Heat and Mass Transport Investigation in Radiative and Chemically Reacting Fluid Over a Differentially Heated Surface and Internal Heating

Open Physics 2020; 18: 842–852 Research Article Yu-Ming Chu, Adnan, Umar Khan, Naveed Ahmed, Syed Tauseef Mohyud-Din, and Ilyas Khan* Heat and mass transport investigation in radiative and chemically reacting fluid over a differentially heated surface and internal heating https://doi.org/10.1515/phys-2020-0182 received March 20, 2020; accepted August 17, 2020 1 Introduction Abstract: The aim of this study is to investigate the heat Thermal analysis of Newtonian fluids by considering the and mass transport over a stretchable surface. The ana- influence of external magnetic field, radiative heat flux, lysis is prolonged to the concept of thermal radiations internal heating, cross-diffusion and chemical reaction and chemical reaction over a differentially heated surface over an unsteady convectively heated stretchable surface with internal heating. This is significant from an engi- is a potential area of interest because of its potential neering and industrial point of view. The nonlinear model applications in industrial and engineering areas. The is successfully attained by adopting the similarity trans- applications include cooling electronic devices, fiber pro- forms and then further computation is done via a hybrid duction, power generation and wire glass. Runge–Kutta algorithm coupled with shooting technique. In 1970, Crane [1] reported flow over a nonporous The behavior of fluid velocity and heat and mass transport stretchable surface. Afterward, Gupta and Gupta [2] stu- are then furnished graphically for feasible ranges of died thermal and mass transfer over a bilaterally stretch- parameters. A comprehensive discussion of the results is able surface. They investigated the influence of suction or provided against multiple parameters. Foremost, the local injection parameters on the flow characteristics. Later on, thermal performance and mass transport rate are explained Grubka and Booba [3] discussed thermal transportation via numerical computation. The major outcomes of the in fluid over a surface which is capable of stretching study are described in the end. continuously. Wang [4] extended the flow of carrier fluid in a rotating frame. He focused on the parameter λ and Keywords: convective flow, thermal radiations, heat transfer, observed the fluid properties due to fluctuating λ. This stretchable surface, cross diffusion parameter is the quotient of rotating surface and the rate of stretching sheet. Furthermore, he found the solution of the model by applying regular perturbation technique. Unsteady Newtonian flow model in a porous sheet was reported by Shafie et al. [5]. Combined effects of cross- diffusion on three-dimensional (3D) time-dependent flow over a stretching sheet were studied by Reddy et al. [6] in 2016. Influence of externally applied magnetic * Corresponding author: Ilyas Khan, Faculty of Mathematics and fi Statistics, Ton Duc Thang University, Ho Chi Minh City, 72915 eld, thermophoresis and internal heat source was part Vietnam, e-mail: [email protected] of their discussion. Yu-Ming Chu: Department of Mathematics, Huzhou University, Recently, Khan et al. [7] explored 3D squeezed flow of Huzhou 313000, People’s Republic of China; Hunan Provincial Key Newtonian fluid between two parallel plates. They con- Laboratory of Mathematical Modeling and Analysis in Engineering, sidered a rotating system in which both plates and fluid Changsha University of Science & Technology, Changsha 410114, China rotate together in a counter clockwise pattern. They Adnan: Department of Mathematics, Mohi-ud-Din Islamic treated a nonlinear flow model numerically. The heat University, Nerian Sharif, AJ&K 12080, Pakistan transfer investigation under the effects of externally Umar Khan: Department of Mathematics and Statistics, imposed magnetic field over bi-latterly stretchable sheet Hazara University, Mansehra 21120, Pakistan in porous medium explored by Ahmad et al. [8]. For Naveed Ahmed: Department of Mathematics, Faculty of Sciences, mathematical analysis of the model, they employed the HITEC University Taxila Cantt, Taxila Cantt 47070, Pakistan - Syed Tauseef Mohyud-Din: University of Multan, Multan 66000, homotopy analysis method. Heat transportation in mag Pakistan netohydrodynamic flow over a porous stretching sheet Open Access. © 2020 Yu-Ming Chu et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. Influence of convective and internal heating on the heat and mass transfer 843 which is capable of stretching in horizontal and vertical taken over a surface which is extendable in horizontal directions was discussed in ref. [9]. In 2017, Ullah et al. and vertical directions. The surface meets at z = 0 and [10] explored dissipative flow of unsteady fluid (which is fluid flow in the region z > 0. The surface is capable non-Newtonian in nature) over a linearly stretchable of stretching bi-directionally with forces which are equal sheet. They reported the impacts of cross-diffusion and in magnitude. Furthermore, components of the velocity heat generation or absorption in the flow field. In 2016, along are as follows: Oyelakin et al. [11] discussed unsteady flow of the non- along x-axis: fl - Newtonian ow model for a stretchable surface and en −1 uw()=(−)xt,1 ax αt , countered the impacts of thermal heat flux, convective and slip flow conditions. Hydromagnetic flow model for along y-axis: Casson fluid over an inclined stretchable plane by consid- −1 vw()=(−)yt,1 by αt , ering the influences of radiative heat flux and chemical reaction was reported by Reddy [12].Shafieetal.[13] stu- where a and b both are constants and greater than zero. died unsteady Falkner–Skan flow for a stretching sheet. The surface is thermally invariable. The auxiliary condi- The effects of cross-diffusion, thermal radiation and che- tions at and far from the surface are mentioned in the mical reaction on a flow over a stretchable surface were flow geometry. Figure 1 elucidates the configuration of reported in ref. [14] and [15],respectively.Forfurther the model. useful analysis of the various fluid mechanics problems, The set of equations that govern the streamline flow we can study the work presented in ref. [16–25]. over a stretching surface by considering the influence of Recently, Kumar et al. [26] conducted an analytical externally imposed Lorentz forces, radiative heat flux, investigation of Cauchy reaction–diffusion equations. cross thermal and concentration gradients and chemical They successfully tackled the model by the said tech- reaction is defined as follows: nique and explained comprehensively. The tumor and ∂u⁎ ∂v⁎ ∂w⁎ + + = 0 (1) immune cell analysis in immunogenetic tumor model ∂x⁎ ∂y⁎ ∂z⁎ under the influence of nonsingular fraction derivative ∂u⁎ ∂u⁎ ∂u⁎ ∂u⁎ μ ∂2⁎u σB2 was carried out in ref. [27].Moreover,significant inves- u⁎ v⁎ w⁎ u⁎(2) ⁎ + ⁎ + ⁎ + ⁎ = ⁎2 − tigation of various mathematical models under certain ∂t ∂x ∂y ∂z ρ ∂z ρ conditions by adopting multiple techniques was reported in ref. [28–44]. To the best of authors’ literature survey, no studies regarding bi-laterally stretchable surface in the existence of externally imposed magnetic field, thermal heat flux, resistive heating, thermo-diffusion, diffusion-thermo and chemical reactions have been noted so far. The flow model is formulated effectively by employing feasible self-similar variables. Section 3 contains mathematical analysis of the model. Then, influence of pertinent dimensionless parameters on the momentum, thermal and concentration profiles is highlighted in Sec- tion 4. Numerical computations are carried out for quantities of engineering interest such as shear stresses, local Nusselt and Sherwood numbers. Finally, key observa- tions of the study are given in the end. 2 Description of the problem Unsteady radiative and chemically reacting flow of Newtonian fluid in the existence of Lorentz forces, ohmic heating, cross-diffusion and convective flow condition is Figure 1: Interpretation of the model. 844 Yu-Ming Chu et al. ⁎ ⁎ ⁎ ⁎ 2⁎ 2 ∂v ⁎ ∂v ⁎ ∂u ⁎ ∂v μ ∂ v σB ⁎ By means of self-similar transformations embedded + u + v + w = − v (3) ∂t⁎ ∂x⁎ ∂y⁎ ∂z⁎ ρ ∂z⁎2 ρ in equation (8), we arrive with the following dimensionless nonlinear flow model: ∂T ⁎ ∂T ⁎ ∂T ⁎ ∂T ⁎ + u⁎ + v⁎ + w⁎ t⁎ x⁎ y⁎ z⁎ η ∂ ∂ ∂ ∂ F‴+FF ″[ + G ]− F ′22 − SF ′+FMF ″ − ′=0, (9) 2⁎ 2⁎ ⁎ 3 2⁎ 2 k ∂ T DKT ∂ C 16σT∞ ∂ T = ⁎2 + ⁎2 − ⁎ ⁎2 (4) ρcppsp ∂z ρc C ∂z 3kρc ∂z η G‴+GF ″[ + G ]− G ′22 − SG ′+GMG ″ − ′ = 0, (10) ⁎ 2 ⁎ 2 2 2 μ ∂u ∂v σB ⁎2 ⁎2 + + +(+)uv ρc ∂z⁎ ∂z⁎ ρc pp η 2 (+1Rd )″+βS Pr −ββFGDϕ ′+′(+ )+fx ″+Ec F ″ 2 ∂C⁎ ∂C⁎ ∂C⁎ ∂C⁎ + u⁎ + v⁎ + w⁎ +″+(′+′)]=EcGMF22 Ec 2 Ec G 2 0, (11) ∂t⁎ ∂x⁎ ∂y⁎ ∂z⁎ yxy (5) ∂2⁎C DK ∂2⁎T η D T kC⁎ C⁎ . ϕS″+Sc −ϕϕFG ′+ ′( + ) +SrSc β ″− γ Scϕ = ⁎2 + ⁎2 −(1 −∞ ) ( ) ∂z Tm ∂z 2 12 = 0. Equation (1) shows the mass conservation which gra- tifies automatically. Furthermore, equations of motion, Self-similar boundary conditions for particular flow thermal and concentration equations are described in equa- model are the following: tions (2)–(5) in the existence of externally imposed magnetic Fη( )↓ηηη===000 =0, F ′( η )↓ = 1, Gη ( )↓ = 0, field, radiative heat flux, cross-diffusion and chemical Gη′( )↓ηηi00 = cβη,1, ′( )↓ = Bβη ( ( ) − ) (13) reaction. The various physical quantities embedded in the == ϕη()↓η=0 =1 aforementioned flow model are specificheatcapacity()cp , dynamic viscosity μ ,density ρ ,massdiffusivity D , () () () Fη′( )↓ηη→∞ →0, Gη ′( )↓ →∞ → 0, (14) thermal conductivity ()k , mean temperature ()Tm ,thermal

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