# Effects of Homogeneous and Heterogeneous Chemical Features on Oldroyd-B Fluid Flow between Stretching Disks with Velocity and Temperature Boundary Assumptions

Authors: Nargis Khan; Muhammad Sadiq Hashmi; Sami Ullah Khan; Faryal Chaudhry; Iskander Tlili; Mostafa Safdari Shadloo
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This research endeavors the rheological features of Oldroyd-B fluid configured by infinite stretching disks in presence of velocity and thermal slip features. Additionally, the effects of homogeneous and heterogeneous chemical features are also considered. The transmuted flow equations are analytically solved with help of the homotopy analysis method (HAM). It is observed that the homogeneous chemical reaction parameter enhances the concentration distribution, while the heterogeneous reaction reduces the concentration profile. With implementations of temperature jump conditions, the heat transfer from the surfaces of both disks can be effectively controlled. The impacts of various dimensionless parameters are elaborated through graphs and tables.


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## Body

## 1. Introduction

The fluid flow between stretching disks is the main motivation of investigators in recent years due to its leading applications in turbine engines, compression, mechanical components transient loading, semiconductor manufacturing, rotating wafers, injection modeling, power transmission, viscometer, lubrications, radial diffusers, geophysics, biomechanics, geothermal, oceanography, thrust bearings, etc. The usage of microdevices has many practical applications in different scientific areas such as surgery, biotechnology, electronic cooling, microchannels, heat pipes, and pumps. The heat and fluid flow characteristics are different for both microdevices and macroscale counterparts. This difference is constituted by velocity slip and temperature jump. The velocity slip is an important feature to analyze the behavior of microflows because no-slip boundary conditions are not applicable to the fluid flow in microelectro-mechanical-systems (MEMS). Also, no-slip boundary conditions show the impractical behavior for the cases such as corner flow, spreading of liquid on a solid substrate, and extrusion of polymer melts from a capillary tube. Therefore, no-slip boundary condition is replaced by slip boundary condition. Further, in the slip flow regime, temperature jump is significantly used to determine the heat transfer. Because of such applications of slip flow, many interesting contributions have been made by investigators in recent years. For instance, Zheng et al. [1] investigated the stretched flow of viscous fluid in presence of velocity as well as thermal slip features. The peristaltic transport of Carreau fluid through a channel with various flow features with application of velocity slip, temperature, and concentration jump has been inspected by Vajravelu et al. [2]. Khan et al. [3] discussed the double diffusion slip flow of viscous fluid over a vertical plate. Xiao et al. [4] presented a mathematical model for fully developed slip flow in a microtube gas problem. This interesting continuation contains the velocity slip of order two and the assumptions of temperature jump constraints. They claimed an effective change up to 15% in the local Nusselt number at room temperature. Similar slip effects have been performed by Rooholghdos and Roohi [5] for a nanoscale flat plate and a microscale cylinder. Another useful contribution regarding the gas flow associated with thermal slip conditions was examined by Le and Roohi [6]. The peristaltic transport of viscous fluid in an asymmetric channel in presence of velocity and temperature boundary conditions has been discussed by Sinha et al. [7]. El-Aziz and Afify [8] examined the heat transfer characteristics for slip flow of Casson fluid subjected to the induced magnetic field. Khan et al. [9] determined the analytical solution based on the Galerkin technique for an upper convected flow of Maxwell fluid in presence of slip features. Muhammad et al. [10] examined the entropy generation aspects in the flow of nanofluid under the action of the second-order slip. The investigation for fractional Maxwell fluid in presence of slip effects and porous medium was performed by Aman et al. [11].The fluid flow encountered the heat transportation process conveying a diverse engineering and industrial significance in the metal cooling, petroleum engineering, chemical processing, food industries, thermophysical systems, fiber spinning, manufacturing of metallic sheets, and various nuclear processes. Besides this, the thermal performance of disc-shaped bodies had engaged many scholars because of its practical applications in the era of aeronautical sciences. Many engineering and mechanical processes like thermal power generation and heat transfer to automatic control systems encountered the applications of these phenomena. Due to such recurrent applications, several researchers investigate the flow over or flow between two disks. The initial contribution on this topic was led by Kármán and Uber [12] by considering viscous fluid flow between two infinite disks. This study was further extended by many researchers with different flow features. Hayat et al. [13] studied the heat transfer characteristics based on the Fourier law of conduction in third-grade liquid configured by two porous disks. Turkyilmazoglu [14] simulated the numerical solution of hydromagnetic fluid flow near the stagnation point subject to disk rotation. Heat transfer analysis in the hydromagnetic fluid flow caused by a rotating shrinking disk was also performed numerically by Turkyilmazoglu [15]. Soid et al. [16] applied the numerical technique to observe heat transfer phenomenon in viscous fluid for a radially stretching disk. Yin et al. [17] examined the flow thermal characteristics of nanofluid flow due to a rotating disk. Turkyilmazoglu [18] numerically examined the flow of Newtonian fluid through a vertically moving disk. Hashmi et al. [19] analytically explored the mixed convection flow of Oldroyd-B fluid placed between isothermal stretching disks. The idea of flow over stretching surfaces is extremely useful and involved a large number of practical applications in manufacturing processes [20–23]. The spontaneous idea of flow due to a moving surface was originally advised by Sakiadis [24, 25] which encouraged the investigators to pay attention in this direction. The exact solution for a stretching flow problem was successfully provided by Wang [26]. Another investigation in this direction has been suggested by Fang [27] which conferred the viscous fluid flow induced due to a stretched disk. In another attempt, Fang and Zhang [28] derived an exact solution based on the mathematical formulation of Navier Stokes equations modeled in cylindrical coordinates. In fact, such type of flow between two infinite stretching disks arises due to accelerated stretching velocity. Gorder et al. [29] discussed the axisymmetric flow between two infinite stretching disks. Mohyud-Din and Khan [30] implemented effects of nonlinear thermal radiation in flow of Casson fluid concedes between two stretching disks. Slip flow in presence of thermo-diffusion effects in flow of viscous fluid between stretching disks was suggested by Rashidi and his coworkers [31]. Analytical solution based on the homotopy analysis method for flow of viscous fluid through a stretchable disk has been depicted by Khan et al. [32]. In another investigation, Khan et al. [33] examined the viscous dissipation and joule heating effects on the axisymmetric flow of viscous fluid between stretching disks. Khan et al. [34] studied the entropy generation effects on flow of carbon nanotubes between two rotating and stretching disks. The heat transfer analysis based on Cattaneo–Christov heat flux expressions for the flow of micropolar fluid induced by a nonlinear stretching disk was focused by Doh et al. [35]. Renuka et al. [36] computed an analysis solution for the flow of nanofluid, additionally featuring entropy generation features induced by a stretchable spinning disk.In the recent decade, the study of combined heat and mass transportation has inspired the scientists to examine various aspects of the simultaneous phenomenon due to its arising applications in the real-world problems like reacting systems, cooling towers, marine engineering, distillation columns, hydrometallurgical industry, crop damage via freezing, and copse of trees. The collaboration amongst homogeneous and heterogeneous responses happening on some catalytic surfaces is correlated with the production and employment of chemical species at diverse rates within the fluid and on the catalytic surfaces. Merkin [37] developed a very useful mathematical model to explore the relationship between a surface-based reaction and homogeneous and heterogeneous reactions. Another useful contribution is from Kameswaran et al. [38] where flow of nanoparticles is immersed in a porous medium with additional features of binary chemical reactions. Rashidi et al. [31] address the effects of homogeneous/heterogeneous on a peristaltic transport in a channel. Hayat et al. [39] implemented the effects of second-order velocity slip to examine the flow of chemical reactive viscous nanofluid induced by a permeable stretching surface.In this present analysis, our focus is to evaluate the driven transport of Oldroyd-B fluid considered within two infinite stretching disks in presence of homogeneous and heterogeneous reactions. Unlike typical studies, here the idea of second-order velocity slip and temperature jump boundary conditions has been implemented. According to the literature survey, no attempt has been made by researchers for such analysis and is presented for the first time. The present flow problem is utilized in presence of applied magnetic field effects which are useful in the industry of metal-working, chemical reactors, plasma materials, modern metallurgical, oil exploration, and extraction of geothermal energy. The analytical solutions of such transmuted flow equations are determined by employing the homotopy analysis method [40–45]. The accuracy of this method is successfully obtained and expressed in a tabular form. Finally, the important feature effective parameters are graphically underlined and discussed for some velocity, temperature, and concentration profiles with technical relevance.


## 2. Mathematical Modeling

We consider a two-dimensional flow of Oldroyd-B due to infinite stretching disks. Let flow be axisymmetric and considered fluid be incompressible. The velocity slip and temperature jump are also considered at the walls of stretchable disks. A magnetic field with strengthB0 is imposed in z-direction. The effects of electric and induced magnetic fields are neglected. It is assumed that both lower and upper disks are maintained at temperature T1 and T2, respectively. Following Merkin and Chaudhary [46], the mathematical expressions repressing the homogeneous-heterogeneous reactions are expressed as(1)A+2B⟶3B,rate=kcαβ2.The isothermal, first-order reaction associated with a catalyst surface is represented as(2)A⟶B,rate=ksα,where α and β stand for concentrations of chemical species and A, B, kc, and ks denote the rate constants. In the present analysis, both reactions are treated as processes which are isothermal. The analysis is performed by opting a cylindrical coordinate (r, θ, z). All the involved expressions are independent of θ due to axisymmetry. The constitutive partial differential equations for Oldroyd-B fluid in presence of chemical reactions are expressed as(3)1r∂∂rru+∂w∂z=0,(4)u∂u∂r+w∂u∂z=−1ρ∂ρ∂r+ν2∂2u∂r2+∂2w∂r∂z+∂2u∂z2+2r∂u∂r−2ur2−λ1′w2∂2u∂z2+2uw∂2u∂r∂z+∂2u∂r2+vλ2′4u2r3−2wr2∂u∂z−1r∂u∂z2−2∂u∂z∂2w∂z2+w∂3u∂z3−2ur2∂u∂r−∂2u∂z2∂u∂r−2∂u∂r2−1r∂u∂z∂w∂r+2wr∂2u∂r∂z−∂u∂z∂2u∂r∂z−∂u∂r∂2w∂r∂z+u∂3u∂r∂z2+w∂3w∂r∂z2+2ur∂2u∂r2−2∂u∂r∂2u∂r2−∂u∂z∂2w∂r2+2w∂3u∂r2∂z+u∂3w∂r2∂z+2u∂3u∂r3+σB02ρ−u−λ1′w∂u∂z,(5)u∂w∂r+w∂w∂z=−1ρ∂ρ∂z+v∂2w∂r2+∂2u∂r∂z+2∂2w∂z2+1r∂w∂r+1r∂u∂r−λ1′w2∂2u∂z2+2uw∂2u∂r∂z+u2∂2u∂r2+vλ2′−ur2∂u∂z−1r∂u∂z∂w∂z+wr∂2u∂z2−2∂w∂z∂2w∂z2+2w∂3w∂z3+ur2∂w∂r−1r∂w∂r∂u∂z−∂2u∂z2∂w∂r−2r∂u∂r∂w∂r+ur∂2u∂r∂z−∂w∂z∂2u∂r∂z+wr∂2w∂r∂z−∂w∂r∂2w∂r∂z+w∂3w∂r∂z2+2u∂2w∂r∂z2−2∂w∂r∂2u∂r2+ur∂2w∂r2−∂w∂z∂3w∂r2+u∂3u∂r2∂z+w∂3w∂r2∂z+u∂3w∂r3,(6)u∂T∂r+w∂T∂z=K∂2T∂r2+1r∂T∂r+∂2T∂z2,(7)u∂α∂r+w∂α∂z=DA∂2α∂r2+1r∂α∂r+∂2α∂z2−kcαβ2,(8)u∂β∂r+w∂β∂z=DB∂2β∂r2+1r∂β∂r+∂2β∂z2+kcαβ2,where u and w are the radial and axial components of velocities, respectively, p is the pressure, ρ is the fluid density, μ stands for dynamic viscosity of fluid, ν=μ/ρ represents the kinematic viscosity, a and c are the stretching constants, λ1′ is the constant of relaxation, λ2′ is the retardation time, T is the temperature, K is the thermal diffusivity, and DA,DB are the diffusion species coefficient of A and B.


### 2.1. Slip Boundary Conditions

As it has been mentioned earlier that the present flow problem is assisted with slip boundary conditions. For the velocity profile, the derivation of second-order velocity slip is based on the expansion of Taylor series from the first-order Maxwell conditions which are generally expressed as(9)u=uw+2−σuσu∂u∂nτ1+2−σuσuτ12∂2u∂n2.For the present analysis, we propose the following second-order boundary conditions:(10)u=ar+2−σuσuτ1∂u∂z+2−σuσuτ12∂2u∂z2,w=0,p=aμβ1r24d2,atz=0,u=cr−2−σuσuτ1∂u∂z−2−σuσuτ12∂2u∂z2,w=0,p=0,atz=d,where a and b represent the stretching rates, σu is the tangential momentum accommodation coefficient, and τ1 denotes the molecular mean-free path. It is a well-established fact that the molecular mean-free path is assumed positive, i.e., ϵ1>0 and. ϵ2<0.


### 2.2. Temperature Jump Boundary Conditions

By using Taylor series second-order expansion forKn from the first order, Smoluchowski jump condition second-order jump conditions are proposed in [6] as follows:(11)T=Tw+2−σTσT2ξξ+11Prτ2∂T∂n+2−σTσT2ξξ+11Prτ222∂2T∂n2.The second-order temperature jump boundary conditions associated with the governing equations are(12)T=T1+2−σTσT2ξξ+11Prτ2∂T∂z+2−σTσT2ξξ+11Prτ222∂2T∂z2,atz=0,T=T2−2−σTσT2ξξ+11Prτ2∂T∂z+2−σTσT2ξξ+11Prτ222∂2T∂z2,atz=d,where σT is the thermal accommodation coefficient and ξ is the specific heat ratio. The other boundary conditions for the flow problem are prescribed by(13)α=α0atz=0,DA∂α∂z=ksαatz=d,β=0atz=0,DB∂β∂z=−ksαatz=d.Introducing the similarity variables,(14)u=−ar2H′η,w=adHη,p=aμPη+β1r24d2,η=zd,T=T1+T2−T1θη,α=α0φη,β=α0gη,T=T1+T2−T1θη,α=α0φη,β=α0gη.In view of the above similarity variables, equations (4)–(10) yield(15)R2H′2−2HH″=−β1+H‴+λ1′aRHH′H″−H2H‴+λ2′aHHiv−H″2+RMH′+λ1′aHH″,(16)θ″−RPrHθ′=0,(17)φ″−RScK1φg2+Hφ′=0,(18)δg″+RScK1φg2+Hg′=0,(19)P′=3H″2−RHH′−λ1RH2H″,(20)H0=0,H1=0,H′0=−2+ϵ1H″0+ϵ2H‴0,(21)H′1=−2γ−ϵ1H″0+ϵ2H‴0,P0=0,(22)θ0=ϵ3θ′0+ϵ4θ″0,θ1=1−ϵ3θ′1+ϵ4θ″1,(23)φ0=1,φ′1=K2φ1,g0=0,δg′1=−K2g1,where γ is the wall stretching parameter, R stands for the Reynolds number, Pr is the Prandtl number, ϵ1 is the first-order velocity slip parameter, M is the Hartmann number, ϵ2 is the second-order velocity slip parameter, ϵ3 is the first-order temperature jump parameter, ϵ4 stands for temperature jump parameter of the second order, Sc represents the Schmidt number, δ is the ratio of the diffusion coefficient,Kn denotes the Knudsen number, K1 is the strength of the homogeneous reaction, and K2 is the strength of the heterogeneous reaction and is defined as(24)γ=ca,R=ad2v,Pr=να,Kn=τ1,2d,ε1=Kn2−σuσu,ε2=Kn222−σuσu,ε3=Kn2−σTσT2ξξ+11Pr,ε4=Kn222−σTσT2ξξ+11Pr,δ=DBDA.The constantβ1 has been eliminated from equation (15) as the following procedure:(25)Hiv=RHH‴−λ1R−HH′H‴−H2Hiv+HH″2+H′2H″−MRH″+λ1H′H″+HH‴−λ2−2H″H‴+H′Hiv+HHv,in which λ1=λ1′a is the Deborah number for relaxation time and λ2=λ2′a for the retardation time. It is pointed out here that the diffusion coefficients of chemical species A and B are not equal in general. So, we remarked that constants A and B are of comparable size as a special case and subsequently DA and DB are equal, i.e., δ=1. Equations (16) and (17) lead to the following relation:(26)φ+g=1,φ″−RScK1φ1−φ2+Hφ′=0,φ0=1,φ′1=K2φ1.Following mathematical expressions are suggested for the wall skin friction coefficient, local Nusselt number, and local Sherwood number at both surfaces of disks:(27)C1f,2f=τrzη=0,11/2ρδr2=2R−1H″ηη=0,1,N1u,2u=−dkT∂T/∂zη=0,1kTT2−T1=−θ′ηη=0,1,Sh=−D∂C/∂zη=0,1DC1−C2=−φ′ηη=0,1.


## 2.1. Slip Boundary Conditions

As it has been mentioned earlier that the present flow problem is assisted with slip boundary conditions. For the velocity profile, the derivation of second-order velocity slip is based on the expansion of Taylor series from the first-order Maxwell conditions which are generally expressed as(9)u=uw+2−σuσu∂u∂nτ1+2−σuσuτ12∂2u∂n2.For the present analysis, we propose the following second-order boundary conditions:(10)u=ar+2−σuσuτ1∂u∂z+2−σuσuτ12∂2u∂z2,w=0,p=aμβ1r24d2,atz=0,u=cr−2−σuσuτ1∂u∂z−2−σuσuτ12∂2u∂z2,w=0,p=0,atz=d,where a and b represent the stretching rates, σu is the tangential momentum accommodation coefficient, and τ1 denotes the molecular mean-free path. It is a well-established fact that the molecular mean-free path is assumed positive, i.e., ϵ1>0 and. ϵ2<0.


## 2.2. Temperature Jump Boundary Conditions

By using Taylor series second-order expansion forKn from the first order, Smoluchowski jump condition second-order jump conditions are proposed in [6] as follows:(11)T=Tw+2−σTσT2ξξ+11Prτ2∂T∂n+2−σTσT2ξξ+11Prτ222∂2T∂n2.The second-order temperature jump boundary conditions associated with the governing equations are(12)T=T1+2−σTσT2ξξ+11Prτ2∂T∂z+2−σTσT2ξξ+11Prτ222∂2T∂z2,atz=0,T=T2−2−σTσT2ξξ+11Prτ2∂T∂z+2−σTσT2ξξ+11Prτ222∂2T∂z2,atz=d,where σT is the thermal accommodation coefficient and ξ is the specific heat ratio. The other boundary conditions for the flow problem are prescribed by(13)α=α0atz=0,DA∂α∂z=ksαatz=d,β=0atz=0,DB∂β∂z=−ksαatz=d.Introducing the similarity variables,(14)u=−ar2H′η,w=adHη,p=aμPη+β1r24d2,η=zd,T=T1+T2−T1θη,α=α0φη,β=α0gη,T=T1+T2−T1θη,α=α0φη,β=α0gη.In view of the above similarity variables, equations (4)–(10) yield(15)R2H′2−2HH″=−β1+H‴+λ1′aRHH′H″−H2H‴+λ2′aHHiv−H″2+RMH′+λ1′aHH″,(16)θ″−RPrHθ′=0,(17)φ″−RScK1φg2+Hφ′=0,(18)δg″+RScK1φg2+Hg′=0,(19)P′=3H″2−RHH′−λ1RH2H″,(20)H0=0,H1=0,H′0=−2+ϵ1H″0+ϵ2H‴0,(21)H′1=−2γ−ϵ1H″0+ϵ2H‴0,P0=0,(22)θ0=ϵ3θ′0+ϵ4θ″0,θ1=1−ϵ3θ′1+ϵ4θ″1,(23)φ0=1,φ′1=K2φ1,g0=0,δg′1=−K2g1,where γ is the wall stretching parameter, R stands for the Reynolds number, Pr is the Prandtl number, ϵ1 is the first-order velocity slip parameter, M is the Hartmann number, ϵ2 is the second-order velocity slip parameter, ϵ3 is the first-order temperature jump parameter, ϵ4 stands for temperature jump parameter of the second order, Sc represents the Schmidt number, δ is the ratio of the diffusion coefficient,Kn denotes the Knudsen number, K1 is the strength of the homogeneous reaction, and K2 is the strength of the heterogeneous reaction and is defined as(24)γ=ca,R=ad2v,Pr=να,Kn=τ1,2d,ε1=Kn2−σuσu,ε2=Kn222−σuσu,ε3=Kn2−σTσT2ξξ+11Pr,ε4=Kn222−σTσT2ξξ+11Pr,δ=DBDA.The constantβ1 has been eliminated from equation (15) as the following procedure:(25)Hiv=RHH‴−λ1R−HH′H‴−H2Hiv+HH″2+H′2H″−MRH″+λ1H′H″+HH‴−λ2−2H″H‴+H′Hiv+HHv,in which λ1=λ1′a is the Deborah number for relaxation time and λ2=λ2′a for the retardation time. It is pointed out here that the diffusion coefficients of chemical species A and B are not equal in general. So, we remarked that constants A and B are of comparable size as a special case and subsequently DA and DB are equal, i.e., δ=1. Equations (16) and (17) lead to the following relation:(26)φ+g=1,φ″−RScK1φ1−φ2+Hφ′=0,φ0=1,φ′1=K2φ1.Following mathematical expressions are suggested for the wall skin friction coefficient, local Nusselt number, and local Sherwood number at both surfaces of disks:(27)C1f,2f=τrzη=0,11/2ρδr2=2R−1H″ηη=0,1,N1u,2u=−dkT∂T/∂zη=0,1kTT2−T1=−θ′ηη=0,1,Sh=−D∂C/∂zη=0,1DC1−C2=−φ′ηη=0,1.


## 3. Solution Methodology

To start our simulations, first we introduce the following initial guesses for velocity, temperature, and concentration profiles:(28)H0η=18ϵ1+12ϵ12−2η1+η−1+η1+γ+2ϵ1−2+η+γ1+η−6ϵ21+η,θ0η=η+ϵ31+2ϵ3,φ0η=−1+K21−η−1+K2,with auxiliary linear operators:(29)LH=d4dη4,Lθ=d2dη2,Lφ=d2dη2.The mathematical expressions associated with the zeroth-order deformation problem are defined as(30)1−qLHHη;q−H0η=qℏHNHHη;q,1−qLθθη;q−θ0η=qℏθNθθη;q,1−qLφφη;q−φ0η=qℏφNφφη;q,where ℏH, ℏθ, and ℏφ denote the auxiliary parameters and qϵ0,1 represents the embedding parameter. And,(31)NHHη;q=∂4Hη,q∂η4−RHη,q∂3Hη,q∂η3−Rλ1Hη,q∂Hη,q∂η∂3Hη,q∂η3+H2η,q∂4Hη,q∂η4−∂Hη,q∂η2∂2Hη,q∂η2−Hη,q∂2Hη,q∂η22+λ2−2∂2Hη,q∂η2∂3Hη,q∂η3+∂Hη,q∂η∂4Hη,q∂η4+Hη,q∂5Hη,q∂η5+MR∂2Hη,q∂η2+λ1∂Hη,q∂η∂2Hη,q∂η2+Hη,q∂3Hη,q∂η3,Nθθη;q=∂2θη,q∂η2−RPrHη,q∂θη,q∂η,Nφφη;q=∂2φη,q∂η2−RScK1φη,q1−φη,q2+Hη,q∂φη,q∂η.The equations for them-th deformations of the problem are(32)LHHmη−χmHm−1η=ℏHR1mη,Lθθmη−χmθm−1η=ℏθR2mη,Lφφmη−χmφm−1η=ℏφR3mη,Hm0=Hm1=0,Hm′0=ϵ1Hm″0+ϵ2Hm′′′0,Hm′1=−ϵ1Hm″1+ϵ2Hm′′′1,θm0=ϵ3θm′0+ϵ4θm″0,θ1=−ϵ3θm′1+ϵ4θm″1,φm0=0,φm′1=K2φm1,χm=0,m≤1,1,m>0.The series solution is computed iteratively form=1,2,3,... using MATHEMATICA software.


## 4. Convergence of Solution

In order to obtain the comfortable accuracy of the homotopic solution, the significance of auxiliary parameters cannot be denied. This task has been completed by preparing threeh-curves, organized for velocity, temperature, and concentration profiles for some dignified values of emerging parameters. The admissible values of such parameter guaranteed the convergence of the solution. The convergence of the derived series solution is controlled by auxiliary parameters ℏH, ℏθ, and ℏφ. Therefore, we have sketched the ℏ-curves in Figure 1 to determine the admissible values of ℏH, ℏθ, and ℏφ. These figures reveal that the convergence region lies within the domain −0.8≤ℏH≤−0.2,−1.5≤ℏθ≤−0.4, and −1.4≤ℏφ≤−0.7.Figure 1
ℏ-curves for (a) ℏH, (b) ℏθ, and (c) ℏφ with R=2,γ=0.5,M=0.3,K2=0.2,Pr=0.5,λ1=λ2=0.2,K1=0.5,Sc=0.2,ϵ1=0.2,ϵ2=0.3,ϵ3=0.3, and ϵ4=0.5..
(a)(b)(c)In Table1, the computations have been performed to illustrate the convergence of the obtained solution for H″0,θ′0, and φ′0 at various approximations. Close observations to the table suggest that accuracy of the solution has been obtained at the 15th order of approximations.Table 1
The HAM convergence at different order of approximations.
ApproximationH″0θ′0φ′0076.891320.4999080.214794106.891230.4999070.215223136.891220.4999060.215228146.891210.4999050.215215156.891200.4999050.215215


## 5. Physical Interpretations of Results

In this section, the effects of various arising parameters on radial and vertical velocity components, pressure, temperature, and concentration fields are discussed with relevant physical significances.


### 5.1. Dimensionless Velocity and Pressure Profiles

Figure2(a) shows the impact of the Hartmann number M on the velocity vertical component by keeping other parameters fixed. The interface of stronger magnetic force is more valuable to decay the motion of fluid particles. A small increment in velocity was observed first which decreases up to a certain height. Physically, as M increases, the Lorentz force boosts up which resists the flow of liquid due to which velocity decay occurs. Therefore, the presence of magnetic field combats the transport phenomena and subsequently diminishes the vertical velocity. The effects of wall stretching parameter γ on the velocity profile are shown in Figure 2(b). The vertical velocity component rises up with a variation of γ. However, a change in the radial component is not similar to vertical components. Here, velocity increases at a specific range and then gradually decreases. Figure 2(c) delineates the significance of the Deborah number in terms of relaxation time λ1 on vertical and radial component of velocities. A rise in the vertical component of velocity is observed for larger values of the Deborah number; however, the radial component of velocity decreases smoothly after a small increment. The variation of material parameter λ2 on both vertical and radial velocity components is illustrated in Figure 2(d). The reverse trend is observed as compared to λ1 for both components. We observe from Figures 2(e) and 2(f) that when we increase of first- and second-order velocity slip constants ϵ1,ϵ2, the vertical velocity component also increases. Physically, with increase of velocity slip parameters, the stretching velocity affects the movement of fluid so velocity profiles get maximum values. Moreover, the amplitude of radial velocity increases up to a specific range due to the difference of the stretching rate. Figures 2(g) and 2(h) show that the skin friction coefficient increases with increase of both slip parameters. It is scrutinized from Figure 2(i) that pressure decreases in the whole domain by increasing values of the Hartmann number M. It is found from Figure 2(j) that decay in pressure is observed by increasing the velocity slip parameter.Figure 2
(a)–(f) Graphs of vertical and radial components of velocity, (g)-(h) graphs of for skin friction, and (i)-(j) graphs of pressure for different values ofγ=0.5,ℏH=−0.5,ℏθ=−1.2,ℏφ=−1.0,M=0.3,R=5,ϵ1=0.5,ϵ2=0.2,λ1=0.2,andλ2=0.5. (a) Effects of the Hartmann number, (b) effects of the stretching parameter, (c) effects of the Deborah number of relaxation, (d) effects of the Deborah number of retardation, (e) effects of the first-order velocity slip parameter, (f) effects of the second-order velocity slip parameter, (g) influence of the first-order velocity slip parameter on the skin friction coefficient, (h) influence of the second-order velocity slip parameter on the skin friction coefficient, (i) Influence of Hartmann number on pressure (j) Influence of first-order velocity slip parameter on pressure.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)


### 5.2. Dimensionless Temperature Profile

In Figures3(a) and 3(b), the dimensionless temperature θη is plotted to study the impact of the velocity slip parameter. The temperature decreases by increasing both velocity slip parameters. It is elucidated from Figures 3(c) and 3(d) that the distribution of temperature θ boosts up due to alteration of the first- and second-order temperature jump parameters. Physically, due to slip effect, more flow penetrates through the thermal boundary with an increase in temperature jump parameters. Figure 3(e) accomplishes the significance of the Prandtl number Pr on the temperature profile. The impression Pr declined the temperature of the fluid effectively. The dimensionless number Pr depends upon thermal diffusivity which decreases by increasing Pr. Therefore, a decline in the temperature field is observed. Thus, higher values of Pr correspond to lower thermal diffusivity and subsequently declining temperature distribution. Figure 3(f) exhibits the dominant effect of the Hartmann number M on the temperature profile. As expected, the temperature of fluid increases by increasing M. Physically, the applied magnetic field produces the Lorentz force, which creates a drag force which has a tendency to enhance the temperature of the fluid between both disks.Figure 3
Temperature profile withγ=0.5,ℏH=−0.5,ℏθ=−1.2,ℏφ=−1.0,M=0.3,R=5,ϵ1=ϵ2=0.2,ϵ3=0.3,ϵ4=0.5,λ1=0.2,andλ2=0.5. (a) Influence of the first-order velocity slip parameter, (b) influence of the second-order velocity slip parameter, (c) variation of the first-order temperature jump parameter, (d) variation of the second-order temperature jump parameter, (e) variation of the Hartmann number, and (f) variation of the Prandtl number.
(a)(b)(c)(d)(e)(f)


### 5.3. Dimensionless Concentration Profile

Taking into account of the concentration profileφ, the effects for various parameters are encountered. First, we consider the variation of the homogeneous reaction K1 on φ. An increase in K1 results in diminishing of the concentration profile (Figure 4(a)). Figure 4(b) shows the consequence of heterogeneous reaction parameter K2 on the concentration profile. The rate of mass transfer is enhanced by increasing K2. Figure 4(c) shows that the rate of mass transfer solely decreases by varying Schmidt number Sc. Sc has an inverse relation with molecular diffusivity which decreases by increasing Sc. The variation of different values of the strengths of the homogeneous parameter K1 and heterogeneous reaction parameter K2 on wall concentration on both disks is shown in Figures 5 and 6, respectively. These figures indicate that values of φ′0 and φ′1 increase by increasing K1 while contradictory behavior is noted for K2.Figure 4
Concentration distribution forγ=0.5,ℏH=−0.5,ℏθ=−1.2,ℏφ=−1.0,M=0.3,R=2,ϵ1=ϵ2=0.2,ϵ3=0.3,ϵ4=0.5,λ1=0.2,andλ2=0.2. (a) Influence of strength of the homogeneous reaction, (b) influence of strength of the heterogeneous reaction, and (c) influence of strength of the Schmidt number.
(a)(b)(c)Figure 5
Influence of strength of the homogeneous reaction.
(a)(b)Figure 6
Influence of strength of the heterogeneous reaction.
(a)(b)


### 5.4. Local Nusselt Number

Table2 aims to elaborate the iterative numerical variation in the local Nusselt number against involved fluid parameters. We found that with the increase in the velocity slip parameter, the temperature profile at the lower disk increases. The heat transfer rate decreases by increasing the Hartmann number M at the lower disk. However, opposite values for M are observed for the upper disk. Such observations are made as both disks are stretched with different velocities.Table 2
Variation in the local Nusselt number at lower and upper disks.
ϵ1ϵ2ϵ3ϵ4PrMLower diskUpper disk0.10.50.50.21.00.3−0.66799−0.759610.20.50.50.21.00.3−0.65734−0.730760.30.50.50.21.00.3−0.64662−0.721700.20.10.50.21.00.3−0.66799−0.759610.20.20.50.21.00.3−0.50683−0.619610.20.30.50.21.00.3−0.17056−0.257600.20.50.10.21.00.3−0.70753−0.661510.20.50.20.21.00.3−0.89202−0.738290.20.50.30.21.00.3−0.90542−0.845910.20.50.51.01.00.3−0.52624−0.467330.20.50.52.01.00.3−0.52626−0.467360.20.50.53.01.00.3−0.52628−0.467390.20.50.50.20.10.3−0.49842−0.501660.20.50.50.20.20.3−0.49684−0.503320.20.50.50.20.30.3−0.49527−0.504980.20.50.50.21.00.1−0.46169−0.538290.20.50.50.21.00.2−0.46974−0.530520.20.50.50.21.00.3−0.48423−0.51662


## 5.1. Dimensionless Velocity and Pressure Profiles

Figure2(a) shows the impact of the Hartmann number M on the velocity vertical component by keeping other parameters fixed. The interface of stronger magnetic force is more valuable to decay the motion of fluid particles. A small increment in velocity was observed first which decreases up to a certain height. Physically, as M increases, the Lorentz force boosts up which resists the flow of liquid due to which velocity decay occurs. Therefore, the presence of magnetic field combats the transport phenomena and subsequently diminishes the vertical velocity. The effects of wall stretching parameter γ on the velocity profile are shown in Figure 2(b). The vertical velocity component rises up with a variation of γ. However, a change in the radial component is not similar to vertical components. Here, velocity increases at a specific range and then gradually decreases. Figure 2(c) delineates the significance of the Deborah number in terms of relaxation time λ1 on vertical and radial component of velocities. A rise in the vertical component of velocity is observed for larger values of the Deborah number; however, the radial component of velocity decreases smoothly after a small increment. The variation of material parameter λ2 on both vertical and radial velocity components is illustrated in Figure 2(d). The reverse trend is observed as compared to λ1 for both components. We observe from Figures 2(e) and 2(f) that when we increase of first- and second-order velocity slip constants ϵ1,ϵ2, the vertical velocity component also increases. Physically, with increase of velocity slip parameters, the stretching velocity affects the movement of fluid so velocity profiles get maximum values. Moreover, the amplitude of radial velocity increases up to a specific range due to the difference of the stretching rate. Figures 2(g) and 2(h) show that the skin friction coefficient increases with increase of both slip parameters. It is scrutinized from Figure 2(i) that pressure decreases in the whole domain by increasing values of the Hartmann number M. It is found from Figure 2(j) that decay in pressure is observed by increasing the velocity slip parameter.Figure 2
(a)–(f) Graphs of vertical and radial components of velocity, (g)-(h) graphs of for skin friction, and (i)-(j) graphs of pressure for different values ofγ=0.5,ℏH=−0.5,ℏθ=−1.2,ℏφ=−1.0,M=0.3,R=5,ϵ1=0.5,ϵ2=0.2,λ1=0.2,andλ2=0.5. (a) Effects of the Hartmann number, (b) effects of the stretching parameter, (c) effects of the Deborah number of relaxation, (d) effects of the Deborah number of retardation, (e) effects of the first-order velocity slip parameter, (f) effects of the second-order velocity slip parameter, (g) influence of the first-order velocity slip parameter on the skin friction coefficient, (h) influence of the second-order velocity slip parameter on the skin friction coefficient, (i) Influence of Hartmann number on pressure (j) Influence of first-order velocity slip parameter on pressure.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)


## 5.2. Dimensionless Temperature Profile

In Figures3(a) and 3(b), the dimensionless temperature θη is plotted to study the impact of the velocity slip parameter. The temperature decreases by increasing both velocity slip parameters. It is elucidated from Figures 3(c) and 3(d) that the distribution of temperature θ boosts up due to alteration of the first- and second-order temperature jump parameters. Physically, due to slip effect, more flow penetrates through the thermal boundary with an increase in temperature jump parameters. Figure 3(e) accomplishes the significance of the Prandtl number Pr on the temperature profile. The impression Pr declined the temperature of the fluid effectively. The dimensionless number Pr depends upon thermal diffusivity which decreases by increasing Pr. Therefore, a decline in the temperature field is observed. Thus, higher values of Pr correspond to lower thermal diffusivity and subsequently declining temperature distribution. Figure 3(f) exhibits the dominant effect of the Hartmann number M on the temperature profile. As expected, the temperature of fluid increases by increasing M. Physically, the applied magnetic field produces the Lorentz force, which creates a drag force which has a tendency to enhance the temperature of the fluid between both disks.Figure 3
Temperature profile withγ=0.5,ℏH=−0.5,ℏθ=−1.2,ℏφ=−1.0,M=0.3,R=5,ϵ1=ϵ2=0.2,ϵ3=0.3,ϵ4=0.5,λ1=0.2,andλ2=0.5. (a) Influence of the first-order velocity slip parameter, (b) influence of the second-order velocity slip parameter, (c) variation of the first-order temperature jump parameter, (d) variation of the second-order temperature jump parameter, (e) variation of the Hartmann number, and (f) variation of the Prandtl number.
(a)(b)(c)(d)(e)(f)


## 5.3. Dimensionless Concentration Profile

Taking into account of the concentration profileφ, the effects for various parameters are encountered. First, we consider the variation of the homogeneous reaction K1 on φ. An increase in K1 results in diminishing of the concentration profile (Figure 4(a)). Figure 4(b) shows the consequence of heterogeneous reaction parameter K2 on the concentration profile. The rate of mass transfer is enhanced by increasing K2. Figure 4(c) shows that the rate of mass transfer solely decreases by varying Schmidt number Sc. Sc has an inverse relation with molecular diffusivity which decreases by increasing Sc. The variation of different values of the strengths of the homogeneous parameter K1 and heterogeneous reaction parameter K2 on wall concentration on both disks is shown in Figures 5 and 6, respectively. These figures indicate that values of φ′0 and φ′1 increase by increasing K1 while contradictory behavior is noted for K2.Figure 4
Concentration distribution forγ=0.5,ℏH=−0.5,ℏθ=−1.2,ℏφ=−1.0,M=0.3,R=2,ϵ1=ϵ2=0.2,ϵ3=0.3,ϵ4=0.5,λ1=0.2,andλ2=0.2. (a) Influence of strength of the homogeneous reaction, (b) influence of strength of the heterogeneous reaction, and (c) influence of strength of the Schmidt number.
(a)(b)(c)Figure 5
Influence of strength of the homogeneous reaction.
(a)(b)Figure 6
Influence of strength of the heterogeneous reaction.
(a)(b)


## 5.4. Local Nusselt Number

Table2 aims to elaborate the iterative numerical variation in the local Nusselt number against involved fluid parameters. We found that with the increase in the velocity slip parameter, the temperature profile at the lower disk increases. The heat transfer rate decreases by increasing the Hartmann number M at the lower disk. However, opposite values for M are observed for the upper disk. Such observations are made as both disks are stretched with different velocities.Table 2
Variation in the local Nusselt number at lower and upper disks.
ϵ1ϵ2ϵ3ϵ4PrMLower diskUpper disk0.10.50.50.21.00.3−0.66799−0.759610.20.50.50.21.00.3−0.65734−0.730760.30.50.50.21.00.3−0.64662−0.721700.20.10.50.21.00.3−0.66799−0.759610.20.20.50.21.00.3−0.50683−0.619610.20.30.50.21.00.3−0.17056−0.257600.20.50.10.21.00.3−0.70753−0.661510.20.50.20.21.00.3−0.89202−0.738290.20.50.30.21.00.3−0.90542−0.845910.20.50.51.01.00.3−0.52624−0.467330.20.50.52.01.00.3−0.52626−0.467360.20.50.53.01.00.3−0.52628−0.467390.20.50.50.20.10.3−0.49842−0.501660.20.50.50.20.20.3−0.49684−0.503320.20.50.50.20.30.3−0.49527−0.504980.20.50.50.21.00.1−0.46169−0.538290.20.50.50.21.00.2−0.46974−0.530520.20.50.50.21.00.3−0.48423−0.51662


## 6. Conclusions

In this work, a chemically reactive flow of Oldroyd-B fluid subject to stretchable disks is considered in presence of homogeneous and heterogeneous chemical reactions. The homogeneous-heterogeneous reactions are considered in the concentration equation. The physical features are visualized for various involved parameters graphically. The important observations are summarized as follows:(i)
The velocity distribution increases with variation of slip parameters while it decreases with the Deborah number for retardation time.(ii)
The concentration distribution declines with increment of the Schmidt number and the homogeneous reaction while effects of the heterogeneous reaction parameter are quite reverse.(iii)
The temperature distribution increases by increasing the Hartmann number while lower temperature distribution is observed for larger values of the Prandtl number.(iv)
The presence of first- and second-order velocity slip results in an increment in the wall shear stress.



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