Muhammad Faisal

@ajku.edu.pk

Assistant Professor of Mathematics, Department of Mathematics, Faculty of Science
The University of Azad Jammu and Kashmir



                                   

https://researchid.co/muhammadfaisal

Muhammad Faisal is a Faculty Member in the Department of Mathematics, Faculty of Science, Azad Jammu & Kashmir University, Muzaffarabad, Pakistan. He received his PhD, MPhil and MSc degrees from Department of Mathematics UAJK in 2021, 2013 and 2010, respectively. His research is in the field of computational fluid dynamics, thermal engineering, engineering mathematics and physics, nanocomposites, computational methods, nanofluids, hybrid nanofluids, etc.

EDUCATION

PhD in Mathematics
MPhil in Mathematics
MSc in Mathematics
Specialization: Applied Mathematics (Fluid Mechanics and Numerical Methods)

RESEARCH, TEACHING, or OTHER INTERESTS

Computational Mathematics, Fluid Flow and Transfer Processes, Materials Science, General Physics and Astronomy

FUTURE PROJECTS

Dynamics of Nanofluids: Modeling and Simulation

To gain an understanding of the heat transfer performance  Applications of extending surfaces in nanotechnology  Stability analysis of the solution  Numerical Inspections of Newtonian and non-Newtonian nanofluid flows


Applications Invited
Collaborators

Dynamics of Hybrid nanofluids: Modeling and Simulation

1. Inspection of heat transport in hybrid nanofluids. 2. Discussion on thermo-physical properties of various nanoparticles. 3. Thermal conductivity of heterogeneous two-component systems. 4. Unsteady flows towards stretching devices. 5. Numerical Simulation via Keller-Box method.


Applications Invited
Collaborators

Dynamics of Ternary Hybrid nanofluids: Modeling and Simulation

1. To study the aspects of random motion and thermo-migration of nanoparticles. 2. Discussion on the shapes of nanoparticles. 3. Numerical Simulation via Keller-Box method. 4. Unsteady flows towards stretching devices.


Applications Invited
Collaborators
46

Scopus Publications

Scopus Publications

  • Entropy optimization in squeezed nanofluidic dissipative transport of radiative water conveying aluminum alloys
    Muhammad Faisal, Kanayo Kenneth Asogwa, F. Mabood, and I. A. Badruddin

    World Scientific Pub Co Pte Ltd
    In this paper, squeezing transport of radiative water conveying aluminum alloys (i.e., AA7072 and AA7075) mobilized by entropy generation and dissipative energy is analyzed. Problem is formulated in a rotating frame with the consideration of magnetohydrodynamics and Joule heating aspects. Maxwell model for nanofluid has been used to incorporate the thermophysical properties of nanoelements. Formulated governing expressions have been transformed into system of ODEs by introducing similarity variables. The transformed system of ODEs is then numerically solved by Runge–Kutta–Fehlberg (RKF) method based on shooting background. The physical quantities (i.e., skin-friction coefficient, Nusselt and Bejan numbers) of scientific interest are formulated and illustrated via various plots. Graphical representations of squeezing function, temperature profile and velocity profile have been made to examine the effects of involved parameters. Streamlines and isotherms patterns have been formed and discussed. To authenticate the validity of model, skin-friction values have been compared with published literature for limited version of the model. Entropy and temperature of the system are improved with the involvement of aluminum alloys in water. Symmetrical behavior of streamlines is observed for positive approach of squeezing parameter.

  • Entropic behavior with activation energy in the dynamics of hyperbolic-tangent mixed-convective nanomaterial due to a vertical slendering surface
    Muhammad Faisal, F. Mabood, I.A. Badruddin, Muhammad Aiyaz, and Faisal Mehmood Butt

    Emerald
    PurposeNonlinear mixed-convective entropy optimized the flow of hyperbolic-tangent nanofluid (HTN) with magnetohydrodynamics (MHD) process is considered over a vertical slendering surface. The impression of activation energy is incorporated in the modeling with the significance of nonlinear radiation, dissipative-function, heat generation/consumption connection and Joule heating. Research in this area has practical applications in the design of efficient heat exchangers, thermal management systems or nanomaterial-based devices.Design/methodology/approachSuitable set of variables is introduced to transform the PDEs (Partial differential equations) system into required ODEs (Ordinary differential equations) system. The transformed ODEs system is then solved numerically via finite difference method. Graphical artworks are made to predict the control of applicable transport parameters on surface entropy, Bejan number, Sherwood number, skin-friction, Nusselt number, temperature, velocity and concentration fields.FindingsIt is noticed from present numerical examination that Bejan number aggravates for improved estimations of concentration-difference parameter a_2, Eckert number E_c, thermal ratio parameter ?_w and radiation parameter R_d, whereas surface entropy condenses for flow performance index n, temperature-difference parameter a_1, thermodiffusion parameter N_t and mixed convection parameter ?. Sherwood number is enriched with the amplification of pedesis-motion parameter N_b, while opposite development is perceived for thermodiffusion parameter. Lastly, outcomes are matched with formerly published data to authenticate the present numerical investigation.Originality/valueTo the best of the authors' knowledge, no investigation has been reported yet that explains the entropic behavior with activation energy in the flowing of hyperbolic-tangent mixed-convective nanomaterial due to a vertical slendering surface.

  • Analysis of entropy generation and activation energy in unsteady double diffusive flow over a stretching cylinder
    Muhammad Faisal, Iftikhar Ahmad, and Muhammad Awais Awan

    Wiley
    AbstractEntropy generation with activation energy has garnered worldwide attention from researchers due to its extensive applications in thermodynamic design, chemical engineering, and optimization processes. One of the primary reasons for focusing on entropy is the global energy crisis, driven by massive consumption against limited resources. This study elucidates the double diffusion process in the dynamics of a viscous fluid over a stretching cylinder, incorporating considerations of entropy generation and activation energy. The non‐Fourier heat flux model is employed to describe the thermodynamics of the thermal system, while non‐Fick's law is used to elucidate the mass diffusion process. The unsteady phases of axisymmetric flow of the viscous fluid are thoroughly discussed. Transport equations in cylindrical configuration are transformed into ordinary differential equations, and an efficient mathematical method (i.e., homotopy analysis method) is applied to solve the transformed system. The graphical examination reveals the impact of flow parameters on velocity configuration, entropy configuration, temperature configurations, and concentration configurations. Additionally, a comparative benchmark is established with the results from previously published work for authentication and validation purposes. It is noted that concentration and temperature configurations are directly related to the unsteady stretching parameter. Entropy generation exhibits an inverse relationship with the unsteady stretching parameter when close to the stretching cylinder, whereas it shows a direct relationship when the fluid is flowing away from the surface.

  • Entropic behavior in bidirectional flow of CeO<inf>2</inf>-ZnO/water hybrid nanofluid with prescribed surface temperature/heat flux aspects
    Muhammad Faisal, Iftikhar Ahmad, and Abdur Rashid

    Emerald
    PurposeThe present study aims to encompass the bidirectional magnetized flowing of a hybrid-nanofluid over an unsteady stretching device with the inclusion of thermal radiation and entropy generation. Brick-shaped nanoparticles (zinc-oxide and ceria) are suspended in water, serving as the base-fluid to observe the performance of the hybrid mixture. The Maxwell thermal conductivity relation is employed to link the thermophysical attributes of the hybrid mixture with the host liquid. Additionally, a heat source/sink term is incorporated in the energy balance to enhance the impact of the investigation. Both prescribed-surface-temperature (PST) and prescribed-heat-flux (PHF) conditions are applied to inspect the thermal performance of the hybrid nanofluid.Design/methodology/approachThe transport equations in Cartesian configuration are transformed into ordinary differential equations (ODEs), and an efficient method, namely the Keller-Box method (KBM), is utilized to solve the transformed system. Postprocessing is conducted to visually represent the velocity profile, thermal distribution, skin-friction coefficients, Bejan number, Nusselt number and entropy generation function against the variations of the involved parameters.FindingsIt is observed that more entropy is generated due to the increases in temperature difference and radiation parameters. The Bejan number initially declines but then improves with higher estimations of unsteadiness and Hartmann number. Overall, the thermal performance of the system is developed for the PST scenario than the PHF scenario for different estimations of the involved constraints.Originality/valueTo the best of the authors' knowledge, no investigation has been reported yet that explains the bidirectional flow of a CeO2-ZnO/water hybrid nanofluid with the combined effects of prescribed thermal aspects (PST and PHF) and entropy generation.

  • Insight into the bidirectional dynamics of hyperbolic tangent hybrid nanomaterial subject to Nield’s conditions using numerical approach
    Muhammad Faisal, Qazi Zan-Ul-Abadin, Irfan Anjum Badruddin, Abdul Hamid Ganie, Iftikhar Ahmad, and Mohamed Hussien

    Informa UK Limited

  • Stagnation point flow of third-order nanofluid towards a lubrication surface using hybrid homotopy analysis method
    M. Ahmad, Basharat Bashir, Taseer Muhammad, M. Taj, and Muhammad Faisal

    World Scientific Pub Co Pte Ltd
    In recent times, the interaction of nanoparticles has significantly enhanced the thermal association of heat transport. This phenomenon plays a crucial role in hydraulic systems, particularly in the context of lubrication and its associated consequences on mass and heat transport. Current studies have focused on investigating the thermal effects of a third-order nanofluid on a lubricated stretched surface near an analytical stagnation point. The lubrication process involves the use of a thin, adjustable coating of lubricant fluid. To analyze this complex system, we employ the Buongiorno model and explore thermophoresis and the Brownian motion phenomenon. For deriving analytical results of updated boundary layer ordinary differential equations, we rely on the dependable and effective hybrid homotopy analysis method (HHAM). To exhibit the effectiveness of our study, we provide a numerical comparison. Based on theoretical flow assumptions, we establish a range of flow parameters. In the presence of lubrication, we physically examine how these parameters affect temperatures, velocities, concentration, and other relevant quantities of thermal interest. These new findings have practical applications in polymer production, heat transmission, and hydraulic systems.


  • Prescribed thermal effects on the dynamics of radiative and chemically reactive hyperbolic tangent nanofluid
    Muhammad Faisal, Muhammad Bilal Riaz, Iftikhar Ahmad, and Syed Basit Ali Kazmi

    Informa UK Limited

  • On convective heat and mass transport of radiative double diffusive Williamson hybrid nanofluid by a Riga surface
    Muhammad Faisal, F. Mabood, K. K. Asogwa, and I. A. Badruddin

    World Scientific Pub Co Pte Ltd
    Convective heat and mass transport of radiative Williamson hybrid [Formula: see text] nanofluid (NF) by a Riga surface with the novel features of Cattaneo–Christov double-diffusion has been investigated. Thermal contributions of internal heat mechanism and Arrhenius energy in Darcy–Forchheimer medium have also been incorporated in the modeling. Mathematical modeling has been completed by using suitable mathematical expressions for thermophysical features of hybrid nanofluid (HNF). Transport partial differential equations (PDEs) have been transformed into ordinary differential equations (ODEs) by means of similarity variables. Numerical approximation of the transformed system has been obtained by using shooting-based Runge–Kutta–Fehlberg approach. Results have been presented through various graphs and discussed physically in detail. Solution is validated for limited cases. Concentration of the hybrid mixture is reduced for progressive concentration-relaxation parameter. Temperature is alleviated for developing thermal-relaxation parameter. Nusselt number is observed to be higher for Williamson HNF than simple ordinary NF.



  • Natural convective behavior of hybrid nanofluid (Al<inf>2</inf>O<inf>3</inf>–Cu/water) in an isosceles triangular cavity with bifurcation analysis
    Muhammad Faisal, Farah Javed, K. Loganathan, Reema Jain, and Rifaqat Ali

    Springer Science and Business Media LLC

  • Darcy–Forchheimer dynamics of hybrid nanofluid due to a porous Riga surface capitalizing Cattaneo–Christov theory
    Muhammad Faisal, Kanayo Kenneth Asogwa, Fazle Mabood, and Irfan Anjum Badruddin

    Informa UK Limited

  • Numerical analysis of heat and mass transfer with viscous dissipation, Joule dissipation, and activation energy
    G. Dharmaiah, B. Shankar Goud, Nehad Ali Shah, and Muhammad Faisal

    Informa UK Limited

  • Morphological nanolayer impact on hybrid nanofluids flow due to dispersion of polymer/CNT matrix nanocomposite material
    M Zubair Akbar Qureshi, M Faisal, Qadeer Raza, Bagh Ali, Thongchai Botmart, and Nehad Ali Shah

    American Institute of Mathematical Sciences (AIMS)
    &lt;abstract&gt; &lt;p&gt;The objective of this study is to explore the heat transfer properties and flow features of an MHD hybrid nanofluid due to the dispersion of polymer/CNT matrix nanocomposite material through orthogonal permeable disks with the impact of morphological nanolayer. Matrix nanocomposites (MNC) are high-performance materials with unique properties and design opportunities. These MNC materials are beneficial in a variety of applications, spanning from packaging to biomedical applications, due to their exceptional thermophysical properties. The present innovative study is the dispersion of polymeric/ceramic matrix nanocomposite material on magnetized hybrid nanofluids flow through the orthogonal porous coaxial disks is deliberated. Further, we also examined the numerically prominence of the permeability ($ {\\mathrm{A}}_{\\mathrm{*}} $) function consisting of the Permeable Reynold number associated with the expansion/contraction ratio. The morphological significant effects of these nanomaterials on flow and heat transfer characteristics are explored. The mathematical structure, as well as empirical relations for nanocomposite materials, are formulated as partial differential equations, which are then translated into ordinary differential expressions using appropriate variables. The Runge–Kutta and shooting methods are utilized to find the accurate numerical solution. Variations in skin friction coefficient and Nusselt number at the lower and upper walls of disks, as well as heat transfer rate measurements, are computed using important engineering physical factors. A comparison table and graph of effective nanolayer thermal conductivity (ENTC) and non-effective nanolayer thermal conductivity are presented. It is observed that the increment in nanolayer thickness (0.4−1.6), enhanced the ENTC and thermal phenomena. By the enhancement in hybrid nanoparticles volume fraction (2% to 6%), significant enhancement in Nusselt number is noticed. This novel work may be beneficial for nanotechnology and relevant nanocomponents.&lt;/p&gt; &lt;/abstract&gt;

  • Dynamics of MHD tangent hyperbolic nanofluid with prescribed thermal conditions, random motion and thermo-migration of nanoparticles
    Muhammad Faisal, Iftikhar Ahmad, and Tariq Javed

    Informa UK Limited
    Investigation of expanding sheet flow in the existence of tiny particles is an interesting field of research and this type of mathematical model is usually expressed in the form of partial differen...

  • Nonlinear Radiative Nanofluidic Hydrothermal Unsteady Bidirectional Transport with Thermal/Mass Convection Aspects
    Muhammad Faisal, Kanayo Kenneth Asogwa, Nazek Alessa, and Karuppusamy Loganathan

    MDPI AG
    The collective effect of thermal and mass convection along with the significance of thermal radiation, heat source/sink, and magneto-nanofluid are considered. A bi-directional stretching device is used to generate the symmetry of the flowing structure. Nonlinear behavior of thermal radiation is considered here. The magnetic field is considered non-uniform and vertically upward. Significances of pedesis motion and Ludwig–Soret are also revealed in an innovative way with heat source/sink effects. The concept of symmetry is used to transmute the transport equations from PDE type to nonlinear ODE type. We solved the transformed setup numerically by adopting Keller-box method criteria with the targeted accuracy rate. Graphical interpretations are explored with code verification. It is important to conclude that friction coefficients decline for incremental values of stretching parameter (0.1≤α≤0.9), magnetic field (0.3≤M≤0.9), and unsteady parameter (0.2≤Λ≤0.9) along with the bidirectional velocity components, and the rate of heat transmission rises with temperature ratio (1.3≤Γ≤1.7) and temperature Biot number (0.3≤BiT≤0.9) amplification. Moreso, the rate of mass transfer is enhanced with growing values of pedesis motion (0.2≤Nb≤0.6), unsteady parameter and concentration Biot number (0.3≤BiC≤0.9) with opposite effect when the Ludwig–Soret parameter (0.3≤Nt≤0.6) is boosted.

  • Activation Energy Performance through Magnetized Hybrid Fe<inf>3</inf>O<inf>4</inf>–PP Nanofluids Flow with Impact of the Cluster Interfacial Nanolayer
    M. Zubair Akbar Qureshi, Qadeer Raza, Aroosa Ramzan, M. Faisal, Bagh Ali, Nehad Ali Shah, and Wajaree Weera

    MDPI AG
    The current work investigated the mass and heat transfer of the MHD hybrid nanofluid flow subject to the impact of activation energy and cluster interfacial nanolayer. The heat transport processes related to the interfacial nanolayer between nanoparticles and base fluids enhanced the base fluid’s thermal conductivity. The tiny particles of Fe3O4 and PPy were considered due to the extraordinary thermal conductivity which is of remarkable significance in nanotechnology, electronic devices, and modern shaped heat exchangers. Using the similarity approach, the governing higher-order nonlinear coupled partial differential equation was reduced to a system of ordinary differential equations (ODEs). Fe3O4–PPy hybrid nanoparticles have a considerable influence on thermal performance, and when compared to non-interfacial nanolayer thermal conductivity, the interfacial nanolayer thermal conductivity model produced substantial findings. The increase in nanolayer thickness from level 1 to level 5 had a significant influence on thermal performance improvement. Further, the heat and mass transfer rate was enhanced with higher input values of interfacial nanolayer thickness.

  • Insight into the motion of water conveying three kinds of nanoparticles shapes on a horizontal surface: Significance of thermo-migration and Brownian motion
    S. Saleem, I.L. Animasaun, Se-Jin Yook, Qasem M. Al-Mdallal, Nehad Ali Shah, and Muhammad Faisal

    Elsevier BV


  • Dynamics of Nanoplatelets in Mixed Convective Radiative Flow of Hybridized Nanofluid Mobilized by Variable Thermal Conditions
    Iftikhar Ahmad, Muhammad Faisal, Qazi Zan-Ul-Abadin, K. Loganathan, Tariq Javed, and Balachandra Pattanaik

    Hindawi Limited
    The present mathematical model discloses the effects of Boussinesq and Rosseland approximations on unsteady 3D dynamics of water-driven hybridized nanomaterial with the movements of nanoplatelets (molybdenum disulfide, MoS 2 and graphene oxide, GO ). Variable thermal conditions, namely, VST (variable surface temperature) and VHF (variable heat flux), are opted to provide temperature to the surface. MHD effects have also been used additionally to make the study more versatile. In order to transmute the transportation equations into nondimensionlized forms, similarity transformations have been adopted. The Keller-Box technique has been applied to obtain a numerical simulation of the modeled problem. The convergence of the solution for both VST and VHF cases is presented via the grid independence tactic. Thermal setup against escalating choices of power indices and nonlinear thermal radiation parameter is discussed via graphical illustrations. The rate of heat transaction has been discussed with the growing choices of mixed convection, thermal radiation, and unsteady parameters through various tabular arrangements. It is observed through the present analysis that mixed convection parameter, radiation parameter, temperature maintaining indices r , s , and unsteady parameter magnify the rate of heat transference under the control of platelet-shaped nanoparticles.

  • Convective Heat Transport in Bidirectional Water Driven Hybrid Nanofluid Using Blade Shaped Cadmium Telluride and Graphite Nanoparticles under Electromagnetohydrodynamics Process
    Iftikhar Ahmad, Qazi Zan-Ul-Abadin, Muhammad Faisal, K. Loganathan, Tariq Javed, and Ngawang Namgyel

    Hindawi Limited
    The recent progress in nanotechnology provides the concept of hybrid-class nano-fluids having advanced thermal features comparing to regular nanofluids. The idea of a hybrid nanofluid has motivated many researchers because of its convincible performance in thermal systems. The novel theme of the present effort is to scrutinize the consequences of convective heat transfer in bidirectional water driven hybridclass nanofluid involving blade shaped cadmium telluride CdTe and graphite C nanoparticles with electromagnetohydrodynamics (EMHD) process. The transport equations representing the aforementioned topic are firstly nondimensionalized by using scaling-group transformation and then tackled by the Keller-box method, numerically. The significant results for pertinent parameters have been simulated and presented graphically as well as in tabular forms. Coefficients of drag forces are diminished with the more loadings of cadmium telluride and graphite nanoparticles and opposite results are noticed in the case of the Nusselt number. Heat transport has been improved significantly with more loadings of nanoparticles from 1 wt% to 10 wt%. A comparison benchmark for a limited version of the investigation is made with the previously published data.

  • Prescribed Thermal Activity in the Radiative Bidirectional Flow of Magnetized Hybrid Nanofluid: Keller-Box Approach
    I. Ahmad, Qazi Zan-Ul-Abadin, M. Faisal, K. Loganathan, T. Javed and D. Chaudhary


    In this exploration, we decided to investigate the significance of prescribed thermal conditions on unsteady 3D dynamics of water-based radiative hybrid nanofluid with the impact of cylindrical-shaped nanosized particles (alumina ( Al 2 O 3 ) and titania ( Ti O 2 )). For physical relevancy, the impact of the Lorentz force is also included. The combination of suitable variables has been used to transform the transport equations into the system of ordinary differential equations and then numerically solved via the Keller-Box approach. Graphical illustrations have been used to predict the impact of the involved parameters on the thermal setup. Convergence analysis is presented via the grid independence approach. Skin frictions and local Nusselt numbers against various choices of involved parameters are plotted and arranged in tabular forms. It is observed through the present investigation that temperature distribution is increased with the higher choices of radiation parameter 0.0 ≤ R d ≤ 2.0 and decreased with the improvement in the choices of temperature maintaining indices (i.e., − 2.0 ≤ r , s ≤ 2.0 ). Moreover, the thermophysical properties except specific heat for hybrid nanofluid are improved with the involvement of cylindrical-shaped nanoparticles. The temperature of the hybrid nanofluid is observed to be higher for variable thermal conditions as compared to uniform thermal conditions. Outfalls for a limited version of the report have been compared with a previous published paper.

  • Entropy Analysis in Bidirectional Hybrid Nanofluid Containing Nanospheres with Variable Thermal Activity
    I. Ahmad, Qazi Zan-Ul-Abadin, M. Faisal, K. Loganathan, T. Javed and Sonam Gyeltshen


    An investigation of the thermal performance of water-conveying nanospheres (magnetite ( Fe 3 O 4 ) and silver ( Ag )) subject to variable thermal controls, namely, variable surface temperature and variable surface normal heat flux, has been made. A bidirectionally elongating surface is used to generate an unsteady flow mechanism with the action of the Lorentz force. Derived equations of basic laws are firstly nondimensionalized and then numerically solved by applying the Keller-Box method. The local Nusselt number for both the thermal cases is calculated and discussed. Percent-wise enhancement in the rate of heat transport has also been included in the analysis. It was concluded through the present exploration that at lower volume fractions of magnetite and silver, the rate of heat transport is observed to be dominant. The rate of heat transference has attained identical values for both the provided thermal conditions at the surface. Moreover, intensities of velocity and thermal profiles diminish with the appreciation of the choice of unsteadiness. The temperature-controlling indices also affect the thermal profile, and it is reduced with the intensification in the considerations of these indices. The values of thermal conductivity, density, and electrical conductivity have been improved with the inclusion of nanospheres (magnetite ( Fe 3 O 4 ) and silver ( Ag )), whereas the value of specific heat is reduced with the mixture of these nanospheres. The Nusselt number is increased up to 5% with the involvement of magnetite nanospheres, and it is enhanced up to 4% with the involvement of silver nanospheres.

  • Nonlinear Mixed Convective Bidirectional Dynamics of Double Stratified Radiative Oldroyd-B Nanofluid Flow with Heat Source/Sink and Higher-Order Chemical Reaction
    Iftikhar Ahmad, Muhammad Faisal, K. Loganathan, Muhammad Zaheer Kiyani, and Ngawang Namgyel

    Hindawi Limited
    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 of 0 ≤ β t ≤ 100 and is reduced with the growth of 0 ≤ β c ≤ 100 . Nusselt number is declined with the upgraded amounts of 0 ≤ N b ≤ 3 and 0 ≤ N t ≤ 5 .

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