the investigation of mhd williamson nanofluid over

Research ArticleThe Investigation of MHD Williamson Nanofluid over StretchingCylinder with the Effect of Activation Energy

Wubshet Ibrahim 1 and Mekonnen Negera2

1Department of Mathematics, Ambo University, Ambo, Ethiopia2Department of Mathematics, Wollega University, Nekemte, Ethiopia

Correspondence should be addressed to Wubshet Ibrahim; [email protected]

Received 15 March 2020; Revised 18 May 2020; Accepted 3 June 2020; Published 25 June 2020

Academic Editor: Emilio Turco

Copyright © 2020Wubshet Ibrahim andMekonnen Negera. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

In this paper, we discussed the effect of activation energy on mixed convective heat and mass transfer of Williamson nanofluid withheat generation or absorption over a stretching cylinder. Dimensionless ordinary differential equations are obtained from themodeled PDEs by using appropriate transformations. Numerical results of the skin friction coefficient, Nusselt number, andSherwood number for different parameters are computed. The effects of the physical parameter on temperature, velocity, andconcentration have been discussed in detail. From the result, it is found that the dimensionless velocity decreases whereastemperature and concentration increase when the porous parameter is enhanced. The present result has been compared withpublished paper and found good agreement.

1. Introduction

The heat and mass transfer of boundary layer flow of non-Newtonian fluids over a stretching cylinder is of greatinterest for scientists, engineers, and researchers due toits wide applications. Some examples of its applicationsare the extrusion process, extraction of metals, annealing,thinning of copper wire, and pipe industry. Nowadays, inmost industries, the significance of non-Newtonian fluidsgoverns the Newtonian fluids. The rheological propertiesof non-Newtonian fluids cannot be illuminated by theclassical Naiver-Stokes equations. Also, non-Newtonianfluid characteristics cannot be modeled by a single model.To overcome this difficulty, ample models have come intobeing. The rheological models that were projected wereWilliamson, Cross, Ellis, power law, Carreau fluid model,etc. Typical of a non-Newtonian fluid model with shearretreating property is the Williamson fluid model andwas first expected by Williamson [1]. Some recent investi-gations concerning the flow and heat and mass transfer ofMHD flow of Williamson fluid can be mentioned throughinvestigations [2–6].

Heat and mass transfer in non-Newtonian fluid flowthrough porous medium in engineering has extensive appli-cation, such as ventilation procedure, oil production, solarcollection, cooling of nuclear reactors, and electronic cooling.Accordingly, Maripala and Kishan [7] investigated the effectof thermal radiation and chemical reaction on time-dependent MHD flow and heat transfer of nanofluid over apermeable shrinking sheet. Their result indicates that withan increase of the suction parameter, the temperature profilesdecrease whereas the concentration profiles upsurge. More-over, Kairi and RamReddy [8], Vijaya et al. [9], Ambreenet al. [10], Aurangzaib et al. [11], Bal Reddy et al. [12],Al-Mamun et al. [13], and Ahmed et al. [14] studiedmixed convective heat and mass transfer of MHD nano-fluids embedded in porous medium over stretching sur-faces. From their result, it can be seen that temperaturedistribution and thermal boundary layer reduce with anincrease of dimensionless thermal free convection parame-ter, dimensionless mass free convection parameter, andPrandtl number.

The miracle of heat generation/absorption in workingfluid plays an important role in several engineering and

HindawiAdvances in Mathematical PhysicsVolume 2020, Article ID 9523630, 16 pageshttps://doi.org/10.1155/2020/9523630

https://orcid.org/0000-0003-2281-8842

https://creativecommons.org/licenses/by/4.0/

https://creativecommons.org/licenses/by/4.0/

https://doi.org/10.1155/2020/9523630

industries. The influence of heat generation may increase thetemperature distribution in moving fluids and consequentlyimpacts the heat transfer rate. Some of its application inmany industries is the geothermal system, thermal absorp-tion, removal of heat from nuclear wreckage, food storage,microelectronics manufacturing, and cooling of nuclear reac-tors. Consequently, heat generation or absorption in MHDflow of Casson fluid over a stretching wedge with viscous dis-sipation and Newtonian heating was investigated by Ullahet al. [15]. Moreover, Khan et al. [16] and Padmavathi andAnitha Kumari [17] examined the effect of heat generationor absorption on mixed heat and mass transfer nanofluidsembedded over stretching surfaces. Some researchers havestudied the impact of heat source/sink on MHD non-Newtonian nanofluids over stretching surfaces [18–23].

Activation energy is the smallest amount of energyneeded by chemical reactants to endure a chemical reaction.The influence of activation energy on convective heat andmass transfer in the region of boundary layers was initiallyinspected by Bestman [24]. Later, many researchers havestudied the impact of activation energy on heat and masstransfer of boundary layer flow of the fluids. Among thoseresearchers, Awad et al. [25], Dhlamini et al. [26], Anuradhaand Sasikala [27], and Hamid et al. [28] scrutinized theinfluence of activation energy on heat and mass transfer inunsteady fluid flow under different geometry. Moreover,Huang [29] and Mustafa [30] studied the effect of activationenergy on MHD boundary layer flow of nanofluids past avertical surface and permeable horizontal cylinder. Furtherinvestigations on the effect of activation energy on non-Newtonian fluid under different surfaces are stated in[31–34]. From their plots, it can be seen that as the valuesof the activation energy parameter upsurges, the concentra-tion nanoparticles increase.

In view of all the above-revealed studies, it is decided thatthe effect of activation energy on mixed convective heat andmass transfer of Williamson nanofluid over a stretchingcylinder embedded in a porous medium with heat generatio-n/absorption is not examined yet. Thus, to fill this gap, weaim to explore the effect of activation energy on mixedconvective heat and mass transfer of Williamson nanofluidover a stretching cylinder embedded in a porous mediumwith heat generation/absorption. For information, our workis innovative in terms of the proposed fluids and incorpo-rated parameters in the boundary layer flow. A numericalsolution is obtained by using the Runge-Kutta method inconjugation with the shooting technique. Numerical resultsof the Nusselt number, skin friction coefficient, andSherwood number for different values of the physicalparameter are computed in tables. The effects of differentphysical parameters on dimensionless velocity, temperature,and concentration are presented in graphs.

2. Mathematical Formulation

Two-dimensional time-independent Williamson nanofluidover a stretching cylinder in the presence of a boundary layerslip has been considered. A porous cylinder is chosen asshown in Figure 1. The magnetic field of constant strength

B0 is applied normal to the flow along the r-axis in a radialdirection with the assumption of a small Reynolds numberso that the induced magnetic field is neglected. The fluid isinfinite in magnitude of positive x-direction. The mixedconvective mass and heat transfer phenomenon are includedin the existence of heat generation/absorption. Tw and Cw aresurface temperature and concentration, respectively. Theambient fluid temperature and concentration are denotedby T∞ and C∞, respectively. Further, u and v are the velocitycomponents in the direction of x and r, respectively. All thephysical properties of the fluid are considered constant.

Using the assumptions given above, the governingequations of continuity, momentum, energy, and mass arespecified as follows:

∂∂x

ruð Þ + ∂∂y

rvð Þ = 0, ð1Þ

u∂u∂x

+ v∂u∂r

= ν1r∂u∂r

+ ∂2u∂r2

!

+ νΓffiffiffi2

pr

∂u∂r

� �2+

ffiffiffi2

pΓ∂u∂r

∂2u∂r2

!−

ν

k′u

−σ1B

20

ρu + gβt T − T∞ð Þ + gβc C − C∞ð Þ

= 0,ð2Þ

u∂T∂x

+ v∂T∂r

= kρcp

1r∂T∂r

+ ∂2T∂r2

!

+ τ DB∂C∂r

∂T∂r

+ DT

T∞

∂T∂r

� �2( )

+ Qρcp

T − T∞ð Þ,

ð3Þ

r

Tw

T∞

v

u x

2R

Figure 1: Physical model of the problem.

2 Advances in Mathematical Physics

u∂C∂x

+ v∂C∂r

=DB1r∂C∂r

+ ∂2C∂r2

!+ DT

T∞

1r∂T∂r

+ ∂2T∂r2

!

− k2r C − C∞ð Þ TT∞

� �e

exp Ea

k1T

� �:

ð4Þ

The boundary conditions are

u = u0xL+ b0ν

∂u∂r

,

v = −vw,

T = Tw,

C = Cw at r = R,

u⟶ u∞,

T ⟶ T∞,

C⟶ C∞, as r⟶∞,

ð5Þ

where Tw = T∞ + T0ðx/LÞβ andCw = C∞ + C0ðx/LÞς are sur-face temperature and surface concentration, respectively; u0, T0, andC0 are reference velocity, temperature, and concen-tration, respectively; β is temperature exponent; and ς is con-centration exponent signifying the change of amount ofsolute in x-direction.

The partial differential equations given in equations(2)–(4) are converted to a system of ordinary differentialequations by inspiring the following similarity transforma-tions:

η = r2 − R2

2Ru0xν

� �1/2,

ψRf ηð Þ,

θ ηð Þ = T − T∞Tw − T∞

,

ϕ ηð Þ = C − C∞Cw − C∞

,

u = 1r∂ψ∂r

,

v = −1r∂ψ∂x

:

ð6Þ

Thus, the system of ordinary differential equations is

1 + 2ηAð Þf ′′′ + 2Af ′′ + 32 1 + 2ηAð Þ1/2λf ′′2

+ λ 1 + 2ηAð Þ3/2 f ′′ f ′′′ − f ′2 − d +Mð Þf ′+ pθ + qϕ = 0,

ð7Þ

1 + 2ηAð Þθ′′ + PrNb 1 + 2ηAð Þθ′ϕ′ + 2Aθ′

+ Pr f ′θ′ + PrNt 1 + 2ηAð Þθ′2

+ Pr Sθ − Pr βf ′θ = 0,ð8Þ

1 + 2ηAð Þϕ′′ + Scf ϕ′ + 2Aϕ′ + 2 NtNbAθ′ + Nt

Nb 1 + 2ηAð Þθ′′

− Scσ 1 + ζθð Þe exp −E1 + ζθ

� �− Scςf ′ϕ = 0:

ð9Þ

With the boundary conditions

f 0ð Þ = δ, ð10Þ

f ′ 0ð Þ = 1 + γf ′′ 0ð Þ, ð11Þ

θ 0ð Þ = 1, ð12Þ

ϕ 0ð Þ = 1, ð13Þ

f ′ ∞ð Þ = 0, ð14Þ

θ ∞ð Þ = 0, ð15Þ

ϕ ∞ð Þ = 0, ð16Þ

where M = ðσ1B20L/ρu0Þ1/2 is the magnetic parameter, A =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

νL/u0R2p

is the curvature parameter, γ = B0ffiffiffiffiffiffiffiffiffiffiffiffiνu0/L

pis the

slip parameter, λ = Γffiffiffiffiffiffiffiffiffiffiffiffiffiffi2u30/νL

pis the Weissenberg number

(Williamson parameter), p = gβtðTw − T∞Þ/Rexν is the localtemperature buoyancy parameter, q = gβcðCw − C∞Þ/Rexν isthe concentration buoyancy parameter, d = νL/u0k is theporosity parameter, Nb = τDBðCw−C∞Þ/v is the Brownianmotion parameter, Nb = τDTðTw−T∞Þ/νT∞ is the thermo-phoresis parameter, S =QL/ρcu0 is the heat source/sinkparameter, Pr = μcp/k is the Prandtl number, Sc = ν/DB is

the Schmidt number, σ = k2r /c is the ratio rate parameter,E = Ea/k1T∞ is the activation energy parameter, and ζ =Tw − T∞/T∞ is the temperature difference parameter.

Local skin friction coefficient ðCf Þ, local Nusselt numberðNuxÞ, and local Sherwood number ðShxÞ are defined asfollows:

Cf =2τwρu2w

,

Nux =xpw

k Tw − T∞ð Þ ,

Shx =xqw

DB Cw − C∞ð Þ ,

ð17Þ

3Advances in Mathematical Physics

where

τw = μ∂u∂r

+ Γffiffiffi2

p ∂u∂r

� �2 !

r=R

,

pw = −k∂T∂r

� �r=R

,

qw = −DB∂C∂r

� �r=R

:

ð18Þ

The nondimensional forms of the local Nusselt number,local skin friction coefficient, and local Sherwood number are

Cf

ffiffiffiffiffiffiffiRexp2 = f ′′ 0ð Þ + λ

2 f′′2 0ð Þ,

Nux Re− 1/2ð Þx = −θ 0ð Þ,

Shx Re−1/2x = −ϕ 0ð Þ,

ð19Þ

where Rex = u20x/v is the local Reynolds number.

3. Solution Methodology

The coupled nonlinear ordinary differential equations(7)–(9) subjected to the boundary conditions (10) and (14)are solved numerically using the Runge-Kutta method withshooting technique. For the minor change of initial guessesu1, u2, and u3, the Newton-Raphson method is applied sub-

ject to the tolerance ε = 10−5. On the origin of a number ofcomputational experiments, there is no important changein the results after η = 5, so we are considering ½0, 5� as thedomain of the problem in terms of ½0,∞Þ. To solve thisproblem by using this method, equations (7)–(9) areconverted to a system of first order ordinary differentialequations. The system of first order ordinary differentialequations is defined:

f = y1,

f ′ = y2,

f ″ = y3,

f ‴ = y′3,

θ = y4,

θ′ = y5,

θ″ = y′5,

ϕ = y6,

ϕ′ = y7,

ϕ″ = y7′:

ð20Þ

Thus, the systems of first order concurrent ODEs are

Here, the prime represents the derivative with respect toη, and the transformed boundary conditions are

y1 0ð Þ = δ,

y2 0ð Þ = 1 + γu1,

y3 0ð Þ = u1,

y4 0ð Þ = 1,

y5 0ð Þ = u2,

y6 = 1,

y7 = u3: ð22Þ

4. Result and Discussion

Figures 2(a)–2(c) are plotted to show the distribution ofvelocity, temperature, and concentration for different valuesof the suction ðδÞ parameter, respectively. From these figures,

y′1 = y2,y′2 = y3,

y′3 =−2Ay3 − 3/2ð Þ 1 + 2ηAð Þ1/2Aλy23 − y1y3 + y22 + d +Mð Þy2 − py4 − qy6

1 + 2ηAð Þ + λ 1 + 2ηAð Þ3/2y3,

y′4 = y5,

y′5 =−PrNb 1 + 2ηAð Þy5y7 − 2Ay5 − Pr y1y5 − PrNt 1 + 2ηAð Þy25 + Pr Sy4 + Pr βy2y4

1 + 2ηA ,

y′6 = y7,

y′7 =−Scy1y7 + PrNt 1 + 2ηAð Þy5y7 + 2Ay7 + 4 Nt/Nbð ÞAy5 − 2 Nt/Nbð Þ Pr y1y5 + 2 Pr Nt2/Nb

� �1 + 2ηAð Þy25 + 2 Nt/Nbð Þ Pr Sy4 − 2 Nt/Nbð Þ Pr βy2y4 + Scσ 1 + ζy4ð Þe exp −E/ 1 + ζy4ð Þð Þy6 + Scςy2y61 + 2ηA :

ð21Þ


it can be seen that when the values of the suction parameterincreases, the velocity, temperature, and concentrationboundary layer thickness are reduced. This is due to the factthat suction or blowing is the method of controlling theboundary layer. The effect of suction consists in the removalof decelerated fluid particles from the boundary layer beforethey are given a chance to cause separation. Figures 3(a)and 3(b) reveal the profiles of velocity and temperature fordifferent values of injection ð−δÞ. The graph shows that withan increase of the injection parameter, velocity and tempera-ture upsurge.

For different values of slip parameter ðγÞ, the variation ofvelocity and temperature is plotted in Figures 4(a) and 4(b).Figure 4(a) shows that when the value of the slip parameterraises, the velocity profiles reduce. This is because of the factthat when the slip parameter increases, slip velocity increases,and consequently, fluid velocity declines because under theslip condition, the pulling of the stretching wall can only bepartly conveyed to the fluid. Figure 4(b) displayed that thetemperature profiles raise with an increase of the slip param-eter. Figures 5(a), 5(b), and 5(c) are plotted to show the veloc-

ity, temperature, and concentration profiles, respectively, forvarious values of the magnetic field. It is noticed inFigure 5(a).

The graph of velocity, temperature, and concentrationis plotted in Figures 6(a), 6(b), and 6(c), respectively, fordifferent values of the porosity parameter. The porousmedium causes higher restriction to the fluid flow, whichin turn decelerates its motion. Consequently, fromFigure 6(a), we observed that with an increase of the poros-ity parameter, the velocity profiles decrease. But fromFigures 6(b) and 6(c), it can be seen that with an increaseof the permeability parameter, both temperature and con-centration profiles are increased. The distribution of veloc-ity and temperature for different values of curvatureparameter (A) is plotted in Figures 7(a) and 7(b), respec-tively. Figure 7(a) displays that the velocity profiles areincreased with an increase of the curvature parameterwhereas Figure 7(b) speculates that when the values of thecurvature parameter upsurge, the temperature boundarylayer thickness declines. This is due to the fact that whenthe curvature parameter increases, the radius of the cylinder

00

0.1

0.2

0.3

f' (𝜂) 0.4

0.5

0.6

0.7

0.8

1 2 3 4 5

𝛿=0.05𝛿=0.2

𝛿=0.4𝛿=0.6

𝜂

(a)

𝛿=0.2𝛿=0.3

𝛿=0.4𝛿=0.5

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.52 3𝜂

(b)

𝛿=0.1𝛿=0.3

𝛿=0.5𝛿=0.7

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.52 3𝜂

(c)

Figure 2: The graph of velocity, temperature, and concentration for different values of suction (δ).


reduces which deduces that the contact area of the cylinderwith the fluid reduces. Hence, less resistance is offered bythe surface to the fluid motion, and as a result, the fluidvelocity improved.

For different values of temperature buoyancy parameter(p), the graph of velocity, temperature, and concentration ispresented in Figures 8(a)–8(c). From these figures, weobserved that velocity profiles increase with an increase ofthe temperature buoyancy parameter, but temperature andconcentration field decrease with an increase of the tempera-ture buoyancy parameter. Figures 9(a)–9(c) show the graphof temperature for different values of heat source/sinkparameter (S) and temperature exponent ðβÞ. It is seen thatthe temperature field increases with an increase of the heatsource, but an opposite trend is happening for large valuesof heat sink and temperature exponent ðβÞ. Figures 10(a)and 10(b) display the graph of velocity, temperature, and

concentration, respectively, for various values of concentra-tion buoyancy parameter (q). The figures illustrate that thevelocity field increases whereas temperature and concentra-tion fields decline with an increase of the concentrationbuoyancy parameter.

The effect of activation energy (E) and reaction rateparameter ðσÞ on the concentration field is plotted inFigures 11(a) and 11(b). From these figures, it is seen thatwhen activation energy upsurges, the concentration fieldrises, but it diminishes with an increase of the reaction rateparameter. This is due to the fact that large activation energyand low temperature cause a smaller reaction rate constantand thus slow down the chemical reaction. So, the concentra-tion of the solute increases. Figures 12(a), 12(b), and 12(c)execute the impact of the Williamson parameter onvelocity, temperature, and concentration fields, respectively.Figure 12(a) tells us that with an increase of the Williamson

00

0.10.20.3

f' (𝜂)

0.40.50.60.7

0.90.8

1 2 3 4 5

𝛿=-0.05𝛿=-0.2

𝛿=-0.4𝛿=-0.6

𝜂

(a)

𝛿=-0.1𝛿=-0.2

𝛿=-0.3𝛿=-0.4

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

1 2 3 4 5𝜂

(b)

Figure 3: The graph of velocity and temperature for different values of injection (−δ).

00

0.10.20.3

f' (𝜂)

0.40.50.60.7

0.90.8

1 2 3 4 5

𝛾=0.0𝛾=0.2

𝛾=0.3𝛾=0.5

𝜂

(a)

𝛾=0.0𝛾=0.2

𝛾=0.3𝛾=0.4

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

1 2 3 4 5𝜂

(b)

Figure 4: The graph of velocity and temperature for different values of slip parameter γ:


parameter, the velocity profiles decrease. As we know that theWilliamson parameter is the ratio of relaxation time to retar-dation time, hence, as the values of the retardation timedecrease, the Williamson parameter increases which displaysthat there is a reduction in the velocity field, and in addi-tion, boundary layer thickness decreases. On behalf ofFigures 12(b) and 12(c), it is noted that the temperatureand concentration fields upsurge with an enhancement ofthe Williamson parameter.

The variation of concentration profiles for various valuesof fitted rate constant (e) and concentration exponent ðςÞ isshown in Figures 13(a) and 13(b). It is noticed that the distri-bution of concentration profiles is decreased with an increasein both fitted rate constant and concentration exponentparameters. This is because when the values of the fitted rateconstant increased, the factor σð1 + ζθÞe exp ðð−EÞ/ð1 + ζθÞÞis augmented. This finally favors the destructive chemicalreaction which causes concentration gradient rise. So, thediminution in the concentration field is accompanied by ahigher concentration gradient at the wall. The impact ofBrownian motion (Nb), thermophoresis parameter (Nt),and Prandtl number (Pr) on temperature is plotted inFigures 14(a)–14(c). Figure 14(a) indicates that when the

Brownian motion ðNbÞ parameter increases, the movementof nanoparticles from the hot surface to the cold surfacehappens and ambient fluid occurred. Due to this, the temper-ature and thermal boundary layer thickness rise. FromFigure 14(b), it is observed that the temperature field growswith an increase of the thermophoresis parameter. A phe-nomenon in which small particles are pulled away from thehot surface to the cold one is called thermophoresis. So, whenthe surface is heated, the large number of nanoparticles ismoved away which raises the temperature of the fluid. There-fore, the temperature of fluid increases. Figure 14(c) displaysthe effect of the Prandtl number (Pr) on temperature. Fromthis, it can be seen that when the values of Pr rise, the temper-ature profiles decline. This is because fluid with higher Pr hasrelatively low thermal conductivity, which results in heatconduction and thereby thermal boundary layer thicknessand temperature drop. The effect of Brownian motion (Nb),thermophoresis parameterðNtÞ, Schmidt number ðScÞ, andtemperature difference parameter ðζÞ on concentration pro-files is plotted in Figures 15(a)–15(d). Figure 15(a) indicatesthat when Brownian motion ðNbÞ upsurges, volume frac-tion of nanoparticles within the boundary layer upsurges.It is interesting to note that Brownian motion of the

M=0.0M=0.1

M=0.2M=0.3

00

0.10.20.3

f' (𝜂)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.5 3.52 4

0.380.5 0.52 0.54

0.390.4

0.410.420.43

3𝜂

(a)

M=0.0M=0.1

M=0.2M=0.3

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.5 3.52 4

0.330.5 0.51 0.52

0.340.350.36

3𝜂

(b)

M=0.0M=0.1

M=0.2M=0.3

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.5 3.52 4

0.29

0.5 0.52 0.54

0.30.310.320.330.34

3𝜂

(c)

Figure 5: The graph of velocity, temperature, and concentration for different values of magnetic field (M).


nanoparticles at the molecular and nanoscale levels is a keymechanism in governing their thermal behavior. In thenanofluid system, due to the size of the nanoparticles, Brow-

nian motion affects the heat transfer properties. As the par-ticle size scale approaches to the nanometer scale, Brownianmotion on the surrounding liquids plays an important role

00

0.10.20.3

f' (𝜂)

0.40.50.60.7

0.90.8

1 2 3 4 5

A=0.2A=1.0

A=2.0A=5.0

𝜂

(a)

A=0.2A=1.0

A=1.5A=2.5

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

1 42 87653𝜂

(b)

Figure 7: The graph of velocity and temperature for different values of curvature parameter (A).

00

0.1

0.2

0.3f' (𝜂) 0.4

0.5

0.6

0.7

1 2 3 4 5

d=0.0d=0.2

d=0.4d=0.6

𝜂

(a)

d=0.0d=0.2

d=0.4d=0.6

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

1 42 5

0.30.6 0.62 0.64

0.32

0.34

3𝜂

(b)

d=0.0d=0.2

d=0.4d=0.6

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

1 42 5

0.260.6 0.65 0.7

0.28

0.3

3𝜂

(c)

Figure 6: The graph of velocity, temperature, and concentration for different values of porous parameter (d).


in heat transfer. Figure 15(b) shows that with an increase ofthe thermophoresis parameter, concentration profiles rise.From Figure 15(c), it is observed that the concentration pro-files decrease with an increase of the Schmidt number. Thisis due to the fact that weaker molecular diffusivity appearedfor a larger Schmidt number and stronger molecular diffusiv-ity corresponds to a lower Schmidt number. Figure 15(d) fore-casts that with an increase in the temperature differenceparameter, the concentration profiles decline.

Table 1 shows the comparison of the skin friction coeffi-cient with the available published results of Malik et al. [2]and found that it is an excellent agreement. Moreover, thecomparisons of the values of the Nusselt number andSherwood number with the other published result of Hayatet al. [33] are shown in Table 2. Here, from the table, it canbe seen that there is an excellent agreement.

Straightforwardly, the numerical values of the skinfriction coefficient, Nusselt number, and Sherwood numberappear in Table 3 for different values of suction parameterðδÞ, permeability parameter (d), curvature parameter (A),temperature buoyancy parameter (p), concentration buoy-ancy parameter (q), and slip parameter ðγÞ: It is noticed

that the magnitude of the skin friction coefficient increaseswith an increase of suction parameter ðδÞ, permeabilityparameter (d), and curvature parameter (A) whereas withan upsurge of temperature buoyancy parameter (p), concen-tration buoyancy parameter (q), and slip parameter ðγÞ, theskin friction coefficient magnitude was reduced, but itremains constant with an increase of activation energy (E)and reaction rate parameter ðσÞ. Moreover, from this table,it can be seen that when the values of suction parameter ðδÞ,temperature buoyancy parameter (p), concentration buoyancyparameter (q), activation energy (E), and curvature parameter(A) increase, the magnitude of the Nusselt number increaseswhereas with an increase of permeability parameter (d),reaction rate parameter ðσÞ, and slip parameter ðγÞ, there isa magnitude of Nusselt number diminution. Furthermore,from this table, it can be seen that the Sherwood numberdeclines when the values of permeability parameter (d), activa-tion energy (E), and slip parameter ðγÞ rise, but an oppositetrend is happening on the Sherwood number as the values ofsuction parameter ðδÞ, temperature buoyancy parameter (p),concentration buoyancy parameter (q), and reaction rateparameter ðσÞ.

p=0.2p=0.4

p=0.5p=0.7

00

0.10.20.3

f' (𝜂)

0.40.50.6

0.8

1

0.7

0.9

1 42 53𝜂

(a)

p=0.2p=0.4

p=0.6p=0.8

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

1 42 653𝜂

(b)

p=0.2p=0.6

p=1.0p=1.4

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.52 3

0.20.76 0.78 0.8

0.210.220.230.240.25

𝜂

(c)

Figure 8: The graph of velocity, temperature, and concentration for different values of temperature buoyancy parameter (p).


5. Conclusion

This article deals with numerical solution of MHD non-Newtonian nanofluid over a stretching cylinder embedded

in a porous medium with the effect of activation energy andheat generation/absorption. The impact of various dimension-less physical parameters on velocity, temperature, and concen-tration profiles as well as skin coefficient friction ðCf Þ, heat

S=0.0S=0.1

S=0.3S=0.5

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

1 42 53𝜂

(a)

S=0.0S=−0.1

S=−0.3S=−0.5

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

1 42 53𝜂

(b)

𝛽=0.0𝛽=1.0

𝛽=2.0𝛽=3.0

00.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

𝜂

0 0.5 1 1.5 2.52 3

(c)

Figure 9: The graph of temperature for different values of heat source/sink parameter (S) and temperature exponent β.

q=0.0q=0.2

q=0.4q=0.7

00

0.10.20.3

f' (𝜂)

0.40.50.6

0.80.9

0.7

1 2 543𝜂

(a)

q=0.0q=0.2

q=0.4q=0.7

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

1 2 54

0.22 2.1 2.2

0.22

0.24

3𝜂

(b)

q=0.2q=0.6

q=0.8q=1.0

00

0.10.20.3

a\ph

i (\e

t)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.52 3

0.29

0.5 0.52 0.54

𝜂

0.30.310.320.330.340.35

(c)

Figure 10: The graph of velocity, temperature, and concentration for different values of concentration buoyancy parameter (q).


transfer rate ðNuxÞ, and mass transfer rate parameter ðShxÞ isdiscussed in graphs and tables. The summarized conclusion ofnumerical study is as follows:

(i) The suction ðδÞ parameter interaction reduces thevelocity, temperature, and concentration profiles ofthe flow

E=1.0E=3.0

E=4.0E=8.0

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.5 3.52 43𝜂

(a)

𝜎=-1.0𝜎=0.0

𝜎=1.0𝜎=2.0

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

1 42 53𝜂

(b)

Figure 11: The graph of concentration for different values of activation energy (E) and reaction rate parameter σ.

00

0.1

0.2

0.3f' (𝜂) 0.4

0.5

0.6

1 2 3 4 5

𝜆=0.2𝜆=1.0

𝜆=2.0𝜆=3.0

𝜂

0.20.8 0.85 0.9

0.210.220.23

(a)

𝜆=0.0𝜆=1.0

𝜆=3.0𝜆=5.0

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.5 3.52 4

0.20.75 0.8 0.85

3𝜂

0.22

0.24

0.26

0.9

(b)

𝜆=0.0𝜆=1.0

𝜆=3.0𝜆=5.0

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.5 3.52 4

0.250.6 0.65

3𝜂

0.260.270.280.29

0.3

0.7

(c)

Figure 12: The graph of velocity, temperature, and concentration for different values of Williamson parameter λ.


(ii) Curvature parameter (A) improves velocity profiles,but it reduces temperature profiles

(iii) With an increase of porous parameter (d), the pro-files of velocity decline whereas temperature andconcentration profiles are improved

(iv) Raise in temperature and concentration buoyancyparameter enhances velocity whereas temperatureand concentration are reduced

(v) Increase in activation energy (E) enhances concen-tration profiles

e=-0.5e=0.0

e=0.5e=1.0

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.52 3

0.150.6 0.7

𝜂

0.2

0.25

0.8

(a)

𝜍=0.5𝜍=1.0

𝜍=2.0𝜍=3.0

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.52 3𝜂

(b)

Figure 13: The graph of concentration for different values of fitted rate constant (e) and concentration exponent ς.

Nb=0.1Nb=0.2

Nb=0.3Nb=0.5

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

1 2 543𝜂

(a)

Nt=0.0Nt=0.06

Nt=0.12Nt=0.2

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.52 3

0.240.6 0.62 0.64

0.260.28

0.32

𝜂

0.3

(b)

Pr=1.0Pr=1.2Pr=2.0

00

0.10.20.3

𝜃 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

1 2 543𝜂

(c)

Figure 14: The graph of temperature for different values of Brownian motion (Nb), thermophoresis parameter (Nt), and Prandtl number (Pr).


(vi) With an increase of reaction rate parameter ðσÞ, themagnitude of the skin friction coefficient remainsconstant but the magnitude of Nusselt numberdiminution and Sherwood number rises

(vii) Sherwood number declines when the value of slipparameter ðγÞ rises.

Nb=0.1Nb=0.13

Nb=0.15Nb=0.2

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.5 3.52 43𝜂

(a)

Nt=0.01Nt=0.02

Nt=0.03Nt=0.05

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.5 3.52 43𝜂

(b)

Sc=1.0Sc=2.0

Sc=3.0Sc=6.0

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

0.5 1 1.5 2.52 3𝜂

(c)

𝜁=0.0𝜁=0.5

𝜁=0.7𝜁=1.0

00

0.10.20.3

𝜙 (𝜂

)

0.40.50.6

0.8

1

0.7

0.9

2 3 4 5𝜂

(d)

Figure 15: The graph of concentration for different values of Brownian motion (Nb), thermophoresis parameter (Nt), Schmidt number ðScÞ,and temperature difference parameter (ζ); the velocity declines with an increase of the magnetic field parameter. From (b, c), it can be seenthat the temperature and concentration upsurge with an increase of the magnetic field parameter (M). The physical meaning of this behavioris an upsurge in the values of the magnetic field parameter develops the force opposite to the flow, which is known as Lorentz force. This bodyforce reduced the boundary layer flow and congeals the velocity boundary layers. Furthermore, this force creates a resistance force thatopposes the fluid motion. Therefore, for the higher magnetic field, heat is generated leading to a boost in the temperature and thermalboundary layer thickness.

Table 1: Comparison of values of skin friction coefficient ð f ″ð0ÞÞfor various values of λ with earlier scholar results.

ΛMalik et al. [2]

f ″ 0ð ÞPresent results

f ″ 0ð Þ0.1 −0:9776 −0:97790.2 −0:9118 −0:91280.3 −0:8413 −0:8417

Table 2: Comparison of values of the Nusselt number ð−θ′ð0ÞÞ andSherwood number ð−ϕ′ð0ÞÞ for various values of Pr and E withearlier scholar results.

Pr EHayat et al. [33]

−θ′ 0ð Þ� � Present

resultHayat et al. [33]

−ϕ′ 0ð Þ� � Present

result

1 1 0.51903 0.51913

2 0.85109 0.85121

5 1.51553 1.51558

7 1.84540 1.84547

10 2.26033 2.26039

1 0.95064 0.95069

2 0.75333 0.75338

4 0.58015 0.58020

6 0.53528 0.53530

8 0.52368 0.52371


Nomenclature

A: Curvature parameterc: Volumetric volume expansion coefficientCf : Skin friction coefficientd: Porosity parameterDB: Brownian diffusion coefficientDT : Thermophoretic diffusion coefficiente: Fitted rate parameterE: Activation energyf : Dimensionless stream functionM: Magnetic fieldNb: Brownian motion parameterNt: Thermophoresis parameterNux : Nusselt number

p: Temperature buoyancy parameterPr: Prandtl numberpw: Heat flux of the nanofluidq: Concentration buoyancy parameterqw: Mass flux of the nanofluidRe: Reynolds numberS: Heat source/sink parameterSc: Schmidt numberShx: Sherwood numberT : TemperatureT∞: Free stream temperatureUe: Free stream velocityðu, vÞ: Velocity components in x and y coordinatesψ: Stream functionθ : Dimensionless temperatureη: Similarity variableμ: Dynamic viscosityβ: Temperature exponent parameterα: Thermal diffusivity of the nanofluidϕ: Dimensionless concentration functionðρcÞf : Heat capacity of the fluidðρcÞp: Heat capacity of the nanoparticlesρp: Density of nanoparticlesτ: The parameter defined by ðρcÞp/ðρcÞfγ: Stretching parameterι: Slip coefficientδ: Suction/injection parameterλ: Williamson parameterζ: Temperature differenceΓ: Positive time constantσ: Reaction rate parameterσ1: Electrical conductivity of the fluidς: Concentration exponentτw: Skin friction or shear stressρ: Densityν: Kinematic viscosity.

Data Availability

Data sharing is not applicable to this paper as no databasewas generated or analyzed during the contact study.

Conflicts of Interest

Both authors have no competing interests.

References

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Table 3: Numerical results of the skin friction coefficient, Nusseltnumber, and Sherwood number for different values of δ, d, p,q, γ, E, R, andσ when Nt = 0:1, Nb = 0:1,M = 0:3, λ = 0:1, e = 0:1,ς = 1:0, ζ = 0:1, Sc = 3, Pr = 2, β = 1, and S = 0:1:

δ d p Q E A σ γ −f ″ 0ð Þ −θ′ 0ð Þ −ϕ′ 0ð Þ0.0 0.0 0.1 0.1 1.0 0.0 0.1 0.0 1.5944 1.3886 1.8893

0.1 1.7116 1.4823 2.0540

0.2 1.8482 1.5808 2.2277

0.3 2.0143 1.6829 2.4095

0.0 1.6684 1.6015 2.2530

0.2 1.8482 1.5805 2.2277

0.4 2.0500 1.5602 2.2029

0.6 2.3045 1.5402 2.1784

0.1 1.7628 1.5895 2.2387

0.3 1.6121 1.6060 2.2588

0.5 1.4802 1.6209 2.2771

1.0 1.2026 1.6534 2.3467

0.1 1.6992 1.5914 2.2461

0.3 1.5826 1.6066 2.2600

0.5 1.4770 1.6171 2.2731

1.5 1.2469 1.6407 2.3024

1.0 1.8482 1.5509 1.1909

2.0 1.8482 1.5534 1.1781

4.0 1.8482 1.5538 1.1711

8.0 1.8482 1.5539 1.1699

0.0 1.5491 1.5273 2.1991

0.15 1.8482 1.5805 2.2277

0.2 1.9930 1.5967 2.2352

0.25 2.2087 1.6117 2.2413

0.1 1.8482 1.5527 1.1909

0.3 1.8482 1.5504 1.2315

0.7 1.8482 1.5463 1.3080

1.5 1.8482 1.5393 1.4457

0.0 1.8482 1.5805 2.2277

0.2 1.1506 1.4413 2.0311

0.3 0.9958 1.3940 1.9643

0.4 0.8832 1.3544 1.9083


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the investigation of mhd williamson nanofluid over

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