Analysis of Thermal radiation effects on MHD flow of a nanofluid over an

exponentially stretching sheet with heat and mass fluxes in the occurrence

of viscous dissipation

PARAKAPALI ROJA1, THUMMALA SANKAR REDDY1, SHAIK MOHAMMED IBRAHIM3,

GIULIO LORENZINI4*

1Department of Mathematics, Annamacharya Institute of Technology and sciences, Rajmpeta, Kadapa

Andhra Pradesh-516126, INDIA

2Department of Mathematics, Annamacharya Institute of Technology and sciences, C. K. Dinne, Kadapa

Andhra Pradesh-516003, INDIA

3Department of Mathematics, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur, Andhra

Pradesh- 522302, INDIA

4Department of Engineering and Architecture, University of Parma, Parco Area delleScienze 181/A,

Parma 43124, ITALY

Abstract: The numerical analysis of thermal emission on a magneto-hydrodynamics (MHD) stream of a

viscous in-compressible nano-fluid due to an exponentially stretched sheet with warm and mass fluxes

frontier conditions in the occurrence of viscous dissipation were examined in this work. Using self-

similarity transformation, the controlling PDEs are altered into a set of ODEs, which are after that

numerically solved via the shooting procedure and a 4th Runge-Kutta method. On the non-dimensional

stream, temperature, nano-particle capacity percent, and confined Nusselt and Sherwood figures, the

consequences of many miscellaneous constraints are illustrated. The mathematical values of the friction

factor coefficient, as well as confined Nusselt and Sherwood statistics, are computed and examined.

Keywords: Thermal Emission; MHD; Nanofluid; heat and mass fluxes; Viscous Dissipation.

Received: August 8, 2021. Revised: April 12, 2022. Accepted: May 15, 2022. Published: June 14, 2022.

1. Introduction

Because of its wide range of applications,

such like continuous casting, exchangers, metal

spinning, bundle wrapping, foodstuff processing,

substance processing, equipment, in addition to

polymer extrusion, the analysis of velocity and heat

transport over a stretched surface has gotten a lot of

awareness in modern years. The Newtonian runny

stream induced by a stretched sheet was initially

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studied by Crane [1]. Many researchers, including

Dutta et al. [2], Chen and Char [3], and Gupta [4],

improved Crane's [1] work by considering the

influence of mass transport in diverse situations. The

exponential stretched sheet was used by Nadeem et

al. [5] to explore the heat transmission phenomenon

of a water-based nanofluid. Bhattacharyya [6]

investigated frontier layer stream and heat

conduction across a sheet that was shrinking rapidly.

The high temperature transport stream

through a permeable exponential stretched sheet with

thermal emission was studied by Mukhopadhyay et

al. [7]. The influence of heat emission on the frontier

layer stream owing to an exponentially stretched

sheet was studied by Sajid and Hayat [8]. Zhang et

al. [9] focuses on the heat transport of a power law

nanofluid thin film caused by a stretched sheet in the

occurrence of stream slip and magnetic field. Majeed

et al. [10] demonstrate ferromagnetic fluid frontier

layer stream across a stretched surface. Pal and Saha

[11] investigated the warm and mass transmission in

a thin liquid film with the influence of non linear

thermal emission using an unstable stretched sheet.

Weidman [12] investigated a unified formulation for

stagnation point streams including stretched surfaces.

MHD stream of an electrically conducting

runny over a stretched sheet has out of the ordinary

uses in modern metallurgical and metal-working

techniques. Many professional polymer processes

include pulling continuous strips and filaments from

a moving fluid to cool them. The pace of cooling,

which is determined by the structure of the

frontierlayer adjacent to the stretched sheet, has a

significant impact on the final result. MHD stream of

Casson fluid due to exponentially stretched sheet

with heat emission was explored by Mukhopadhyay

et al. [13]. Hayat et al. [14] describe the consequence

of magnetahydrodynamics on bidirectional nanofluid

stream with second order slip stream and

homogeneous–heterogeneous reactions.

Lin et al. [15] investigated the stream and

heat transport of an unsteady MHD pseudo-plastic

nanofluid in a finite thin film over a stretched surface

with internal heat generation. Sheikholeslami et al.

[16] used a two-phase model to investigate the

consequence of thermal emission on

magnetohydrodynamics nanofluid stream and heat

transmission. Farooq et al. [17] demonstrate the use

of the HAM-based Mathematica tool BVP h 2.0 on

MHD Falkner–Skan nanofluid stream. Shehzad et al.

[18] conducted an analytical study to examine

thermal emission consequences in three-dimensional

Jeffrey nanofluid stream with internal heat generation

and magnetic field.

In procedures carried out at extremely high

temperatures, the importance of emission cannot be

overstated. Gas turbines, missiles, aircraft, space

vehicles, and nuclear power plants all use radiative

consequences. Moradi et al. [19] investigate the

interplay of emission in a thermally convective

viscous liquid stream over an inclined surface.

Sheikholeslami et al. [20] used the two phasemodel

to study the influence of emission in viscous

nanofluid stream. Hayat et al. [21] investigate the

laminar stream of an Oldroyd-B liquid with

nanoparticles and emission. The radiative three-

dimensional stream of Maxwell fluid with

thermophoresis and convective condition was

researched by Ashraf et al. [22]. Hayat et al. [23]

investigated the heat emission in a Powell-Eyring

nanofluid laminar stream over a stretched sheet.

Bidin and Nazar [24] investigated the

influence of thermal emission on the constant laminar

two-dimensional frontierlayer stream and heat

transport over an exponentially stretched sheet. Using

the Runge–Kutta fourth-order approach, Hady et al.

[25] investigated the emission influence on viscous

nanofluid over nonlinear stretched sheet. Hayat et al.

[26] investigated the consequences of Joule heating

and thermophoresis in a Maxwell model stretched

stream under convection. Mustafa et al. [27] address

Sakiadis stream of Maxwell fluid with convective

frontiercondition. Hayat et al. [28] investigated the

Maxwell fluid's stagnation point stream in the

occurrence of thermal emission and convection.

Hayat et al. [29] investigated the consequences of an

angled field of magnetic and generating of heat in

nanofluid stream with non-linear thermal emission.

Khan et al. [30] investigate the nonlinear radiative

stream of a three-dimensional Burgers nanofluid with

a new mass flux consequence.

The goal of this explore is to probe the

consequence of thermal emission on the

magnetahydrodynamic (MHD) stream of a viscous

incompressible nano sized particle fluid appropriate

to an exponentially stretched sheet with warm and

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massfluxes conditions in the occurrence of viscous

dissipation numerically. The controlling PDEs are

distorted into self-similar ODEs passing through

similarity transformations, and subsequently solved

numerically via the shooting practice.

2. Mathematical Formulation

An exponentially extending sheet is used to

model the two-dimensional hydromagnetic stream of

an incompressible fluid. The occurrence of thermal

emission, viscous dissipation, a generating of heat,

and a chemical reaction characterises warm and mass

transport analysis. In the y-direction, a non-uniform

magnetic field B(x) = B0exp(x/2l) is applied. For

lower magnetic Reynolds statistics, the induced

magnetic field is ignored. At the sheet's surface, we

imposed warm and mass flux frontier conditions. The

subsequent are the principal equations of movement:

(i) Continuity:

0

uu

xy







(1)

(ii) Momentum:

2

0

2

B

u u u

u v u

x y y





  

  

  

(2)

(iii) Energy:

2

()

1

pT

B

f

r

pp

cD

T T T C T T

u v D

x y y c y y T y

qu

c y c y















     

   





     



















(3)

(iv) Nanoparticle volume fraction:

22

T

B

D

N N N T

u v D

x y y T y



   

  

   

(4)

subject to the frontier conditions:

0

( ) U exp , ( ),

()

, , 0

w

np

w

B

x

u U x v V x

l

qx

TN

at y

y y D





   







    



(5a)

0, , , asu T T N N y



   

(5b)

Here u and v denote the stream mechanism in the x

and y information respectively,



a kinematic

viscosity,

p

k

c





a thermal diffusivity,

k

a fluid

density,



a thermal conductivity,

p

c

a specific heat,

T a fluid temperature, T∞ a ambient temperature, N a

fluid concentration, C∞ a ambient concentration,

/p

kc





a thermal diffusivity, k a thermal

conductivity, cp a specific heat,

*3

*

16

3

r

TT

qkY







a

radiative heat flux, k* a mean absorption coefficient,

*



a Stefan-Boltzmann constant,

()

p

c



a

consequence heat capacity of nanoparticles,

()

f

c



heat capacity of the base fluid. N is

nanoparticle volume, D a mass diffusion

 

0

( ) U exp /

w

U x x l

is a stretched stream of sheet,

U0 a reference stream, l a reference length,

 

0 0 0

( ) T / 2 exp /

ww

q x q U vl x l

the variable

heat flux,

 

0 0 0

( ) / 2 exp /

np np

q x q C U vl x l

a

variable surface nanoparticle flux,

0

U

,

0

T

,

0w

q

,

0np

q

,

0

N

, are the reference stream, temperature

and heat flux, surface nanoparticle flux, nanoparticle

capacityfraction respectively,

 

0

( ) exp /V x V x l

a

special type of stream at the wall is considered

(Bhattacharyya [30]) where

0

V

is a constant. Here

( ) 0Vx

is the stream of suction and

( ) 0Vx

is

the stream of injection.

Introducing similarity transformations as follows:

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   

     

   

121

02

0

00

exp , 2 exp ,

2

exp , exp ,

2

exp , exp

np

w

Uxx

y U x f

x l l

U

xx

u U f v f f

l l l

q

qxx

T T T C C C

ll

   



   

   





    



   

    



   



   



    

   

   

   

   

   

(6)

If the dimensional stream function

( , )xy



then

uy







and

ux





 

.

The continuity equation is automatically satisfied and

using similarity transformation, the system of Eqs.

(2), (3) and (4) becomes:

22

20f ff f Ha f

   

   

(7)

 

2

4

1 Pr Pr 0

3bt

R f f N N Ec f

     



      

      





(8)

( ) 0

t

b

N

Le f f N

   

   

   

(9)

Here primes mean differentiation with respect to



,

2

0()

()

w

B x l

Ha Ux







is a Hartmann statistics,

Pr







is a

Prandtl statistics,

*3

*

4T

Rkk







is a emission

constraint and

B

Le D





is a Lewis statistics,

 

0

np

p

b

f

cq

NN

c







is a Brownian motion

constraint,

2

0

0p

U

Ec Tc





is a Eckert statistics,

and

 

0

w

p

T

t

f

cq

D

NT

Tc



 





is a thermophoresis

constraint, respectively.

The transformed frontier conditions (5a) and (5b) are

given by

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

( ) 0, ( ) 0, ( ) 0

f S f

f





      

     

(10)

Where

0

/2

v

Scl







is suction/injection constraint.

Here the constraint is positive

S

>0 (

0

v

<0) for mass

suction and negative

S

<0 (

0

v

>0) for mass injection.

The substantial quantities of concern are the

confined skin friction coefficient, the wall heat

transport coefficient (or the confined Nusselt

statistics) and the wall deposition flux (or the

confined Stanton statistics) which are defined as

respectively where the skin friction

f

C

, the heat

transport

()

w

qx

and the mass transport

x

Sh

from

the wall are given by

 

0

2 Re (0), ,

exp /

f x f

wy

u du

C f C U x l dy 





 



(11)

From the temperature field, we can study the rate of

heat transport which is given by

 

0

4

1 (0),

23

Re

x

w

xy

Nu x x T

R Nu

l T T y













     

 

 

(12)

From the concentration field, we can study the rate of

mass transport which is given by

 

0

(0),

2

Re

x

w

xy

Sh x x C

Sh

l C C y











    





(13)

where

0

RexUx





the confined Reynolds

statistics.

3. Method of solution:

The structure of ordinary differential

equations (7) – (9) subject to the frontier conditions

(10) are solved numerically using Runge–Kutta

fourth-order integration with shooting technique. A

step size of

0.01





was selected to be

satisfactory for a convergence criterion of 10-6 in all

cases. The grades are obtainable graphically in Figs.

(1) –(7) and conclusions are drawn for stream field

and other physical quantities of interest that have

noteworthy consequences.

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4. Results and discussion:

Eqs. (7)–(9) with frontier conditions (10) are

numerically solved using the Runge–Kutta fourth-

order integration with shooting method, and

numerical values are presented in Figs. (1)–(9). (7).

Throughout the computations, the leading constraints

are kept constant at Ha=1.0, S = 3.0, Le = 1.3, R =

0.1, Pr = 0.71, Ec=0.1, Nt = 0.8, and Nb = 0.5. On

the stream, temperature, and particles of nano sized

capacity friction profiles, the consequence of the

involved constraints suction constraint Hartmann

statistics, Eckert statistics Ec, Lewis statistics Le,

emission constraint R, thermophoresis statistics Nt,

and Brownian motion constraint Nb. Figures 1(a)-(b)

show the stream, temperature, and nanoparticle

capacityfriction profiles for various suction constraint

values.

The stream profiles rise as the suction

constraint is increased, as shown in Fig. 1(a). Figure

1(b) further shows that as the suction constraint is

increased, the temperature drops. In Figs 2(a)-(b), the

consequence of the Hartmann statistics (i.e. magnetic

field constraint Ha) on the stream, temperature, and

nano-particle capacity friction profiles is shown. The

stream profiles rise with higher values of Hartmann

statistics, as shown in Fig. 2(a). The Lorentz force

increases when the magnetic field is increased

physically. The fluid's stream is raised as more

resistance is applied to the fluid's motion. The

temperature drops as the Hartmann statistics rises, as

shown in Fig. 2(b).

The work of the dissipative constraint, Eckert

statistics Ec, on the stream and temperature profiles

is shown in Fig. 3(a)-(b). The stream and temperature

profiles of the stream are observed to grow as the

dissipation constraint Ec is increased. In general,

increasing viscosity increases heat conductivity,

which leads to increased stream profiles. The

consequences of the thermophoresis constraint Nt on

temperature and nano-particle capacity fraction are

shown in Fig. 4(a)-(b). The temperature (Fig. 4(a))

and nano-particle capacity fraction (Fig. 4(b))

profiles both increase as the thermophoresis

constraint is increased. The ratio of nanoparticle

diffusion to thermal diffusion in the nanofluid is the

thermophoresis constraint Nt.

The temperature differential between the sheet and

the fluid grows as Nt increases, and the thermal

frontier layer expands in this situation.

Thermophoresis force increases as Nt increases,

allowing the nano-particle to migrate from hot to

cold regions. The capacity fraction of nano-particles

increases as a result of this migration. The

consequence of the emission constraint R on

temperature and nano-particle capacity fraction

profiles is shown in Fig. 5(a)-(b). It's worth noting

that higher R values improve the temperature profile.

This is owing to the fact that when R increases, the

mean absorption coefficient decreases. The nano-

particle capacity fraction profile increases as R

increases, as shown in Fig. 5(b).

Finally, Figs. (6) and (7) show how the

Lewis statistics Le and the Brownian motion

constraint Nb affect the nano-particle capacity

fraction profiles. The distribution of nano-particle

capacity fraction diminishes as the Lewis statistics

increases, as shown in Fig. (6). This is likely due to

the fact that when Le increases, the Brownian

diffusion coefficient Nb decreases, limiting nano-

particles' ability to penetrate further into the fluid. As

a result, at a larger Lewis statistics Le, the capacity

fraction of nano-particles is thinner. Furthermore, as

the Brownian motion constraint Nb is increased, the

nano-particle capacity fraction profile decreases. This

could cause the thermal frontier layer to thicken.

Physically, an increase in Brownian motion generates

an increase in nano-particle diffusion, which lowers

the concentration inside the frontier layer.

Table 1& Table 2 illustrate mathematical information

on the consequences of assorted constraints on the

friction factor term, Nusselt statistics, and Sherwood

statistics. Table 1 indicates that raising the values of

the emission constraint R and the Eckert statistics Ec

lowers the skin friction coefficient, whereas

increasing the values of R and Ec raises the Nusselt

statistics. Table 2 presences that when the

significance of Ha increases, the confined Nusselt

statistics and Sherwood statistics drop, whereas the

opposite is true for higher values S.

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Table 1: Statistical values of friction factor term and confined Nusselt statistics for different values of Ec and R

when Ha=1.0, Nt=0.8, Nb=0.5, Pr=0.71, R=0.1, Ec=0.1 and Le = 1.3.

Constraints(fixed values) Constraints

(0)f

1/2

Rexx

Nu



Nt=0.8, Nb = 0.5, S=3.0, Pr=0.71, R=0.1, Le=1.3 R=0.10 1.120557 0.460557

0.15 1.116654 0.466654

0.30 1.097774 0.467774

Ec=0.0 1.165307 0.459307

0.1 1.120557 0.460557

0.2 1.108726 0.468726

Table 2: Numerical values of confined Nusselt statistics and confined Sherwood statistics for different values of

Ha and S when Ha=1.0, Nt=0.8, Nb=0.5, Pr=0.71, R=0.1, Ec=0.1 and Le = 1.3.

Constraints(fixed values) Constraints

1/2

Rexx

Nu



1/2

Rexx

Sh



Nt=0.8, Nb = 0.5, S=3.0, Pr=0.71, R=0.1, Le=1.3 Ha=1.0 0.451145 0.294788

1.5 0.438589 0.287824

3.0 0.403746 0.267842

S=0.5 0.415307 0.251433

0.6 0.423403 0.262577

0.8 0.440726 0.278440

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Fig 1(a). Consequence of

s

on

()f





Fig 1(b). Consequence of

s

on

()



Ha 1.0, 1.5, 2.0, 3.0

0

1

2

3

4

1.0

0.8

0.6

0.4

0.2

0.0

f

Fig 2(a). Consequence of

Ha

on

()f





Ha 1.0, 1.5, 2.0, 3.0

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

Fig 2(b). Consequence of

Ha

on

()



Fig 3(a). Consequence of

Ec

on

()f





Fig 3(b). Consequence of

Ec

on

()



Ec 0.0, 1.0, 2.0, 3.0

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

Ec 0.0, 1.0, 2.0, 3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

1.0

0.8

0.6

0.4

0.2

0.0

f

S 3.0, 3.2, 3.5, 4.0

0

1

2

3

4

1.0

0.8

0.6

0.4

0.2

0.0

f

S3.0, 3.2, 3.5, 4.0

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

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Fig 4(a). Consequence of

R

on

()



Fig 4(b). Consequence of

R

on

()



Fig 5(a). Consequence of

Ec

on

()



Fig 5(b). Consequence of

Ec

on

()



Fig 6. Consequence of

Le

on

()



Fig 7. Consequence of

Nb

on

()



5. Conclusion:

The importance of thermal emission and

Magneto-hydro-dynamics (MHD) in the stream of a

viscous dissipating nanofluid, as well as heat

transport analysis using an exponentially starching

sheet with warm and mass flux conditions with

viscous dissipation, are presented in this work. Using

suitable transformations, the leading partial

differential equations were converted into a set of

nonlinear coupled ordinary differential equations,

and the resultant well-posed frontier value problem

was numerically solved using the Runge–Kutta

fourth order based shooting method. The implications

R0.1, 0.2, 0.3, 0.5

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

R0.1, 0.2, 0.3, 0.5

0

1

2

3

4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Nt 0.5, 0.6, 0.7, 1.0

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

Nt 0.5, 0.6, 0.7, 1.0

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

Le 1.0, 1.5, 2.0, 3.0

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

Nb 0.8, 0.9, 1.0, 1.5

0

1

2

3

4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

WSEAS TRANSACTIONS on HEAT and MASS TRANSFER

DOI: 10.37394/232012.2022.17.16

Parakapali Roja, Thummala Sankar Reddy,

Shaik Mohammed Ibrahim, Giulio Lorenzini

E-ISSN: 2224-3461

148

Volume 17, 2022

of relevant constraints on the fields of stream,

temperature, and nano-particle capacity friction, skin

friction, heat, and mass transport coefficients are

addressed and illustrated in graphs and tables. The

following are the key conclusions drawn from this

research:

(i) Stream profile and frontier layer thickness

increase via suction constraint

,S

Hartmann

statistics Ha and Eckert statistics Ec.

(ii) Temperature profile and thermal frontier

layer thickness yields a decrease via larger

suction constraint s, Hartmann statistics Ha,

but they reduces by increasing emission

parameter R and Eckert statistics Ec.

(iii) The nano-particle capacity fraction increases

as the value of emission constraint R and

thermophoresis constraint Nt increases.

(iv) Skin fraction coefficient decrease as the

value of constraints R and Ec.

(v) Confined Nusselt statistics is increasing

function of

,S

R and Ec.

(vi) Confined Sherwood statistics is increasing

function of

.S

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AUTHORS’ CONTRIBUTIONS: All authors

have contributed equally to this.

WSEAS TRANSACTIONS on HEAT and MASS TRANSFER

DOI: 10.37394/232012.2022.17.16

Parakapali Roja, Thummala Sankar Reddy,

Shaik Mohammed Ibrahim, Giulio Lorenzini

E-ISSN: 2224-3461

150

Volume 17, 2022

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