Research ArticleHall and Ion Slip Effects on Mixed Convection Flow ofEyring-Powell Nanofluid over a Stretching Surface

Wubshet Ibrahim 1 and Temesgen Anbessa2

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

Correspondence should be addressed to Wubshet Ibrahim; wubshetib@yahoo.com

Received 27 June 2020; Revised 18 August 2020; Accepted 29 August 2020; Published 9 September 2020

Academic Editor: Shuo Yin

Copyright © 2020 Wubshet Ibrahim and Temesgen Anbessa. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

The purpose of this research is to inspect the mixed convection flow of Eyring-Powell nanofluid over a linearly stretching sheetthrough a porous medium with Cattaneo–Christov heat and mass flux model in the presence of Hall and ion slip, permeability,and Joule heating effects. Proper similarity transforms yield coupled nonlinear differential systems, which are solved using thespectral relaxation method (SRM). The story audits show that the present research problem has not been studied until thispoint. Efficiency of numerous parameters on velocity, temperature, and concentration curves is exposed graphically. Likewise,the numerical values of skin friction coefficients, local Nusselt, and Sherwood numbers are computed and tabulated for somephysical parameters. It is manifested that fluid velocities, skin friction coefficients, local Nusselt, and Sherwood numberspromote with the larger values of Eyring-Powell fluid parameter ε. It is also noticed that primary velocity promotes with largervalues of mixed convection parameter λ, Hall parameter βe, and ion slip parameter βi, while the opposite condition is observedfor secondary velocity, temperature, and concentration. Furthermore, comparative surveys between the previously distributedwriting and the current information are made for explicit cases, which are examined to be in a marvelous understanding.

1. Introduction

Mixed convection streams emerge in numerous transportprocesses both in nature and in engineering applications.They play a vital role, for instance, in air limit layer streams,heat exchangers, atomic reactors, solar collectors, and in elec-tronic hardware. Such procedures happen when the impactsof buoyancy forces in constrained convection or the impactsof the constrained stream in natural convection becomenoteworthy. The interaction of natural and constrained con-vection is particularly articulated in circumstances where theconstrained stream velocity is low as well as the temperaturecontrasts are huge. This stream is likewise an important kindof stream showing up in numerous mechanical procedures,for example, assembling and extraction of polymer and elas-tic sheets, paper creation, wire drawing and glass-fiber crea-tion, dissolve turning, and consistent throwing. Moreover,mixed convection in permeable media has numerous appli-cations, for example, food handling and storage, metallurgy,

geophysical framework, fibrous insulation, and undergroundremoval of atomic waste. Then again, thermal conductivity ofthe ordinary heat transport liquids, for instance, water, oil,and ethylene glycol are exceptionally low. Thus, expandingthermal conductivity of the traditional liquids promptsimproves the heat transport of these liquids. As of late, nano-fluids are presented with upgraded thermal conductivity. Ananofluid is a suspension of nanoparticles with normal sizesbeneath 100nm in the base liquids. Due to the upgradedthermal conductivity, nanofluids are proposed for somemechanical applications, for example, transportation, atomicreactors, and nourishment. In this manner, numerous ongo-ing analysts have generous enthusiasm for the mixed convec-tion stream of nanofluid over an extending sheet in apermeable medium due to their impressive use in the modernand innovative applications. Accordingly, Ameen et al. [1]investigated a 3D turning stream of carbon nanotubes(CNTs) over a permeable stretchable sheet for warmth andmass exchange with thought of kerosene oil as a base fluid

HindawiAdvances in Mathematical PhysicsVolume 2020, Article ID 4354860, 16 pageshttps://doi.org/10.1155/2020/4354860

and announced that both velocity fields delineate expandingconduct for enormous value of the CNT nanoparticle.Besides, Rasool et al. [2] examined a numerical investigationof the MHD Williamson nanofluid flow maintained to flowthrough permeable medium bounded by a nonlinearlystretching flat surface and reported that an enhancement inthe values of magnetic parameter augments the drag force,whereas a decrement in the drag force can be observed asthe values of Weissenberg number enhances. Also, Ibrahimand Anbessa [3] examined the mixed convection stream ofa Maxwell nanofluid over a growing surface in a penetrablemedium utilizing the spectral relaxation method (SRM) andfound that a climb in Deborah number decreases both thestream and transverse velocity profiles, while the backwardsdesign is seen with enlargement in the mixed convectionparameter. Furthermore, Waini et al. [4] examined the con-sistent mixed convection stream along a vertical surfaceimplanted in a permeable medium with hybrid nanoparticlesutilizing the bvp4c solver in Matlab programming. They con-sidered both assisting and opposing streams and found thatthere exist dual solutions for the case of opposing stream.Further, numerous other ongoing works have been donearound there as found in references [4–12].

Investigation of nonNewtonian liquid has incredible sig-nificance because of its numerous industrial and engineeringapplications. Specifically, these liquids are used in the mate-rial preparing, concoction and atomic ventures, bioengineer-ing, oil repository designing, polymeric fluids, and groceries.A few liquids like paints, paper mash, shampoos, ketchup,fruit purée, slurries, certain oils, and polymer arrangementsare instances of nonNewtonian liquids. NonNewtonian liq-uids are logically perplexed when compared with Newtonianliquids due to nonlinear association among the stress andstrain rate. Various models have been proposed in the writingfor the examination of nonNewtonian fluids; however, not asole model is built up that shows all properties of nonNewto-nian liquids. In general, nonNewtonian fluids are mainlycharacterized into three kinds: to be explicit differential, rate,and integral kinds. Among the nonNewtonian liquid models,the Eyring-Powell model has accomplished exceptional con-sideration of the analysts because of its distinctive attributesin current science. This could be derived from kinetic theoryof liquids rather than from its empirical relation. Likewise,the Eyring-Powell model decreases to Newtonian liquid qual-ities for low and high shear rate. Warmth and mass exchangein the Eyring-Powell liquid model assumes a significant rolein the procedures which include formation and spread ofhaze, plotting of concoction handling instrumentation, envi-ronmental contamination, drying of permeable slides, raisedoil recuperation, warm protection, and underground energytransport. Khan et al. [13] explored blended convectionstream of the Eyring-Powell liquid model with variable vis-cosity and convective limit conditions over a slanted extend-ing sheet utilizing the homotopy analysis method (HAM)and revealed that velocity profile increments by expandingliquid parameterM and Grashof numbers Gm and Gr, whilevelocity profile diminishes by expanding variable viscosityparameter A, liquid parameter k, and magnetic field parame-ter Ha. Moreover, Khan et al. [14] examined warm disper-

sion and dissemination thermo impacts on the wobbly flowof electrically conducting Eyring-Powell liquid over an oscil-latory extending sheet with convective limit conditions utiliz-ing HAM and found that the bigger values of Eyring-Powellliquid parameter improve the amplitude of velocity and limitthe layer thickness. However, inverse impacts are seen intemperature and concentration profiles. Furthermore, Ishaqet al. [15] explored two dimensional nanofluid film streamof Eyring-Powell liquid with variable warmth transmissionin the presence of MHD on a shaky permeable extendingsheet and announced that porosity parameter diminishesthe movement of the fluid movies, and enlarging thenanoparticle concentration effectively expands the rubbingcharacteristic of Eyring-Powell nanofluid. In recent times,various researches have been made in the field of Eyring-Powell nanofluids that can be found in references [16–23].

Heat transport in the presence of strong magnetic field issignificant in different parts of MHD power generation,nanotechnological processing, nuclear energy systemsexploiting fluid metals, and blood stream control. Further,Hall and ion slip impacts become noteworthy in strong mag-netic fields and can impressively influence the current densityin hydromagnetic heat transport. Most recently, Krishna andChamkha [24] explored Hall and ion slip impacts on theMHD free convective pivoting stream of nanofluids in a per-meable medium past a moving vertical semiboundless levelplate and announced the velocity increments with Hall andion slip parameters. Furthermore, Rani et al. [25] exploredHall and ion slip impacts on the MHD natural convectivepivoting stream of Ag-water-based nanofluid past a semi-boundless penetrable moving plate with consistent warmthsource utilizing the perturbation method. They found thatin the limit layer area, liquid velocity diminishes with theexpanding values of rotation parameter, magnetic fieldparameter, and suction parameter, while it increments withthe expanding values of Hall and ion slip parameters. Furtherexaminations identified with the MHD stream issues withHall and ion slip impacts can be found in references [26–31].

The temperature differences between two distinct bodiescause heat transport device. It assumes a significant task inthe creation of energy, cooling of atomic reactors, and bio-medical applications such as medication targeting and heatconduction in tissues. Fourier [32] was the spearheadingindividual who at the outset portrayed the marvels of heattransport, which is the parabolic energy equation for temper-ature field and has disadvantage that initial interruption isfelt right away all through the entire medium. Later on, soas to control this limitation, Cattaneo [33] improved theFourier law of heat conduction by including the thermalrelaxation term which causes heat transportation as thermalwaves with limited speed. Furthermore, Christov [34] uti-lized Oldroyd’s upper convected derivative instead of timederivative so as to achieve the material-invariant detailing.This new model is named as the Cattaneo–Christov heat fluxmodel. After the spearheading work of Christov [34], a num-ber of ongoing investigations in this setting are given in thereferences [35–45].

The aforesaid writing surveys affirmed that no endeavorhas been made at this point to investigate the MHD mixed

2 Advances in Mathematical Physics

convection flow of Eyring–Powell nanofluid past a verticallinearly extending sheet through a porous medium in thepresence of the Cattaneo–Christov heat and mass fluxmodel with the consolidated effects of Hall and ion slip,porosity, and Joule heating. With this perspective, the cur-rent correspondence aims to fill this gap inside the currentwriting. Consequently, the fundamental object of the pres-ent work is to examine the impact of Hall and ion slip onthe mixed convection stream of Eyring–Powell nanofluidpast a vertical linearly extending sheet through a penetra-ble medium in the presence of Joule heating with theCattaneo–Christov heat and mass transition model. Numer-ical clarifications were accomplished employing the spectralrelaxation method [3, 46–48]. The effects of embeddingparameters on the velocity, temperature, and concentrationfields are analyzed graphically. Moreover, numerical estima-tions of skin friction coefficients, local Nusselt, and Sherwoodnumbers for some pertinent parameters are computed andtabulated.

2. Mathematical Formulations

Considering a laminar, steady, viscous, and incompressiblemixed convection stream of an electrically conductingEyring-Powell nanofluid past a vertical linearly extendingsheet through a porous medium with velocity UwðxÞ = ax(where a is a positive constant) towardx-axis, the Cattaneo–Chirstov heat and mass flux model are utilized to explorethe heat and mass transfer qualities. The surface temperatureTw and concentration Cw are assumed consistent at theextending surface. As y extends to infinity, the encompassingvalues of temperature and concentration are, respectively,T∞ and C∞: Besides, a strong uniform magnetic field B0 isapplied in the direction along y − axis as shown in Figure 1.Because of this strong magnetic field force, the electricallyconducting liquid has the Hall and ion slip impacts whichproduces a crossflow in the z-direction, and consequently,the stream becomes three dimensional. The x-axis is alongthe course of movement of the surface, y-axis is perpen-dicular to the surface, and z-axis is normal to the xy-plane. The induced magnetic field can be overlookedconversely with the applied magnetic field on the assump-tion that the stream is steady, and the magnetic Reynoldsnumber is very little.

The extra stress tensor τij of the Eyring–Powell liquidmodel is characterized as [15, 21]:

τij = μ∂ui∂xj

+1βsin−1

1b∂ui∂xj

!, ð1Þ

where μ is the dynamic viscosity, and β and b are thematerial fluid parameters of the Eyring–Powell fluid modelsuch that

sin−11b∂ui∂xj

!≅1b∂ui∂xj

−16

1b∂ui∂xj

!3

and1b∂ui∂xj

����������≪ 1: ð2Þ

Generalized Ohm’s law with Hall and ion slip impactsis given by [20, 23]

J = σ Ε + V × Βð Þð Þ − ωeτeB0

J × Βð Þ + ωeτeβi

Β20

J × Βð Þ × Βð Þ,

ð3Þ

where J = ðJx, Jy, JzÞ is the current density vector, Ε is theintensity vector of the electric field,V is the velocity vec-tor, Β is the magnetic field, ωe is the cyclotron frequency,and τe is the electrical collision time.

Applying the Boussinesq approximations alongsidethe above suppositions, the basic governing equations are[30, 36, 43, 44, 49]

∂u∂x

+∂v∂y

= 0 ð4Þ

u∂u∂x

+ v∂u∂y

= υ +1

βbρ

� �−

12βb3ρ

∂u∂y

� �2" #

∂2u∂y2

−σB2

0

ρ α2e + β2e

� � αeu + βewð Þ + g βT T − T∞ð Þ½

+ βC C − C∞ð Þ� − υ

ku

ð5Þ

u∂w∂x

+ v∂w∂y

= υ +1

βbρ

� �−

12βb3ρ

∂w∂y

� �2" #

∂2w∂y2

+σB2

0

ρ α2e + β2e

� � βeu − αewð Þ − υ

kw

ð6Þ

u∂T∂x

+ v∂T∂y

= α∂2T∂y2

− γT u∂u∂x

+ v∂u∂y

� �∂T∂x

+ u∂v∂x

+ v∂v∂y

� �∂T∂y

�

+ 2uv∂2T∂x∂y

+ u2∂2T∂x2

+ v2∂2T∂y2

#

+ τ DB∂T∂y

� �∂C∂y

� �+

DT

T∞

� �∂T∂y

� �2" #

+σB2

0

ρCp α2e + β2e

� � u2 +w2� �ð7Þ

T∞, C∞Tw, Cw

Boundary layer

w

u

x,u

y,v

vg

Figure 1: Physical representation of the flow problem.

3Advances in Mathematical Physics

u∂C∂x

+ v∂C∂y

=DB∂2C∂y2

+DT

T∞

� �∂2T∂y2

!

− γC u∂u∂x

+ v∂u∂y

� �∂C∂x

+ u∂v∂x

+ v∂v∂y

� �∂C∂y

�

+ 2uv∂2C∂x∂y

+ u2∂2C∂x2

+ v2∂2C∂y2

#,

ð8Þ

with the relating boundary conditions [3, 36]:

u =Uw xð Þ = ax, v =w = 0, T = Tw, C = Cw, at y = 0,

u→ 0, w→ 0, T → T∞, C→ C∞, as y→∞,ð9Þ

where u, v, andw are the velocity components along the x− , y − , and z—directions, respectively, and υ is the kine-matic viscosity, ρ is the density of the fluid, g is the accel-eration due to gravity, βe is the Hall parameter, βi is theion-slip parameter, αe = 1 + βe βi is a constant, BT is thecoefficient of thermal expansion, BC is the solutal coefficientof expansion, κ is the permeability of the porous medium,T and C are the fluid temperature and concentration,respectively, γT and γC are the heat and mass flux relaxa-tion times correspondingly, Cp is the specific heat capacity,τ is the ratio of the effective heat capacity of nanoparticle tothe heat capacity of the fluid, i.e., τ = ðρCÞp/ðρCÞf ,DB is theBrownian diffusion coefficient, and DT is the thermophor-esis diffusion coefficient.

The overseeing equations can be changed to couplednonlinear ordinary differential equations utilizing the subse-quent similarity transformations [3, 36]:

where η is the similarity variable, and ψ is the stream functiondefined as

u =∂ψ∂y

and v = −∂ψ∂x

: ð11Þ

Equation (4) is identically fulfilled, and equations (5)–(9)become

1 + εð Þf ′′′ − εδð Þf ′′2 f ′′′ − f ′2 + f f ′′ − M

α2e + β2e

� � αe f ′ + βeg�

+ λ θ +Nrϕð Þ −Κf ′ = 0,ð12Þ

1 + εð Þg′′ − εδð Þg′2g′′ − f ′g + f g′

+M

α2e + β2e

� � βe f ′ − αeg�

−Κg = 0,ð13Þ

1Pr

θ′′ + f θ′ − γ1 f f ′θ′ + f 2θ′′�

+Nbθ′ϕ′ +Ntθ′2

+M:Ec

α2e + β2e

� � f ′2 + g2�

= 0,ð14Þ

ϕ′′ + Pr:Le f ϕ′ − γ2 f f ′ϕ′ + f 2ϕ′′� h i

+NtNb

θ′′ = 0:

ð15Þ

The transformed boundary conditions are

f 0ð Þ = 0, f ′ 0ð Þ = 1, g 0ð Þ = 0, θ 0ð Þ = 1, ϕ 0ð Þ = 1,

f ′ ⟶ 0, g⟶ 0, θ⟶ 0, ϕ⟶ 0, as η⟶∞:

ð16Þ

The dimensionless parameters showing up in equations(12)–(15) can be characterized as

M = σB20/ρa, αe = 1 + βeβi,λ =Grx/Re2x , Grx = gβTðTw −

T∞Þx3/υ2, Rex = ax2/υ, Nr = βCðCw − C∞Þ/βTðTw − T∞Þ,K = υ/aκ, ε = 1/βbμ,δ = a3x2/2b2υPr = υ/α, Nb = τDBðCw −C∞Þ/υ, Nt = τDTðTw − T∞Þ/υT∞, τ = ðρCÞp/ðρCÞf , Ec =ðaxÞ2/CpðTw − T∞Þ, Le = α/DB,γ1 = γTa and γ2 = γCa:

The skin friction coefficients Cf xandCgz , the localNusselt number Nux, and the Sherwood number Shxare the physical quantities of interest which can be char-acterized as follows:

Cf x =τwxρU2

w

, Cf z =τwzρU2

w

, Nux =xqw

k TW − T∞ð Þ , Shx =xjm

DB CW − C∞ð Þ ,

ð17Þ

where τwx and τwz are the wall shear stresses in thedirections of x and z, respectively, and qw is the surface

η =ffiffiffiaυ

ry, ψ =

ffiffiffiffiffiaυ

pxf ηð Þ, w = axg ηð Þ, θ ηð Þ = T − T∞

Tw − T∞, ϕ ηð Þ = C − C∞

Cw − C∞, ð10Þ

4 Advances in Mathematical Physics

heat flux, and jm is the surface mass flux, which are givenby

From equations (10), (11), (17), and (18), we obtain

3. Method of Solution

Equations (12)–(15) subject to the boundary conditions (16)are explained utilizing the spectral relaxation method. Thespectral relaxation algorithm uses the thought of the Gauss-Seidel method to decouple the arrangement of overseeingequations (12)–(15). The strategy is created by assessing thelinear terms at the current iteration level r + 1 and nonlinearterms at the former iteration level r. The Chebyshev pseudo-spectral technique is used to unravel the decoupled equa-tions. For the subtleties of the technique, interested readerscan refer [3, 46–48].

Hence, to utilize the SRM, we start by decreasing therequest for the momentum (equation (12)) from third to sec-ond order presenting the change f ′ = h so that f ′′ = h′ andf ′′′ = h′′:Along these lines, equations (12)–(15) become

f ′ = h ð20Þ

1 + εð Þh′′ − εδð Þh′2h′′ − h2 + f h′ − M

α2e + β2e

� � αeh + βegð Þ

+ λ θ +Nrϕð Þ −Κh = 0,ð21Þ

1 + εð Þg′′ − εδð Þg′2g′′ − hg + f g′ + M

α2e + β2e

� � βeh − αegð Þ −Κg = 0,

ð22Þ

1Pr

θ′′ + f θ′ − γ1 f hθ′ + f 2θ′′�

+Nbθ′ϕ′ +Ntθ′2

+M:Ec

α2e + β2e

� � h2 + g2� �

= 0,ð23Þ

Table 1: Comparison of results for the Nusselt number −θ′ð0Þ and Sherwood number −ϕ′ð0Þ for various values of Nb and Nt when Le = 1,Pr = 10, and M = λ = βe =βi =Nr =ε = δ =γ1 =γ2 =K =Ec = 0 are fixed.

Nb NtNoghrehabadi et al.

[51]Goyal and Bhargava

[52]Ibrahim and Gadisa

[53]Present results

−θ′ 0ð Þ −ϕ′ 0ð Þ −θ′ 0ð Þ −ϕ′ 0ð Þ −θ′ 0ð Þ −ϕ′ 0ð Þ −θ′ 0ð Þ −ϕ′ 0ð Þ0.1 0.1 0.9523768 2.1293938 0.95244 2.12949 0.95239 2.12938 0.9523768 2.1293938

0.2 0.6931743 2.2740215 0.69318 2.27401 0.69318 2.27401 0.6931743 2.2740216

0.3 0.5200790 2.5286382 0.52025 2.52855 0.52019 2.52855 0.5200790 2.5286387

0.4 0.4025808 2.7951701 0.40260 2.79520 0.40258 2.79519 0.4025823 2.7951628

0.5 0.3210543 3.0351425 0.32105 3.03511 0.32109 3.03511 0.3210491 3.0351686

0.2 0.1 0.5055814 2.3818706 0.50561 2.38186 0.50557 2.38190 0.5055815 2.3818706

0.3 0.2521560 2.4100188 0.25218 2.41009 0.25217 2.41002 0.2521562 2.4100188

0.4 0.1194059 2.3996502 0.11940 2.39970 0.11940 2.39966 0.1194058 2.3996502

0.5 0.0542534 2.3835712 — — — — 0.0542528 2.3835713

τwx = μ +1βb

� �∂u∂y

−1

6βb3∂u∂y

� �3 !�����

y=0

, τwz = μ +1βb

� �∂w∂y

−1

6βb3∂w∂y

� �3 !�����

y=0

,

qw = −k∂T∂y

� �����y=0

, jm = −DB∂C∂y

� �����y=0

:

ð18Þ

Cf x

ffiffiffiffiffiffiffiRex

p= 1 + εð Þf ′′ 0ð Þ − εδ

3f ′′ 0ð Þ� 3

, Cgz

ffiffiffiffiffiffiffiRex

p= 1 + εð Þg′ 0ð Þ − εδ

3g′ 0ð Þ� 3

,

NuxffiffiffiffiffiffiffiRex

p� −1= −θ′ 0ð Þ, Shx

ffiffiffiffiffiffiffiRex

p� −1= −ϕ′ 0ð Þ:

ð19Þ

5Advances in Mathematical Physics

ϕ′′ + Pr:Le f ϕ′ − γ2 f hϕ′ + f 2ϕ′′� h i

+NtNb

θ′′ = 0:

ð24ÞThe transformed boundary conditions are

f 0ð Þ = 0, h 0ð Þ = 1, g 0ð Þ = 0, θ 0ð Þ = 1, ϕ 0ð Þ = 1,

h⟶ 0, g⟶ 0, θ⟶ 0, ϕ⟶ 0, as η⟶∞:

ð25Þ

Executing the SRM to equations (20)–(25), we get the fol-lowing iteration scheme:

1 + εð Þhr+1′′ − εδð Þhr′2hr′′− h2r + f rhr+1′

−M

α2e + β2e

� � αehr+1 + βegrð Þ + λ θr +Nrϕrð Þ −Κhr+1 = 0,

ð26Þ

f r+1′ = hr+1, ð27Þ

1 + εð Þgr+1′′ − εδð Þgr′2gr′′− hr+1gr+1 + f r+1gr+1′

+M

α2e + β2e

� � βehr+1 − αegr+1ð Þ −Κgr+1 = 0,ð28Þ

1Pr

θr+1′′ + f r+1θr+1′ − γ1 f r+1hr+1θr+1′ + f 2rθr+1′′�

+Nbθr+1′ ϕr+1′

+Ntθr′2 +

M:Ec

α2e + β2e

� � h2r + g2r� �

= 0,

ð29Þ

ϕr+1′′ + Pr:Le f r+1ϕr+1′ − γ2 f r+1hr+1ϕr+1′ + f 2rϕr+1′′� h i

+NtNb

θr+1′′ = 0:

ð30Þ

0 1 2 3 4 5 6 7 8 9 100

0.10.2

0.30.4

0.5

0.60.7

0.80.9

1

𝜃 (𝜂

)

𝜀 = 0.0𝜀 = 0.5𝜀 = 1.0

𝜂

Figure 4: Effect of ε on the temperature θðηÞ.

0 5 10 150

0.10.2

0.30.4

0.5

0.60.7

0.80.9

1

𝜀 = 0.0𝜀 = 0.5𝜀 = 1.0

𝜂

(𝜂)

ϕ

Figure 5: Effect of ε on the concentration ϕðηÞ.

0 1 2 3 4 5 6 7 8 9 100

0.0010.002

0.0030.004

0.0050.0060.0070.0080.009

0.01

𝜀 = 0.0𝜀 = 0.5𝜀 = 1.0

𝜂

g (𝜂

)

Figure 3: Effect of ε on the secondary velocity gðηÞ.

0 1 2 3 4 5 6 70

0.10.2

0.30.4

0.5

0.60.7

0.80.9

1

𝜀 = 0.0𝜀 = 0.5𝜀 = 1.0

𝜂

f´ (𝜂)

Figure 2: Effect of ε on the primary velocity f ′ðηÞ:

6 Advances in Mathematical Physics

The transformed boundary conditions are

f r+1 0ð Þ = 0, hr+1 0ð Þ = 1, gr+1 0ð Þ = 0, θr+1 0ð Þ = 1, ϕr+1 0ð Þ = 1,

hr+1 ⟶ 0, gr+1 ⟶ 0, θr+1 ⟶ 0, ϕr+1 ⟶ 0, as η⟶∞:

ð31Þ

Actualizing the Chebyshev spectral collocation methodto equations (26)–(30), we get the ensuing matrix equations:

A1hr+1 = B1, hr+1 ξNð Þ = 1, hr+1 ξ0ð Þ = 0, ð32Þ

A2f r+1 = B2, f r+1 ξNð Þ = 0, ð33Þ

A3gr+1 = B3, gr+1 ξNð Þ = 0, gr+1 ξ0ð Þ = 0, ð34Þ

0 1 2 3 4 5 6 7 8 9 100

0.10.20.30.40.50.60.70.80.9

1

𝜃 (𝜂

)

M = 5M = 10M = 15

𝜂

Figure 8: Effect of M on the temperature θðηÞ.

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M = 5M = 10M = 15

𝜂

(𝜂)

ϕ

Figure 9: Effect of M on the concentration ϕðηÞ.

0 1 2 3 4 5 6 70

0.10.20.30.40.50.60.70.80.9

1

0.3 0.4 0.50.6

0.65

0.7

0.75

𝛽e = 5𝛽e = 10𝛽e = 15

𝜂

f´ (𝜂)

Figure 10: Effect of βe on the primary velocity f ′ðηÞ.

0 1 2 3 4 5 6 7 8 9 100

0.002

0.004

0.006

0.008

0.01

0.012

g (𝜂

)

M = 5M = 10M = 15

𝜂

Figure 7: Effect of M on the secondary velocity gðηÞ.

0 1 2 3 4 5 6 70

0.10.20.30.40.50.60.70.80.9

1f´ (𝜂)

𝜂

M = 5M = 10M = 15

Figure 6: Effect of M on the primary velocity f ′ðηÞ.

7Advances in Mathematical Physics

A4θr+1 = B4, θr+1 ξNð Þ = 1, θr+1 ξ0ð Þ = 0, ð35Þ

A5ϕr+1 = B5, ϕr+1 ξNð Þ = 1, ϕr+1 ξ0ð Þ = 0, ð36Þwhere

A1 = 1 + εð ÞD2 + diag f rð ÞD −Mαe

α2e + β2e

+ K

!I,

B1 = εδð Þhr′2hr′′+ h2r +

Mβe

α2e + β2e

� �gr − λ θr +Nrϕrð Þ,ð37Þ

A2 =D, B2 = hr+1,

A3 = 1 + εð ÞD2 + diag f r+1ð ÞD − diag hr+1 +Mαe

α2e + β2e

+ K

!I

B3 = εδð Þgr′2gr′′−

Mβe

α2e + β2e

hr+1,

A4 = diag1Pr

− γ1 f2r

� �D2 + diag f r+1 − γ1 f r+1hr+1 +Nbϕr′

� D,

B4 = −M:Ec

α2e + β2e

� � h2r + g2r� �

−Ntθr′2,

A5 = diag 1 − PrLeγ2 f2r

� �D2 + PrLe diag f r+1 − γ2 f r+1hr+1ð ÞD,

B5 = −NtNb

θr+1′′ : ð38Þ

Here,D = 2D/η∞ where D is the Chebyshev differentia-tion matrix (Trefethen [50]), η∞ is a limited length which isadequately big so that we can simply incorporate the condi-tion at perpetuity in this point, I and diag [.] are the identityand diagonal matrices of order ðN + 1Þ × ðN + 1Þ, N is the

0 1 2 3 4 5 6 7 8 9 100

0.10.20.30.40.50.60.70.80.9

1

0.6 0.8 10.6

0.65

0.7

0.75

0.8

𝛽e = 5𝛽e = 10𝛽e = 15

𝜂

𝜃 (𝜂

)

Figure 12: Effect of βe on the temperature θðηÞ.

0 5 10 150

0.10.20.30.40.50.60.70.80.9

1

1.6 1.8 2

0.65

0.7

0.75

𝛽e = 5𝛽e = 10𝛽e = 15

𝜂

𝜃 (𝜂

)

Figure 13: Effect of βe on the concentration ϕðηÞ.

0 1 2 3 4 5 6 70

0.10.20.30.40.50.60.70.80.9

1

0.3 0.4 0.50.55

0.6

0.65

0.7

𝛽i = 5𝛽i = 10𝛽i = 15

𝜂

f´ (𝜂)

Figure 14: Effect of βi on the primary velocity f ′ðηÞ.

0 1 2 3 4 5 6 7 8 9 100

0.0010.0020.0030.0040.0050.0060.0070.0080.009

0.01

𝛽e = 5𝛽e = 10𝛽e = 15

𝜂

g (𝜂

)

Figure 11: Effect of βe on the secondary velocity gðηÞ.

8 Advances in Mathematical Physics

number of grid points, and f , h, g, θ and ϕ are, respectively,the values of f, h, g , θ, and ϕ, when assessed at the grid points.

The decoupled equations (32)–(36) can be solved sepa-rately by picking appropriate introductory suppositions:

f0 ηð Þ = 1 − e−η, g0 ηð Þ = ηe−η, θ0 ηð Þ = e−η = ϕ0 ηð Þ: ð39Þ

4. Results and Discussions

The nonlinear differential equations (12)–(15) with the limitconditions (16) have been explained numerically via thespectral relaxation method, and consequences of the numer-ical calculations for the velocity, temperature, and concentra-tion profiles as well as skin friction coefficients, local Nusseltnumber, and Sherwood number have been obtained for dif-ferent input parameters and presented through graphs andtables. Also, we contrasted our outcomes and those of thecurrent writing as appeared in Table 1, and the outcomes give

a generally excellent understanding which gives a certainty ofour current numerical outcomes. In this study, for ournumerical computations, we fixed the numerical values ofthe physical parameters as = 0:1, Le = 1:0, M = 10:0, βe =βi = 5:0, Pr = 0:72, K = 0:5, Nr =Nb =Nt = 0:1, Ec = 0:02,δ = 0:2, and ε = γ1 = γ2 = 0:3 unless otherwise stated.

Figures 2–5 reveal the impact of the Eyring-Powell fluidparameter ðεÞ on the primary velocity, secondary velocity,temperature, and concentration distributions, respectively.By expanding the liquid parameter ε, the primary velocityprofile rises and on the contrary, the temperature and con-centration profiles diminish. Because the fluid parameter εð= 1/μβbÞ has an inverse relation with viscosity of fluid ðμÞ ;thus, fluid turns out to be less viscous for bigger values of εwhich upsurges fluid velocity. Moreover, the secondaryvelocity profile reduces near the surface and enhances faroff from the surface. This is possibly due to the fact that the

0 1 2 3 4 5 6 7 8 9 100

0.10.20.30.40.50.60.70.80.9

1

0.6 0.7 0.80.65

0.7

0.75

𝛽i = 5𝛽i = 10𝛽i = 15

𝜂

𝜃 (𝜂

)

Figure 16: Effect of βi on the temperature θðηÞ.

0 5 10 150

0.10.20.30.40.50.60.70.80.9

1

1.4 1.6 1.80.7

0.75

0.8

𝛽i = 5𝛽i = 10𝛽i = 15

𝜂

(𝜂)

ϕ

Figure 17: Effect of βi on the concentration ϕðηÞ.

0 1 2 3 4 5 6 70

0.10.2

0.30.4

0.5

0.60.7

0.80.9

1

𝜆𝜆𝜆

𝜂

= 0.2= 0.4= 0.6

f´ (𝜂)

Figure 18: Effect of λ on the primary velocity f ′ðηÞ.

0 1 2 3 4 5 6 7 8 9 100

0.0010.0020.0030.0040.0050.0060.0070.0080.009

0.01

𝛽i = 5𝛽i = 10𝛽i = 15

𝜂

g (𝜂

)

Figure 15: Effect of βi on the secondary velocity gðηÞ.

9Advances in Mathematical Physics

viscosity can be varied as the Eyring-Powell fluid parametervaries resulting in an increasing viscous force as a resultdecreasing its velocity up to some maximum value, and thenthe velocity starts to increase due to decrease in viscosity. Theoptimum value is obtained around the point ≈1.

Figures 6–9 divulge the impact of magnetic field parame-ter M on primary velocity, secondary velocity, temperature,and concentration profiles, respectively. The transverse mag-netite field which is applied vertical to the direction of flowprovides a resistive force known as the Lorentz force. Physi-cally, the Lorentz force opposes the flow of nanofluid; hence,the velocity diminishes. Moreover, this force tends to hinderthe movement of the liquid in the boundary layer. This assistswith declining the primary velocity profile and upgrades thesecondary velocity, thermal, and concentration profiles asM increments as appeared in Figures 6–9. It is additionallyseen that an augmentation in M prompts a lower in primaryvelocity boundary layer thickness and superior in thermal

and concentration boundary layer thicknesses, though thesecondary velocity boundary layer thickness diminishes nearthe surface and upsurges far off from the surface.

Figures 10–13 illustrate the impact of the Hall parameterβe on primary velocity, secondary velocity, temperature, andconcentration profiles, respectively. It is seen that the pri-mary velocity profile increments with an expansion in βe,though the contrary condition is watched for secondaryvelocity, thermal, and concentration profiles. Besides, as theHall parameter βe enlarges, the primary velocity boundarylayer thickness upgrades, and thermal and concentrationboundary layer thicknesses lessen, while the secondary veloc-ity boundary layer thickness improves near the surface anddiminishes distant from the surface. Physically, the consider-ation of Hall parameter decreases the efficient conductivityand subsequently drops the magnetic resistive force. Thus,the Joule heating impact is condensed, and quantity of heat

0 1 2 3 4 5 6 7 8 9 100

0.10.20.30.40.50.60.70.80.9

1

𝜆𝜆𝜆

𝜂

= 0.2= 0.4= 0.6

𝜃 (𝜂

)

Figure 20: Effect of λ on the temperature θðηÞ.

0 1 2 3 4 5 60

0.10.20.30.40.50.60.70.80.9

1

K = 1.0K = 2.0K = 3.0

𝜂

f´ (𝜂)

Figure 22: Effect of K on the primary velocity f ′ðηÞ.

0 1 2 3 4 5 6 7 8 9 100

0.10.20.30.40.50.60.70.80.9

1

𝜆𝜆𝜆

𝜂

= 0.2= 0.4= 0.6

(𝜂)

ϕ

Figure 21: Effect of λ on the concentration ϕðηÞ.

0 1 2 3 4 5 6 7 8 9 100

0.002

0.004

0.006

0.008

0.01

0.012

𝜆𝜆𝜆

𝜂

= 0.2= 0.4= 0.6

g (𝜂

)

Figure 19: Effect of λ on the secondary velocity gðηÞ.

10 Advances in Mathematical Physics

owing to ohmic dissipation can be abridged by utilizing par-tially ionized fluid presented to magnetic field. The compara-ble conduct is seen with expanding ion slip parameter βi asrevealed in Figures 14–17. Physically, the expansion in βiimproves the efficient conductivity and therefore diminishesthe damping force on the dimensionless velocity, and hencethe dimensionless velocity expands.

Figures 18–21 portray the impact of blended convectionparameter λ on primary velocity, secondary velocity, temper-ature, and concentration profiles, respectively. It is deducedthat as λ upgrades, the primary velocity and secondary veloc-ity profiles ascend. In contrast, the temperature and concen-tration profiles reduce. These outcomes physically hold sinceupgrade in blended convection parameter causes upsurge inbuoyancy forces which speed up fluid motion.

Figures 22–25 outline the impact of penetrability param-eter K on the primary velocity, secondary velocity, thermal,

and concentration distributions, respectively. It is obviousfrom the figures that as K augments, both primary and sec-ondary velocity profiles decline while the opposite conditionis watched for thermal and concentration profiles. Physically,the porousness builds obstruction of the permeable mediumwhich will in general, diminish the fluid velocity. It is normalthat an expansion in the porousness of the permeablemedium prompts the ascent in the flow of fluid through it.When the outlets of the permeable medium become big, theobstruction of the medium might be ignored.

Figure 26 proves the impact of the thermal relaxationparameter γ1 on temperature distribution. It uncovers thatthe fluid temperature and energy boundary layer decreasewith expanding value of γ1. Physically, thermal relaxationtime is the time required by the fluid particles to move heatenergy to its neighboring particles. Subsequently, as γ1ascends, the material particles need additional time to trans-fer heat to its adjoining particles, and this prompts less

0 1 2 3 4 5 6 7 8 9 10

×10–3

0

1

2

3

4

5

6

7

K = 1.0K = 2.0K = 3.0

𝜂

g (𝜂

)

Figure 23: Effect of K on the secondary velocity gðηÞ.

0 5 10 150

0.10.20.30.40.50.60.70.80.9

1

K = 1.0K = 2.0K = 3.0

𝜂

𝜃 (𝜂

)

Figure 24: Effect of K on the temperature θðηÞ.

0 2 4 6 8 10 12 14 16 180

0.10.2

0.30.4

0.5

0.60.7

0.80.9

1

K = 1.0K = 2.0K = 3.0

𝜂

(𝜂)

ϕ

Figure 25: Effect of K on the concentration ϕðηÞ.

0 2 4 6 8 10 120

0.10.20.30.40.50.60.70.80.9

1

𝛾1 = 0.0𝛾1 = 0.5𝛾1 = 1.0

𝜂

𝜃 (𝜂

)

Figure 26: Effect of γ1 on the temperature θðηÞ.

11Advances in Mathematical Physics

transport of heat from the surface to the fluid. Similarly, theconcentration relaxation parameter γ2emerges from therelaxation time of mass flux. Less relaxation time of mass fluxprompts superior concentration. and the fluid with increas-ingly mass flux relaxation time brings about smaller concen-tration. Therefore, increasingly mass flux relaxation timehappens owing to the bigger value of γ2 which diminishesthe concentration profile as revealed in Figure 27.

Figure 28 depicts the impact of Eckert number Ec on thetemperature profile. It is seen that the temperature profileupsurges with an expansion in Ec. Physically, by expandingthe Eckert number, the heat energy is hoarded in the fluidas a result of the frictional or drag forces. Accordingly, thefluid temperature field enhances.

Figure 29 demonstrates the effect of Lewis number Le onthe concentration profile ϕðηÞ. By expanding Le, the concen-tration and the boundary layer thicknesses decline. This isbecause of the way that the mass transfer rate enhances by

expanding Le which improves the volume fraction ofnanoparticles.

Table 2 specifies the impact of physical parameters on theskin fraction coefficients, local Nusselt, and Sherwood num-bers. The primary skin friction coefficient −

ffiffiffiffiffiRe

pCf x aug-

ments as magnetic parameter M , Eyring-Powell fluidparameter ε, thermal relaxation parameter γ1, concentrationrelaxation parameter γ2, and permeability parameter K riseand decreases for increasing values of mixed convectionparameter λ, Hall parameter βe, ion slip parameter βi, andEckert number Ec. Also, it is observed that the secondary skinfriction coefficient

ffiffiffiffiffiRe

pCgz amplifies as magnetic parameter

M, mixed convection parameter λ, and Eyring-Powell fluidparameter ε increase and declines as Hall parameter βe, ionslip parameter βi, thermal relaxation parameter γ1, concen-tration relaxation parameter γ2, and permeability parameterK increase, while it remains constant as the Eckert numberEc enhances. Similarly, both the local Nusselt number –θ′ð0Þand Sherwood number –ϕ′ð0Þ enhance for larger values ofmixed convection parameter λ, Hall parameter βe, ion slipparameter βi, and Eyring-Powell fluid parameter ε anddecrease for larger values of magnetic parameter M and per-meability parameter K. Further, opposite behaviors of thelocal Nusselt number –θ′ð0Þ and the local Sherwood number–ϕ′ð0Þ are observed for increasing values of thermal relaxa-tion parameter γ1, concentration relaxation parameter γ2,and Eckert number Ec:

5. Conclusions

This paper looks at the mixed convection stream of Eyring-Powell nanofluid with the Cattaneo-Christov heat and massflux model over linearly stretching sheet through a permeablemedium. The impacts of Hall and ion slip, permeability, andJoule heating are also considered. The governing nonlinearpartial differential equations are changed into joined nonlinear

0 2 4 6 8 10 12 140

0.10.20.30.40.50.60.70.80.9

1

𝛾2 = 0.0𝛾2 = 0.5𝛾2 = 1.0

𝜂

(𝜂)

ϕ

Figure 27: Effect of γ2 on the concentration ϕðηÞ.

0 2 4 6 8 10 12 14 16 18 200

0.10.20.30.40.50.60.70.80.9

1

Ec = 0.0Ec = 0.2Ec = 0.4

𝜂

𝜃 (𝜂

)

Figure 28: Effect of Ec on the temperature θðηÞ.

0 1 2 3 4 5 6 7 8 9 100

0.10.20.30.40.50.60.70.80.9

1

Le = 2.0Le = 2.5Le = 3.0

𝜂

(𝜂)

ϕ

Figure 29: Effect of Le on the concentration ϕðηÞ.

12 Advances in Mathematical Physics

ordinary differential equations utilizing similitude transforma-tions. These nonlinear ordinary differential equations subjectto boundary conditions are handled numerically by means ofSRM. Numerical results are presented for different parametersof interest, and results are discussed with the assistance ofgraphs and tables. The study has a momentous applicationin the material processing technology. From the introducedinvestigation, the accompanying ends were drawn:

(i) Primary velocity profile upsurges by increasing thevalues of Eyring-Powell fluid parameter ε, Hallparameter βe, ion slip parameter βi, and mixed con-vection parameter λ, while opposite condition isnoticed for mounting the values of magnetic fieldparameter M and porosity parameter K

(ii) Secondary velocity profile increases by increasingthe values of Eyring-Powell fluid parameter ε andmagnetic field parameter M, while reverse behavioris observed for increasing values of Hall parameter

βe, ion slip parameter βi, mixed convection parame-ter λ, and porosity parameter K

(iii) Both temperature and concentration profiles enhancewith increasing values of magnetic field parameterMand porosity parameter K, whereas reverse trend isnoticed for increasing values of Eyring-Powell fluidparameter ε, Hall parameter βe, ion slip parameterβi, mixed convection parameter λ, thermal relaxationparameter γ1, and concentration relaxation parame-ter γ2. Moreover, temperature profile increases asEckert numberEc increases, while concentration pro-file decreases as Lewis numberLe increases

(iv) Both skin friction coefficients increase with increas-ing values of magnetic field parameter M andEyring-Powell fluid parameter ε and decrease withincreasing values of Hall parameter βe and ion slipparameter βi: On the contrary, opposite behavior isobserved for the primary and secondary skin friction

Table 2: Numerical values of the skin friction coefficientsffiffiffiffiffiRe

pCf x = ð1 + εÞf ′′ð0Þ − ðεδ/3Þð f ′′ð0ÞÞ3 and

ffiffiffiffiffiRe

pCgz = ð1 + εÞg′ð0Þ − ðεδ/3Þ

ðg′ð0ÞÞ3, local Nusselt number –θ′ð0Þ, and local Sherwood number –ϕ′ð0Þ for various values of M, λ, βe, βi, ε, γ1, γ2, K , and Ec when Nr = 1:0, Pr = 0:72, Nb = 0:2, Nt = 0:1, Le = 1, and δ = 10 are fixed.

M λ βe βi ε γ1 γ2 K Ec −ffiffiffiffiffiRe

pCf x

ffiffiffiffiffiRe

pCgz −θ′ 0ð Þ −ϕ′ 0ð Þ

5.0 0.5 5.0 5.0 0.3 0.5 0.5 2.0 0.02 1.47604 0.01432 0.43751 0.33050

10.0 1.53447 0.02757 0.43011 0.32389

15.0 1.59039 0.03987 0.42296 0.31755

10.0 0.2 1.77436 0.02470 0.37874 0.27471

0.4 1.61585 0.02671 0.41540 0.30992

0.6 1.45292 0.02836 0.44328 0.33631

0.5 5.0 1.53447 0.02757 0.43011 0.32389

10.0 1.47713 0.01485 0.43740 0.33037

15.0 1.45699 0.01016 0.43993 0.33265

5.0 5.0 1.53447 0.02757 0.43011 0.32389

10.0 1.47883 0.00763 0.43719 0.33018

15.0 1.45839 0.00350 0.43976 0.33249

5.0 0.0 1.38279 0.02496 0.42130 0.31614

0.5 1.63596 0.02914 0.43581 0.32901

1.0 1.88555 0.03270 0.44940 0.34156

0.3 0.0 1.53132 0.02769 0.40768 0.33589

0.5 1.53447 0.02757 0.43011 0.32389

1.0 1.53813 0.02743 0.45628 0.31008

0.5 0.0 1.53050 0.02771 0.43601 0.30351

0.5 1.53447 0.02757 0.43011 0.32389

1.0 1.53912 0.02740 0.42355 0.34772

0.5 1.0 1.18778 0.03474 0.47314 0.36359

2.0 1.53447 0.02757 0.43011 0.32389

3.0 1.79830 0.02303 0.39431 0.29262

2.0 0.0 1.53447 0.02757 0.43019 0.32385

0.1 1.53446 0.02757 0.42980 0.32404

0.2 1.53444 0.02757 0.42941 0.32422

13Advances in Mathematical Physics

coefficients as mixed convection parameter λ, poros-ity parameter K , thermal relaxation parameter γ1,and concentration relaxation parameter γ2 upsurges.An increase in Eckert number Ec shows insignificantchange in the primary skin friction coefficient, whilethe secondary skin friction coefficient remainsconstant

(v) Both local Nusselt and Sherwood numbers increaseby enhancing the values of mixed convection param-eter λ, Hall parameter βe, ion slip parameter βi, andEyring-Powell fluid parameter ε, while oppositebehavior is noticed for increasing values of magneticfield parameter M and permeability parameter K .On the contrary, opposite behavior is observed forlocal Nusselt and Sherwood numbers as EckertnumberEc, thermal relaxation parameter γ1, andconcentration relaxation parameter γ2 increase

Nomenclature

x, y, z: Cartesian coordinates [m]u, v,w: Velocity components [m/s]Uw: Velocity of the stretching sheet [m/s]B0: Constant magnetic field [Wb/m2]g: Acceleration due to gravity [m/s2]T : Fluid temperature [K]Tw: Surface temperature [K]T∞: Ambient temperature [K]C: Concentration of fluid [kg/m3]Cw: Surface concentration [kg/m3]C∞: Ambient concentration [kg/m3]M: Magnetic field parameterGrx: Local Grashof numberRex: Local Reynolds numberNr: Buoyancy ratioK : Permeability parameterPr: Prandtl numberNb: Brownian motion parameterNt: Thermophoresis parameterEc: Eckert numberLe: Lewis numberCf x: Skin friction coefficient in x-directionCgz : Skin friction coefficient in z-directionNux: Local Nusselt numberShx: Local Sherwood numberf , g: Dimensionless stream functions

Greek letters

ρ: Density of fluid [kg/m3]υ: Kinematic viscosity of the fluid [m2/s]ε, δ: Eyring-Powell fluid parametersλ: Mixed convection parameterα: Thermal diffusivity [m2/s]κ: Thermal conductivity [W/(mK)]σ: Electrical conductivity [m2/s]βe: Hall parameterβi: Ion slip parameter

βT : Coefficient of thermal expansion[1/K]βC : Coefficient of concentration expansion [1/K]ψ: Stream functionη: Dimensionless similarity variableθ: Dimensionless temperatureϕ: Dimensionless concentrationγ1: Thermal relaxation parameterγ2: Concentration relaxation parameter

Subscripts

f : Fluidp: Nanoparticlew: Condition at the surface∞: Ambient condition

Superscripts

′: Differentiation w. r. t. η.

Data Availability

Availability of data: The data available in this paper is avail-able without any restriction.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All the authors have made substantive contributions to thearticle and assume full responsibility for its content. Theauthors read and approved the final manuscript.

Acknowledgments

The authors would like to express their deep Thanks are dueto the referee for his/her careful reading and many valuablesuggestions towards the improvement of the paper.

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