Dynamics of chemical reactions and temperature-dependent thermal conductivity in a viscous fluid over an oscillating curved surface

Abstract

This study addresses the critical analysis of heat and mass transfer within a time-dependent, viscous fluid flow over an oscillatory stretched curved surface, incorporating variable thermal conductivity and both homogeneous and heterogeneous chemical reactions. The research highlights the influence of heat generation and magnetic fields on energy and momentum equations, employing a curvilinear coordinate system to transform a complex physical problem into a tractable mathematical model. A homotopy analysis method is utilized to derive a convergent series solution for the resulting partial differential equations, enabling a comprehensive parametric investigation of temperature, velocity, concentration, and pressure fields. Extensive graphical and tabular analyses reveal how specific parameters, such as reaction strengths, radius of curvature, and Schmidt number, influence the surface concentration, heat and mass transfer rates, and skin friction coefficient. The insights provided are essential for applications in engineering and industrial processes involving curved surfaces under dynamic thermal and magnetic conditions, such as in chemical reactors, biomedical devices, and materials processing, where precise control of heat and mass transfer is critical for system efficiency and performance optimization.

Introduction

Heat transmission analysis as well as heat production phenomena by incorporating constant and temperature-dependent thermal conductivity has apprehended the attention of scholars and scientists in the last few years due to its widespread employment in the disciplines of mechanical manufacturing and certain physical procedures such as cooling towers, refrigeration, heat exchangers, equipment power collectors, electric motors, gas turbines, and solar power machinery. Abbas et al. ¹ evaluated heat production effects along with constant thermal conductivity in nanofluid flow across a stretched curved wall. Saba et al. ² have conducted heat transport analysis for the motion of viscous nanofluid toward a stretchable curved surface and computed the effects of heat generation. Hydromagnetic flow and heat transfer analysis with temperature-dependent thermal conductivity over a curvy surface was considered by Murtaza et al. ³ Afridi et al. ⁴ analyzed entropy production in dissipative liquid motion toward a curved sheet by considering the impacts of varying thermal conductivity. Kempannagari et al. ⁵ have considered the influence of temperature-dependent thermal conductivity and Joule heating on non-Newtonian liquid flow across a curved exponential sheet. Kumar et al. ⁶ have detected the significance of Soret–Dufour's effects on radiative flow across a porous plate in the presence of a chemical reaction.

Catalysis, biological procedures, and combustion are examples of chemically reactive structures that include heterogeneous and homogeneous (HH) chemical reactions. The correlation between these chemical reactions is extremely complicated. Dispersal and fog deposition, ceramics manufacture, food production, polymers, and hydrometallurgical procedures are examples of physical systems involving HH chemical reactions. Hayat et al. ⁷ have considered HH reactions in the flow of a micropolar liquid in a curvy channel. Imtiaz et al. ⁸ have scrutinized the consequences of HH chemical reactions in magnetohydrodynamics convective movement of ferro liquid over a stretched curved sheet. Inspection of HH chemical reactions in boundary layer motion toward a nonlinear convectively heated curvy stretchable surface was conducted by Saif et al. ⁹ Hamid ¹⁰ detected the consequences of HH reactions and variable thermal conductivity on the flow of Williamson fluid. Waini et al. ¹¹ have studied equally diffusive chemically reactive hybrid nanoliquid motion over a stretched wall. Abbas et al. ¹² have scrutinized the impacts of equally diffusive reactions on the oscillatory and radiative motion of Casson nanoliquid toward a curved stretched sheet.

The research of boundary layer liquid motion accompanied by curved stretched surfaces has aroused the concentration of scientists and researchers in recent years because of its broad applications in the sectors of manufacturing, engineering, and industries. Such industrial and manufacturing sectors are the expulsion of polymer sheets from a dye, elongation of plastic films, production of paper, flow generated in hot rolling, and blowing of glass. In all of these procedures, the superiority of the resulting product is greatly affected by the stretching rate of the surface. First, Sajid et al. ¹³ evaluated the numerical results of viscous fluid across a stretching curved surface. After the revolutionary work of Sajid et al., ¹³ many researchers have performed different research studies on curved surfaces by considering different stretching velocities. Naveed et al. ¹⁴ have studied hydromagnetic flow toward an unsteady curvy stretchable surface. The examination of viscous liquid motion due to a nonlinear curved stretched sheet was conducted by Sanni et al. ¹⁵ Okechi et al. ¹⁶ analyzed the flow of Newtonian fluid over an exponentially curved stretchable wall. Analysis of heat transmission for viscous fluid motion across a generalized stretchable or shrinkable curved sheet was conducted by Naveed et al. ¹⁷ Narla et al. ¹⁸ have evaluated the impacts of entropy creation in magnetohydrodynamic flow caused by the stretching of a curved surface. Yasmin et al. ¹⁹ have detected the consequences of velocity and thermal slips on the radiative motion of hybrid nanoliquid toward a stretched curved sheet. Recently, Imran and Naveed ²⁰ conducted a numerical inspection to scrutinize the effects of activation energy and heat production in the curvilinear flow of Carreau nanofluid over a curved wall.

All of the research cited above addresses the flow problems due to either linear or nonlinear and exponential curved stretching surfaces. However, another attractive area of research is in which a surface performs a stretching response across its designated plane with consistent cyclic velocity and is titled as an oscillatory stretchable sheet. In such a situation, the flow mechanism is due to the periodic motion of the surface. The first examination of the motion of Newtonian fluid toward a flat oscillatory stretched sheet was performed by Wang. ²¹ Abbas et al. ²² have extended Wang's ²¹ work on a curvy oscillatory stretched sheet and evaluated the stimulus of thermal radiation in the magnetized flow of viscous liquid above on a convectively heated stretched oscillatory curved sheet. Imran et al. ²³ conducted analytical research for the flow of nanomaterials over a curved oscillating stretching wall and detected the effects of Joule heating and melting heat transport. Naveed et al. ²⁴ have considered the impacts of the Cattaneo–Christov heat flux model on the oscillatory motion of micropolar fluid over a curvy stretchable sheet. A detailed and comprehensive literature about various flow problems over curved oscillatory geometries is referenced in the reference section.^25–28

Building on previous research, the present study emphasizes the critical importance of investigating flow and heat transfer in viscous fluids with temperature-dependent thermal conductivity, specifically under the influence of HH chemical reactions across a curved oscillating stretched sheet. Such transport phenomena are crucial in various chemical and mechanical engineering applications, including thermal management systems, chemical reactors, polymer processing, and energy conversion technologies. Despite the encouraging relevance of such phenomena, a comprehensive literature review exposes a notable research gap: although some researchers have examined the individual effects of temperature-dependent thermal conductivity, chemical reactions, or their combined influence over flat or curved stretched surfaces, the simultaneous impact of heat and mass transfer in viscous fluids with variable thermal conductivity over a curved oscillating stretchable surface remains largely unexplored. To address this gap, the present study examines the coupled heat and mass transport features in a magnetized flow of a viscous fluid with temperature-dependent thermal conductivity subjected to an oscillating stretchable curved surface. The energy equation incorporates internal heat generation, while the concentration equation accounts for both HH chemical reactions, thereby capturing the complex interplay between thermal and chemical transport mechanisms. An analytical approach namely the homotopy analysis method (HAM) is employed to solve the resulting nonlinear partial differential equations. The findings, presented through comprehensive graphical and tabular analysis, provide valuable insights into the influence of key flow parameters and establish a theoretical foundation for the design and optimization of practical systems such as catalytic reactors, electronic cooling units, and dynamically operated heat exchangers.

Problem description

Consider a laminar boundary layer and incompressible electrically conducting unsteady motion of viscous fluid across a stretched curved oscillatory sheet of radius ${\stackrel{⌢}{R}}_1$ (see Figure 1). The oscillatory velocity of the sheet which is regarded to be the cause of flow mechanism is specified as $\stackrel{⌢}{v}=\stackrel{⌢}{s}\stackrel{⌢}{a}\sin \stackrel{⌢}{t}\stackrel{⌢}{\omega },$ with $\stackrel{⌢}{a}$ and $\stackrel{⌢}{\omega }$ be the stretched rate and the frequency of the oscillatory sheet, respectively. An unvarying magnetic field of amount $B_0$ is forced along the radial direction $\stackrel{⌢}{r}$ . The significances of the induced magnetic field are unnoticed due to the very low Reynolds number. Let ${\stackrel{⌢}{T}}_w$ and ${\stackrel{⌢}{T}}_{\mathrm{\infty }}$ represent the oscillatory sheet and the ambient fluid temperatures, respectively, with ${\stackrel{⌢}{T}}_w>{\stackrel{⌢}{T}}_{\mathrm{\infty }}$ . Moreover, the characteristics of HH chemical reactions are also explored for two species ${\stackrel{⌢}{A}}_1$ and ${\stackrel{⌢}{A}}_2$ . The homogeneous chemical process for the cubic auto catalyst is specified as

$${\stackrel{⌢}{A}}_1+2{\stackrel{⌢}{A}}_2\to 3{\stackrel{⌢}{A}}_2,\mathrm{rate}={\stackrel{⌢}{l}}_c{\stackrel{⌢}{C}}_{\alpha }{\stackrel{⌢}{C}}_{\beta }^2$$

(1)

Figure 1.

Schematic flow geometry.

The heterogeneous chemical reaction on the outer surface is

$${\stackrel{⌢}{A}}_1\to {\stackrel{⌢}{A}}_2,\mathrm{rate}={\stackrel{⌢}{l}}_s{\stackrel{⌢}{C}}_{\alpha }$$

(2)

In the above equations ${\stackrel{⌢}{C}}_{\alpha }$ , ${\stackrel{⌢}{C}}_{\beta }$ indicate the strengths of the chemical species ${\stackrel{⌢}{A}}_1$ and ${\stackrel{⌢}{A}}_2$ , and ${\stackrel{⌢}{l}}_c$ , ${\stackrel{⌢}{l}}_s$ are the rate constants, respectively. Governing equations for the present boundary layer time-dependent viscous liquid motion problem with an applied magnetic field can be written as^7,23

$$\mathrm{\nabla }\cdot \stackrel{⌢}{\mathbf{U}}=0$$

(3)

$${\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{\mathbf{U}}}{\mathrm{\partial }\stackrel{⌢}{t}}\mathbf{+}\left(\stackrel{⌢}{\mathbf{U}}\cdot \mathrm{\nabla }\right)\stackrel{⌢}{\mathbf{U}}=-{\displaystyle \frac{1}{\stackrel{⌢}{\rho }}\left(\mathrm{\nabla }\stackrel{⌢}{p}\right)+\stackrel{⌢}{\nu }\left({\mathrm{\nabla }}^2\stackrel{⌢}{\mathbf{U}}\right)+{\displaystyle \frac{1}{\stackrel{⌢}{\rho }}\left(\stackrel{⌢}{\mathbf{J}}\times \stackrel{⌢}{\mathbf{B}}\right)}}}$$

(4)

$${\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{T}}{\mathrm{\partial }\stackrel{⌢}{t}}\mathbf{+}\left(\stackrel{⌢}{\mathbf{U}}\cdot \mathrm{\nabla }\right)\stackrel{⌢}{T}={\displaystyle \frac{-1}{\left(\stackrel{⌢}{\rho }{\stackrel{⌢}{c}}_{\stackrel{⌢}{p}}\right)}\mathrm{\nabla }\cdot \left({\stackrel{⌢}{k}}_{\alpha }\left(\stackrel{⌢}{T}\right)\mathrm{\nabla }\stackrel{⌢}{T}\right)+{\displaystyle \frac{\stackrel{⌢}{Q}}{\left(\stackrel{⌢}{\rho }{\stackrel{⌢}{c}}_{\stackrel{⌢}{p}}\right)}\left(\stackrel{⌢}{T}-{\stackrel{⌢}{T}}_{\mathrm{\infty }}\right)}}}$$

(5)

$${\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{C}}_{\alpha }}{\mathrm{\partial }\stackrel{⌢}{t}}\mathbf{+}\left(\stackrel{⌢}{\mathbf{U}}\cdot \mathrm{\nabla }\right){\stackrel{⌢}{C}}_{\alpha }={\stackrel{⌢}{D}}_{\alpha }{\mathrm{\nabla }}^2{\stackrel{⌢}{C}}_{\alpha }-{\stackrel{⌢}{l}}_c{\stackrel{⌢}{C}}_{\alpha }{\stackrel{⌢}{C}}_{\beta }^2}$$

(6)

$${\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{C}}_{\beta }}{\mathrm{\partial }\stackrel{⌢}{t}}\mathbf{+}\left(\stackrel{⌢}{\mathbf{U}}\cdot \mathrm{\nabla }\right){\stackrel{⌢}{C}}_{\beta }={\stackrel{⌢}{D}}_{\beta }{\mathrm{\nabla }}^2{\stackrel{⌢}{C}}_{\beta }+{\stackrel{⌢}{l}}_c{\stackrel{⌢}{C}}_{\alpha }{\stackrel{⌢}{C}}_{\beta }^2}$$

(7)

Here, $\stackrel{⌢}{p}$ and $\stackrel{⌢}{T}$ are the pressure and temperature, $\stackrel{⌢}{\mathbf{U}}$ is the velocity vector, ${\stackrel{⌢}{D}}_{\beta }$ and ${\stackrel{⌢}{D}}_{\alpha }$ are the diffusion coefficients, ${\mathrm{\nabla }}^2$ is the Laplacian operator, $\stackrel{⌢}{Q}$ and $\stackrel{⌢}{\rho }$ are the volumetric rate of heat production parameter and density, $\stackrel{⌢}{\mathbf{J}}=\stackrel{⌢}{\sigma }\left(\stackrel{⌢}{\mathbf{U}}\times \stackrel{⌢}{\mathbf{B}}\right)$ and $\stackrel{⌢}{\nu }=\stackrel{⌢}{\mu }/\stackrel{⌢}{\rho }$ are the current density and kinematic viscosity, $\stackrel{⌢}{\mathbf{B}}$ and $\stackrel{⌢}{t}$ are the magnetic flux vector and time, ${\stackrel{⌢}{c}}_{\stackrel{⌢}{p}}$ is the specific heat capacity, and ${\stackrel{⌢}{k}}_{\alpha }\left(\stackrel{⌢}{T}\right)$ represent the thermal conductivity of the liquid which is stated as ¹⁰

$${\stackrel{⌢}{k}}_{\alpha }\left(\stackrel{⌢}{T}\right)={\stackrel{⌢}{k}}_{\mathrm{\infty }}\left(1+{\epsilon }_0\stackrel{⌢}{\theta }\right)$$

(8)

in which ${\epsilon }_0\left(={\gamma }_1\left({\stackrel{⌢}{T}}_w-{\stackrel{⌢}{T}}_{\mathrm{\infty }}\right)\right)$ is a small parameter and ${\stackrel{⌢}{k}}_{\mathrm{\infty }}$ is the ambient thermal conductivity of the liquid, respectively.

For the current flow problem, consider the velocity field in the following form:

$$\stackrel{⌢}{\mathbf{U}}=\left(\stackrel{⌢}{v}\left(\stackrel{⌢}{r},\stackrel{⌢}{t},\stackrel{⌢}{s}\right),\stackrel{⌢}{u}\left(\stackrel{⌢}{r},\stackrel{⌢}{t},\stackrel{⌢}{s}\right),0\right)$$

(9)

In the above equation $\stackrel{⌢}{u}$ and $\stackrel{⌢}{v}$ are the parts of velocity toward the $\stackrel{⌢}{r}$ and $\stackrel{⌢}{s}$ directions, respectively. By incorporating equation (9) along with boundary layer approximations, equations (3), (4), (5), (6), and (7) can be rewritten as^7,12

$${\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{v}}{\mathrm{\partial }\stackrel{⌢}{s}}+{\displaystyle \frac{1}{{\stackrel{⌢}{R}}_1}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\stackrel{⌢}{r}}\left\{\stackrel{⌢}{u}\left({\stackrel{⌢}{R}}_1+\stackrel{⌢}{r}\right)\right\}=0}}}$$

(10)

$${\displaystyle \frac{{\stackrel{⌢}{v}}^2\stackrel{⌢}{\rho }}{\left({\stackrel{⌢}{R}}_1+\stackrel{⌢}{r}\right)}-{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{p}}{\mathrm{\partial }\stackrel{⌢}{r}}=0}}$$

(11)

$$\begin{array}{rl}& {\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{v}}{\mathrm{\partial }\stackrel{⌢}{r}}\stackrel{⌢}{u}+\left({\displaystyle \frac{\stackrel{⌢}{u}}{{\stackrel{⌢}{R}}_1+\stackrel{⌢}{r}}}\right)\stackrel{⌢}{v}+{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{v}}{\mathrm{\partial }\stackrel{⌢}{t}}+\left({\displaystyle \frac{\stackrel{⌢}{v}{\stackrel{⌢}{R}}_1}{{\stackrel{⌢}{R}}_1+\stackrel{⌢}{r}}}\right){\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{v}}{\mathrm{\partial }\stackrel{⌢}{s}}}}}\\ & =-{\displaystyle \frac{{\stackrel{⌢}{R}}_1}{\left({\stackrel{⌢}{R}}_1+\stackrel{⌢}{r}\right)}{\displaystyle \frac{1}{\stackrel{⌢}{\rho }}{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{p}}{\mathrm{\partial }\stackrel{⌢}{s}}-{\displaystyle \frac{\stackrel{⌢}{\sigma }}{\stackrel{⌢}{\rho }}{B}_0^2\stackrel{⌢}{v}+\stackrel{⌢}{\nu }\left({\displaystyle \frac{{\mathrm{\partial }}^2\stackrel{⌢}{v}}{\mathrm{\partial }{\stackrel{⌢}{r}}^2}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\stackrel{⌢}{r}}\left({\displaystyle \frac{\stackrel{⌢}{v}}{{\stackrel{⌢}{R}}_1+\stackrel{⌢}{r}}}\right)}}\right)}}}}\end{array}$$

(12)

$$\begin{array}{rl}& \left({\displaystyle \frac{{\stackrel{⌢}{R}}_1\stackrel{⌢}{v}}{\stackrel{⌢}{r}+{\stackrel{⌢}{R}}_1}}\right){\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{T}}{\mathrm{\partial }\stackrel{⌢}{s}}+\stackrel{⌢}{u}{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{T}}{\mathrm{\partial }\stackrel{⌢}{r}}+{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{T}}{\mathrm{\partial }\stackrel{⌢}{t}}}}}\\ & =\left({\displaystyle \frac{1}{\left(\stackrel{⌢}{\rho }{\stackrel{⌢}{c}}_{\stackrel{⌢}{p}}\right)\left({\stackrel{⌢}{R}}_1+\stackrel{⌢}{r}\right)}}\right){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\stackrel{⌢}{r}}\left({\stackrel{⌢}{k}}_{\alpha }\left(\stackrel{⌢}{T}\right)\left(\stackrel{⌢}{r}+{\stackrel{⌢}{R}}_1\right){\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{T}}{\mathrm{\partial }\stackrel{⌢}{r}}}\right)+{\displaystyle \frac{\stackrel{⌢}{Q}}{\left(\stackrel{⌢}{\rho }{\stackrel{⌢}{c}}_{\stackrel{⌢}{p}}\right)}\left({\stackrel{⌢}{T}}_w-{\stackrel{⌢}{T}}_{\mathrm{\infty }}\right)}}\end{array}$$

(13)

$$\begin{array}{rl}& {\displaystyle \frac{{\stackrel{⌢}{R}}_1}{\left({\stackrel{⌢}{R}}_1+\stackrel{⌢}{r}\right)}{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{C}}_{\alpha }}{\mathrm{\partial }\stackrel{⌢}{s}}\stackrel{⌢}{v}+{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{C}}_{\alpha }}{\mathrm{\partial }\stackrel{⌢}{t}}+{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{C}}_{\alpha }}{\mathrm{\partial }\stackrel{⌢}{r}}\stackrel{⌢}{u}={\stackrel{⌢}{D}}_{\alpha }\left({\displaystyle \frac{{\mathrm{\partial }}^2{\stackrel{⌢}{C}}_{\alpha }}{\mathrm{\partial }{\stackrel{⌢}{r}}^2}+{\displaystyle \frac{1}{\left(\stackrel{⌢}{r}+{\stackrel{⌢}{R}}_1\right)}{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{C}}_{\alpha }}{\mathrm{\partial }\stackrel{⌢}{r}}}}}\right)-{\stackrel{⌢}{l}}_c{\stackrel{⌢}{C}}_{\alpha }{\stackrel{⌢}{C}}_{\beta }^2}}}}\end{array}$$

(14)

$${\displaystyle \frac{{\stackrel{⌢}{R}}_1\stackrel{⌢}{v}}{\left({\stackrel{⌢}{R}}_1+\stackrel{⌢}{r}\right)}{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{C}}_{\beta }}{\mathrm{\partial }\stackrel{⌢}{s}}+{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{C}}_{\beta }}{\mathrm{\partial }\stackrel{⌢}{t}}+{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{C}}_{\beta }}{\mathrm{\partial }\stackrel{⌢}{r}}\stackrel{⌢}{u}={\stackrel{⌢}{D}}_{\beta }\left({\displaystyle \frac{{\mathrm{\partial }}^2{\stackrel{⌢}{C}}_{\beta }}{\mathrm{\partial }{\stackrel{⌢}{r}}^2}+{\displaystyle \frac{1}{\left({\stackrel{⌢}{R}}_1+\stackrel{⌢}{r}\right)}{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{C}}_{\beta }}{\mathrm{\partial }\stackrel{⌢}{r}}}}}\right)+{\stackrel{⌢}{l}}_c{\stackrel{⌢}{C}}_{\alpha }{\stackrel{⌢}{C}}_{\beta }^2}}}}$$

(15)

with boundary conditions

$$\begin{array}{c}\stackrel{⌢}{v}=\stackrel{⌢}{a}\stackrel{⌢}{s}\sin \stackrel{⌢}{\omega }\stackrel{⌢}{t},\stackrel{⌢}{T}={\stackrel{⌢}{T}}_{\mathrm{w}},\stackrel{⌢}{u}=0,{\stackrel{⌢}{D}}_{\alpha }{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{C}}_{\alpha }}{\mathrm{\partial }\stackrel{⌢}{r}}={\stackrel{⌢}{l}}_s{\stackrel{⌢}{C}}_{\alpha },{\stackrel{⌢}{D}}_{\beta }{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{C}}_{\alpha }}{\mathrm{\partial }\stackrel{⌢}{r}}=-{\stackrel{⌢}{l}}_s{\stackrel{⌢}{C}}_{\alpha }\mathrm{at}\stackrel{⌢}{r}=0,}}\\ {\stackrel{⌢}{C}}_{\alpha }\to {\stackrel{⌢}{C}}_{\mathrm{\infty }},{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{v}}{\mathrm{\partial }\stackrel{⌢}{r}}\to 0,{\stackrel{⌢}{C}}_{\beta }\to 0,\stackrel{⌢}{T}\to {\stackrel{⌢}{T}}_{\mathrm{\infty }},\stackrel{⌢}{v}\to 0,\mathrm{at}\stackrel{⌢}{r}\to \mathrm{\infty }}\end{array}\}$$

(16)

Employing the following transformations ¹²

$$\begin{array}{rl}& \sqrt{{\displaystyle \frac{\stackrel{⌢}{a}}{\stackrel{⌢}{\nu }}}}\stackrel{⌢}{r}=\xi ,{\stackrel{⌢}{f}}_{\xi }(\xi ,\tau )\stackrel{⌢}{a}\stackrel{⌢}{s}=\stackrel{⌢}{v},{\stackrel{⌢}{C}}_{\alpha }={\stackrel{⌢}{C}}_{\mathrm{\infty }}\stackrel{⌢}{\varphi }(\xi ,\tau ),\stackrel{⌢}{T}-{\stackrel{⌢}{T}}_{\mathrm{\infty }}=\left({\stackrel{⌢}{T}}_w-{\stackrel{⌢}{T}}_{\mathrm{\infty }}\right)\stackrel{⌢}{\theta }(\xi ,\tau ),\\ & {\displaystyle \frac{-{\stackrel{⌢}{R}}_1}{{\stackrel{⌢}{R}}_1+\stackrel{⌢}{r}}\sqrt{\stackrel{⌢}{a}\stackrel{⌢}{\nu }}\stackrel{⌢}{f}(\xi ,\tau )=\stackrel{⌢}{u},{\stackrel{⌢}{a}}^2{\stackrel{⌢}{s}}^2\stackrel{⌢}{\rho }\stackrel{⌢}{P}(\xi ,\tau )=\stackrel{⌢}{p},\stackrel{⌢}{\omega }\stackrel{⌢}{t}=\tau ,{\stackrel{⌢}{C}}_{\beta }={\stackrel{⌢}{C}}_{\mathrm{\infty }}\stackrel{⌢}{\phi }(\xi ,\tau )}\end{array}$$

(17)

and equation (8), the continuity equation (10) is identically verified, and the rest of equations (11)–(16) takes the following non-dimensional form:

$${\stackrel{⌢}{P}}_{\xi }={\displaystyle \frac{{\stackrel{⌢}{f}}_{\xi }^2}{\delta +\xi }}$$

(18)

$$\begin{array}{rl}\stackrel{⌢}{P}(\xi ,\tau )=& {\displaystyle \frac{\delta +\xi }{2\delta }\left({\stackrel{⌢}{f}}_{\xi \xi \xi }-M^2{\stackrel{⌢}{f}}_{\xi }+{\displaystyle \frac{{\stackrel{⌢}{f}}_{\xi \xi }}{(\delta +\xi )}-{\displaystyle \frac{{\stackrel{⌢}{f}}_{\xi }}{{(\delta +\xi )}^2}-S{\stackrel{⌢}{f}}_{\tau \xi }}}\right)}\\ & +{\displaystyle \frac{\delta +\xi }{2}\left(-{\displaystyle \frac{{\stackrel{⌢}{f}}_{\xi }^2}{(\delta +\xi )}+\stackrel{⌢}{f}{\displaystyle \frac{{\stackrel{⌢}{f}}_{\xi }}{{(\delta +\xi )}^2}+\left({\displaystyle \frac{\stackrel{⌢}{f}}{(\delta +\xi )}}\right){\stackrel{⌢}{f}}_{\xi \xi }}}\right)}\end{array}$$

(19)

$$\left({\epsilon }_0\stackrel{⌢}{\theta }+1\right)\left({\stackrel{⌢}{\theta }}_{\xi \xi }+{\displaystyle \frac{{\stackrel{⌢}{\theta }}_{\xi }}{(\delta +\xi )}}\right)+Pr\left(\gamma \stackrel{⌢}{\theta }+{\displaystyle \frac{\stackrel{⌢}{f}\stackrel{⌢}{\mathrm{K}}{\stackrel{⌢}{\theta }}_{\xi }}{(\delta +\xi )}-S{\stackrel{⌢}{\theta }}_{\tau }}\right)+{\epsilon }_0{\stackrel{⌢}{\theta }}_{\xi }^2=0$$

(20)

$${\displaystyle \frac{1}{Sc}\left({\stackrel{⌢}{\varphi }}_{\xi \xi }+{\displaystyle \frac{{\stackrel{⌢}{\varphi }}_{\xi }}{(\delta +\xi )}}\right)+{\displaystyle \frac{\stackrel{⌢}{\mathrm{K}}}{(\delta +\xi )}\stackrel{⌢}{f}{\stackrel{⌢}{\varphi }}_{\xi }-S{\stackrel{⌢}{\varphi }}_{\tau }-{\mathrm{\Gamma }}_1\stackrel{⌢}{\varphi }{\stackrel{⌢}{\phi }}^2=0}}$$

(21)

$${\displaystyle \frac{{\epsilon }_1}{Sc}\left({\stackrel{⌢}{\phi }}_{\xi \xi }+{\displaystyle \frac{{\stackrel{⌢}{\phi }}_{\xi }}{(\delta +\xi )}}\right)+{\displaystyle \frac{\stackrel{⌢}{\mathrm{K}}}{(\delta +\xi )}\stackrel{⌢}{f}{\stackrel{⌢}{\phi }}_{\xi }-S{\stackrel{⌢}{\phi }}_{\tau }-{\mathrm{\Gamma }}_1\stackrel{⌢}{\varphi }{\stackrel{⌢}{\phi }}^2=0}}$$

(22)

$$\begin{array}{rl}& \stackrel{⌢}{f}(0,\tau )=0,{\stackrel{⌢}{\varphi }}_{\xi }(0,\tau )={\mathrm{\Gamma }}_2\stackrel{⌢}{\varphi }(0,\tau ),\stackrel{⌢}{\theta }(0,\tau )=1,{\stackrel{⌢}{f}}_{\xi }(0,\tau )=\sin \tau ,\\ & \delta {\stackrel{⌢}{\phi }}_{\xi }(0,\overline{\tau })=-{\mathrm{\Gamma }}_2{\stackrel{⌢}{\varphi }}_{}(0,\tau ),{\stackrel{⌢}{f}}_{\xi }(\mathrm{\infty },\tau )=0,{\stackrel{⌢}{f}}_{\xi \xi }(\mathrm{\infty },\tau )=0,\\ & \stackrel{⌢}{\theta }(\mathrm{\infty },\tau )=0,\stackrel{⌢}{\varphi }(\mathrm{\infty },\tau )=1,\stackrel{⌢}{\phi }(\mathrm{\infty },\tau )=0\end{array}$$

(23)

In the above equations $M\left(=\sqrt{\stackrel{⌢}{\sigma }{B}_0^2/\stackrel{⌢}{\rho }\stackrel{⌢}{a}}\right)$ is the magnetic parameter, $\delta \left(=\sqrt{\stackrel{⌢}{a}/\stackrel{⌢}{\nu }}{\stackrel{⌢}{R}}_1\right)$ is the radius of curvature, $Pr\left(=\stackrel{⌢}{\nu }/\stackrel{⌢}{\alpha }\right)$ and $S\left(=\stackrel{⌢}{\omega }/\stackrel{⌢}{a}\right)$ are the Prandtl number and the fraction of sheet oscillating frequency to stretched rate of the sheet, ${\epsilon }_1\left(={\stackrel{⌢}{D}}_{\beta }/{\stackrel{⌢}{D}}_{\alpha }\right)$ is the proportion of diffusion coefficient, ${\mathrm{\Gamma }}_1\left(={\stackrel{⌢}{l}}_c{\stackrel{⌢}{C}}_{\mathrm{\infty }}^2/\stackrel{⌢}{a}\right)$ is the homogeneous reaction strength, $Sc\left(=\stackrel{⌢}{\nu }/{\stackrel{⌢}{D}}_{\alpha }\right)$ is the Schmidt number, $\gamma \left(=\stackrel{⌢}{Q}/\stackrel{⌢}{\rho }{\stackrel{⌢}{c}}_{\stackrel{⌢}{p}}\stackrel{⌢}{a}\right)$ is the heat production parameter, and ${\mathrm{\Gamma }}_2\left(={\stackrel{⌢}{l}}_s/{\stackrel{⌢}{D}}_{\alpha }\sqrt{\stackrel{⌢}{a}/\stackrel{⌢}{\nu }}\right)$ is the strength rate of heterogeneous reaction parameter, respectively.

After omission of the pressure $\stackrel{⌢}{P}$ term between equations (18) and (19), we attain the fluid velocity equation as

$$\begin{array}{rl}& {\stackrel{⌢}{f}}_{\xi \xi \xi \xi }+{\displaystyle \frac{2{\stackrel{⌢}{f}}_{\xi \xi \xi }}{\delta +\xi }-{\displaystyle \frac{{\stackrel{⌢}{f}}_{\xi \xi }}{{(\delta +\xi )}^2}-{\displaystyle \frac{{\stackrel{⌢}{f}}_{\xi }\delta {\stackrel{⌢}{f}}_{\xi \xi }}{(\delta +\xi )}+{\displaystyle \frac{{\stackrel{⌢}{f}}_{\xi }}{{(\delta +\xi )}^3}-{\displaystyle \frac{\stackrel{⌢}{f}\delta {\stackrel{⌢}{f}}_{\xi }}{{(\delta +\xi )}^3}-{\displaystyle \frac{S{\stackrel{⌢}{f}}_{\xi \tau }}{\delta +\xi }+{\displaystyle \frac{{\stackrel{⌢}{f}}_{\xi \xi }\delta \stackrel{⌢}{f}}{{(\delta +\xi )}^2}}}}}}}}\\ & +S{\stackrel{⌢}{f}}_{\xi \xi \tau }-{\displaystyle \frac{\delta {\stackrel{⌢}{f}}_{\xi }^2}{{(\xi +\delta )}^2}-\left({\displaystyle \frac{{\stackrel{⌢}{f}}_{\xi }}{\delta +\xi }+{\stackrel{⌢}{f}}_{\xi \xi }}\right)M^2+{\displaystyle \frac{{\stackrel{⌢}{f}}_{\xi \xi \xi }\delta \stackrel{⌢}{f}}{(\delta +\xi )}=0}}\end{array}$$

(24)

Here, it is presumed that ${\stackrel{⌢}{D}}_{\alpha }={\stackrel{⌢}{D}}_{\beta }$ , i.e. ${\epsilon }_1=1$ , and by letting

$$1-\stackrel{⌢}{\varphi }(\xi ,\tau )=\stackrel{⌢}{\phi }(\xi ,\tau )$$

(25)

Now equations (21) and (22) yield

$${\stackrel{⌢}{\varphi }}_{\xi \xi }+{\displaystyle \frac{{\stackrel{⌢}{\varphi }}_{\xi }}{(\xi +\delta )}+Sc\left({\displaystyle \frac{\delta }{(\delta +\xi )}\overline{f}{\stackrel{⌢}{\varphi }}_{\xi }-S{\stackrel{⌢}{\varphi }}_{\tau }}\right)-{\mathrm{\Gamma }}_{1}Sc\stackrel{⌢}{\varphi }{\left(1-\stackrel{⌢}{\varphi }\right)}^2=0}$$

(26)

subject to reduced boundary conditions:

$$\begin{array}{rl}& {\stackrel{⌢}{\varphi }}_{\xi }(0,\overline{\tau })={\mathrm{\Gamma }}_2\stackrel{⌢}{\varphi }(0,\tau ),{\stackrel{⌢}{f}}_{\xi }(0,\tau )=\sin \tau ,\stackrel{⌢}{\theta }(0,\tau )=1,\stackrel{⌢}{f}(0,\tau )=0,\\ & \stackrel{⌢}{\theta }(\mathrm{\infty },\tau )=0,{\stackrel{⌢}{f}}_{\xi }(\mathrm{\infty },\tau )=0,\stackrel{⌢}{\varphi }(\mathrm{\infty },\tau )=1,{\stackrel{⌢}{f}}_{\xi \xi }(\mathrm{\infty },\tau )=0\end{array}$$

(27)

The drag wall force and the rate of heat transportation along the curvy surface is defined as

$${\stackrel{⌢}{C}}_{\stackrel{⌢}{f}}={\displaystyle \frac{{\stackrel{⌢}{\tau }}_{\stackrel{⌢}{r}\stackrel{⌢}{s}}}{\stackrel{⌢}{\rho }{\stackrel{⌢}{u}}_{\mathrm{w}}^2},\stackrel{⌢}{N}{\stackrel{⌢}{u}}_{\stackrel{⌢}{s}}=\left({\displaystyle \frac{\stackrel{⌢}{s}}{\left({\stackrel{⌢}{T}}_{\mathrm{w}}-{\stackrel{⌢}{T}}_{\mathrm{\infty }}\right){\stackrel{⌢}{k}}_{\mathrm{\infty }}}}\right){\stackrel{⌢}{q}}_{\stackrel{⌢}{w}}}$$

(28)

where ${\stackrel{⌢}{\tau }}_{\stackrel{⌢}{r}\stackrel{⌢}{s}}$ represents the shear stress and ${\stackrel{⌢}{q}}_{\stackrel{⌢}{w}}$ denotes the heat flux and are depicted as

$${\stackrel{⌢}{\tau }}_{\stackrel{⌢}{r}\stackrel{⌢}{s}}=\stackrel{⌢}{\mu }{\left({\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{v}}{\mathrm{\partial }\stackrel{⌢}{r}}-{\displaystyle \frac{\stackrel{⌢}{v}}{\stackrel{⌢}{r}+{\stackrel{⌢}{R}}_1}}}\right)|}_{\stackrel{⌢}{r}=0},{\stackrel{⌢}{q}}_{\stackrel{⌢}{w}}={-\left({\stackrel{⌢}{k}}_{\alpha }\left(\stackrel{⌢}{T}\right){\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{T}}{\mathrm{\partial }\stackrel{⌢}{r}}}\right)|}_{\stackrel{⌢}{r}=0}$$

(29)

By employing equation (17) in equations (28) and (29), we attain

$${\stackrel{⌢}{C}}_{\stackrel{⌢}{f}}R{e}_{\stackrel{⌢}{s}}^{1/2}={\displaystyle \frac{1}{\delta }\left[\delta {\stackrel{⌢}{f}}_{\xi \xi }(0,\tau )-{\stackrel{⌢}{f}}_{\stackrel{⌢}{\xi }}(0,\tau )\right]}$$

(30)

$$\stackrel{⌢}{N}{\stackrel{⌢}{u}}_{\stackrel{⌢}{s}}R{e}_{\stackrel{⌢}{s}}^{-1/2}=-{\stackrel{⌢}{\theta }}_{\xi }(0,\tau )$$

(31)

here, ${\mathrm{Re}}_{\stackrel{⌢}{s}}^{1/2}=\stackrel{⌢}{a}{\stackrel{⌢}{s}}^2/\stackrel{⌢}{\nu }$ characterizes the Reynolds number.

HAM solution

In this section, we proposed HAM to solve the governing equations (20), (24), (26), and (27) by incorporating the auxiliary linear operators and initial guesses indicated below:

$$\begin{array}{rl}& {\stackrel{⌢}{\theta }}_0(\tau ,\xi )=\exp (-\xi ),{\stackrel{⌢}{f}}_0(\xi ,\tau )=\sin \tau -\sin \tau \exp (-\xi )\\ & {\stackrel{⌢}{\varphi }}_0(\xi ,\tau )=\left(1-{\displaystyle \frac{1}{2}\exp (-{\mathrm{\Gamma }}_2\xi )}\right)\end{array}$$

(32)

$${\stackrel{⌢}{N}}_{\stackrel{⌢}{\theta }}\left(\stackrel{⌢}{\theta }\right)={\displaystyle \frac{{\mathrm{\partial }}^2\stackrel{⌢}{\theta }}{\mathrm{\partial }{\xi }^2}-\stackrel{⌢}{\theta },{\stackrel{⌢}{N}}_{\stackrel{⌢}{f}}\left(\stackrel{⌢}{f}\right)={\displaystyle \frac{{\mathrm{\partial }}^4\stackrel{⌢}{f}}{\mathrm{\partial }{\xi }^4}-{\displaystyle \frac{{\mathrm{\partial }}^3\stackrel{⌢}{f}}{\mathrm{\partial }{\xi }^3}-{\displaystyle \frac{{\mathrm{\partial }}^2\stackrel{⌢}{f}}{\mathrm{\partial }{\xi }^2}+{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{f}}{\mathrm{\partial }\xi },{\stackrel{⌢}{N}}_{\stackrel{⌢}{\varphi }}\left(\stackrel{⌢}{\varphi }\right)={\displaystyle \frac{{\mathrm{\partial }}^2\stackrel{⌢}{\varphi }}{\mathrm{\partial }{\xi }^2}-\stackrel{⌢}{\varphi }}}}}}}$$

(33)

with

$$\begin{array}{rl}& {\stackrel{⌢}{N}}_{\stackrel{⌢}{f}}[(c_3\xi +c_2)\exp (\xi )+c_4\exp (-\xi )+c_1]=0\\ & {\stackrel{⌢}{N}}_{\stackrel{⌢}{\theta }}[c_5\exp (\xi )+c_6\exp (-\xi )]=0\\ & {\stackrel{⌢}{N}}_{\stackrel{⌢}{\varphi }}[c_7\exp (\xi )+c_8\exp (-\xi )]=0\end{array}$$

(34)

Here, $c_i,$ $(i=1\text{–}8)$ are the arbitrary constants. Let $x\in [0,1]$ characterizes an embedding variable with ${\hslash }_{\stackrel{⌢}{f}},{\hslash }_{\stackrel{⌢}{\theta }}$ , and ${\hslash }_{\stackrel{⌢}{\varphi }}$ are non-zero auxiliary variables. One can construct the zeroth-order deformation expressions as

$$(1-x){\stackrel{⌢}{N}}_{\stackrel{⌢}{f}}\left(\stackrel{⌢}{f}(\tau ,\xi ;x)-{\stackrel{⌢}{f}}_0(\tau ,\xi )\right)=x{\mathrm{K}}_{\stackrel{⌢}{f}}\left(\stackrel{⌢}{f}(\tau ,\xi ;x)\right){\hslash }_{\stackrel{⌢}{f}}$$

(35)

$$(1-x){\stackrel{⌢}{N}}_{\stackrel{⌢}{\theta }}\left(\stackrel{⌢}{\theta }(\tau ,\xi ;x)-{\stackrel{⌢}{\theta }}_0(\tau ,\xi )\right)=x{\mathrm{K}}_{\stackrel{⌢}{\theta }}\left(\stackrel{⌢}{\theta }(\tau ,\xi ;x)\right){\hslash }_{\stackrel{⌢}{\theta }}$$

(36)

$$(1-x){\stackrel{⌢}{N}}_{\stackrel{⌢}{\varphi }}\left(\stackrel{⌢}{\varphi }(\tau ,\xi ;x)-{\stackrel{⌢}{\varphi }}_0(\tau ,\xi )\right)=x{\mathrm{K}}_{\stackrel{⌢}{\varphi }}\left(\stackrel{⌢}{\varphi }(\tau ,\xi ;x)\right){\hslash }_{\stackrel{⌢}{\varphi }}$$

(37)

subject to boundary conditions:

$$\begin{array}{c}\begin{array}{c}{{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{f}(\tau ,\xi ;x)}{\mathrm{\partial }\xi }}|}_{\xi =0}=\sin \tau ,{{\mathrm{\Gamma }}_2\stackrel{⌢}{\varphi }(\tau ,\xi ;x)|}_{\xi =0}={{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{\varphi }(\tau ,\xi ;x)}{\mathrm{\partial }\xi }}|}_{\xi =0},\\ {\stackrel{⌢}{f}(\tau ,\xi ;x)|}_{\xi =0}=0,{\stackrel{⌢}{\theta }(\tau ,\xi ;x)|}_{\xi =0}=1,\end{array}\}\\ {{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{f}(\tau ,\xi ;x)}{\mathrm{\partial }\xi }}|}_{\xi =\mathrm{\infty }}={\stackrel{⌢}{\theta }(\tau ,\xi ;x)|}_{\xi =\mathrm{\infty }}={{\displaystyle \frac{{\mathrm{\partial }}^2\stackrel{⌢}{f}(\tau ,\xi ;x)}{\mathrm{\partial }{\xi }^2}}|}_{\xi =\mathrm{\infty }}=0,{\stackrel{⌢}{\varphi }(\tau ,\xi ;x)|}_{\xi =\mathrm{\infty }}=1\end{array}\}$$

(38)

The associated non-linear operators are designated as

$$\begin{array}{rl}{\mathrm{K}}_{\stackrel{⌢}{f}}\stackrel{⌢}{f}(\xi ,\tau ;x)=& {\displaystyle \frac{{\mathrm{\partial }}^4\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }{\xi }^4}+{\displaystyle \frac{2}{\xi +\delta }{\displaystyle \frac{{\mathrm{\partial }}^3\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }{\xi }^3}}}}\\ & -{\displaystyle \frac{1}{{(\xi +\delta )}^2}{\displaystyle \frac{{\mathrm{\partial }}^2\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }{\xi }^2}+{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }\xi {(\xi +\delta )}^3}}}}\\ & -{\displaystyle \frac{\delta }{(\xi +\delta )}{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }\xi }{\displaystyle \frac{{\mathrm{\partial }}^2\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }{\xi }^2}+{\displaystyle \frac{\delta \stackrel{⌢}{f}(\xi ,\tau ;x)}{(\xi +\delta )}{\displaystyle \frac{{\mathrm{\partial }}^3\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }{\xi }^3}}}}}}\\ & -{\displaystyle \frac{\delta \stackrel{⌢}{f}(\xi ,\tau ;x)}{{(\xi +\delta )}^3}{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }\xi }-S\left({\displaystyle \frac{{\mathrm{\partial }}^3\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }{\xi }^2\mathrm{\partial }\tau }+{\displaystyle \frac{{\mathrm{\partial }}^2\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }\tau \mathrm{\partial }\xi (\xi +\delta )}}}\right)}}\\ & +{\displaystyle \frac{\delta \stackrel{⌢}{f}(\xi ,\tau ;x)}{{(\xi +\delta )}^2}{\displaystyle \frac{{\mathrm{\partial }}^2\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }{\xi }^2}-{\displaystyle \frac{\delta }{{(\xi +\delta )}^2}{\left({\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }\xi }}\right)}^2}}}\\ & -M^2\left({\displaystyle \frac{{\mathrm{\partial }}^2\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }{\xi }^2}+{\displaystyle \frac{1}{(\xi +\delta )}{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }\xi }}}}\right)\end{array}$$

(39)

$$\begin{array}{rl}{\mathrm{K}}_{\stackrel{⌢}{\theta }}\left[\stackrel{⌢}{\theta }(\tau ,\xi ;x),\stackrel{⌢}{f}(\tau ,\xi ;x)\right]=& {\displaystyle \frac{{\mathrm{\partial }}^2\stackrel{⌢}{\theta }(\tau ,\xi ;x)}{\mathrm{\partial }{\xi }^2}\left({\epsilon }_0\stackrel{⌢}{\theta }(\tau ,\xi ;x)+1\right)}\\ & +{\displaystyle \frac{\left({\epsilon }_0\stackrel{⌢}{\theta }(\tau ,\xi ;x)+1\right)}{(\delta +\xi )}{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{\theta }(\tau ,\xi ;x)}{\mathrm{\partial }\xi }}}\\ & +{\displaystyle \frac{\delta \stackrel{⌢}{f}(\tau ,\xi ;x)}{(\delta +\xi )}{\displaystyle \frac{Pr\mathrm{\partial }\stackrel{⌢}{\theta }(\tau ,\xi ;x)}{\mathrm{\partial }\xi }-PrS{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{\theta }(\tau ,\xi ;x)}{\mathrm{\partial }\tau }}}}\\ & +\gamma Pr\stackrel{⌢}{\theta }(\tau ,\xi ;x)+{\epsilon }_0{\left({\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{\theta }(\tau ,\xi ;x)}{\mathrm{\partial }\xi }}\right)}^2\end{array}$$

(40)

$$\begin{array}{rl}{\mathrm{K}}_{\stackrel{⌢}{\theta }}\left[\stackrel{⌢}{\varphi }(\tau ,\xi ;x),\stackrel{⌢}{f}(\tau ,\xi ;x)\right]=& {\displaystyle \frac{{\mathrm{\partial }}^2\stackrel{⌢}{\varphi }(\tau ,\xi ;x)}{\mathrm{\partial }{\xi }^2}+{\displaystyle \frac{1}{(\delta +\xi )}{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{\varphi }(\tau ,\xi ;x)}{\mathrm{\partial }\xi }-ScS{\displaystyle \frac{\mathrm{\partial }\stackrel{⌢}{\varphi }(\tau ,\xi ;x)}{\mathrm{\partial }\tau }}}}}\\ & +{\displaystyle \frac{\delta \stackrel{⌢}{f}(\tau ,\xi ;x)}{(\delta +\xi )}{\displaystyle \frac{Sc\mathrm{\partial }\stackrel{⌢}{\varphi }(\tau ,\xi ;x)}{\mathrm{\partial }\xi }}}\\ & -Sc{\mathrm{\Gamma }}_1\stackrel{⌢}{\varphi }(\tau ,\xi ;x){\left(1-\stackrel{⌢}{\varphi }(\tau ,\xi ;x)\right)}^2\end{array}$$

(41)

put $x=0$ and $x=1$ in equations (35), (36), and (37), the following results are accomplished:

$$\stackrel{⌢}{f}(\tau ,\xi ;1)=\stackrel{⌢}{f}(\tau ,\xi ),\stackrel{⌢}{f}(\tau ,\xi ;1)={\stackrel{⌢}{f}}_0(\tau ,\xi )$$

(42)

$$\stackrel{⌢}{\theta }(\tau ,\xi ;1)=\stackrel{⌢}{\theta }(\tau ,\xi ),\stackrel{⌢}{\theta }(\tau ,\xi ;1)={\stackrel{⌢}{\theta }}_0(\tau ,\xi )$$

(43)

$$\stackrel{⌢}{\varphi }(\tau ,\xi ;1)=\stackrel{⌢}{\varphi }(\tau ,\xi ),\stackrel{⌢}{\varphi }(\tau ,\xi ;1)={\stackrel{⌢}{\varphi }}_0(\tau ,\xi )$$

(44)

Making use of the Taylor series expansion in equations (42), (43), and (44), the following results are accomplished:

$$\stackrel{⌢}{f}(\xi ,\tau ;x)={\stackrel{⌢}{f}}_0(\xi ,\tau )+\sum _{r=1}^{\mathrm{\infty }}{\stackrel{⌢}{f}}_r(\xi ,\tau )x^r,{\stackrel{⌢}{f}}_r(\xi ,\tau )={\displaystyle \frac{1}{r!}{{\displaystyle \frac{{\mathrm{\partial }}^r\stackrel{⌢}{f}(\xi ,\tau ;x)}{\mathrm{\partial }{x}^r}}|}_{x=0}}$$

(45)

$$\stackrel{⌢}{\theta }(\xi ,\tau ;x)={\stackrel{⌢}{\theta }}_0(\xi ,\tau )+\sum _{r=1}^{\mathrm{\infty }}{\stackrel{⌢}{\theta }}_r(\xi ,\tau )x^r,{\stackrel{⌢}{\theta }}_r(\xi ,\tau )={\displaystyle \frac{1}{r!}{{\displaystyle \frac{{\mathrm{\partial }}^r\stackrel{⌢}{\theta }(\xi ,\tau ;x)}{\mathrm{\partial }{x}^r}}|}_{x=0}}$$

(46)

$$\stackrel{⌢}{\varphi }(\xi ,\tau ;x)={\stackrel{⌢}{\varphi }}_0(\xi ,\tau )+\sum _{r=1}^{\mathrm{\infty }}{\stackrel{⌢}{\varphi }}_r(\xi ,\tau )x^r,{\stackrel{⌢}{\varphi }}_r(\xi ,\tau )={\displaystyle \frac{1}{r!}{{\displaystyle \frac{{\mathrm{\partial }}^r\stackrel{⌢}{\varphi }(\xi ,\tau ;x)}{\mathrm{\partial }{x}^r}}|}_{x=0}}$$

(47)

To ensure the convergence of equations (45), (46), and (47) at $x=1$ , the variables ${\hslash }_{\stackrel{⌢}{f}},{\hslash }_{\stackrel{⌢}{\theta }}$ , and ${\hslash }_{\stackrel{⌢}{\varphi }}$ must be carefully adjusted. At $x=1$ , equations (45), (46), and (47) are transmuted as

$$\stackrel{⌢}{f}(\xi ,\tau ;1)={\stackrel{⌢}{f}}_0(\tau ,\xi )+\sum _{r=1}^{\mathrm{\infty }}{\stackrel{⌢}{f}}_r(\xi ,\tau )$$

(48)

$$\stackrel{⌢}{\theta }(\xi ,\tau ;1)={\stackrel{⌢}{\theta }}_0(\tau ,\xi )+\sum _{r=1}^{\mathrm{\infty }}{\stackrel{⌢}{\theta }}_r(\xi ,\tau )$$

(49)

$$\stackrel{⌢}{\varphi }(\xi ,\tau ;1)={\stackrel{⌢}{\varphi }}_0(\xi ,\tau )+\sum _{r=1}^{\mathrm{\infty }}{\stackrel{⌢}{\varphi }}_r(\xi ,\tau )$$

(50)

Differentiate equations (35), (36), (37), and (38) by r times to embed parameter x, then put $x=0$ and finally divided by $r!$ , the following results are obtained:

$${\stackrel{⌢}{N}}_{\stackrel{⌢}{f}}\left[{\stackrel{⌢}{f}}_r(\xi ,\tau )-{\lambda }_r{\stackrel{⌢}{f}}_{r-1}(\xi ,\tau )\right]={\hslash }_{\stackrel{⌢}{f}}{R}_r^{\stackrel{⌢}{f}}(\xi ,\tau )$$

(51)

$${\stackrel{⌢}{N}}_{\stackrel{⌢}{\theta }}\left[{\stackrel{⌢}{\theta }}_r(\xi ,\tau )-{\lambda }_r{\stackrel{⌢}{\theta }}_{r-1}(\xi ,\tau )\right]={\hslash }_{\stackrel{⌢}{\theta }}{R}_r^{\stackrel{⌢}{\theta }}(\xi ,\tau )$$

(52)

$${\stackrel{⌢}{N}}_{\stackrel{⌢}{\varphi }}\left[{\stackrel{⌢}{\varphi }}_r(\xi ,\tau )-{\lambda }_r{\stackrel{⌢}{\varphi }}_{r-1}(\xi ,\tau )\right]={\hslash }_{\stackrel{⌢}{\varphi }}{R}_r^{\stackrel{⌢}{\varphi }}(\xi ,\tau )$$

(53)

$$\begin{array}{c}{{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{f}}_r(\tau ,\xi ;x)}{\mathrm{\partial }\xi }}|}_{\xi =0}={{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{\varphi }}_r(\tau ,\xi ;x)}{\mathrm{\partial }\xi }-{\mathrm{\Gamma }}_2{\stackrel{⌢}{\varphi }}_r(\tau ,\xi ;x)}|}_{\xi =0}={{\stackrel{⌢}{f}}_r(\tau ,\xi ;x)|}_{\xi =0}={{\stackrel{⌢}{\theta }}_r(\tau ,\xi ;x)|}_{\xi =0}=0,\\ {{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{f}}_r(\tau ,\xi ;x)}{\mathrm{\partial }\xi }}|}_{\xi =\mathrm{\infty }}={\stackrel{⌢}{\theta }(\tau ,\xi ;x)|}_{\xi =\mathrm{\infty }}={{\displaystyle \frac{{\mathrm{\partial }}^2{\stackrel{⌢}{f}}_r(\tau ,\xi ;x)}{\mathrm{\partial }{\xi }^2}}|}_{\xi =\mathrm{\infty }}={{\stackrel{⌢}{\varphi }}_r(\tau ,\xi ;x)|}_{\xi =\mathrm{\infty }}=0\end{array}\}$$

(54)

where

$$\begin{array}{rl}{R}_r^{\stackrel{⌢}{f}}=& {\displaystyle \frac{{\mathrm{\partial }}^4{\stackrel{⌢}{f}}_{r-1}}{\mathrm{\partial }{\xi }^4}+{\displaystyle \frac{2}{(\xi +\delta )}{\displaystyle \frac{{\mathrm{\partial }}^3{\stackrel{⌢}{f}}_{r-1}}{\mathrm{\partial }{\xi }^3}-{\displaystyle \frac{{\mathrm{\partial }}^2{\stackrel{⌢}{f}}_{r-1}}{\mathrm{\partial }{\xi }^2}{\displaystyle \frac{1}{{(\xi +\delta )}^2}+{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{f}}_{r-1}}{\mathrm{\partial }\xi }{\displaystyle \frac{1}{{(\xi +\delta )}^3}}}}}}}}\\ & -M^2\left({\displaystyle \frac{1}{(\xi +\delta )}{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{f}}_{r-1}}{\mathrm{\partial }\xi }+{\displaystyle \frac{{\mathrm{\partial }}^2{\stackrel{⌢}{f}}_{r-1}}{\mathrm{\partial }{\xi }^2}}}}\right)\\ & +\sum _{i=0}^{r-1}\left(\begin{array}{c}{\displaystyle \frac{\delta }{(\xi +\delta )}{\stackrel{⌢}{f}}_{r-i-1}{\displaystyle \frac{{\mathrm{\partial }}^3{\stackrel{⌢}{f}}_i}{\mathrm{\partial }{\xi }^3}-{\displaystyle \frac{\delta }{(\xi +\delta )}{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{f}}_{r-i-1}}{\mathrm{\partial }\xi }{\displaystyle \frac{{\mathrm{\partial }}^2{\stackrel{⌢}{f}}_i}{\mathrm{\partial }{\xi }^2}-{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{f}}_i}{\mathrm{\partial }\xi }{\displaystyle \frac{{\stackrel{⌢}{f}}_{r-i-1}\delta }{{(\xi +\delta )}^3}}}}}}}}\\ +{\displaystyle \frac{\delta }{{(\xi +\delta )}^2}{\stackrel{⌢}{f}}_{r-i-1}{\displaystyle \frac{{\mathrm{\partial }}^2{\stackrel{⌢}{f}}_i}{\mathrm{\partial }{\xi }^2}-{\displaystyle \frac{\delta }{{(\xi +\delta )}^2}{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{f}}_i}{\mathrm{\partial }\xi }{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{f}}_{r-i-1}}{\mathrm{\partial }\xi }}}}}}\end{array}\right)\\ & -S\left({\displaystyle \frac{{\mathrm{\partial }}^3{\stackrel{⌢}{f}}_{r-1}}{\mathrm{\partial }{\xi }^2\mathrm{\partial }\tau }+{\displaystyle \frac{1}{(\xi +\delta )}{\displaystyle \frac{{\mathrm{\partial }}^2{\stackrel{⌢}{f}}_{r-1}}{\mathrm{\partial }\xi \mathrm{\partial }\tau }}}}\right)\end{array}$$

(55)

$$\begin{array}{rl}{R}_r^{\stackrel{⌢}{\theta }}=& {\displaystyle \frac{{\mathrm{\partial }}^2{\stackrel{⌢}{\theta }}_{r-1}}{\mathrm{\partial }{\xi }^2}+\sum _{i=0}^{r-1}{\displaystyle \frac{{\epsilon }_0{\stackrel{⌢}{\theta }}_{r-i-1}}{(\xi +\delta )}{\displaystyle \frac{{\mathrm{\partial }}^2{\stackrel{⌢}{\theta }}_i}{\mathrm{\partial }{\xi }^2}}}+{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{\theta }}_{r-1}}{(\xi +\delta )\mathrm{\partial }\xi }+\sum _{i=0}^{r-1}{\displaystyle \frac{{\epsilon }_0{\stackrel{⌢}{\theta }}_{r-i-1}}{(\xi +\delta )}{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{\theta }}_i}{\mathrm{\partial }\xi }}}+\sum _{i=0}^{r-1}{\displaystyle \frac{\delta {\stackrel{⌢}{f}}_{r-i-1}}{(\xi +\delta )}{\displaystyle \frac{Pr\mathrm{\partial }{\stackrel{⌢}{\theta }}_i}{\mathrm{\partial }\xi }}}}}\\ & -PrS{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{\theta }}_{r-1}}{\mathrm{\partial }\overline{\tau }}+\gamma Pr{\stackrel{⌢}{\theta }}_{r-1}+{\epsilon }_0{\left({\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{\theta }}_{r-1}}{\mathrm{\partial }\xi }}\right)}^2}\end{array}$$

(56)

$$\begin{array}{rl}{R}_r^{\stackrel{⌢}{\varphi }}=& {\displaystyle \frac{{\mathrm{\partial }}^2{\stackrel{⌢}{\varphi }}_{r-1}}{\mathrm{\partial }{\xi }^2}+{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{\varphi }}_{r-1}}{(\xi +\delta )\mathrm{\partial }\xi }+\sum _{i=0}^{r-1}{\displaystyle \frac{\delta Sc{\stackrel{⌢}{f}}_{r-i-1}}{(\delta +\xi )}{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{\varphi }}_i}{\mathrm{\partial }\xi }}}}}\\ & -{\mathrm{\Gamma }}_{1}Sc\sum _{i=0}^{r-1}\left(-2{\stackrel{⌢}{\varphi }}_{r-i-1}{\stackrel{⌢}{\varphi }}_i+{\stackrel{⌢}{\varphi }}_{r-i-1}\sum _{l=0}^{r-1}{\stackrel{⌢}{\varphi }}_{i-l}{\stackrel{⌢}{\varphi }}_i\right)\\ & -ScS{\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{\varphi }}_{r-1}}{\mathrm{\partial }\tau }-Sc{\mathrm{\Gamma }}_1{\stackrel{⌢}{\varphi }}_{r-1}.}\end{array}$$

(57)

and

$${\lambda }_r=\{\begin{array}{c}0r\le 1,\\ 1r>1\end{array}$$

(58)

The following expressions designate the general form of the solution:

$${\stackrel{⌢}{f}}_r(\xi ,\tau )={\stackrel{⌢}{f}}_r*(\xi ,\tau )+c_1+c_2\exp (\xi )+c_3\xi \exp (\xi )+c_4\exp (-\xi )$$

(59)

$${\stackrel{⌢}{\theta }}_r(\xi ,\tau )={\stackrel{⌢}{\theta }}_r*(\xi ,\tau )+c_5\exp (\xi )+c_6\exp (-\xi )$$

(60)

$${\stackrel{⌢}{\varphi }}_r(\xi ,\tau )={\stackrel{⌢}{\varphi }}_r*(\xi ,\tau )+c_7\exp (\xi )+c_8\exp (-\xi )$$

(61)

here ${\stackrel{⌢}{f}}_r*(\xi ,\tau ),{\stackrel{⌢}{\theta }}_r*(\xi ,\tau )$ , and ${\stackrel{⌢}{\varphi }}_r*(\xi ,\tau )$ denote the particular solution. By incorporating boundary conditions equation (54), the arbitrary constants $c_i$ , $(i=1-8)$ are calculated as

$$\begin{array}{rl}& c_1=-\left(c_4+{\stackrel{⌢}{f}}_r*(0,\tau )\right),c_2=c_3=c_5=c_7=0,c_4={\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{f}}_r*(0,\tau )}{\mathrm{\partial }\overline{y}},}\\ & c_6=-{\stackrel{⌢}{\theta }}_r*(0,\overline{\tau }),c_8={\displaystyle \frac{1}{({\mathrm{\Gamma }}_2+1)}\left({\displaystyle \frac{\mathrm{\partial }{\stackrel{⌢}{\varphi }}_r*(0,\tau )}{\mathrm{\partial }\xi }-{\mathrm{\Gamma }}_2{\stackrel{⌢}{\varphi }}_r*(0,\tau )}\right)}\end{array}$$

(62)

Validation of results

Table 1 is organized to justify the validity of the accomplished results by comparing them with the existing published results^26,29 by considering $S=1.0$ , $M=12$ , and $\delta \to \mathrm{\infty }$ as a limiting case. The table displays that the current findings show remarkable agreement with the published solutions.

Table 1.

Comparison of arithmetical outcomes of ${\stackrel{⌢}{f}}^{\mathrm{″}}(\tau ,0)$ with existing data for dissimilar values of variable $\tau$ when $M=12,S=1.0$ , and $\delta \to \mathrm{\infty }.$

$\tau$	Imran et al. 26	Abdelmalek et al. 29	Present results
$\tau =(3/2)\pi$	$11.678656$	$11.678657$	$11.678656$
$\tau =(11/2)\pi$	$11.678706$	$11.678708$	$11.678706$
$\tau =(19/2)\pi$	$11.678656$	$11.678656$	$11.678655$

Results and discussion

The outcomes of distinct concerned parameters such as homogeneous reaction strength parameter $({\mathrm{\Gamma }}_1)$ , the ratio of oscillatory frequency to the stretched rate of the sheet $(S),$ heterogeneous reaction strength parameter $({\mathrm{\Gamma }}_2)$ , Schmidt number $(Sc),$ magnetic parameter $(M)$ , thermal conductivity parameter $({\epsilon }_0)$ , Prandtl number $(Pr)$ , and radius of curvature $(\delta )$ on the interested profiles are illustrated in detail in this section.

Table 2 demonstrates the impacts of $(\delta ),$ $(M)$ , and $(S)$ on skin friction coefficient $\left({\mathrm{Re}}_{\stackrel{⌢}{s}}^{1/2}{\stackrel{⌢}{C}}_{\stackrel{⌢}{f}}\right)$ when $\tau =\pi /2$ is fixed. It is observed that the magnitude of $\left({\mathrm{Re}}_{\stackrel{⌢}{s}}^{1/2}{\stackrel{⌢}{C}}_{\stackrel{⌢}{f}}\right)$ shows the diminishing response with improving $(\delta )$ , $(S)$ , and it grows with an enhancement of $(M)$ . Table 3 shows the change in the magnitude of $\left(\stackrel{⌢}{N}{\stackrel{⌢}{u}}_{\stackrel{⌢}{s}}{\mathrm{Re}}_{\stackrel{⌢}{s}}^{-1/2}\right)$ for unalike values of $(Pr)$ , $(\delta )$ , $({\epsilon }_0)$ , and $(\gamma )$ at $\tau =\pi /2$ . It is detected that the arithmetic magnitude of the Nusselt number $\left(\stackrel{⌢}{N}{\stackrel{⌢}{u}}_{\stackrel{⌢}{s}}{\mathrm{Re}}_{\stackrel{⌢}{s}}^{-1/2}\right)$ rises for advanced values of $(Pr)$ , whereas it is reduced with growing $(\delta )$ , $({\epsilon }_0)$ , and $(\gamma )$ .

Table 2.

Assessment of $\left({\stackrel{⌢}{C}}_{\stackrel{⌢}{f}}{\mathrm{Re}}_{\overline{s}}^{1/2}\right)$ for various values of $(S),$ $(M)$ , and $(\delta )$ at $\tau =0.5\pi$ fixed.

$S$	$M$	$\delta$	$\left({\stackrel{⌢}{C}}_{\stackrel{⌢}{f}}{\mathrm{Re}}_{\overline{s}}^{1/2}\right)$
0.5	0.2	1.0	−2.01304
2.0			−2.01072
3.5			−2.00562
0.1	0.5		−2.02046
	1.5		−2.09248
	2.5		−2.25134
	0.2	1.0	−2.01318
		2.0	−1.51683
		3.0	−1.35297

Table 3.

Assessment of $\left(\stackrel{⌢}{N}{\stackrel{⌢}{u}}_{\stackrel{⌢}{s}}{\mathrm{Re}}_{\stackrel{⌢}{s}}^{-1/2}\right)$ for some values of $({\epsilon }_0)$ , $(\gamma )$ , $(\delta )$ , and $(Pr)$ at $\tau =\pi /2$ when $M=0.1$ and $S=0.2$ fixed.

${\epsilon }_0$	$Pr$	$\delta$	$\gamma$	$\left(\stackrel{⌢}{N}{\stackrel{⌢}{u}}_{\stackrel{⌢}{s}}{\mathrm{Re}}_{\stackrel{⌢}{s}}^{-1/2}\right)$
0.1	1.0	1.0	0.2	0.994010
0.2				0.991370
0.6				0.988738
1.0	1.5			0.994835
	3.0			0.995329
	4.5			0.995829
	1.0	3.0		0.994672
		5.0		0.977576
		1.0		0.960775
		0.1	0.5	0.990171
			1.0	0.982648
			1.5	0.975098

Figure 2(a) and (b) shows the effects of $(\delta )$ and parameter $(M)$ on velocity profile ${\stackrel{⌢}{f}}^{\mathrm{\prime }}(\tau ,\xi )$ for time interval $0\le \tau \le 10\pi$ . The amplitude of ${\stackrel{⌢}{f}}^{\mathrm{\prime }}(\xi ,\tau )$ gradually declines with both parameters $(\delta )$ and $(M)$ . The mounting values of the parameter $(M)$ originate a significant amount of Lorentz force in the boundary layer section of the liquid flow. The Lorentz force, being a resistive force in nature, plays a key role in opposing the motion of the fluid particles. Consequently, the amplitude of the velocity profile decreases gradually as values of parameter $(M)$ enlarges. Figure 3(a) and (b) characterizes the consequences of $(\delta )$ and $(M)$ on ${\stackrel{⌢}{f}}^{\mathrm{\prime }}(\xi ,\tau )$ at fixed time $\tau =\pi /2$ respectively. The velocity profile ${\stackrel{⌢}{f}}^{\mathrm{\prime }}(\xi ,\tau )$ increases by increasing $(\delta )$ and it shows the opposite trend for advanced values of $(M)$ . The mounting trend in the velocity profile due to the radius of curvature constant indicates that as the radius of curvature increases, the boundary layer effects reduce, resulting to a more stable flow within the boundary layer region. Consequently, the surface resistance decreases which allows the fluid velocity to increase.

Figure 2.

Alteration in ${\stackrel{⌢}{f}}^{\mathrm{\prime }}(\xi ,\tau )$ for the period $0\le \tau \le 10\pi$ : (a) effects of $\delta$ and (b) effects of magnetic parameter M.

Figure 3.

Alteration of dissimilar fluid parameters on ${\stackrel{⌢}{f}}^{\mathrm{\prime }}(\xi ,\tau )$ at $\tau =\pi /2$ : (a) effects of $\delta$ and (b) effects of M.

The change in pressure distribution $\stackrel{⌢}{P}(\xi ,\tau )$ in interval $\tau \in (0,10\pi )$ is illustrated in Figure 4(a) and (b). Figure 4(a) depicts that the $\stackrel{⌢}{P}(\xi ,\tau )$ amplitude is diminished with increasing values of $(\delta )$ . While it is perceived from Figure 4(b) that $\stackrel{⌢}{P}(\xi ,\tau )$ amplitude is developed with $(M).$ Figure 5(a) and (b) shows the alteration in $\stackrel{⌢}{P}(\xi ,\tau )$ for $(\delta )$ and $(M)$ at $\tau =(0.5)\pi$ . It is witnessed that $\stackrel{⌢}{P}(\xi ,\tau )$ is a decaying function of $(\delta )$ and it is upward by growing the parametric values of $(M)$ .

Figure 4.

Variation in $\stackrel{⌢}{P}(\xi ,\tau )$ profile: (a) impacts of $\delta$ and (b) impacts of M.

Figure 5.

Change in $\stackrel{⌢}{P}(\xi ,\tau )$ at $\tau =\pi /2$ : (a) influences of the radius of curvature variable $\delta$ and (b) impacts of M.

Figure 6(a) to (d) confirms that the magnitude of temperature profile and thermal boundary layer thickness is improved with increment of $(\delta )$ , $({\epsilon }_0)$ , and $(\gamma )$ . However, both the thermal boundary layer width and temperature distribution profile decline with the Prandtl number $(Pr)$ , this is because the Prandtl number is inversely related to thermal diffusivity. As the Prandtl number grows, thermal diffusivity drops. The reduction in thermal diffusivity leads to a decrease in the rate of heat transfer. Consequently, both the temperature profile and the thermal boundary layer thickness decrease as the Prandtl number increases.

Figure 6.

Variation in temperature outline $\stackrel{⌢}{\theta }(\xi ,\tau )$ : (a) effects of $\delta$ , (b) effects of Prandtl number $Pr$ , (c) impacts of ${\epsilon }_0$ , and (d) effects of $\gamma$ .

Figure 7(a) to (d) shows the alteration in the concentration field $\stackrel{⌢}{\varphi }(\xi ,\tau )$ under the influences of different parameters likely $(\delta )$ , $({\mathrm{\Gamma }}_1)$ , $({\mathrm{\Gamma }}_2)$ , and $(Sc)$ at $\tau =0.5\pi$ . This figure clearly depicts that $\stackrel{⌢}{\varphi }(\xi ,\tau )$ shows a decaying response with $({\mathrm{\Gamma }}_1)$ and it increases with the improving values of $(\delta )$ , $({\mathrm{\Gamma }}_2)$ , and $(Sc).$

Figure 7.

Influences of dissimilar variables on the profile $\stackrel{⌢}{\varphi }(\xi ,\tau )$ when $\tau =\pi /2$ : (a) effects of $\delta$ , (b) effects of ${\mathrm{\Gamma }}_1$ , (c) influences of ${\mathrm{\Gamma }}_2$ , and (d) impacts of $Sc$ .

Figure 8(a) to (d) exemplifies the consequences of dissimilar variables likely $(\delta )$ , $({\mathrm{\Gamma }}_1)$ , $({\mathrm{\Gamma }}_2)$ , and $(Sc)$ on the surface concentration profile $\stackrel{⌢}{\varphi }(0,\tau )$ at time instants $\tau \in (0,10\pi ).$ It is observed from this figure that the amplitude of $\stackrel{⌢}{\varphi }(0,\tau )$ is reduced with $({\mathrm{\Gamma }}_1)$ , $(\delta )$ , and $({\mathrm{\Gamma }}_2)$ and is amplified with $(Sc).$

Figure 8.

Deviation in $\stackrel{⌢}{\varphi }(\tau ,0)$ at $\tau \in [0,10\pi ]$ : (a) influences of $\delta$ , (b) effects of ${\mathrm{\Gamma }}_1$ , (c) effects of ${\mathrm{\Gamma }}_2$ , and (d) influences of $Sc$ .

Figure 9(a) and (b) presents the consequences of varying values of homogeneous reaction parameter $({\mathrm{\Gamma }}_1)$ versus radius of curvature $(\delta )$ and heterogeneous reaction parameter $({\mathrm{\Gamma }}_2)$ versus Schmidt number $(Sc)$ respectively on the profile of $\stackrel{⌢}{\varphi }(0,\tau )$ for $\tau =\pi /2$ . It is evident from Figure 9(a) and (b) that $\stackrel{⌢}{\varphi }(0,\tau )$ gradually declines with uplifting values of $({\mathrm{\Gamma }}_1)$ , $(\delta )$ , and $({\mathrm{\Gamma }}_2)$ and it enhances with an enhancement in the values of $(Sc)$ .

Figure 9.

Effects of diverse flow variables on surface concentration distribution $\stackrel{⌢}{\varphi }(0,\tau )$ : (a) effects of ${\mathrm{\Gamma }}_1$ versus $\delta$ and (b) effects of ${\mathrm{\Gamma }}_2$ versus $Sc$ when $\tau =0.5\pi .$

Figures 10(a) to (c) and 11(a) to (c) illustrate the behavior of the streamlines corresponding to the axial and the radial velocity components at various time points respectively. Figures 10(a) and 11(a) elucidate the symmetric distribution of streamlines radiating outward from the origin. In contrast, Figures 10(b), (c) and 11(b), (c) exhibit the oscillating response of the streamlines. Figure 12(a) to (c) demonstrates the behavior of streamlines and isotherms when $\tau =0.5\pi$ fixed. It is evident from these figures that the streamlines and isotherms show a symmetrical pattern.

Figure 10.

Alterations in streamlines $\stackrel{⌢}{v}\left(\stackrel{⌢}{r},\stackrel{⌢}{t}\right)$ by letting $\stackrel{⌢}{r}\to 0$ with $M=S=0.1$ and $\delta =1.0$ .

Figure 11.

Alterations in streamlines of $\stackrel{⌢}{u}\left(\stackrel{⌢}{r},\stackrel{⌢}{t}\right)$ by letting $\stackrel{⌢}{r}\to 0$ with $M=S=0.1$ and $\delta =1.0$ .

Figure 12.

Expressions of streamlines and isotherms by keeping $\tau =0.5\pi$ $M=S={\epsilon }_0=0.1,Pr=1.0,\gamma =0.2$ , and $\delta =1.0$ .

Concluding remarks

This study explored the transport phenomena of heat by considering the impacts of heat production along with temperature-dependent thermal conductivity and HH chemical reactions in the time-dependent viscous fluid motion on a stretchable curved oscillatory sheet. The analytical solution of the developed nonlinear partial differential equations in the convergent series form has been attained by employing the HAM. The significant results have been recognized on the evidence of the recent study, with significant implications for engineering and industrial applications involving heat and mass transfer over curved surfaces.

The liquid velocity amplitude is a diminishing function of $(\delta )$ and $(M)$ . This reduction in liquid velocity amplitude highlights its potential role in augmenting flow dynamics in biomedical devices such as controlling blood flow in artificial organs and in industrial coating applications where precise fluid control is vital.

The fluid velocity declines with improving values of the magnetic parameter $(M)$ at $\tau =\pi /2$ . This declining response in fluid velocity due to mounting values of the magnetic parameter is significant for electromagnetic casting and plasma-assisted material processing where precise flow control plays a crucial role in ensuring uniform material properties.

For upward values of $(M)$ , the pressure profile amplitude is enhanced, while for $(\delta )$ it is reduced. These alterations in pressure amplitude offer valuable insights for precisely modulating pressure in the design of both microfluidic and macrofluidic systems including drug delivery systems and lab-on-a-chip technologies.

The temperature profile is increases with uplifting values of $(\delta )$ , $(\gamma )$ , and $({\epsilon }_0)$ , while it reduces gradually with $(Pr).$ Understanding the temperature trend due to the above-defined parameters is fundamental for thermal management systems, such as advanced cooling systems for electronic devices and heat exchangers in industrial procedures.

The profile of concentration distribution depicts a decreasing manner with $({\mathrm{\Gamma }}_1)$ , $(\delta )$ , and $({\mathrm{\Gamma }}_2)$ . However, it shows upward behavior for $(Sc)$ . This diminishing trend against the upward values of homogeneous reaction parameter, radius of curvature variable, and heterogeneous reaction parameter highlights the significance of optimizing concentration distribution for chemical reactors, catalytic converters, and pollution control systems.

The amplitude of $\stackrel{⌢}{\varphi }(0,\tau )$ is enhanced with $(Sc)$ and it displays the opposite trend for $({\mathrm{\Gamma }}_1)$ , $(\delta )$ , and $({\mathrm{\Gamma }}_2).$ The increase in surface concentration amplitude with Schmidt number emphasizes its importance in filtration and separation technologies.

The numerical magnitude of the skin friction coefficient increases by improving $(M)$ , and it declines against $(S)$ and $(\delta ).$ The alteration in the magnitude of the skin friction coefficient due to the above-defined variables has widespread practical implications for aerodynamic and hydrodynamic areas, particularly in reducing drag in transportation and marine engineering.

The magnitude of the Nusselt number is a rising function of $(Pr)$ , while it reduces for growing parametric values of $({\epsilon }_0)$ , $(\delta )$ , and $(\gamma ).$ The growing response of the Nusselt number with the Prandtl number $(Pr)$ and declining trend with $({\epsilon }_0)$ , $(\delta )$ , and $(\gamma )$ highlights strategies for exploiting heat transference efficiency in industrial heating and cooling systems ensuring optimal thermal regulation in energy-intensive processes.

The present findings offer valuable insights into real-world implications across manufacturing and engineering disciplines, including liquid transport, thermal energy management, and chemical reaction engineering. The study provides a foundation for optimizing system performance in industries where precise control over heat and mass transfer is essential, such as chemical processing, biomedical engineering, and energy production.