# Nonlinear Mixed Convective Bidirectional Dynamics of Double Stratified Radiative Oldroyd-B Nanofluid Flow with Heat Source/Sink and Higher-Order Chemical Reaction

Authors: Iftikhar Ahmad; Muhammad Faisal; K. Loganathan; Muhammad Zaheer Kiyani; Ngawang Namgyel
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Little is known in the literature about the concept of nonuniform heat source/sink and higher-order chemical reaction for the dynamics of Oldroyd-B nanoparticles. Therefore, the present article addresses the nonuniform heat source/sink and higher-order chemical reaction features in nonlinear mixed convection bidirectional MHD dynamics of Oldroyd-B nanoparticles with thermal radiation aspects through porous space. Stratification effects for both the temperature and concentration setups are also used in the mathematical model with the significance of random movement and thermodiffusion of nanoparticles. Shape-preserving transformations have been employed to convert the transport equations into solvable forms. An innovative analytical tactic, namely, homotopy analysis method, has been adopted to find the solution of the modeled problem. Behaviors of pertinent parameters on thermal and concentration profiles have been discussed through various graphs. Inspection of heat/mass transport against appropriate varieties of pertinent parameters has been made and explained physically. Thermal profile is augmented with the higher estimations of space and temperature-dependent heat source/sink links. Concentration profile is diminished with the augmentation of higher-order chemical reaction parameter. Sherwood number is improved with the estimation of0≤βt≤100 and is reduced with the growth of 0≤βc≤100. Nusselt number is declined with the upgraded amounts of 0≤Nb≤3 and 0≤Nt≤5.


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## Body

## 1. Introduction

Rheology of non-Newtonian fluids is based on their multiphase nature, and they have got great importance on account of their wide spread involvement in industrial and technological applications. Shear-thinning, shear thickening, and viscoelasticity are the main characteristics of non-Newtonian liquids. These characteristics are described by nonlinear relationship between shear force and shearing deformation. Several models of non-Newtonian materials have been suggested, like Maxwell, Jeffery, and Oldroyd-B models. An Oldroyd-B model is a rate type non-Newtonian material having relaxation and retardation time effects.Thermal energy has a significant role in almost every field of science, engineering, industries, biomedicine, plant processing, transports, power houses, generating stations, and many others. The effectiveness of all the above cited applications is greatly dependent upon thermal conductivity of the fluids involved in the heat interchanging processes. Many fluids like water, ethylene, and oil bear the restrictions in applications due to low thermal conductivity. Hence, nanoparticles of average size of about 1–100 nm are inserted into the base fluid to augment the thermophysical and concentration circumstances of the fluid, and the obtained colloidal suspension is regarded as nanofluid. The novel attributes of nanofluids make them positively functional in the numerous applications of heat transference along with microelectronics, energy cells, and domiciliary refrigeration and heating equipment, in disintegrating and cracking units, machining, hybrid powered engines, medications, and heat exchange units. Choi and Eastman [1] firstly proposed the term nanofluid through a permeable prolonging sheet along with consumption/injection. Then, Xuan and Li [2] extended the concept of nanofluids by proposing that fluids with solidified particles of metalloid and polymeric origin in a base fluid can also be termed as nanofluids. Buongiorno [3] formulated a mathematical tool for the analysis of convective transference in nanofluids with the consideration of Brownian movement and thermal diffusion properties. Copper, titanium, silicon, aluminum, zinc, magnesium, and graphene oxide are the best well-known particulates used for the development of nanofluids, while water, oil, glycol, and ethylene are the most regular working liquids.Boundary-layer flows through stretching surfaces have been considered in studies because of their countless applications from household practices to aerospace such as in industries, irrigation waterways, environmental production, aerodynamics, sports, enhancement of heat transference, enhancement of mixing, transport of species, formation of rubber and plastic sheets, and fiber glass manufacturing. Boundary-layer flow produced by continuous elongation of obstacle was primarily studied by Sakiadis [4]. Crane [5] introduced the development of boundary-layer stream through stretching of surface and found exact solutions for the flow field. The bidirectional flow caused as a result of linearly plane bidirectional stretching of sheet was studied by Wang [6]. Ariel [7] deduced approximate analytical and numerical solutions of bidirectional steady flow over an elongating sheet. Sajid and Hayat [8] premeditated the aspects of thermal deposition and heat transportation in the boundary-layer dynamics of a viscid fluid influenced by an exponential elongation of obstacle. An analysis of the motion of a nanofluid produced by stretching of surface using connective model was developed by Khan and Pop [9].Bhattacharyya and Layek [10] investigated the effects of chemically receptive solute convergence in magnetohydrodynamic boundary-layer stream. Ahmad et al. [11] continued the work accomplished by Liu and Anderson [12] by estimating the Darcy resistance impact and applied magnetic field. Stratification is a predominant phenomenon which has gained remarkable consideration because of its inclusion in the geophysical flows such as waterways, within lakes, oceans, ground-water supplies, warm energy stockpiling frameworks, and so on. A lot of researchers experimented with stratification with different effects. Ibrahim and Makinde [13] considered the doubly stratified boundary-regime flow of nanomaterials through an upright plate. Loganathan et al. [14] inspected the effect of second-order slip in a convective dynamic of an Oldroyd-B material with the aspects of chemical reaction and thermal radiation. Sandeep and Reddy [15] studied magnetohydrodynamic (MHD) flow having double stratified and cross diffusion aspects for an Oldroyd-B fluid. Waqas et al. [16] described the features of mixed convection progression of an Oldroyd-B nanomaterial via accumulating the impact of heat creation as well as heat/mass stratification.The notion of heat production/consumption is convenient in many practices including heat disposal of atomic fuel wreckage, underground removal of radiative garbage substances, food storages, and disconnecting materials in packed bed reactors. In this way, nanocomposites have a capacity to assemble incident radiation. Turkyilmazoglu and Pop [17] explored the concurrent effects of heat and mass transport with thermal radiation through an unsteady nanomaterial progression. Rashidi et al. [18] analyzed the effect of second law of thermodynamics in magnetohydrodynamic dynamics of nanoliquid over a revolving disk. Moradi et al. [19] performed an analysis to check the effects of nanoparticles in Jeffery–Hamel flow. Makinde et al. [20] analyzed the buoyancy-driven dynamics of nanoliquid near a stagnated domain with connective boundary conditions. Mahanthesh et al. [21] worked toward approximate solution for the dynamics of an Oldroyd-B material with the significance of thermal deposition and heat consumption/generation impacting upon a nonlinearly expanding obstacle.The subject of fluid motion with the involvement of porous medium has developed significantly in many fields of science and engineering like flow through water rocks, skin pronouncement, liquid purification processes, chemical trash, crude oil production, porous insulation, grain storage, mastic transport modeling, and underground removal of atomic waste. The interdependence of the voids in a porous medium permits certain liquids to flow through material. Ghosh and Sana [22] studied the time dependent dynamics of an Oldroyd-B material influenced by reformed sine pulses. Mukhopadhyay [23] considered and studied the time dependent heat transference and mixed connective dynamics produced by a penetrable stretching obstacle. The magnetohydrodynamic dynamics of tiny-sized nanoparticles passed over a permeable wedge is scrutinized by Kandasamy et al. [24]. Tripathy et al. [25] analyzed the impact of chemical reaction on magnetohydrodynamic free connective sheet over an upright moving plate through permeable material. Ali et al. [26] presented the analytical solution for flowing fluid through exponential stretching of permeable surface including heat flux in permeable medium utilizing homotopy analysis method (HAM). Goud [27] introduced the magnetohydrodynamic flow through an upright appendicular plate including radiation and chemical reaction effects through a permeable medium. Some more impactful scientific contributions related to the bidirectional dynamics of nanofluids with various mechanical and thermal aspects have been made by some researchers [28–36].The above literature survey reveals that the study of non-Newtonian nanofluid performs a vital role in the industrial and engineering advancements. Moreover, a huge gap regarding the combined study of nonlinear aspects of mixed convection, chemical reaction, and heat consumption/generation is found during literature review. Hence, the novelty of present exploration is to discuss the steady dynamics of 3D radiative non-Newtonian (Oldroyd-B) nanofluid with the nonlinear aspects of mixed convection, chemical reaction, and heat source/sink. Additionally, significance of porous medium, magnetohydrodynamics, and double stratification has been addressed to achieve the thermal engineering relevancy of the mathematical model. To the best of our knowledge, no such attempt has been made previously. Mathematical modeling of the physical model has been completed with the help of boundary-layer approximations and basic laws of fluid dynamics. With the functionality of similarity transformations, transport equations have been converted into one parameter family of solvable equations based on the physical domain. Analytical analysis of the mathematical model has been presented by following the procedure of homotopy analysis method [37–44]. Some more solution techniques (Keller-Box method, Lie group analysis, etc.) have also been offered by some researchers/scholars [45–50], but here preference is given to homotopy analysis method because of its compatibility with the modeled problem. Finally, the obtained outcomes have been discussed physically with the support of various charts and tables against the fluctuating choices of involved parameters.


## 2. Model Development

An incompressible, doubly stratified, and three-dimensional steady dynamic of radiative Oldroyd-B magneto-nanomaterial impacting upon an expanding obstacle through porous space is considered with the nonlinear aspects of mixed convection, chemical reaction, and heat consumption/generation. Effects of random motion and thermodiffusion of nanoparticles have also been provided via Buongiorno nanofluid model. The Lorentz force is perpendicular to thexy-plane, whereas gravitational force is parallel to the x-direction. Influence of chemical reaction has been included in the nanoparticle mass transport equation, whereas radiation effects have been involved in the heat equation of the nanofluid. The expansion velocity along x-direction is explained as uw=ax, whereas expansion velocity along the y-direction is elaborated as vw=by. The relations for wall and free stream temperatures are settled according to the rule of thermal stratification, whereas the relations for wall and free stream nanoparticle mass concentration are established according to the regulation of solutal stratification. The graphical abstract of the present investigation is presented via Figure 1. The governing equations for the present analysis supported by [16, 22, 27, 31, 32, 44] are composed as(1)∂u∂x+∂v∂y+∂w∂z=0,(2)u∂u∂x+v∂u∂x+w∂u∂z+λ1u2∂2u∂x2+v2∂2u∂y2+w2∂2u∂z2+2uv∂2u∂x∂y+2vw∂2u∂y∂z+2uw∂2u∂x∂z,=ϑ∂2u∂z2+ϑλ2u∂3u∂x∂z2+v∂3u∂x∂z2+w∂3u∂z3−∂u∂x∂2u∂z2−∂u∂y∂2u∂z2−∂u∂z∂2w∂z2,+gα1T−T∞+α2T−T∞2+gα3C−C∞+α4C−C∞2−μfϕ1ρfp1u−σB°2ρfu+λ1w∂u∂z,(3)u∂v∂x+v∂v∂y+w∂v∂z+λ1u2∂2v∂x2+v2∂2v∂y2+w2∂2v∂z2+2uv∂2v∂x∂y+2vw∂2v∂y∂z+2uv∂2v∂x∂z,=ϑ∂2v∂z2+ϑλ2u∂3v∂x∂z2+v∂3v∂x∂z2+w∂3v∂z3−∂v∂x∂2v∂z2−∂v∂y∂2v∂z2−∂v∂z∂2w∂z2−μfϕ1ρfp1v−σB°2ρfv+λ1w∂v∂z,(4)u∂T∂x+v∂T∂y+w∂T∂z=αm∂2T∂z2+ρcpρcfDB∂T∂z∂C∂z+DTT∞∂T∂z2−1ρcf∂qr∂z+q‴ρcf,(5)u∂C∂x+v∂C∂y+w∂C∂z=DB∂2C∂z2+DTT∞∂2T∂z2−kc1C−C∞+kc2C−C∞2.Figure 1
Graphical abstract of the mathematical model.Here,q‴=kuw/xϑA∗Tw−T∞f′+B∗T−T∞.The relevant boundary conditions supported by [15, 16, 44] are(6)u=uw,v=vw,w=0,T=Tw,C=Cwatz=0,u⟶0,v⟶0,T⟶T∞,C⟶C∞asz⟶∞.Using the relation of Rosseland approximation, radiative heat-fluxqr is determined as (see [48]) qr=−4σ∗/3K∗∂T4/∂z with T4≅4T∞3T−3T∞4.Here,x,y, and z describe the space coordinates; u, v, and w express parts of velocity in the direction of x, y, and z, respectively; λ1 and λ2 are used to represent relaxation time and retardation time, respectively; ϑ=μf/ρf denotes kinematic viscosity with μf as dynamic viscosity and ρf as density; g represents the gravitational acceleration; α1 exhibits the involvement of linear thermal expansion; α2 represent involvement of nonlinear thermal expansion; α3 and α4 depict, respectively, linear and nonlinear concentration coefficients; σ∗ denotes Stefan–Boltzmann constant; B0 stands for magnetic field strength with electrical conductivity σ; ϕ1 is used to express the porosity constant; p1 the permeability constant; T is the temperature, C is the concentration of nanoparticles; αm=k/ρcf denotes the thermal diffusivity with thermal conductivity k; and DB and DT are the symbols of random and thermodiffusion coefficients of nanocomposites, respectively. Efficient heat capability of nanoparticles and efficient heat capability of the fluid are symbolized by cp and cf, respectively; kc1 and kc2 represent the first- and second-order chemical reaction parameters, respectively; qr is thermal deposition; q′'′ is nonuniform heat consumption/absorption; Tw is temperature at the surface; T∞ denotes the ambient temperature; and Cw and C∞ serve as nanoparticle concentrations at the obstacle and far from the obstacle. Here, the expansion velocities, temperature, and nanoparticle mass concentration at the surface are described as (see [15, 16, 44])(7)uwx=ax,vwy=by,Twx=T∞,0+M1x,Cwx=C∞,0+N1x,T∞x=T∞,0+A1x,C∞x=C∞,0+B1x.Here,a,b,A1,B1,M1,N1,T∞,0, andC∞,0 are dimensional constants.We utilize the accompanying similarity transformations supported by [37, 38, 44] as follows:(8)u=axf′η,v=ayg′ηw=−aϑ1/2f+g,η=aϑ1/2zθη=T−T∞,0T−A1xTΔT=Twx−T∞,0=M1xϕη=C−C∞,0C−B1xCΔC=Cwx−C∞,0=N1x.Via using the above-mentioned local similarity set, (1) is straightforwardly fulfilled and simplified forms of (2-5) are mentioned below:(9)f‴+1+M2β1f+gf″−f′2+β12f+gf′f″−f+g2f‴,+β22f″+g″f″−f+gfiv−δ+M2f′+λ1+βtθθ+N∗1+βcϕϕ=0,(10)g‴+1+M2β1f+gg″−g′2+β12f+gg′g″−f+g2g‴,+β22f″+g″g″−f+ggiv−δ+M2g′=0,(11)1+43Rdθ″+Prf+gθ′−θ+εtf′+Nbθ′ϕ′+Ntθ′2+A∗1−εtf′+B∗θ=0,(12)ϕ″+PrLef+gϕ′−ϕ+εsf′−γ1+kcϕϕ+NtNbθ″=0.The boundary conditions are confined as(13)f=0,g=0,f′=1,g′=α,θ=1−εt,ϕ=1−εsatη=0,(14)f′⟶0,g′⟶0,f″⟶0,g″⟶0,θ⟶0,ϕ⟶0,asη⟶∞.In the above-mentioned equations,f′ and g′ are the dimensionless velocities along x- and y-direction, respectively; θ and ϕ are the dimensionless temperature and concentration, respectively; β1=λ1a and β2=λ2a are the Deborah numbers corresponding to relaxation and retardation times, respectively; βt=α2/α1T and βc=α4/α3C indicate the nonlinear convection constraints for temperature and concentration, respectively; M2=σB°2/ρfa is the magnetic constraint; δ=ϑϕ1/p1a is used to express the porosity constraint; N∗=λ∗/λ stands for buoyancy ratio parameter; λ=Gr/Rex2 and λ∗=Gr∗/Rex2 denote the mixed convective parameters for thermal and concentration setups, respectively; Le=αm/DB is the Lewis number; Pr=ϑ/αm signifies the Prandtl number; Nb=ρcp/ρcfDBΔC/ϑ denotes the random movement involvement; Nt=ρcp/ρcfDTΔT/ϑT∞ corresponds to thermodiffusion parameter; A∗ and B∗ denote, respectively, the quantities of space-dependent and temperature-dependent heat consumption/generation; γ=kc1/a is the rate of chemical reaction; kc=kc2/kc1C is the dimensionless chemical reaction parameter; Rd=4T∞3σ∗/kk∗ represents radiation parameter; k∗ represents the Rosseland mean absorption constant; α=b/a is the expansion ratio parameter; εt=A1/M1 and εs=B1/N1 are the representatives of the heat and mass stratified parameters, respectively; and prime represents differentiation taking η as an independent similarity variable.The amount of mass transference is presented by local Sherwood numberShx while the amount of heat transference is described by local Nusselt number Nux, and these numbers can be expressed in the form of Reynolds number Rex=uwx/ϑ as below:(15)Shx=−xCw−C∞∂C∂z|z=0,=−Rex1/2ϕ′0,Nux=−xTw−T∞∂T∂z−qrk|z=0,=−Rex1/21+43Rdθ′0.


## 3. Homotopic Series Solution

This section deals with the series solutions of the mathematical modeled dimensionless equations (9)–(12) based on boundary restrictions via (8) and (9) with the procedure of homotopy analysis method (HAM). For this purpose, primary guesses are approximated in accordance with boundary conditions (8) and (9) and are shown as(16)f0η=1−e−η,g0η=α1−e−η,θ0η=1−εte−η,ϕ0η=1−εce−η.The linear operators are denoted byℒf,ℒg, ℒθ, and ℒϕ as follows:(17)ℒf=f‴−f′,ℒg=g‴−g′,ℒθ=θ″−θ′,ℒϕ=ϕ″−ϕ′,ℒfC1+C2eη+C3e−η=0ℒgC4+C5eη+C6e−η=0ℒθC7eη+C8e−η=0ℒϕC9eη+C10e−η=0,1−Pℒff^η,P−f0η=PℏfNff^η,P,g^η,P,θ^η,P,ϕ^η,P,1−Pℒgg^η,P−g0η=PℏgNgf^η,P,g^η,P,θ^η,P,ϕ^η,P,1−Pℒθθ^η,P−θ0η=PℏθNθf^η,P,g^η,P,θ^η,P,ϕ^η,P,1−Pℒϕϕ^η,P−ϕ0η=PℏϕNϕf^η,P,g^η,P,θ^η,P,ϕ^η,P.In the above relations,ℏf, ℏg, ℏθ, and ℏϕ are auxiliary constraints, whereas Nf, Ng, Nθ, and Nϕ serve as nonlinear operators, and P is the embedding parameter. The BCs (boundary conditions) in the coding language can be expressed as(18)f^0,P=0,f^′0,P=1,f^′∞,P=0,g^0,P=0,g^′0,P=α,g^′∞,P=0,θ^0,P=1−εt,θ^∞,P=0,ϕ^0,P=1−εc,ϕ^∞,P=0,(19)Nff^η,P,g^η,P,θ^η,P,ϕ^η,P=∂3f^∂η3+1+M2∗β1f^+g^∂2f^∂η2−∂f^∂η2,+β12f^+g^∂f^∂η∂2f^∂η2−f^+g^2∂3f^∂η3+β22∂2f^∂η2+∂2g^∂η2∂2f^∂η2−f^+g^∂4f^∂η4−δ+M2∂f^∂η,+λ1+βtθ^θ^+N∗1+βtϕ^ϕ^,(20)Ngf^η,P,g^η,P,θ^η,P,ϕ^η,P=∂3g^∂η3+1+M2β1f^+g^∂2g^∂η2−∂g^∂η2,+β12f^+g^∂g^∂η∂2g^∂η2−f^+g^2∂3g^∂η3+β22∂2f^∂η2+∂2g^∂η2∂2g^∂η2−f^+g^∂4g^∂η4−δ+M2∂g^∂η,(21)Nθf^η,P,g^η,P,θ^η,P,ϕ^η,P=1+43Rd∂2θ^∂η2+Prf^+g^∂θ^∂η,−Prθ^+εt∂f^∂η+Pr∗Nb∂θ^∂η∂ϕ^∂η+Pr∗Nt∂θ^∂η2+A∗1−εt∂f^∂η+B∗θ^,(22)Nϕf^η,P,g^η,P,θ^η,P,ϕ^η,P=∂2ϕ^∂η2+Le∗Prf^+g^∂ϕ^∂η,−Le∗Prϕ^+εc∂f^∂η−Le∗Pr∗γϕ^−Le∗Pr∗γ∗kcϕ^2+NtNb∂2θ^∂η2,(23)ℒffmη−χmfm−1η=ℏfRfmη,(24)ℒggmη−χmgm−1η=ℏgRgmη,(25)ℒθθmη−χmθm−1η=ℏθRθmη,(26)ℒϕϕmη−χmϕm−1η=ℏϕRϕmη,(27)fm0=fm′0,=fm′∞,=0,gm0=gm′0,=gm′∞,=0,(28)θm0=θm∞,=0,ϕm0=ϕm∞,=0,(29)Rfmη=fm−1′″η+1+M2β1∑k=0m−1fm−1−kfk″+gm−1−kfk″−∑k=0m−1fm−1−k′+fk′,+2β1∑k=0m−1fm−1−k∑l=0kfk−1′fl″+2β1∑k=0m−1gm−1−k∑l=0kfk−1′fl″−β1∑k=0m−1fm−1−k″′∑l=0kfk−1fl,−β1∑k=0m−1fm−1−k″′∑l=0kgk−1gl−∑k=0m−1fm−1−k′″∑l=0kgk−1fl+2β1∑k=0m−1fm−1−k″fk″+2β1∑k=0m−1gm−1−k″fk″,−β2∑k=0m−1fm−1−kfk″″−β2∑k=0m−1gm−1−kfk″″−δ+M2fm−1η,+λθm−1η+βt∑k=0m−1θm−1−kθk+N∗ϕm−1η+N∗∗βc∑k=0m−1ϕm−1−kϕk,(30)R^gmη=gm−1″′η+1+M2β1∑k=0m−1fm−1−kgk″+gm−1−kgk″−∑k=0m−1gm−1−k′gk′,+2β1∑k=0m−1fm−1−k∑l=0kgk−l′gl″+2β1∑k=0m−1gm−1−k∑l=0kgk−lgl″−m,β1∑k=0m−1gm−1−k″′∑l=0kfk−1fl−β1∑k=0m−1gm−1−k″′∑l=0kgk−1f+2β1∑k=0m−1gm−1−k″∑l=0kgk−1gl+,2β1∑k=0m−1fm−1−k″gk″+2β1∑k=0m−1gm−1−k″gk″−β2∑k=0m−1fm−1−kgk″″−β2∑k=0m−1gm−1−kgk″″−δ+M2gm−1η,(31)R^θmη=1+43Rdθm−1″η+Pr∑k=0m−1fm−1−kθk′+gm−1−kθk′−Pr∑k=0m−1fm−1−k′θk′,−Pr∗εtfm−1′η+Pr∗Nt∑k=0m−1θm−1−k′θk′+Pr∗Nb∑k=0m−1ϕm−1−k′θk′+A∗1−εtfm−1′η+B∗θm−1η,(32)R^ϕmη=ϕm−1″η+Le∗Pr∑k=0m−1fm−1−kϕk′+gm−1−kϕk′−Le∗Pr∑k=0m−1fm−1−k′ϕk,−Le∗Pr∗εc∗fm−1′η−Le∗Pr∗γ∗ϕm−1η−γ∗Le∗Pr∗kc∑k=0m−1ϕm−1−kϕk+NtNbθm−1″η,(33)χm=0,m≤1,1,m>1.The general solutionsfm, gm, θm, and ϕm of (23)–(26) with regard to particular solutions fm∗,gm′∗θm∗,ϕm∗ are expressed by(34)fmη=fm∗η+C1+C2eη+C3e−η,(35)gmη=gm∗η+C4+C5eη+C6e−η,(36)θmη=θm∗η+C7eη+C8e−η,(37)ϕmη=ϕm∗η+C9eη+C10e−η.The general constantsCii=1−10 in the above equations are calculated by applying boundary conditions and are stated as(38)C1=C2=C5=C7=C9,C3=∂fm∗η∂η|η=0C1=−C3−fm∗0,C6=∂gm∗η∂η|η=0C4=−C6−gm∗0,C1=−C3−fm∗0C8=−θm∗0,C10=−ϕm∗0.


## 4. Convergence Analysis

The admissible range for unknown boundary restrictions is the first task to obtain the solution via homotopy analysis method. This range can be traced by identifying those portions of curves lying exactly parallel to the horizontal axis as sketched in Figure2. The solid-green line curve expresses f″0 with admissible range of −0.7≤ℏf≤−0.1. The admissible range for the curve of g″0 is −0.8≤ℏg≤−0.1, and it is depicted by the black dashed line. Similarly, the admissible ranges of θ′0 and ϕ′0 are estimated as −0.6≤ℏθ≤−0.1 and −0.4≤ℏϕ≤−0.1, and they are represented by red and blue curves, respectively. For convenience, throughout the analysis we have chosen ℏf=ℏg=ℏϕ=ℏθ=−0.3=ℏ. Hence, the convergence table for appropriate amounts of emerging parameters is generated, and the corresponding results are shown through Table 1. Table 1 illustrates that at least sixty-four orders of approximation are needed to obtain the missing boundary conditions for θη and ϕη, whereas at least seventy orders of approximation are required to attain the momentum boundary conditions. To maintain the convergence and stability of the solution, the numbers of approximations are increased up to eight.Figure 2
Estimation of convergent region viaℏ-curve sketching.Table 1
Convergence check of series solutions by considering different orders of deformationm when ℏ=−0.3 with the values of involved parameters kept the same as Figure 1.
m−f''0−g''0−θ'0−ϕ'010.671800.330000.604800.9956680.544870.221330.410781.45734160.542940.228970.407511.45982220.542200.225460.410261.46269280.545560.226970.409221.45788340.541600.227880.408931.46162400.543990.225060.410251.46070460.542360.227020.410481.46132520.543110.226880.410571.46149580.542860.226970.410651.46158640.542920.226920.410671.46159700.542880.226950.410671.46159800.542880.226950.410671.46159


## 5. Results and Discussion

This section describes the graphical illustrations of temperature fluctuation, concentration fluctuation, local Nusselt number, and local Sherwood number under the consequences of important involved parameters. Figure3 explains the impact of thermal stratification parameter 0≤εt≤1 on dimensionless temperature fluctuation when the similarity variable is increased from η=0 to η=4 with the constant choices of other involved parameters. Maximum temperature fluctuation occurs at εt=0, minimum temperature fluctuation is achieved at εt=1, and thermal stratification phenomenon is generated for 0<εt<1. Temperature profile is reduced with the improvement in the choice of εt. Physically, higher choice of εt develops the ambient temperature and reduces the temperature at the surface. As a result, the temperature profile is reduced. The thickness of thermal layer is improved with the improvement in the value of the similarity variable. Figure 4 illustrates the trend of solutal stratification parameter 0≤εs≤1 on dimensionless concentration fluctuation when the similarity variable is increased from η=0 to η=1.8 with the fixed amounts of other influential parameters. Maximum concentration is achieved at εs=0, minimum concentration is obtained at εs=1, and solutal stratification phenomenon is traced for 0<εs<1. Concentration setup is diminished with the higher estimation of εs. The physical reason behind this outcome is the downfall of nanoparticle concentration at the wall and improvement of nanoparticle concentration away from the stretching device. Concentration layer thickness is also increased with the improvement of the similarity variable.Figure 3
Interference ofεt on thermal layer θη.Figure 4
Interference ofεs on concentration layer ϕη.Figure5 indicates the role of space-dependent heat source/sink parameter −1.5≤A∗≤1.5 in dimensionless temperature fluctuation with the increase of the similarity variable from η=0 to η=8. Minimum temperature is attained for heat consumption case, −1.5≤A∗<0−1.5≤A∗<0; maximum temperature is achieved for heat generation case, 0<A∗≤1.50<A∗≤1.5; and moderate temperature is reached in the absence of heat generation/consumption, A∗=0. Mathematically, A∗ is involved as a multiple of velocity profile f′η in (7); therefore, the velocity profile also increases with the higher estimation of A∗, and hence temperature profile is improved. Therefore, it is called space-dependent heat source/sink influence on thermal profile. The thickness of thermal layer is also improved with the growth of the similarity variable. Figure 6 illuminates the potential of temperature-dependent heat source/sink parameter −1.5≤B∗≤1.5 on dimensionless temperature fluctuation with the growth of the similarity variable from η=0 to η=8. Minimal temperature is accomplished for heat ingestion case, −1.5≤B∗<0−1.5≤B∗<0; maximal temperature is realized for heat production case, 0<B∗≤1.50<B∗≤1.5; and modest temperature is touched in the nonappearance of heat production/consumption, B∗=0. Precisely, B∗ is involved as a multiple of θη in (7); therefore, the thermal profile is also augmented with the greater estimation of B∗, and hence it is called temperature-dependent heat source/sink influence on thermal profile. The depth of thermal layer is also enriched with the progress of similarity variable. Figure 7 establishes the connection between dimensionless concentration fluctuation and chemical reaction parameter 0≤kc≤20 with the growth of the similarity variable up to η=3.2. Decreasing trend in concentration fluctuation is observed with the growth in the estimation of kc, maximal concentration is achieved for smaller values of kc, and minimal concentration is generated for greater estimations of kc. Physically, higher estimation of kc increases the second-order chemical reaction and reduces the first-order chemical reaction at the same time. As an outcome, negative trend in the concentration profile is attained. The thickness of concentration layer is also developed with the improvement in the similarity variable. Figure 8 explores the effect of thermal radiation parameter 0≤Rd≤4 on thermal fluctuation with the evolution of the similarity variable from η=0 to η=7. A huge improvement in the thermal fluctuation is attained with the intensification in Rd. Maximal temperature is measured for higher Rd, and minimal temperature is obtained for basic/lower Rd. The thickness of thermal layer is also improved and it is found to be higher for the middle values of Rd. Physically, radiation is basically the transference of energy in the form of electromagnetic waves. More electromagnetic waves are produced with the improvement of Rd, and hence dominant temperature fluctuation is observed when Rd reaches maximal estimation. Figure 9(a) deals with the relationship between relaxation time Deborah number 0≤β1≤10 and temperature distribution for the similarity variable 0≤η≤7. Here, thermal environment is upgraded with the higher estimation of β1. Basically, β1 is generated in the momentum equation due to the inclusion of Oldroyd-B nanoparticles and defined as the product of expansion rate a and the relaxation time λ1. Hence, higher estimation of β1 enhances the value of λ1 as well as the expansion rate. As an outcome, improvement in thermal setup is noticed with the upgraded width of thermal layer. Figure 9(b) shows the effect of β1 on concentration fluctuation. Concentration is tremendously promoted with the advanced values of β1. Physically, upgraded value of β1 enhances the expansion rate that increases the movement of Oldroyd-B nanoparticles, and so concentration setup is boosted. Figure 10(a) establishes the link between retardation time Deborah number 0≤β2≤4 and thermal fluctuation for the similarity variable 0≤η≤4. At this time, thermal environment is degraded with the refined approximation of β2. Fundamentally, β2 involves in the momentum equation due to the presence of Oldroyd-B nanoparticles and is defined as the mathematical product of the stretching rate a and the retardation time λ2. Therefore, estimation of β2 augments the value of λ2 as well as the expansion rate. As a conclusion, reduction in thermal setup is perceived with the progressive width of thermal stream. Figure 10(b) illustrates the influence of β2 on concentration flux. Concentration is enormously decreased with the advancement of β2. Actually, improved value of β2 develops the retardation rate, and so concentration setup is reduced.Figure 5
Interference ofA∗ on thermal layer θη.Figure 6
Interference ofB∗ on thermal layer θη.Figure 7
Interference ofkc on concentration layer ϕη.Figure 8
Interference ofRd on thermal layer θη.Figure 9
(a) Interference ofβ1 on thermal layer θη. (b) Interference of βt on concentration layer ϕη.
(a)(b)Figure 10
(a) Interference ofβ2 on thermal layer θη. (b) Interference of β2 on concentration layer ϕη.
(a)(b)Nusselt number measures the rate of heat transference whereas Sherwood number measures the rate of mass transference from hot region to cold region. Figure11 describes the rate of heat transport with the combined inspiration of random motion 0≤Nb≤3 and thermodiffusion 0≤Nt≤5 parameters. Rate of heat transport is declined with the upgraded amounts of these parameters related to the nanofluid. Maximal rate of heat transport is measured for smaller amounts of these nanoparticle parameters, whereas minimal rate of heat transport is attained for greater amounts of Nb and Nt. Maximum value of Nusselt number is approximated as 1.2, whereas minimum value of Nusselt number is estimated as 0.3 for the adopted ranges of Brownian and thermodiffusion parameters. Physically, a higher value of Nb enhances the Brownian diffusion coefficient, and a higher value of Nt enhances the thermodiffusion coefficient, whereas kinematic viscosity of Oldroyd-B nanofluid is reduced with the escalation of these parameters. As a whole, these physical changes are the main causes for the reduction of heat transport. Figure 12 shows the rate of mass transport with the collective stimulation of random motion 0≤Nb≤1 and thermodiffusion 0≤Nt≤6 parameters. Rate of mass transport is degenerated with the progress in the amount of Nt, whereas it is improved with the advancement in the amount of Nb. Highest rate of mass transport is stated for smaller choice of Nt and higher choice of Nb. Maximum value of Sherwood number is formulated as 1.4, whereas minimum value of Sherwood number is predicted as −0.4 for the implemented ranges of Brownian and thermodiffusion factors. Physically, a higher value of Nb enhances the heat capacity induced by nanoparticles, and a higher value of Nt develops the temperature difference, whereas density of Oldroyd-B nanofluid is upgraded with the intensification of these forces. Overall, these physical variations are the main reasons for the improvement of Sherwood number with the positive tendency in Nb and reduction of Sherwood number with the positive approach of Nt. Figure 13(a) indicates the rate of heat transport with the communal effect of thermal convection parameter 0≤βt≤100 and concentration convection parameter 0≤βc≤100. The maximum value of heat transport is reported as 0.84 for smaller communal effect of convection parameters, whereas the minimum value of heat transport is calculated as 0.7 for larger communal effect of convection parameters. Nusselt number is reduced with the higher estimation of βc, and it is improved with the growth of βt. On the other hand, Figure 13(b) explains the rate of mass transport with the combined effect of convection parameters. Sherwood number is improved with the estimation of βt, and it is reduced with the growth of βc. Maximum mass transport is observed at the right top of the figure and minimal mass transport is noticed at the left bottom of the figure. Physically, thermal convection parameter depends on the temperature difference, and thermal environment is reduced with its escalation; hence, rate of heat transport is improved. Moreover, concentration convection parameter relies on the concentration difference, and thermal environment is boosted with its growth; hence, rate of heat transport is reduced for this situation.Figure 11
Nusselt number in the view ofNb and Nt.Figure 12
Sherwood number in the view ofNb and Nt.Figure 13
(a) Nusselt number in the view ofβt and βc. (b) Sherwood number in the view of βt and βc.
(a)(b)Table2 is arranged to compare the present outcomes with the previously published work in the absence of porosity, magnetic environment, dual stratifications, radiation, chemical reaction, and space-dependent heat source/sink factors. The reduced results for thermal and concentration gradients are best matched with the previously published report by Khan et al. [47] via homotopy analysis method.Table 2
Comparison establishment with the previously published report of Azeem Khan et al. [44] for δ=M=λ=βt=βc=A∗=εt=εs=Rd=γ=kr=0.
αPrB∗NbNtLePresent outcomesPublished outcomes [44]−θ′0−ϕ′0−θ′0−ϕ′00.01.20.20.10.11.00.3518530.4742100.3518530.4742100.30.5098370.4828380.5098370.4828380.40.5494380.4889390.5494380.4889390.51.00.5137280.4309300.5137280.4309301.10.5512380.4632160.5512380.4632161.30.6173930.5277230.6173930.5277230.00.7572080.3566700.7572080.3566700.10.6769070.4221430.6769070.4221430.40.3353900.6902210.3353900.6902210.20.5405140.6833620.5405140.6833620.30.4975860.7452810.4975860.7452810.40.4568410.7757260.4568410.7757260.20.5757950.1993430.5757950.1993430.40.5053940.2777620.5053940.2777620.50.4807570.4644100.4807570.4644100.80.5911650.3525480.5911650.3525480.90.5882000.4264140.5882000.4264141.10.5834270.5605780.5834270.560578


## 6. Conclusions

The significance of higher-order chemical reaction and nonuniform heat source/sink influence on the dynamics of Oldroyd-B nanoparticles have been analytically investigated in this contribution. For physical relevancy, thermal radiation, mixed convection, and double stratification have also been incorporated in the bidirectional flow of the nanoliquid with the help of Buongiorno nanofluid model. Chief findings/outcomes of the ongoing exploration are listed below:(i)
Level of heat exposure is improved with the intensification in thermal stratification parameterεt, whereas level of mass exposure is reduced with the improvement in solutal stratification parameter εs.(ii)
Level of mass exposure is developed with the higher selections of Brownian motionNb and chemical reaction kr parameters, whereas level of heat exposure is condensed with the escalation in the choices of heat source/sink parameters A∗&B∗ and radiation factor Rd.(iii)
Nusselt and Sherwood numbers are decreased with greater amounts of concentration convection parameter, whereas opposite trend is noticed for thermal convection parameter.(iv)
Higher estimations of Brownian and thermodiffusion parameters reduce the rate of heat transport.(v)
Temperature and concentration profiles are reduced with the positive tendency of retardation time Deborah numberβ2, and opposite trends are detected for relaxation time Deborah number β1.Finally, the outcomes obtained through this investigation are helpful in the improvement of thermal instruments. This mathematical model has some applications in biomedicine and chemical and mechanical engineering like coating a sheet with nanomaterial, paper production process, and treatment of cancer.



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