1 2 2

1M (1 )

(1 )

iv

f f

bf b

b

f fb ff f b

ff

b f f f ff

b Ha Da

( ) 0 (13)f

**'(0) 1 ''(0) '''(0), (0) 0, '( )f f f f f (14)

where0P f

LDa

K U

is the Darcy parameter, 1 0

f

U

L

is

the viscoelastic parameter, 0UA

L

is the first order slip

parameter and ** 0U

BL

is the second order slip

parameter,

20

0

f

f

B LHa

U

is the Hartmann parameter,

Re

z

z

Gr is the mixed convection parameter,

3( )T f

z

g TzGr

is the local thermal Grashof number

and

20Rez

U z

L is the local Reynolds number,

0

e

U

Uis

the velocity ratio and

1 31( )

, 1( )

s

f

T s

T f

M M

2

3 1

1

2 1

s

f

s s

f f

M

are constants obtained from the thermophysical parameters in

Equation (11).

2.3 Energy equation

The energy balance equation, as dictated by the first law of

classical thermodynamics, takes the form Lukaszewicz,[43]:

: ,nf

TT

t

v q vτ (15)



where the second term on the right of (15) represents the

energy due to various sources such as body forces. It is taken

in this case as heat generation or absorption in the nanofluid

and modelled as 0Q Q T T . Additionally, ( )ijτ τ is

the Cauchy stress tensor, q is the thermal flux, : vτ is the

dyadic notation for ,v ,ij j iτ called the scalar product of τ

and v . Lastly,,

vv

j

j iix

v is a tensor of second rank.

Assuming the non-Fourier Cattaneo-Christov thermal flux,

Cattaneo [16], and Christov [17] instead of the conventional

Fourier’s type:

o nfk T Tt

qv q q v v q qτ ,

0nfk T , (16)

where oτ is the thermal flux relaxation time, ( , r)T z is

nanofluid temperature and nfk T is the temperature-

dependent thermal conductivity. The case of the Fourier's law

arises when 0o τ in equation (16). Considering the

steadiness of the flow and applying the criterion of

incompressibility i.e 0 v , equation (16) reduces to

o nfk T T v q q v qτ (17)

On further simplification, the energy equation yields

1 1nf r

P Pnf nf

Tk T r rq

r r rC r C r

0

P nf

Q T TT T w u

z rC

2 2 22 2

2 2

0

2

o

P nf

T T Tw u wu

z rz r

w w T u u Tw u w u

z r z z r r

Q T Tw u

z rC

(18)

with boundary conditions

1( ) , at , (19)w

TT T z e r T T asa r

r

where 0( ) Tw

zT z T

L

.

It is noteworthy to mention that most recent authors (see

Dogonchi and Ganji [44], Kahn et al. [45], Mahanthesh et al.

[46]) incorporated the heat generation/absorption and/or

thermal radiation into their studies without due consideration

for mathematical derivation from the first principle. In

consequence, their energy balance equations seemed a bit of

misnomer. They lost the last term on the RHS of equation

(18).

0T is a constant which is a measure of temperature. The

nanofluid temperature which was suggested by Ostrach [47]

and used by Sharma [48] is given as ( )

Nfk T T

k T

, while

the temperature difference ratio /T T . ( )fk T could

be recast as

( ) 1N

fk T k . (20)

Here, k is the freestream thermal conductivity. For 1N ,

the range of variation of value of takes the form Elbarbary

and Elgazery [49]: 0 6, for air; 0 0.12, for

water and 0.1 0, for lubrication oil.

The heat radiation flux approximated by Rosseland is given

as

* * 44

* *

4 4

3 3r

Tq T

rk k

(21),

as the flux is radial from the surface of the cylinder, and

dependent on r only. Here * 2 4Wm K

is the Stefan-

Boltzmann constant and * 1k m

is the Rosseland mean

spectral absorption coefficient. It is further assumed that the

term 4T due to radiation within the flow can be expressed as

a linear function of temperature itself. Hence, 4T can be

expanded as Taylor series about T and can be approximated

after neglecting the higher order terms as,

4 3 4 3T T T T , (22)

so that

434

T TT

r r

.

It is gainful and instructive to follow Magyari &

Pantokratoras [50], wherein the effective heat flux

accounting for both thermal conduction and radiation heat

transfer is

( )eff nfeff

Tk T

r

q , (23)



where *

3

*

16( ) ( )

3nfeff nfk T k T T

k

(24)

is the nanofluid effective thermal conductivity. It is worthy to

mention that in this work, one drops the subscript of effq

2Wm

in equation (23) without dropping the meaning as

per effective conductivity.

In order to reflect the influence of radiation, equation (16)

now looks like

o effnfk T Tt

qv q q v v q qτ

In view of this modification of energy equation (18), the

first two terms on the left take the form

1 1

1( ) (25)

nf r

P Pnf nf

nfeff

P nf

Tk T r rq

r r rC r C r

Trk T

r rC r

where 1 1( )nfeffk T Wm K

is given by (24) and by virtue

of equation (25), equation (18) takes the form

2 2 22 2

2 2

0

0

2

1( ) (26)

o

P nf

nfeff

P Pnf nf

T T Tw u wu

z rz r

w w Tw u

z r z T Tw u

u u T z rw u

z r r

Q T Tw u

z rC

QTrk T T T

r rC r C

Employing (20) and w

T T

T T

, dimensionless forms of

equation (26) and (19) is given as

1 2

5

(1 2 ) 1 '' 1 '(27)

2 1 '

n n

n

nM

*

4

4 2 2

' ' ' '

Pr 0' '' ''

*' '

eff

Qf f f f

MQ M

f f ff

ff

1(0) 1 (0), ( ) 0 , (28)

where

4 51

2 2,

2

P s

P f

s f f s

s f f s

C

C

k k k kM M

k k k k

316

3

TR

k k

is the thermal radiation parameter,

Pr

pf

C

k

is the Prandtl number,

5

Pr 1Pr

1

n

eff nM R

is the effective Prandtl number,

0

0P f

Q LQ

C

U is the zeroth-order heat source/sink

parameter and * 0oU

L

is the thermal relaxation

parameter.

Also, originating from this study is

* 0 0

P f

QQ

C

, which is

the heat source/sink parameter of first order.

In addition, the skin friction coefficient is defined as

2,w

f

f w

CU

(29)

2

1 2

where

(30)w nf

r a

w w w w uu w

r z r r rr

3. METHOD OF SOLUTION

The general nonlinear, strongly coupled system of ordinary

differential equations defined by (13) and (27), with

conditions (14) and (28) are to be solved using the SQLM

technique, as described by Motsa [51] and Otegbeye [52].

The Taylor series for univariate function is applied to

linearize the nonlinear system, thereafter, integrated by the

Chebyshev Spectral Collocation method. An initial guess which satisfies the imposed boundary conditions is taken and

this ensures the nippy convergence and accuracy of the

obtained solutions.

Equations (13) and (27) can be expressed in their

decomposed form as a sum of both the linear and non-linear



constituents. Employing Taylor's series to linearize, the

iterative schemes (31) and (32) are obtained as

1, 1 2, 1 3, 1 4, 1 5, 1

6, 1 7, 1 (31)

'''' ''' '' '

'

n n n n n n n n n n

fn n n n n

f f f f f

R

8, 1 9, 1 10, 1 11, 1

12, 1 13, 1

'' ' ''

'

n n n n n n n n

n n n n n

f

f f R

(32)

with corresponding boundary conditions:

**1 1 1 1

1 1 1 1 1

(0) 0, (0) 1 (0),

( ) 0, (0) 1 (0), ( ) 0 (33)

' '' '''

' '

n n n n

n n n n

f f f f

f

where the coefficients , ( 1, ...,13)i n i , are known functions

from previous iterations and are given by,

1, ( 1)(2 1)'n n nf b ,

2, 2.5

1(2 1) 4( 1) 2 ( 1)(2 1)

1

'n n n n nb f b f

3, 2.5 2.5

1

(2 1)1 12

11 1

( 1) 4 2 (2 1)

'

' ''

n nn

n n n n

b

b

b M f f f

,

4, 1 2

2.5

( 1) 2 4 2 (2 1)

1

1

' '' '''n n n n nb M f f f M Ha

Da

5, 1 1( 1) 4 (2 1)'' ''' ''''n n n n nb M f f f

.

''b(2 1)16, 2.5 11

fn

n bn

22 '' ' ''' ' ''

7, 1

2'''' ' ''' ''

3 3

2'' ' '

22.5 2

4

(2 1) 2 ( 1)

(2 1)1,

1 1

n n n n n n n n

n n n n n n n

n n n

n

M b f f f b f f f f

b f f f f f M b M b

bf bM Ha f

b

2*

8, 5 4(2 1)( 1) Prn n eff nM b M f

19, 5 2(2 1) ( 1) 2 ( 1)'n n

n n n nM n

*

*4 4

4

Pr Pr Pr' neff n eff n n

Q fM f M f f

M

212 '''

10, 5

'

* '2

* ' ''4 4

4

1 ( 1)1(2 1)

1 1

2 1

1

PrPr Pr Pr'

nnn nn n

n n

n

n

n n

n

eff n

eff eff n eff n n n

n nn

M

n

Q fQ M f M f f f

M

*

11, 4 Prn eff n nM f

*

*12, 4 4

4

PrPr Pr 2 ' ' eff n

n eff n eff n n n n

QM M f f

M

*

*13, 4 4

4

PrPr Pr 2

'' '' '' ' ' eff n

n eff n eff n n n n n n

QM M f f f

M

'''' ''' '' '1, 2, 3, 4, 5,

'' ,6, 7,

'' ' ''8, 9, 10, 11,

' , (34)12, 13,

fR f f f f f

n n n n n n n n n n n

n n n n f

R fn n n n n n n n n

f fn n n n

1 '''1 22.5(1 )

1 ' ''1 22.51 (1 )

2 ''(1 ) (1 )3 1

' ''1 ''4 1 2

2.5''' (1 )

' ''' 2 '''1 1 2 2

(1 )2

ff n

bf

n nbn

b M b M f fn n n n n

f fn n

b fn n

f fn n

b f f f f fn n n n n

M bn

(35)

1 '

2.5(1 )

''1(1 2 )

1 ' 216

'2 1

Preff

Ha Da fn

n

n n

nM n

n n

n

n n

Qn



Equations (31) – (33) form the SQLM iterative scheme,

which is a coupled linear system of equations, where

n terms are initial (previous) approximations and ( 1)n

terms are next (current) approximations. These would, in

turn, be solved by the Chebyshev pseudo-spectral technique

Canuto et al [53]. The results for 1nf and 1n , when

1, 2, 3, ..n are iteratively computed by initializing the

iterative algorithm with initial approximations. At this point,

the method of Chebyshev pseudo-spectral collocation is

applied and equations (31) and (32) are discretized using the

steps listed below:

truncate the semi-infinite domain 0, by replacing

it with 0, , where ¢ .

using the transformation 1

12

, transform the

interval 0, 1,1 a

compute the variables ( )f and ( ) using Chebyshev

differentiation matrix D , at the collocation point as a

matrix-vector product, that is:

0

( ) F, 0,1, 2, ..., ,

N

ij i

i

dfD f j N

d

D (36)

where 1N is the number of collocation points, 2D D

and 0 1( ), ( ), ..., ( )T

NF f f f is a vector function at the

collocation point. The Gauss-Lobatto points are selected to

define the nodes in 1, 1 as:

cos , 0,1, ..., ; 1 1k

kk N

N

(37)

In the same vein, we represent with alike vector function

. Higher-order derivatives of f and are evaluated as

powers of D , that is

( )s sf F D and ( )s s D , (38)

where s is the order of the derivative. Substituting equations

(35), (36) and (37) into (30) and (31), the following SQLM

scheme is obtained in a matrix form as:

1,1 1,2 1

2,1 2,2 1

fn n

n n

F R

R

(39)

where ( , 1, ..., 2)ij i j are 1 1N N matrices and

fnR and nR

are 1 1N vectors, defined by

4 3 21,1 1, 2, 3,

4, 5,

1,2 6, 7,

22,1 11, 12, 13,

22,2 8, 9,

n n n

n n

n n

n n n

n n

diag diag diag

diag diag

diag diag

diag diag diag

diag diag diag

= D D D

D

=

= D D

= D D

I,

D I

I,

10,

(40)

n

I,

subject to the boundary conditions

2 ** 31 1

0

0 1 0

0

1 1 1 0

0

( ) 0, ( ) 1,

( ) , (41)

( ) 1, ( ) 0

N

n nN Ni Ni Ni N

i

N

i n

i

N

n nNi Ni N

i

F F

F

D D D

D

I D

where I is the identity matrix and

1, 13,...n ndiag diag are diagonal matrices.

Below is an initial approximation for the SQLM scheme

which is found appropriate

0 **

01

1( ) 1 ,

1

1( )

1

f e

e

(42)

4. RESULTS AND DISCUSSIONS

With the aid of graphical illustrations and tables, the obtained

numerical results are crystallized for the parametric study of

the considered problem to have a good grasp of the physics

of the problem. For several values of the embedded

controlling parameters, the dimensionless fluid velocity

profiles and fluid temperature profiles have been presented in

Fig. (2) to (22), whereas table 3 shows their influence on the

skin friction coefficient. The accuracy of the numerical scheme is tested and shown in table 2, when compared with

the results of Ishak et al. [54].

4.1 Effects of variation of parameters on the velocity profiles

Fig. 2 depicts the effect of solid volumetric fraction of

nanoparticles on the fluid velocity. The fluid velocity together

with the momentum boundary layer thickness increased for

increasing values of . Fig. 3 shows the deportment of the

curvature parameter γ on the nanofluid velocity. Ordinarily,

any increase in curvature parameter would reduce the cylinder’s radius, which in turn reduces the cylinder’s surface

area which is in direct contact with the nanofluid particles.

Hence, the resistance offered to the fluid particles decreases

and velocity profiles increase accordingly. The effect of



viscosity variation parameter on the dimensionless velocity is

presented in Fig. 4. It is important to note that an increase in

viscosity parameter represents a decrease in the nanofluid

viscosity. The same escalating convention was observed as

regards the boundary layer thickness due to higher

temperature difference between the surface and the ambient fluid. The trend observed when the velocity ratio parameter

is being increased is of two-fold as pictured in Fig. 5. A

stratified pattern which showed increament in the velocity

profile of the nanofluid but thinning out of the momentum

boundary layer was ascertained. Fig. 6 is plotted to show the

influence of viscoelastic parameter β on the velocity profile.

Both velocity and associated momentum boundary layer

thickness increase when β increases. This is coherent with the

expression for β in its dimensionless expression, which

shows that β increases as the viscosity decreases. So, fluid

moves easily and as a result, velocity profile increases. From

Fig. 7, the velocity and momentum boundary layer thickness were observed to increase for large values of λ. The ratio of

buoyancy to inertial forces defines the Mixed Convection

parameter. It is, however, noteworthy that when λ = 0, the

mixed convection parameter is absent, implying a forced

convection flow and flow is opposed when 0. Also,

0 indicates that flow is aided when heat is convected

from the surface of the cylinder to the fluid flow i.e. cooling

of the cylinder surface or heating the fluid. With an increase

in , buoyancy forces increase. Therefore, the velocity of

the fluid increases. The effects of Magnetic field parameter,

Ha , on the velocity is limned in Fig. 8. It is right to

anticipate that increasing hydromagnetic drag bottles up the

flow (subjugates velocity values). The resistive Lorentz force

opposed the velocity at which fluid flowed and caused a

decrease in the boundary layer. Fig. 9 epitomizes the effects

of reducing how porous the medium is, by making some

increament in the porousity parameter. A directly

consequential resistance against the flow of the nanofluid was

observed in the flow system and the resultant effect is a

reduction in fluid velocity. As reported by Singh and

Makinde [55], the same trend is seen in Fig. 11 for the

variation of the First Order Slip parameter. It showed that both the fluid velocity and boundary layer thickness decrease

with an increase in , which happens because fluid

experiences less drag with increase in . However, the trend

was just the opposite for the Second Order Slip parameter in

Fig. 10.

4.2 Effects of variation of parameters on the temperature

profiles

All profiles decay smoothly from maximum values at the wall

to zero in the free stream (edge of the boundary layer). Like

the report of Makinde et al. [36], Fig. 12 expatiates the import

of heightening the volumetric fraction of the nanoparticle on the temperature. Fluid temperature and boundary layer

thickness rise with an increase in the values of . This is

imputable to the enhanced thermal conductivity of the base

fluid. Study proves that thermal conductivity of base fluid can

be substantially bettered by ducking of copper nanoparticles.

The upshot of variable thermal conductivity parameter 𝜖 is

evinced in Fig. 13. Fluid temperature is maximized with

larger values of 𝜖 in the case of injection. This is ascribable to the physical fact that thermal conductivity of the nanofluid

increases due to a large amount of heat which is transferred

from the cylinder surface to the nanofluid. In Fig. 15 and Fig.

16, the temperature field increases with the increment of heat

source parameter. Due to exothermic reaction, the heat

releases more rapidly which increases the temperature

profile. The effect of the effective Prandtl number on

is outlined in Fig. 17. A rise in the effective Prandtl number

from 0.02 (liquid metal) through and through to 0.7 (air) to

1.0 and lastly 7.0 (water). When effective Prandtl number is

increased, there is a thinner thermal boundary layer thickness

and more uniform temperature distributions across the

boundary layer is seen. Generally, fluids which have lower Prandtl number possess higher thermal conductivities and so

heat would diffuse away from the cylinder surface at a faster

rate. The thermal relaxation time is the delay(lag) required

for the beginning of heat flux at some point once a

temperature gradient is commenced. Thus, for larger values

of thermal relaxation, particles required more time to transfer

energy to adjacent fluid particles and therefore temperature

profiles decline. Fig. 18 expatiates this. Due to increased

tightness of the porous medium as the porousity parameter is

raised, there is more resistance to the fluid flow and thus the

heat is transferred from the hot surface of the cylinder and finally, the temperature gets higher, as shown in Fig. 19. In

Fig. 20, the thermal slip parameter is seen to increase the

temperature and the boundary layer.

The effects of the emergent parameters on the skin-friction

coefficient is shown in Table 3. The obtained values are

mostly negative, which measures the drag of the fluid on the

cylinder wall. It was ascertained that increase in 𝜑, 𝜆, 𝑏 and 휀

all increased the skin-friction coefficient but increasing

values of 𝐷𝑎 and 𝐻𝑎 reduced it.



Table II

Comparison data of the Nusselt number reported by Ishak et al.[54] with SQLM for distinct values of 𝑃𝑟 when 𝛾 = 𝑏 = 𝛽 = 𝑁 = 𝐻𝑎 = 𝐷𝑎 = 𝛾∗∗ = 𝜖 =𝑛 = 𝛿 = 𝑅 = 𝑄 = 𝛾∗ = 𝑄∗ = 𝛬1 = 𝛬2 = 0, 𝜆 = 휀 = 𝑀1 = 𝑀2 = 𝑀3 = 𝑀4 = 𝑀5 = 𝑀6 = 1

𝑃𝑟 Ishak et al. [54]

𝑓′′(0)

Present (SQLM)

𝑓′′(0)

Absolute Difference Ishak et al. [54]

−𝜃′(0)

Present (SQLM)

−𝜃′(0)

Absolute Difference

0.72 0.3645 0.3645 0 1.0931 1.0931 0

6.8 0.1804 0.1804 0 3.2902 3.2895 7 × 10−4

40 0.0873 0.0872 1 × 10−4 7.9463 7.9383 8 × 10−3

60 0.0729 0.0727 2 × 10−4 9.7327 9.7180 1.47 × 10−2

100 0.0578 0.0587 9 × 10−4 12.5726 12.5411 3.15 × 10−2

Table III

Numerical values of Skin friction coefficient for distinct values of φ, b, γ∗∗, δ, Da, Ha, λ and ε when Λ1 = Q = Pr = R = Q∗ = γ∗ = n = ϵ = 0.1

𝜑 𝑏 𝛽 𝐷𝑎 𝐻𝑎 𝜆 휀 𝑓′′(0)

0.1 0.1 0.1 0.1 0.1 0.1 0.1 -1.52455507

0.5 -1.34270035

0.6 -1.11152139

0.7 -0.87146927

1.9 -2.67909042

2.0 -2.65337032

2.1 -2.63795113

0.1 -1.52455507

0.2 -1.70461236

0.35 -1.83520588

0.5 -1.73978129

1.0 -2.00118374

1.5 -2.26099116

0.2 -1.54293550

0.5 -1.66978531

1.0 -2.10963477

0.05 -1.56285503

0.09 -1.53214172

0.15 -1.48710704

0.20 -1.43767682

0.22 -1.42804568

0.25 -1.39736064

Fig. 2. Effect of Volumetric Fraction of Nanoparticle on the velocity

profile

Fig. 3. Effect of Curvature parameter on the velocity profile



Fig. 4. Effect of Variable Viscosity parameter on the velocity profile

Fig. 5. Effect of Velocity Ratio parameter on the velocity profile

Fig. 6. Effect of Viscoelastic parameter on the velocity profile

Fig. 7. Effect of Mixed Convection parameter on the velocity profile

Fig. 8. Effect of Magnetic parameter on the velocity profile

Fig. 9. Effect of Porousity parameter on the velocity profile



Fig. 10. Effect of Second-Order Velocity Slip parameter on the velocity

profile

Fig. 11. Effect of First Order Velocity Slip parameter on the velocity

profile

Fig. 12. Effect of Volumetric Fraction of Nanoparticle on the

temperature profile

Fig. 13. Effect of variable thermal conductivity parameter on the

temperature profile

Fig. 14. Effect of Index of temperature ratio on the temperature profile

Fig. 15. Effect of Heat Source/Sink parameter of zeroth-order on the

temperature profile



Fig. 16. Effect of Heat Source/Sink parameter of first order on the

temperature profile

Fig. 17. Effect of the effective Prandtl number on the temperature profile

Fig. 18. Effect of Thermal Relaxation parameter on the temperature

profile

Fig. 19. Effect of Porousity parameter on the temperature profile

Fig. 20. Effect of Temperature Slip parameter on the temperature profile

5. CONCLUSION

The problem of stagnation point MHD flow of viscoelastic nanofluid past a circular cylinder with non-Fourier heat

flux and variable properties has been addressed

numerically with the aid of the iterative Spectral

Quasilinearization method. More importantly, the briny

upshots are summarized as follows:

This appears, as far as we know, that the “heat

source/sink parameter of first order” is new; and

perhaps it could be tested experimentally.

Velocity profiles were substantively enhanced

with increasing values of the volumetric fraction

of the nanoparticle and the curvature parameter

but was cut back by the variable viscosity parameter and porousity parameter.

The variable thermal conductivity parameter had

depreciating effects on the temperature profile,

but the temperature slip parameter supported it.

An increase in both the velocity ratio parameter

and first order velocity slip increased the skin-



friction coefficient but the variable viscosity

parameter and porousity parameter decreased it.

The case of Fourier law which arises when 𝛾∗ =0 has a higher temperature than that of non-

Fourier heat flux model

REFERENCES [1] Choi, S. (1995): Enhancing thermal conductivity of fluids with

nanoparticles. ASME Int Mech Eng, 66:99–105.

[2] Xuan, Y., Roetzel, W. (2000): Conceptions for Heat Transfer

Correlation of Nanofluids. International Journal of Heat and

Mass Transfer, 43: 3701-3707.

http://dx.doi.org/10.1016/S0017-9310(99)00369-5.

[3] Buongiorno, J., Hu, L., (2009): A benchmark study on the

thermal conductivity of nanofluids. Journal of Applied

Physics, 106: 094312-1.

[4] Eastman, J. A., Choi, S. U. S., Li, S., Thompson, L. J., Lee, S.

(1997): Enhanced thermal conductivity through the

development of nanofluids, in: S. Komarneni, J.C. Parker, H.J.

Wollenberger (Eds.), Nanophase and Nanocomposite

Materials II, Materials Research Society, Pittsburgh.

[5] Masuda, H., Ebata, A., Teramae, K., Hishinuma, M. (1993):

Alteration of thermal conductivity and viscosity of liquids by

dispersing ultrafine particles. Netsu Busse, 7: 227–233.

[6] Tiwari, R. K., Das, M. K. (2007): Heat transfer augmentation

in a two-sided lid-driven differentially heated square cavity

utilizing nanofluids. International Journal of Heat and Mass

Transfer 50: 2002–2018.

[7] Das, S., Jana, R. N. (2015): Natural convective magneto-

nanofluid flow and radiative heat transfer past a moving

vertical plate. Alexandria Engineering Journal 54: 55 – 64.

[8] Massoudi, M., Rameza, M. (1990): Boundary layer heat

transfer analysis of a viscoelastic fluid at a stagnation point.

ASME J Heat Transfer, 130:81–86.

[9] Adeniyan, A., Adigun, J.A. (2013): Analysis of convective

plane stagnation point chemically reactive MHD flow past a

vertical porous plate with a convective boundary condition in

the presence of a uniform magnetic field. American Journal of

Engineering Research, 2(9): 234 – 243.

[10] Bachok, N., Ishak, A., Pop, I. (2011): Stagnation-point flow

over a stretching/shrinking in a nanofluid. Nano Scale Res

letter, 6:623–630.

[11] Hayat, T., Kiyani, M. Z., Ahmad, I., Khan, M. I., Alsaedi, A

(2018): Stagnation point flow of viscoelastic material over a

stretched surface. Results in Physics, 9: 518 – 526.

[12] Ogunseye, H. A., Sibanda, P., Mondao, H., (2019): MHD

mixed convective stagnation-point flow of Eyring-Powell

nanofluid over stretching cylinder with thermal slip

conditions. J. Cent. South Univ. 26: 1172−1183.

[13] Aman, A., Ishak, A., Pop, I. (2013): Magnetohydrodynamic

stagnation-point flow towards a stretching/shrinking sheet

with slip effects. International Communications in Heat and

Mass Transfer, 47: 68–72.

[14] Fourier, J. B. J. (1822): Théorie Analytique De La Chaleur,

Paris.

[15] Cattaneo, C. (1948): Sulla conduzione del calore, Atti Semin.

Mat Fis Univ Modena Reggio Emilia, 3: 83–101.

[16] Christov, C. I. (2009): On frame indifferent formulation of the

Maxwell-Cattaneo model of finite-speed heat conduction.

Mechanics Research Communications, 36: 481 – 486.

[17] Han. S., Zheng, L., Li, C., Zhang, X. (2014): Coupled flow and

heat transfer in viscoelastic fluid with Cattaneo–Christov heat

flux model. Applied Mathematics Letters. 38: 87–93,

http://dx.doi.org/10.1016/j.aml.2014.07.013

[18] Hayat, T., Khan, M. I., Waqas, M., Alsaedi, A. (2017): On

Cattaneo-Christov heat flux of variable thermal conductivity

Ering-Powell fluid. Results in Physics, 7: 446 – 450.

[19] Dil, T., Khan, M. S. (2018): A non-Fourier approach towards

the analysis of heat transfer enhancement with water-based

nanofluids through a channel. AIP Advances 8, 055311

(2018). https://doi.org/10.1063/1.5005870.

[20] Beg, O, A., Bakier, A. Y., Prasad, V. R. (2009): Numerical

study of free convection magnetohydrodynamic heat and mass

transfer from a stretching surface to a saturated porous

medium with Soret and Dufour effects. Computational

Material Science 46: 57 – 65.

[21] Makinde, O. D. (2009): On MHD boundary-layer flow and

mass transfer past a vertical plate in a porous medium with

constant heat flux. International Journal of Numerical Methods

for Heat and Fluid Flow, 19 (3 – 4): 546 – 564.

[22] Mukhopadhyay, S. (2012): Analysis of boundary-layer flow

and heat transfer along a stretching cylinder in a porous

medium. International Scholarly Research Network (ISRN),

Thermodynamics, 2012: 1 – 7.

[23] Abdel-Wahed, E. (2018): MHD flow and heat transfer over a

moving cylinder in a nanofluid under convective boundary

conditions and heat generation. Thermal Science International

Scientific Journal, pp 1 – 14,

doi.org/10.2298/TSCI170911279A.

[24] Mabood, F., Das, K. (2015): Melting heat transfer on

hydromagnetic flow of a nanofluid over a stretching sheet with

radiation and second-order slip. The European Physical

Journal, Eur. Phys. J. Plus (2016), 131: 3.

https://doi.org/10.1140/epjp/i2016-16003-1.

[25] Nandeppanavar, M. M., Vajravelu, K., Abel, M. S.,

Siddalingappa, M. N. (2012): Second order slip flow and heat

transfer over a stretching sheet with non-linear Navier

boundary condition. International Journal of Thermal

Sciences, 58: 143 - 150.

[26] Goyal, M., Bhargava, R. (2013): Numerical solution of

MHD Viscoelastic nanofluid flow over a stretching sheet

with partial slip and heat source/sink. Hindawi Publishing

Corporation, Volume (2013), Article ID 931021, 11 pages.

http://dx.doi.org/10.1155/2013/931021.

[27] Hayat, T., Hussain, T., Shehzad, S. A., Alsaedi, A. (2014):

Thermal and concentration stratifications effects in radiative

flow of Jeffrey fluid over a stretching sheet. PLoS ONE, 9

(10): 1 - 15.

[28] Makinde, O. D., Mabood, F., Khan, W. A., Tshehla, M. S.

(2016): MHD flow of a variable viscosity nanofluid over a

radially stretching convective surface with radiative heat.

Journal of Molecular Liquids 219 (2016) 624–630,

http://dx.doi.org/10.1016/j.molliq.2016.03.078.

[29] Hayat, T., Waqas, M., Shehzad, S. A., Alsaedi, A. (2016):

Mixed convection flow of viscoelastic nanofluid by a cylinder

with variable thermal conductivity and heat source/sink.

International Journal of Numerical Methods for Heat & Fluid

Flow, Vol. 26 Iss 1 pp. 214 – 234.

[30] Khan, M. I., Zia, Q. M. Z., Alasedi, A., Hayat, T. (2018):

Thermally stratified flow of second-grade fluid with non-

Fourier heat flux and temperature-dependent thermal

conductivity. Results in Physics 8 (2018), 799–804.

Doi.org/10.1016/j.rinp.2018.01.015.

[31] Hayat, T., Anwar, M. S., Farooq, M., Alsaedi, A. (2015):

Mixed convection flow of viscoelastic fluid by a stretching

cylinder with heat transfer, PLoS ONE 10 (3): 1 – 20.

[32] Mahat, R., Rawl, N. A., Kasim, A. R.M., Shafie, S. (2017):

Mixed convection boundary layer flow of viscoelastic

nanofluid past a horizontal circular cylinder: Case of

constant heat flux, IOP Conf. series: Journal of Physics

Conf. Series 890 (2019) 012052.

[33] Rivlin, R. S., Ericksen, J. L. (1955): Stress-deformation

relations for isotropic materials.

[34] Journal of Rational Mechanics and Analysis, 4 (2): 1-103.

[35] Sobamowo, M. G. (2018): Combined effects of thermal

radiation and nanoparticles on free convection flow and heat

transfer of Casson fluid over a vertical plate. International

http://dx.doi.org/10.1016/S0017-9310(99)00369-5

https://doi.org/10.1063/1.5005870

https://doi.org/10.1140/epjp/i2016-16003-1

http://dx.doi.org/10.1155/2013/931021

http://dx.doi.org/10.1016/j.molliq.2016.03.078



Journal of Chemical Engineering, Volume 2018, Article ID

7305973, 25 pages. https://doi.org/10.1155/2018/7305973.

[36] Makinde, O. D., Mahanthesh, B., Gireesha, B. J.,

Shashikumar, N. S., Monaledi, R. L., Tshehla, M. S. (2018):

MHD nanofluid flow past a rotating disk with thermal

radiation in the presence of Aluminium and Titanium alloy

nanoparticles, defect and diffusion Forum, Vol. 384: 60 – 79,

doi:10.4028/www.scientific.net/DDF.384.69.

[37] Pandey, K. A., Kumar, M., (2017): Natural convection and

thermal radiation influence on nanofluid flow over a stretching

cylinder in a porous medium with viscous dissipation.

Alendria Enginering Journal., 56: 55 – 62.

[38] Lings, J. X., Dybbs, A. (1987): Forced convection over a flat

plate submersed in a porous medium: variable viscosity case.

Paper 87-WA/HT-23. ASME, New York.

[39] Animasaun, I. L. (2015): Dynamics of unsteady MHD

convective flow with thermophoresis of particles and variable

thermo-physical properties past a vertical surface moving

through binary mixture. Open Journal of Fluid dynamics, 5,

106 – 120. http://dx.doi.org/10.4236/ojfd.2015.52013.

[40] Choudhury, M., Hazarika, G. C. (2006): The effects of

variable viscosity and thermal conductivity on MHD flow

due to a point sink. Matematicas: Ensenanza Universitaria,

vol. XVI, num 2, diciembre, 2008, pp. 21 – 28.

[41] Prasad, K.V., Vajravelu, K., Vaidya, H., Van Gorder, R. A

(2017): MHD flow and heat transfer in a nanofluid over a

slender elastic sheet with variable thickness. Results in

Physics, 7 (2017) 1462–1474.

[42] Eldabe, N. T. M., Elbashbeshy, E. M. A., Youssef, I. K.

(2014): The effects of temperature-dependent viscosity and

viscous dissipation on MHD convection flow from an

isothermal horizontal circular cylinder in the presence of

stress work and heat generation. European scientific Journal,

10(36): 1857 – 7881.

[43] Elbarbary, E. M. E., Elgazery, N. S. (2005): Flow and heat

transfer of a micropolar fluid in an axisymmetric stagnation

flow on a cylinder with variable properties and suction

(numerical study). Acta Mechanica 176, 213–229 (2005) DOI

10.1007/s00707-004-0205-z.

[44] Dogonchi, A. S. and Ganji, D. D. (2017): Effect of Cattaneo–

Christov heat flux on buoyancy MHD nanofluid flow and heat

transfer over a stretching sheet in the presence of Joule heating

and thermal radiation impacts. Indian J of Phys.

https://doi.org/10.1007/s12648-017-1156-2

[45] Khan, M. S., Rahman, M. M., Arifuzzaman, S. M., Biswas, P.

and Karim, I (2017): Williamson fluid flow behaviour of MHD

convective radiative Cattaneo–christov heat flux type over a

linearly stretched-surface with heat generation and Thermal-

diffusion. Frontiers in Heat and Mass Transfer (FHMT), 9, 15

(2017). DOI: 10.5098/hmt.9.15

[46] Mahanthesh, B., Gireesha, B. J. and Raju, C. S. K. (2017):

Cattaneo-Christov heat flux on UCM nanofluid flow across a

melting surface with double stratification and exponential

space dependent internal heat source. Informatics in Medicine

Unlocked. Informatics in Medicine Unlocked, 9: 26–34.

[47] Ostrach, S. (1952): An analysis of laminar free-convection

flow and heat transfer about a flat plate parallel to the direction

of the generating body force. NACA-TN 2635: 1-40.

[48] Sharma, P. R., Singh, G. (2010): Effects of variable thermal

conductivity, viscous dissipation on steady MHD natural

convection flow of low Prandtl fluid on an inclined porous

plate with Ohmic heating. Meccanica 45: 237–247,

DOI 10.1007/s11012-009-9240-0

[49] Elbarbary, E. M. E., Elgazery, N. S. (2005): Flow and heat

transfer of a micropolar fluid in an axisymmetric stagnation

flow on a cylinder with variable properties and suction

(numerical study). Acta Mechanica 176, 213–229 (2005).

DOI 10.1007/s00707-004-0205-z

[50] Magyari, E. and Pantokratoras, A. (2011): Note on the effect

of thermal radiation in the linearized Rosseland approximation

on the heat transfer characteristics of various boundary layer

flows. International Communications in Heat and Mass

Transfer 38: 554–556.

[51] Motsa, S. S. (2013): A new spectral local linearization method

for non-linear boundary layer flow problems. Hinduwai

Publishing Group: Journal of Applied Mathematics, Vol.

2013, 15 pages, http://dx.doi.org/10.1155/2013/423628.

[52] Otegbeye. O. (2014): On decoupled quasi-linearization

methods for solving systems of nonlinear boundary value

problems. M.Sc. Thesis, University of Kwazulu-Natal.

[53] Canuto, C., Hussaini, M. Y., Quarteroni, A., Thomas Jr, A.

(2012): Spectral methods in fluid dynamics. Springer Science

& Business Media, Berlin.

[54] Ishak, A. Nazar, R. and Pop. I. (2006). Mixed convection

boundary layers in the stagnation-point flow toward a

stretching vertical sheet. Meccanica 41:509–518.

DOI10.1007/s11012-006-0009-4.

[55] Singh, G., Makinde, O. D. (2013): MHD slip flow of viscous

fluid over an isothermal reactive stretching sheet. Annals of

Faculty of Engineering Hunedoara, Intl J of Eng, Vol 11(2):

41- 46.

http://dx.doi.org/10.1155/2013/423628

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MHD Stagnation Point Flow of Viscoelastic Nanofluid Past a …ijens.org/Vol_19_I_06/192506-7373-IJMME-IJENS.pdf · 2020. 1. 16. · stagnation point chemically reactive MHD flow past