Numerical exploration of thermal and mass transportation

Pramana – J. Phys. (2021) 95:185 © Indian Academy of Scienceshttps://doi.org/10.1007/s12043-021-02220-y

Numerical exploration of thermal and mass transportation byutilising non-Fourier double diffusion theories for Casson modelunder Hall and ion slip effects

MUHAMMAD SOHAIL1 ,∗, HUSSAM ALRABAIAH2,3, UMAIR ALI1, FATEMA TUZ ZOHRA4,MAHMOUD M SELIM5,6,∗ and PHATIPHAT THOUNTHONG7

1Department of Applied Mathematics and Statistics, Institute of Space Technology, P.O. Box 2750,Islamabad 44000, Pakistan2College of Engineering, Al Ain University, Al Ain, UAE3Department of Mathematics, Tafila Technical University, Tafila, Jordan4Department of Mathematics, Faculty of Science and Technology, American International University,Dhaka 1229, Bangladesh5Department of Mathematics, Al-Aflaj College of Science and Humanities Studies, Prince Sattam Bin AbdulazizUniversity, Al-Aflaj 710-11912, Saudi Arabia6Department of Mathematics, Suez Faculty of Science, Suez University, Suez 34891, Egypt7Department of Teacher Training in Electrical Engineering, Renewable Energy Research Centre, Faculty ofTechnical Education, King Mongkut’s University of Technology North Bangkok, 1518 Pracharat 1 Road, Bangsue,Bangkok 10800, Thailand∗Corresponding authors. E-mail: [email protected]; [email protected]

MS received 13 March 2021; revised 1 July 2021; accepted 1 July 2021

Abstract. Non-Newtonian materials have attracted the attention of scientists and engineers due to their manyapplications in the current era. This endeavour is conducted to utilise the generalised Ohm law with thermaland mass transportation. Phenomena of heat and mass transfer are based on generalised Fourier and Fick’s lawsrespectively. Present analysis examines magnetohydrodynamic (MHD) three-dimensional flow of the Casson liquid.Flow is assumed to be over a stretched surface which is stretched in two directions. Contribution of Hall and ion slipeffects are included. Diffusion phenomenon is captured using the Boungrino model. Convergent series solutionsby homotopy algorithm is also derived. Physical quantities of interest are discussed with respect to the involvedvariables. Convergence of the applied scheme is presented in the form of error analysis. Also convergence is shownby computing dimensionless stresses, heat and mass transfer rates. Authenticity of the achieved result is shown bycomparing the obtained results with those from the open literature and excellent similarity is attained and recorded.Diffusion of mass and heat can be controlled by enhancing the thermal, solutal factors and Prandtl and Schmidtnumbers.

Keywords. Three-dimensional flow; stretched surface; Boungrino model; Cattaneo–Christov heat flux; erroranalysis.

PACS Nos 47.10.−g; 47.10.ad; 47.15.G−; 47.10.A−

1. Introduction

Dynamics of non-Newtonian fluids has inspired scien-tists and engineers because such materials have manyapplications in engineering processes. Examples of suchmaterials are mud, glues, apple sauce, soaps, printingink, shampoos, condensed milk, lubricant oil, paints,sugar solutions, tomato paste, polymeric solutions etc.

Moreover, these non-Newtonian materials are used inbiology, geophysics, petroleum industry and chemicalengineering. It is now recognised that behaviours of non-Newtonian fluids are different from viscous materials.Thus, traditional Navier–Stokes equations do not predictthe rheological characteristics of the non-Newtonianliquids. Further, there is no single relation which cancapture diverse properties of all the non-Newtonian

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185 Page 2 of 12 Pramana – J. Phys. (2021) 95:185

liquids. That is why many constitutive relations of non-Newtonian liquids are available. The fluid model underconsideration is the Casson model. Casson fluid exhibitsyield stress [1–5]. Khan et al [1] studied the MHD Cas-son fluid (CF) under the homogeneous–heterogeneousreaction over a stretching sheet. They solved the bound-ary layer equations (BLEs) using the built-in shootingmethod in MATHEMATICA 11.0 computational soft-ware. They reported that velocity field varies inverselywith magnetic parameter and thermal profile variesagainst Prandtl numbers. Zia et al [2] explored theinvolvement of mixed convection with heat and masstransport for the Casson model in three dimensions.They used the exponential heat source which is thefunction of spatial coordinates for thermal profile. Theycomputed the solution analytically via homotopy analy-sis method (HAM). They have shown that heat transferrate varies directly with Biot number and radiationparameter. Tamoor et al [3] studied various dissipationcontributions for the MHD flow of Casson liquid in arotating cylinder. They have considered the influenceof Joule heating in their research. They established theinverse bearing of Prandtl number upon thermal fieldand heat transfer rate. Mahanthesh et al [4] studied theflow of Casson liquid in a rotating disk with Marangoniconvection and cross diffusion effects. They have shownthat velocity field varies directly with Marangoni effect.Also, they plotted the direct variation in the thermalprofile for dissipation and Newtonian heating effects.Nadeem et al [5] investigated the contribution of stag-nation point on the Casson model with heat and masstransmission over a stretching heated surface. Theyshowed that augmentation in stretching ratio parame-ter enhances the heat transfer rate, but opposite impactis noticed with the Brownian motion parameter. Pra-manik [6] studied the transport of heat in the Casson fluidover on exponentially stretching porous surface with theinvolvement of thermal radiation. He showed that aug-mentation in radiation parameter enhances the thermaldistribution. Khalid et al [7] reported time-dependantconvection flow of the Casson fluid over a heated platewith constant wall temperature. They computed theexact solution of the transformed equations with thehelp of Laplace procedure. They have shown that thevelocity field varies inversely with Prandtl number andit has a direct variation for Grashof number. Hayat etal [8] discussed the MHD flow of the Casson liquidover a stretching surface under the Soret and Dufoureffects. They solved the governing equations analyti-cally. They tabulated and prepared the graphs againstemerging parameters for the computed solutions in theirresearch. They reduced the opposite impact of fluid andmagnetic parameter on dimensionless stress. Hussananet al [9] addressed heat transfer in unsteady boundary

layer flow of the Casson fluid by an oscillatory plate.They computed the solution for the governing expres-sions. The obtained results are presented in the form ofgraphs. They established that time parameter enhancesthe thermal profile and Prandtl number decreases itsignificantly. Nadeem et al [10] analysed MHD flowof a Casson fluid by an exponential stretching sheet.They handled the governing equations via Adomiandecomposition scheme by involving Pade approxima-tion. They have showed that magnitude of the velocitydecreases with magnetic parameter and suction injec-tion parameter. Nadeem et al [11] also analysed thethree-dimensional flow of the Casson fluid and theyhave assumed that the flow is generated in a poroussurface which is stretched linearly. They presented self-similar solutions to the governing modelled equations.They recorded that escalating values of porosity param-eter and Hartmann number are favourable parametersfor dimensionless stress. Three-dimensional MHD flowof a Casson fluid in a porous medium is studied by She-hzad et al [12]. They modelled the energy equation inthe presence of an external heat source. They mentionedthat the heat source parameter enhances the fluid tem-perature. Mukhopadhyay et al [13] examined the impactof thermal radiation in MHD boundary layer flow of theCasson fluid by an exponential stretching sheet. Theyconsidered the contribution of radiation factor in theenergy equation and solved the modelled problem viashooting scheme. They observed that thermal bound-ary layer (TBL) increases compared to the momentumboundary layer (MBL) by increasing the values of Cas-son parameter (CP). Kumar et al [14] inspected theconsequences of non-Fourier heat flux in nonlinearradiative MHD flow of the Casson fluid. They used theCattaneo–Christov (CC) heat flux with thermal radiationfor thermal transport. They recorded that the buoyancyparameter boost the rate of heat transfer and dimen-sionless stress. Raju and Sandeep [15] addressed MHDboundary layer flow of the Casson–Carreau liquid. Theydiscussed the involvement of non-uniform heat sourceand homogeneous–heterogeneous for both the models.They solved the resulting equations using the shootingscheme. They recorded that mass and heat transmissionrate in the Casson model (CM) is significantly higherthan Carreau fluid model (CFM). Similarity solution forthe three-dimensional Casson nanofluid flow with con-vective boundary conditions is reported by Sulochana etal [16]. They considered the convective boundary con-ditions (CBCs) and influence of nanoparticles using thesingle phase model. They solved the coupled bound-ary layer equations (CBLEs) numerically. Their findingsdepict that rate of heat and mass transmission in thenon-Newtonian model is significantly higher than theclassical case. Hayat et al [17] studied the heat and

Pramana – J. Phys. (2021) 95:185 Page 3 of 12 185

mass transport for the Prandtl fluid model (PFM). Theyhave engaged the double diffusion theory for heat andmass fluxes. They solved the flow equations via OHAM.They mentioned that augmenting values of Schmidtnumber increase the concentration field. In another sur-vey, Hayat et al [18] modelled a second-grade fluidmodel over a stretched surface. Contributions of chem-ical reactions are considered along with modified heatand mass fluxes. They handled the stretched flow prob-lem by applying OHAM. They presented that heatand mass transmission rates are higher against thermalsolutal relaxation factors. Power–energy liquid (PEL)was studied for the generalised fluxes by Waqas et al[19]. They considered the temperature-dependent ther-mal conductivity. They tabulated the numerical valuesagainst emerging parameters for heat transfer rate. Theyreported that fifth-order homotopic solution is sufficientfor velocity and tenth-order approximation is sufficientfor thermal and concentration fields. Khan et al [20]studied heat and mass transmission in Jaffery model(JM) over a wedge. They considered the phenomena ofraised curvature of the modified fluxes. They reportedthat Lewis number reduces the mass transfer rate andPrandtl number enhances the heat transfer rate. Bilalet al [21] reported double diffusion models for heatand mass transport for Prandtl model. They solved theboundary layer expressions numerically. They men-tioned that the heat transfer rate reduces against ther-mophoresis and Brownian motion parameters. Khan etal [22] studied the mass heat transfer in Seiko fluidmodel (SFM) with variable thermal conductivity overa nonlinear stretching surface (NSS). They solved theresulting problem using the bvp4c procedure. Theyanticipated that increasing values of Schmidt numberdiminishes the concentration profile. Sohail et al [23]studied the couple stress fluid (CSF) by considering thevariable properties over a bidirectional stretched surfaceand they solved the model problem analytically. Theypresented the comparative study too. They found thatescalating values of fluid parameters (FP) are favourablefor the velocity field. Hayat et al [24] solved thelinear three-dimensional nanofluid model numerically.They showed that mounting values of rate of parameterenhances the velocity along the x- and y-axes and veloc-ity decreases with the ratio parameters. Farooq et al [25]discussed the application and effectiveness of BVPh2.0 by handling the physical problem of thermal andspecies transportation. Naz et al [26] solved the highlynonlinear and complex fourth-grade fluid problem in aporous medium analytically. They portrayed the conver-gence region of the achieved solution too. They reportedthat augmentation in permeability parameter has simi-lar impact on fluid velocity and temperature. Hayat et al[27] computed the solution of third-grade fluid (TGF)

analytically in stationary plates. They mentioned thattwentieth-order appreciations are sufficient for the con-vergent solution of fluid velocity. They reported thataugmentation in porosity parameter creates a hurdle influid flow. Modelling of Sakiadis flow with heat andmass transport was presented by Awais et al [28]. Theysolved the resulting questions numerically and theyfound that higher values of Deborah number retard theflow. Arqub and El-Ajou [29] examined the epidemicmodel (EM) via homotopy analysis procedure (HAP)by considering the fractional derivative (FD). They pre-sented the sensitivity analysis for the considered modeland convergence region is shown through graphs. Arquband El-Ajou [29] examined the epidemic and Rasooland Wakif [30] examined the EMHD flow of second-grade fluid (SGF) past a Riga plate in the presenceof heat generation effect (HDE) and mixed convection(MV). They computed the solution of the physical sys-tem by engaging spectral local linearisation procedure(SLLP). They reported that the dimensionless stress isan increasing function of fluid parameters and Hart-man number. Phenomena of bioconvection, chemicalreaction and buoyancy effects for the viscoelastic fluidwere explained by Shafiq et al [31]. Their contributionshowed that the bioconvective Lewis number and Pecletnumber are responsible for controlling the diffusion ofdensity profile. Din et al [32] investigated the impactof variable properties, chemical reaction, Joule heatingand viscous dissipation for the stress yield exhibitingmaterial. They solved the resulting transformed equa-tions via wavelets procedure. Temperature-dependentthermal conductivity and diffusion coefficient for themixed convective flow of the yield exhibiting liq-uid flowing over a nonlinear stretching sheet wasreported by Sohail et al [33] analytically via OHAM.They presented error analysis of the OHAM in tab-ular form. They found that augmentation in Schmidtnumber reduces the species profile significantly. Somerecently attempted contributions have been reported in[34–43].

A survey of literature reveals that no one has so farattempted to explore the combined effect of Hall and ionslip effects, modified heat flux model and generalisedFick’s law for the flow of three-dimensional Casson liq-uid (TDCL) over a stretching surface which is stretchedalong x- and y-axes. The flow is produced due to thestretching of the elongating surface and it occupies theregion z ≥ 0. The whole study is organised as follows:Comprehensive literature survey is reported in §1, mod-elling of the flow with conservation law is included in§2, utilising the scheme [23,33,44–46] is discussed in§3, §4 contains the physical interpolation of the obtainedsolution in detail and §5 covers the critical findings ofthe performed research.


2. Flow analysis and mathematical modellingunder boundary layer analysis

Consider a steady three-dimensional flow of an incom-pressible magnetohydrodynamic boundary layer flow ofCasson liquid [1,16,33,34,45] over a linear stretchablesurface as shown in figure 1. In our examination, weselect the Cartesian coordinate system in such a fash-ion that the surface coincides with x1, x2-plane andfluid flow occupies the region x3 ≥ 0. Magnetic fieldB = [0, 0, B0] is with constant strength. Heat andmass transfer are examined by using Cattaneo–Christovheat and mass fluxes.

The relevant equations are of the following forms:

∂u1

∂x1+ ∂u2

∂x2+ ∂u3

∂x3= 0, (1)

u1∂u1

∂x1+ u2

∂u1

∂x2+ u3

∂u1

∂x3

= v∗(

1 + 1

β

)∂2u1

∂x23

+ σ B20

ρ(α2e + β2

e

) (δeu2 − γeu1) , (2)

u1∂u2

∂x1+ u2

∂u2

∂x2+ u3

∂u2

∂x3

= v∗(

1 + 1

β

)∂2u2

∂x23

− σ B20

ρ(α2e + β2

e

) (δeu2 + γeu1) , (3)

ρ∗cp[u1

∂T

∂x1+ u2

∂T

∂x2+ u3

∂T

∂x3

]= −∇·q∗, (4)

u1∂C

∂x1+ u2

∂C

∂x2+ u3

∂C

∂x3= −∇·J∗. (5)

Here u1, u2 and u3 are velocity components in x1-, x2-and x3-directions, T and C are temperature and concen-tration fields respectively, ρ* is the the density, cp isthe specific heat, q* is the heat flux, J* is the mass fluxand β is the Casson fluid parameter which is the ratioof plastic dynamic viscosity to yield stress. The heatflux (q∗) and the generalised mass flux (J∗) here satisfy[14,17–19,22,24,35–37,44]

q∗+γ ∗1

[∂q∗

∂t+ V · ∇q∗ − q∗ · ∇V + (∇ · V)q∗

]

= −K ∗∗∇ · T, (6)

J∗+γ ∗2

[∂J∗

∂t+ V · ∇J∗ − J∗ · ∇V + (∇ · V)J∗

]

= −D∗∗∇ · C, (7)

in which γ1, γ2 are the thermal and solutal relaxationtimes, K** is the thermal conductivity and D** isthe molecular diffusivity. Using incompressibility andsteady-state conditions, eqs (6) and (7) are reduced asfollows:

q∗+γ ∗1

[V · ∇q∗ − q∗ · ∇V

] = −K ∗∗∇ · T, (8)

J∗+γ ∗2

[V · ∇J∗ − J∗ · ∇V

] = −D∗∗∇·C. (9)

Note that for γ1 = γ2 = 0, eqs (8) and (9) are convertedto the traditional Fourier’s and Fick’s laws respectively.Eliminating heat flux (q∗) between eqs (4) and (8) andmass flux (J∗) between eqs (5) and (9), we get the fol-lowing equations:

u1∂T

∂x1+ u2

∂T

∂x2+ u3

∂T

∂x3+ γ ∗

1 λ∗T = KT

ρ∗cp∂2T

∂x23

+δβ

[Dγ

∂C

∂x3

∂T

∂x3+ DT

T∞

(∂T

∂x3

)2]

, (10)

u1∂C

∂x1+ u2

∂C

∂x2+ u3

∂C

∂x3+ γ ∗

2 λ∗C

= D∗∗ ∂2C

∂x23

+ DT

T∞∂2T

∂x23

, (11)

where

λ∗T = γ ∗

1 + γ ∗2 + γ ∗

3 , (12)

γ ∗1 = u2

1∂2T

∂x21

+ u22∂2T

∂x22

+ u23∂2T

∂x23

+ u1∂u1

∂x1

∂T

∂x1

+ u2∂u1

∂x2

∂T

∂x1, (13)

γ ∗2 = u3

∂u1

∂x3

∂T

∂x1+ u1

∂u2

∂x1

∂T

∂x2+ u2

∂u2

∂x2

∂T

∂x2

+ u3∂u2

∂x2

∂T

∂x2+ u1

∂u3

∂x1

∂T

∂x3, (14)

γ ∗3 = u2

∂u3

∂x2

∂T

∂x3+ u3

∂u3

∂x3

∂T

∂x3+ 2u1u2

∂2T

∂x1∂x2

+ 2u2u3∂2T

∂x3∂x2+ 2u1u3

∂2T

∂x1∂x3(15)

and

λ∗T = φ∗

1 + φ∗2 + φ∗

3 , (16)

φ∗1 = u2

1∂2C

∂x21

+ u22∂2C

∂x22

+ u23∂2C

∂x23

+ u1∂u1

∂x1

∂C

∂x1+ u2

∂u1

∂x2

∂C

∂x1, (17)

φ∗2 = u3

∂u1

∂x3

∂C

∂x1+ u1

∂u2

∂x1

∂C

∂x2+ u2

∂u2

∂x2

∂C

∂x2

+ u3∂u2

∂x2

∂C

∂x2+ u1

∂u3

∂x1

∂C

∂x3, (18)


Figure 1. Geometry of the physical problem.

φ∗3 = u2

∂u3

∂x2

∂C

∂x3+ u3

∂u3

∂x3

∂C

∂x3

+2u1u2∂2C

∂x1∂x2+ 2u2u3

∂2C

∂x3∂x2

+2u1u3∂2C

∂x1∂x3. (19)

Boundary conditions are

u1 = Uω (x1) = ax1 , u2 = Vω (x2) = bx2,

u3 = 0, T = Tω, C = Cω at x3 = 0,

u1 → 0, u2 → 0, C → C∞, T → T∞ as x3 → ∞.

(20)

Set

u1 = ax1 f′ (ξ) , u2 = bx2g

′ (ξ) ,

u3 = −√cv∗ ( f (ξ) + g (ξ)) ,

ξ =√

c

v∗ x3, θ (ξ) = T − T∞Tω − T∞

,

C (ξ) = C − C∞Cω − C∞

. (21)

The transformed governing problems with the associ-ated boundary conditions are(

1 + 1

β

)f ′′′ − ( f

′)2 + ( f + g) f ′′

+ Ha(γ 2e + δ2

e

)(δeg′ − γe f

′) = 0, (22)

(1 + 1

β

)g′′′ − (g′)2 + ( f + g) g′′

− Ha(γ 2e + δ2

e

)(δe f′ + γeg

′) = 0, (23)

1

Prθ ′′ + ( f + g) θ ′ − λ1 ( f + g)

×[( f ′ + g′)θ ′ + ( f + g) θ ′′] + Nbφ′θ ′

+Nt (θ′)2 = 0, (24)

1

Scφ′′ + ( f + g) φ′ − λ1 ( f + g)

×[( f ′ + g′)φ′ + ( f + g) φ′′] + Nt

Nbθ ′′ = 0, (25)

f = 0, f ′ = 1, g = 0, g′ = α, θ = 1,

φ = 1 at ξ = 0,

f ′ → 0, g′ → 0, θ → 0, φ → 0 as ξ → ∞.

(26)

3. Analytical solutions

In the available literature, many numerical and analyti-cal methods are available to approximate the solutionsof nonlinear differential equations. In this research, wehave engaged the optimal homotopic scheme [17,18,


Table 1. Convergence of the obtained solution via OHAM.

m εfm ε

gm εθ

m εmφ CPU time (s)

2 0.0000146 0.0000146 0.0001598 0.0002879 2.14356 2.93127 × 10−6 2.93127 × 10−6 0.0000273 0.0001329 8.04508 4.72103×10−7 4.72103×10−7 3.47195×10−6 4.69018×10−4 10.7568

12 5.63198×10−8 5.63198×10−8 2.31762×10−6 2.14891×10−5 16.645214 3.12104×10−8 3.12104×10−8 1.39179×10−7 2.14073×10−6 28.601318 2.10974×10−10 2.10974×10−10 2.93718×10−8 1.62741×10−7 98.801222 1.72186 × 10−12 1.72186×10−12 1.64128×10−9 1.54108×10−8 116.23424 2.46871 × 10−14 2.46871×10−14 1.28614×10−10 1.37651×10−9 152.10926 1.54137 × 10−16 1.54137 × 10−16 1.19784 × 10−12 1.27941 × 10−11 192.16730 1.43174 × 10−18 1.43174 × 10−18 1.72416 × 10−14 1.98204 × 10−13 354.470

23,33,41–46] to approximate the solutions for the con-sidered model which we have developed by using theboundary layer theory. Here, we approximate the solu-tions by considering Ha = 0.4, Pr = 1.2, Nb = Nt =0.1, δe = 0.3 = γe, α = 0.2, Sc = 0.5, λ1 =λ2 = 0.3 and the optimal values which we haveattained through BVPh2.0 are h f = −0.46159076,

hg = −0.6696954, hθ = −0.86184943, hφ =−0.76782310.

Initial estimates for dimensionless velocities, tem-perature and concentration fields corresponding to thelinear operators are

f0 (ξ) = 1 − 1

eξ, g0 (ξ) = α

(1 − 1

eξ

),

θ0 (ξ) = 1

eξ, φ0 (ξ) = 1

eξ, (27)

L̄ f = d

dξ

(d2

dξ2 − 1

)f, L̄g = d

dξ

(d2

dξ2 − 1

)g,

L̄θ =(

d2

dξ2 − 1

)θ, L̄φ = d

dξ

(d2

dξ2 − 1

)φ. (28)

These linear operators confirm the following features:

L̄ f

[a∗∗

1 + a∗∗2 eξ + a∗∗

31

eξ

]= 0,

L̄g

[a∗∗

4 + a∗∗5 eξ + a∗∗

61

eξ

]= 0, (29)

L̄θ

[a∗∗

7 eξ + a∗∗8

1

eξ

]= 0, L̄φ

[a∗∗

9 eξ + a∗∗10

1

eξ

]= 0,

(30)

where a∗∗s (s = 1...10) are the arbitrary constants. These

constants are computed by using the boundary con-ditions. The convergence of the homotopic procedureis shown in table 1. It is confirmed that higher-orderapproximations reduce the errors. Table 2 presents theanalysis of approximation orders against dimensionless

Table 2. Convergence of the homotopy solutions whenδe = 0.3 = γe, Ha = 0.3, β = ∞, Pr= 1.2, Nb =0.7, α = 0.2, Sc = 0.5, λ1 = λ2 and Nt = 0.3 are fixed.

Order ofapproxima-tion

− f ′′(0) −g′′(0) −θ ′(0) φ′(0)

1 1.0042056 0.337458 0.622301 0.3720005 1.0219634 0.328205 0.5495080 0.483576

10 1.0215579 0.328107 0.544923 0.48014715 1.0215632 0.328595 0.594476 0.48388220 1.0215632 0.328293 0.549285 0.48832926 1.0215632 0.328293 0.549285 0.48832930 1.0215632 0.328293 0.549285 0.48832935 1.0215632 0.328293 0.549285 0.488329

stresses, heat and mass transfer coefficients. This tableclarifies the sufficient approximation order for the con-vergent solution.

4. Results and discussion

The flow model in view of transport of phenomenonusing partially ionised Casson liquid containingCattaneo–Christov heat and mass fluxes due to con-stant magnetic field is solved analytically to computethe performance of thermal energy and mass species.The simulations are done graphically and solution of theflow phenomenon is mentioned in the following sectionsagainst different influential parameters.

4.1 Outcomes regarding the fluid flow

The influence of Casson fluid parameter (β), Hallparameter (δe), ion slip number (γe) and Hartmann num-


Figure 2. Influence of β on f ′ (ξ) .

Figure 3. Influence of β on g′ (ξ) .

Figure 4. Influence of δe on f ′ (ξ) .

ber (Ha) on velocity in the x and y space coordinatesis shown in figures 2–9. Figures 2 and 3 illustrate thebehaviour of β on the horizontal and vertical directionsof velocity. It can be seen from these figures that the flowretards due to β in the denominator of dimensionless eqs(22) and (23) and enhanced values of β slows down themotion of Casson fluid. Eventually, the velocity in spacecoordinates (x, y) is decreased by increasing the values

Figure 5. Influence of δe on g′ (ξ) .

Figure 6. Influence of γe on f ′ (ξ) .

Figure 7. Influence of γe on g′ (ξ) .

of β. The opposite situation is found in figures 4 and 5and motion of the fluid is enhanced due to the increasein the values of Hall parameter (δe). Moreover, constantmagnetic force is opposite to the force of Hall parameterwhich brings a significant reduction in the Hall force.Therefore, motion of the fluid becomes faster by enhanc-ing the values of Hall parameter (δe). An enhancement in


Figure 8. Influence of Ha on f ′ (ξ) .

Figure 9. Influence of α on g′ (ξ) .

velocity in space coordinates (x, y) is seen for large val-ues of ion slip parameter (γe) and this is due to Lorentzforce. Basically, Lorentz force is against magnetic forcewhich brings friction in the flow. Therefore, the flowbecomes faster for larger numerical values of γe. Theretardation in flow versus increasing values of magneticfield is observed in figures 8 and 9. It can be estimatedthat motion of the fluid becomes slow when Ha variesas 0.0, 0.02, 0.05, 0.06, 0.09. This fact is based on thelast terms of eqs (22) and (23) where the Lorentz forceis identified in terms of negative force. So, this negativeLorentz force is the reason for the slowdown of the flowin both horizontal and vertical directions.

4.2 Outcomes regarding transport of temperature

The impacts of Brownian motion (Nb), thermophoresisparameter (Nt ), Prandtl number (Pr), Hall parame-ter (δe), ion slip number (γe) and thermal relaxationtime (λ1) on the transport of thermal energy are dis-played in figures 10–15. Figure 10 shows the influenceof Nb on temperature. Further, temperature distributionis increased when Nb is increased. Eventually, motion

Figure 10. Influence of Nb on θ (ξ) .

Figure 11. Influence of Pr on θ (ξ) .

Figure 12. Influence of Nt on θ (ξ) .

of the fluid particles becomes fast and this phenomenongenerates more heat. The role of Pr on the transport ofthermal energy is shown in figure 11. The reduction isfound in thermal boundary layer (TBL) by increasing thevalues of Pr and this decreasing phenomenon has hap-pened due to the enhancement in the values of Pr . Theimpact of Nt on temperature is illustrated in figure 12. Itis known that higher values of Nt is responsible for the


Figure 13. Influence of λ1 on θ (ξ) .

Figure 14. Influence of δe on θ (ξ) .

enhancement of thermal performance. Figure 13 showsthe impact of λ1 on the transport of thermal energy. Thisfigure shows that the temperature is decreased when λ1increases. Hence, a characteristic of fluid exhibits ther-mal relaxation time which plays the role to slow down.Figures 14 and 15 demonstrate the role of Hall and ion

Figure 15. Influence of γe on θ (ξ) .

Figure 16. Influence of γe on φ (ξ) .

slip currents on temperature. From these figures, it isfound that thickness of the thermal boundary layers canbe controlled by enhancing the values of Hall and ionnumbers because δe and γe are appeared in Joule heatingterm of dimensionless energy equation and an increase

Table 3. Application for the velocity gradients for different values of α when δe = 0.0 =γe, β = ∞ are fixed.

α HPM [47] HPM [47] Exact [47] Exact [47] Present result Present result

− f ′′(0) −g′′(0) − f ′′(0) −g′′(0) − f ′′(0) −g′′(0)

0.00 1.0 0.0 1.0 0.0 1.0 0.00.10 1.02025 0.06684 1.020259 0.066847 1.020270 0.0668260.20 1.03949 0.14873 1.039495 0.148736 1.039461 0.1487470.30 1.05795 0.24335 1.05794 0.243359 1.057948 0.2433370.40 1.07578 0.34920 1.075788 0.349208 1.075763 0.3492350.50 1.09309 0.46520 1.093095 0.465204 1.093071 0.4652320.60 1.10994 0.59052 1.109946 0.590528 1.109918 0.5905360.70 1.12639 0.72453 1.126397 0.724531 1.126327 0.7245470.80 1.14248 0.86668 1.142488 0.866682 1.142448 0.8666930.90 1.15825 1.01653 1.158253 1.016538 1.158282 1.0165421.00 1.17372 1.17372 1.173720 1.173720 1.173720 1.173720


Figure 17. Influence of Sc on φ (ξ) .

Figure 18. Influence of Ha on φ (ξ) .

Figure 19. Influence of λ2 on φ (ξ) .

in the values of δe and γe results in the production ofJoule heating.

4.3 Outcomes regarding transport of mass

The parameters involving the effects of Hartmann num-ber (Hα), Schmidt number (Sc), ion slip parameter (γe)and concentration relaxation time (λ2) on transport of

Table 4. Comparison of results for the local Nusselt num-ber −θ ′(0) in the absence of non-Newtonian parameters(β = ∞) and nanoparticles when α = δe = 0.0 = γewith the work of Khan and Pop [49] and Nadeem and Hus-sain [48].

Pr Presentresult

Khan andPop [49]

Nadeem andHussain [48]

0.07 0.064789 0.066 0.0660.20 0.165789 0.169 0.1690.70 0.452683 0.454 0.4542.0 0.912906 0.911 0.911

Table 5. Comparison of results for the local Nusselt num-ber −θ ′(0) and local Sherwood number −φ′(0) in thepresence of nanoparticles when α = 0, δe = 0.0 =γe, λ = 0, Pr = 6 and Sc = 1 are fixed with the work ofNadeem et al [50].

Nt NbPresentresult

Presentresult

Nadeemet al [50]

Nadeemet al [50]

−θ ′(0) φ′(0) −θ ′(0) φ′(0)

0.3 0.3 0.339881 1.83984 0.33984 1.839350.5 0.3 0.240792 1.95853 0.24099 1.958620.3 0.5 0.14824 1.87027 0.14820 1.870350.5 0.5 0.10478 1.94568 0.10486 1.94572

mass are displayed in figures 16–19. Figure 16 showsthat the transport of mass slows down for large valuesof γe. The role of Sc on φ(η) is plotted in figure 17.The relation between momentum diffusion and massdiffusion coefficients is called Schmidt number. It ismentioned that diffusion of a species has direct relationwith mass diffusion coefficient. Therefore, increas-ing function between mass diffusion coefficient andSchmidt number has been investigated. Obviously, thediffusion of a species slows down by increasing the val-ues of Sc. Figure 18 demonstrates the behaviour of Hα

on φ(η) and by enhancing the numerical values of Hα,the transport of mass becomes more. The influence of λ2on the transport of mass are demonstrated in figure 19.The decreasing tendency is identified in the transport ofmass by increasing the values of λ2.

4.4 Outcomes concerning dimensionless stress,Nusselt and Sherwood numbers

The character of involved parameters also has been scru-tinised with the help of tabular data. Important observa-tion is made (table 3). Through table 2, the involve-ment of ratio parameter on dimensionless stresses areshown by comparing the obtained results with the valuereported in [47]. An excellent matching is found. Resultsfor heat transfer coefficient via tabular data are given


Table 6. Variation of the local Nusselt number and localSherwood number with α, Pr, λ, Nb, Nt when δe =0.5 = γe are fixed.

α Pr λ Nb Nt Sc −θ ′(0) −φ′(0)

0.0 12 0.2 0.1 0.1 1.0 0.35185367 0.474210350.3 - – – – – 0.50983738 0.482838460.4 – – – – – 0.54943892 0.488939830.5 1.0 – – – – 0.51372848 0.43093018– 1.1 – – – – 0.55123869 0.46321649– 1.3 – – – – 0.61739337 0.52772337– – 0.0 – – – 0.75720847 0.35667082– – 0.1 – – – 0.67690772 0.42214347– – 0.4 – – – 0.33539048 0.69022194– – – 0.2 – – 0.54051442 0.68336254– – – 0.3 – – 0.49758617 0.74528152– – – 0.4 – – 0.45684147 0.77572634– – – – 0.2 – 0.57579548 0.19934338– – – – 0.4 – 0.50539435 0.27776228– – – – 0.5 – 0.48075727 0.46441049– – – – – 0.8 0.59116558 0.35254838– – – – – 0.9 0.58820028 0.42641426– – – – – 1.1 0.58242762 0.56057368

in table 4 against Prandtl number. With the help oftable 4, it is concluded that the achieved solution fullyagrees with the studies reported in [48] and [49]. Com-bined involvement of Brownian motion parameter andthermophoresis parameter on heat and mass transporta-tion rates are shown in table 5. Excellent agreement isobserved with [50]. The role of α, Pr , λ1, Nt , Sc andNb on Nusselt and Sherwood numbers have been inves-tigated and the outcomes are displayed in table 6. It canbe found from this table that rate of heat and mass trans-port at the wall have greater values versus the variationin α and Pr . Moreover, opposite trend can be seen onNusselt and Sherwood numbers versus large values ofNb, Nt , Sc and λ1. The maximum rate of transport ofmass at the wall is studied in view of α, Pr , Nb, Sc andλ1 while maximum rate of transport of thermal energy atthe wall have been investigated versus increasing valuesof α, Pr , Nt and Nb.

5. Conclusion and important remarks

The simulations regarding the Casson fluid under theHall and ion slip forces via Cattaneo–Christov heat fluxmodel have been carried out by optimal approach. Thesolution related to flow problem has shown several novelkey points which are summarised now.

• Increasing the numerical values of γe results in thereduction in flow of the Casson fluid and transportphenomenon of heat and mass species. Similarly, the

retardation in motion of fluid is occurred by enhanc-ing the values of δe.

• The intensity of magnetic field plays a vital rolein managing the thickness of momentum boundarylayer as well as thermal and concentration boundarylayers. Further, magnetic field works as a barrier inthe flow of Casson liquid.

• Dimensionless velocity is the increasing function ofCasson fluid parameter.

• Thermal performance increases with large numericalvalues of Nb, Nt and δe, whereas it decreases withincreased values of Pr , γe and λ1.

• Higher the values of Sc and λ1, lesser is the transportof mass.

• The rise in α and Pr increases the transport of heatand mass while opposite behaviour can be seen withlarge values of Nb, Nt , Sc and λ1.

• 15th order approximations are sufficient for veloc-ity solution and 20th order approximations are forthermal and concentration solution.

• Special case of the performed analysis is in excellentagreement with the reported materials in the openliterature.

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Numerical exploration of thermal and mass transportation