Investigation of mass and heat transport in bi-directional stretched flow of Sisko fluid with variable magnetic field and radiation effects via OHAM

Abstract

The present work investigates mass and heat transport in non-Newtonian Sisko fluid past a three-dimensional bi-directional stretched heated surface with MHD effect, heat radiation, Joule warming, variable thermic conductivity, and temperature-dependent diffusivity. Comprehension the intricate relationships amongst mass and heat transfer in Sisko fluid across various magnetic field and radiation conditions requires an understanding of this model. It offers information on how to best optimize industrial operations where controlling fluid behavior and thermal characteristics is crucial, such as cooling systems and polymer extrusion. The difficulty is derived from resembling coupled nonlinear PDEs by engaging boundary layer theory and transforming them into systems of ODEs by applying suitable similarity transformations. The nonlinear converted ODEs have been solved using OHAT (optimal homotopy analysis technique) and the results are presented via graphs and tables. The performed analysis shows that escalating values of fluid parameter shows shear-thinning behavior and as a result fluid velocity upsurges. It is observed through findings that the velocity spectrum strengthens along the x-axis and y-axis concerning the Sisko fluid factor $(A)$ . The thermal appearance exhibits improved behavior with higher thermal conductivity $(\epsilon )$ levels. Moreover, the thermal profile indicates enhanced behavior given the enrichment in Eckert number $\mathrm{Ec}$ and Radiation factor $R$ . Previous research indicates that no comprehensive studies have been carried out to examine mass, heat transfer with changing thermal conductivity, diffusive coefficient variation, radiation warmth, and Joule heating for the two-way extended transport of the Sisko nanofluid. This study fills up that void.

Introduction

The tinny surface of fluid that makes instantaneously all over a bound sheet because of fluid circulating along is considered as the “boundary surface” in this context. Due to its location at the boundary of the fluid, engineers refer to this layer as the boundary layer. The boundary layer’s flow properties significantly impact several aerodynamic difficulties, comprising aircraft stalling, skin friction grinding against an item, and heat transmission in speedy airplanes. Sisko fluid with constant and variable viscosity depends on the shear rate. The Sisko fluid model has extensive industrial uses like food production, coatings, and paints. Moreover, wastewater management, chemical processing, oil, and also in gas industry. As a sub-compartment of generalized Newtonian fluids, Khan et al. ¹ studied the Sisko fluid scheme regarded as the ultimate appropriate for lubricating liquids and solids. The Sisko fluid method has substantial value since it adequately explains a few non-Newtonian liquids across the largest kind of shear rates. The concept of mass and heat transport, as its name indicates, is based on calculating the rate at which heat is transmitted across a material, such as by conduction, convection, or radiation. Due to the difference in temperatures between the two mediums. Hatwar ² expressed that several industries use heat and mass transfer analysis, including vehicle, steam-electric authority production, energy schemes, HVAC, electrical equipment refrigeration, and diagnosing infections. The Variable Field Module 4 (VFM4) is best suited for magnetic force microscopy and other uses where the sample depends on the applied magnetic field. The VFM4 has adjustable out-of-plane and in-plane magnetic field strengths of up to 1200 G and 8000 G, respectively. Most fluids that are extremely important in daily life and industry such as paints, oils, oiling lubricants, human blood, honey, biological liquids, etc. do not adhere to the Newtonian expression of viscosity. They are referred to as non-Newtonian fluids. Owing to their significance in everyday life, several recent research studies have examined the features of non-Newtonian fluids analyzed by Malik et al. ³ As energy-efficient heat transfer fluids are needed for such incredibly effective cooling, common base liquids have still-low thermal conductivity, which is a significant concern, and research connected to ultra-high-performance cooling technologies is now quite hard. Pal and Mandal ⁴ addressed that the suspension of solid nanoparticles in common base liquids provided a solution to these problems. Many investigations have been done on the viscous materials created by the traditional Navier-Stokes equations in the literature. Viscoelastic behavior is present in a wide range of compound rheological resources, including polymeric fluids, piercing grimes, shampoos, biological fluids, liquid sparklers, greasing oils, and various further. Hayat et al. ⁵ discussed that simple Newtonian models cannot adequately describe these materials. Many industrial engineering applications include the sensations of momentum and heat transmission in boundary surface movement across a plane animated sheet. Because of its practical significance in several engineering fields, Munir et al. ⁶ observed that the momentum and thermal transport caused by an animated stretched sheet have been of great interest. Due to their novel characteristics, nanofluids are quick to respond to numerous requests for heat transmission. In comparison to the base fluid, Sohail and Naz ⁷ described that these resources improved heat conduction and the convective thermal transference factor. Due to its numerous uses in metallurgical processes, engineering, and industry, coupled mass and thermal transference have become a well-studied spectacle. To fully comprehend the processes of heat and mass movement, Fourier’s regulation of thermal conduction and Fick’s rule of dispersion are used by Khan et al. ⁸ Most research used the traditional Newtonian model to describe the viscous liquids’ flow. Doh et al. ⁹ expressed that the majority of industrial liquids are recognized as non-Newtonian liquids since the generally accepted presumption of a uniform connection between stress and the ratio of strain is not true. A magnetic colloidal solution of single-domain magnetic nanoparticles is known as ferrofluid. As the flow field created by ferrofluid may be appropriately adjusted by adding external magnetic fields, it can be used for heat transference applications. Sheikholeslami et al. ¹⁰ have investigated the treatment of ferrofluid thermal transmission in a fluctuating magnetic field. Due to the vast range of scientific and technological applications, including object damage, lubrication systems, chemical processing units, hydro-dynamical machines, and the processing of polymers and lubricants. The revision of warmth and mass transport in unstable pressing viscous movement among two aligned plates in movement regular to the shells independently of one another and arbitrarily in terms of time was established by Hatami and Ganji ¹¹ as one of the most significant research concerns. Magneto cross tetra hybrid Nano fluidic flow demeanor across a stenosed artery from impacts of heat radiation together with heat source/sink was investigated by Sajid et al. ¹² In this investigation, Tiwari and Das Novel tetra hybrid Nano fluidic model was adopted. The impact of non-Newtonian substances bounded by parallel elastic surfaces under various influences specifically Fourier’s law with implementation of Galerkin finite element methodology was surveyed by Sajid et al. ¹³ Sajid et al. ¹⁴ observed the Maxwell Nano fluidic Darcy-Forchheimer flow demeanor across an extended surface. Mass and heat transmission were observed under factors such as varying thermal conduction, and activating energy together with irregular thermal radiation. The performance of magnetized Maxwell-Sutterby substance across an angled extending sheet was given by Sajid et al. ¹⁵ and influenced by variable heat conductance, an expanding energy source/sink. Sajid et al. ¹⁶ observed the impacts of dynamic molecular diffusion, chaotic radiant heat, convection boundaries, momentum slide, and variable molecular dispersion based on Prandtl liquid across an extended plate.

Heat as well as mass transmission was analyzed by Kho et al. ¹⁷ In this Analysis Buongiorno model was adopted for investigation. The Buongiorno model has many industrial applications in the field of nuclear engineering and the Buongiorno model offers heat transportation phenomena in nuclear reactors and many more. Nasir et al. ¹⁸ discovered Brownian movement and thermophoresis dispersion. Moreover, Heat transmission across the non-uniform energy source and sinks was also observed. The Buongiorno model was adopted in this study. According to Sohail et al.,^19,20 the analysis of species and heat transport will produce non-Newtonian material flowing across the linear stretching sheet. With the help of the Cattaneo-Christov heat flow definition, heat transport phenomena are presented. The importance of temperature-dependent thermal conductivity and coefficient of diffusion for mass and thermal transmission in the Sutterby model was investigated by Waseem et al. ²¹ using OHAM. Sohail et al. ²² explained that this study uses a theoretical and numerical technique to capitalize on the three-dimensional MHD border surface movement of a steady, incompressible Casson fluid via a uniform extensible sheet with Cattaneo-Christov dual diffusion. Numerical calculations for MHD unstable 3D motion through mass and thermal transmission of the Sisko fluid were reported in the attendance of nanomaterials. With this model, Khan et al. ²³ addressed the impacts of the thermophoresis parameter and Brownian movement being considered. Heat and mass transmission under convective boundary conditions were each taken into consideration. The study focuses on non-Newtonian Sisko fluid movement on a uniformly extending surface exposed to heat radiation melting heat transfer studied by Soomro et al. ²⁴ Hazra et al. ²⁵ investigated mixed convection beneath the influence of an externally imposed uniform magnetic field in a vented chamber saturated with a porous material and heated at the top-left and bottom-right corners. Rahaman et al. ²⁶ investigated the linked mixed convective heat transport mechanism inside a grooved channel cavity using a CuO-water nanofluid and an angled magnetic field. From the bottom, the cavity is heated isothermally, causing differences in the locations of the heated walls inside the grooved channel. The influence of nanomaterials on regular convective border layer movement across a vertical sheet has been revisited. Nevertheless, we have now explored a more realistic scenario where the fluid material percentage on the border is inactively rather than dynamically regulated, which is an alternative to the instance of continuous temperature and nanomaterial fraction at the barrier. Kuznetsov and Nield ²⁷ employed Buongiorno’s model, which takes into consideration both Brownian circulation and thermophoresis. This research examines the effects of a changing magnetic field and velocity slip on an incompressible pseudo-plastic EVA fluid flow in a limited film along a stretched exterior expressed by Zhang et al. ²⁸ In this article Bhatti et al. ²⁹ discussed the peristaltic motion of non-Newtonian fluid-comprising gyrotactic microorganisms under the simultaneous stimulus of a changing magnetic field and coagulation (blood clot). Thermal radiation is anticipated as a rising factor owing to its impact, which aids in the removal of hazardous material from blood moving through blood vessels, Thermal transportation and tri hybrid Tiwari and Das Sisko Nano fluidic flow demeanor through a stenosis artery was reported by Fangfang et al. ³⁰ Autocatalyst reactions for thermal efficiency toward the tetra hybrid dual Nano fluidic flow subjected to Riga wedge were discussed by Sajid et al. ³¹ Akbar and Sohail ³² evaluated three-dimensional viscid movement passing via a planar horizontally extended exterior by utilizing a magnetic pitch, radiant heat, and sticky dissolution. Li et al. ³³ discovered the impressions of buoyant and viscosity dispersion on the three-dimensional hybrid nano-liquid motion of titanium oxide $(\mathrm{Ti}{O}_2)$ and magnesium oxide $(\mathrm{MgO})$ . Nazir et al. ³⁴ analyzed the changing characteristics of temperature and momentum transmission in spinning Carreau liquid across the warmed cone. Chakravarty et al. ³⁵ addressed one cooling technique involving side injection of cold fluid using a standard conical heat-generating porous bed positioned centrally within a fluid-filled cylindrical enclosure. Pumping power and heat transmission of a typical system are examined by Biswas et al., ³⁶ taking into account bottom-heating, porous media, and Cu-water nanofluid. In an open room shaped like a butterfly, with the top inverted triangle wall being cold and the bottom triangular wall being hot, Halder et al. ³⁷ studied the thermofluidic phenomena. Calculations based on finite elements are used to analyze the transport processes in this complex geometry.

An optimal homotopy exploration technique is useful for chaotic differential equalities. Fan and You ³⁸ claim that one or more parameters in this technique, which may be determined by minimizing a particular function, influence the convergence of series resolutions. Odibat ³⁹ investigated the approach, developed by Liao, which has the elasticity to regulate the convergence area and rate for series clarifications to unbalanced differential calculations as one of its key benefits. It is harder to describe many nonlinear phenomena mathematically than it is to represent linear ones, even though nonlinear differential equations may properly define many nonlinear phenomena. Such events are often represented by Bahia et al. ⁴⁰ considering two primary types of differential equations: ordinary and partial. Jia et al. ⁴¹ described a process for changing a linear optimum control issue into an ODE system that adheres to the Pontryagins maximal principle (PMP). Meanwhile, a mathematical solution to the linear OCPs was obtained using Mathematica 9.0′s Minimize function to minimize the square residual fault for the 10th or 15th-order estimation and the Optimal HAM method applied to the system of ODEs. The best value for such a convergence control parameter may be found using this method. The diagrammatic results and even abilities for the stream field aspects are obtained using the optimal homotopy investigation approach (OHAM) discussed by Naz et al. ⁴² When it comes to analytical methodology, Sohail et al. ⁴³ examined that the Optimal Homotopy Analysis Method (OHAM) analysis approach provides an outcome to nonlinear flow problems requiring boundary conditions. For higher-order BVPs, several scholars have developed a variety of numerical approaches during the last few decades. Biswal et al. ⁴⁴ studied that the homotopy analysis was a method Liao proposed for BVPs in 1992. OHAM was employed to manage the directing coupled nonlinear DEs analyzed by Biswal et al. ⁴⁴ The convergence controller settings have also been calculated using a cutting-edge method, which might simplify the conventional OHAM.

Existing literature shows that no investigations have been conducted collectively to study mass, heat transmission containing variable thermal conductivity, varying diffusion coefficient, warmth radiation, and Joule warming for the bi-directional stretched movement of Sisko nanofluid. This research covers this gap. This is how the investigation is set up:

• Section 1 covers the detailed literature review on nanofluid, heat, and mass transmission;

• Problem formulation and dimensionless analysis are reported in section 2;

• Nonlinear arising model is handled via the optimal homotopic procedure. A complete narrative of the utilized process is given in segment 3;

• Results are discussed by seeing the underlying physics in segment 4 and the conclusion is presented in sector 5.

Mathematical modeling

Let the smooth, laminar circulation of an inflexible Sisko fluid in three dimensions (3D) over a surface that is capable of bidirectional extension. The Mburu et al. liquid is regarded as electrically conducting in the occurrence of a steady magnetic field $B_0$ ^45,46 employed in the $\tilde{\mathrm{z}}$ -direction (see Figure 1). Additionally disregarded are the effects of the electric and Hall fields. At small magnetic Reynolds numbers, the prompted magnetic field is not considered. Brownian movement and thermophoresis ⁴⁷ have additional effects. The stretching surface is parallel to the z-axis, and the frequently used Cartesian coordinate system is utilized to take the $\tilde{x},\tilde{y}$ axes along the extensible exterior. The $\tilde{x}$ - and $\tilde{y}$ -directional sheet stretching velocities, denoted by $\tilde{U}w$ and $\tilde{V}w$ , respectively. The fluid’s thermophysical characteristics are assumed to be constant. The nanofluid will also be managed as a 2-component system (base fluid with nanoparticles) under the suppositions listed below:

• (1) Lack of chemical reactions;

• (2) Incompressible flow;

• (3) Insignificant outside pressures;

• (4) Diluted mixture $\tilde{C}<<1$ ;

• (5) Insignificant radiative heat transfer;

• (6) Negligible viscous dissipation;

• (7) Local thermal equilibrium exists between the base fluid and the nanoparticles.

Figure 1.

Geometry for the considered flow problem of Sisko fluid structure.

The constitutive equations for the incompressible Sisko fluid’s three-dimensional (3D) flow given by Wang et al. ⁴⁸ are

$$\mathrm{div}(\tilde{V})=0,$$

(1)

$$\rho \frac{d\tilde{v}}{dt}=\mathrm{div}{\tau }^{*}+\rho b^{*}.$$

(2)

In the case of Sisko fluid, the Cauchy stress tensor ${\tau }^{*}$ is

$${\tau }^{*}=-p^{*}\tilde{I}+S^{*}.$$

(3)

Conventions of the extra stress tensor $S^{*}$ induced by Sisko ⁴⁹

$$S^{*}={[a^{*}+b^{*}\left|\sqrt{\frac{1}{2}\mathrm{tr}{\left(A_1\right)}^2}\right|]}^{\tilde{n}-1}{A}_1.$$

(4)

The definition of the first Rivlin-Erickson tensor $A_1$ is given by

$$A_1=\left(\mathrm{grad}\tilde{V}\right)+{\left(\mathrm{grad}\tilde{V}\right)}^T.$$

(5)

The velocity field $\tilde{V}$ is provided by and t denotes the matrix transpose

$$\tilde{V}=[\tilde{u}(\tilde{x},\tilde{y},\tilde{\mathrm{z}}),\tilde{v}(\tilde{x},\tilde{y},\tilde{\mathrm{z}}),\tilde{\mathrm{w}}(\tilde{x},\tilde{y},\tilde{\mathrm{z}})].$$

(6)

With the boundary layer presumptions, the Sohail and Abbas nanofluid’s governing border sheet expressions for heat and mass transmission are assumed as ⁵

$$\frac{\partial \tilde{u}}{\partial \tilde{x}}+\frac{\partial \tilde{v}}{\partial \tilde{y}}+\frac{\partial \tilde{\mathrm{w}}}{\partial \tilde{\mathrm{z}}}=0,$$

(7)

$$\tilde{u}\frac{\partial \tilde{u}}{\partial \tilde{x}}+\tilde{v}\frac{\partial \tilde{u}}{\partial \tilde{y}}+\tilde{\mathrm{w}}\frac{\partial \tilde{u}}{\partial \tilde{\mathrm{z}}}=\frac{a^{*}}{{\rho }_f}\frac{{\partial }^2\tilde{u}}{\partial {\tilde{\mathrm{z}}}^2}-\frac{b^{*}}{{\rho }_f}\frac{\partial }{\partial \tilde{\mathrm{z}}}{(-\frac{\partial \tilde{u}}{\partial \tilde{\mathrm{z}}})}^{\tilde{n}}-\frac{\sigma B_0^2}{{\rho }_f}\tilde{u},$$

(8)

$$\tilde{u}\frac{\partial \tilde{v}}{\partial \tilde{x}}+\tilde{v}\frac{\partial \tilde{v}}{\partial \tilde{y}}+\tilde{\mathrm{w}}\frac{\partial \tilde{v}}{\partial \tilde{\mathrm{z}}}=\frac{a^{*}}{{\rho }_f}\frac{{\partial }^2\tilde{v}}{\partial {\tilde{\mathrm{z}}}^2}-\frac{b^{*}}{{\rho }_f}\frac{\partial }{\partial \tilde{\mathrm{z}}}{(-\frac{\partial \tilde{u}}{\partial \tilde{\mathrm{z}}})}^{\tilde{n}-1}\frac{\partial \tilde{v}}{\partial \tilde{\mathrm{z}}}-\frac{\sigma B_0^2}{{\rho }_f}\tilde{v},$$

(9)

$$\begin{array}{c}\multicolumn{1}{c}{\tilde{u}\frac{\partial \mathrm{T}}{\partial \tilde{x}}+\tilde{v}\frac{\partial T}{\partial \tilde{y}}+\tilde{\mathrm{w}}\frac{\partial T}{\partial \tilde{\mathrm{z}}}=\frac{\partial }{\partial \tilde{\mathrm{z}}}[(K(T)\frac{\partial T}{\partial \tilde{\mathrm{Z}}}]+\frac{\left(p_{\tilde{c}}\right)p}{{\left(p_{\tilde{c}}\right)}_f}(D_B\left(\frac{\partial T}{\partial \tilde{\mathrm{Z}}}\frac{\partial \tilde{C}}{\partial \tilde{\mathrm{Z}}}\right)}\\ +\frac{D_T}{T\infty }{\left(\frac{\partial T}{\partial \tilde{\mathrm{Z}}}\right)}^2)+\frac{16}{{\rho }_f{\tilde{c}}_{P}3K_{\tilde{m}}}{T}_{\infty }^3\frac{{\partial }^2\mathrm{T}}{\partial {\tilde{\mathrm{Z}}}^2}+\frac{\sigma B_0^2}{{\rho }_f{\tilde{c}}_P}[{\left(\tilde{u}\right)}^2+{\left(\tilde{v}\right)}^2],\hfill \end{array}$$

(10)

$$\tilde{u}\frac{\partial \tilde{C}}{\partial \tilde{x}}+\tilde{v}\frac{\partial \tilde{C}}{\partial \tilde{y}}+\tilde{\mathrm{w}}\frac{\partial \tilde{C}}{\partial \tilde{\mathrm{z}}}=\frac{\partial }{\partial \tilde{\mathrm{z}}}((D(T)\frac{\partial \tilde{C}}{\partial \tilde{\mathrm{Z}}})+\frac{D_T}{T_{\infty }}\left(\frac{{\partial }^{2}T}{\partial {\tilde{\mathrm{Z}}}^2}\right).$$

(11)

Boundary conditions that are being considered are

$$\tilde{u}={\tilde{U}}_{\tilde{w}}\left(\tilde{x}\right)=\tilde{c}\tilde{x},\tilde{v}={\tilde{V}}_{\tilde{\mathrm{w}}}\left(\tilde{y}\right)=\tilde{c}\tilde{y},\tilde{\mathrm{w}}=0,T=T_{\tilde{\mathrm{w}}},$$

$$D_B\frac{\partial \tilde{C}}{\partial \tilde{\mathrm{Z}}}+\frac{D_T}{T\infty }\frac{\partial T}{\partial \tilde{\mathrm{Z}}}=0\mathrm{at}\tilde{\mathrm{z}}=0,$$

(12)

$$\tilde{u}\to 0,\tilde{v}\to 0,T\to T_{\infty },\tilde{C}\to {\tilde{C}}_{\infty }\mathrm{as}\tilde{\mathrm{z}}\to \infty .$$

(13)

The subsequent dimension-less variables are now being used

$$\begin{array}{l}\tilde{u}=\tilde{c}\tilde{x}{f}^{\prime }\left(\eta \right),\tilde{v}=\tilde{c}\tilde{y}{g}^{\prime }\left(\eta \right),\\ \tilde{\mathrm{w}}=-\tilde{c}\left(\frac{{\tilde{c}}^{\tilde{n}-2}}{{\rho }_{f/b^{\ast }}}\right)\frac{1}{\tilde{n}+1}\left(\frac{2\tilde{n}}{\tilde{n}+1}f\frac{1-\tilde{n}}{1+\tilde{n}}\tilde{n}{f}^{\prime }+g\right){\tilde{x}}^{\tilde{n}-1/\tilde{n}+1},\end{array}$$

$$\theta \left(\eta \right)=\frac{T-T\infty }{T\tilde{\mathrm{w}}-T\infty },\varphi \left(\eta \right)=\frac{\tilde{C}-{\tilde{C}}_{\infty }}{{\tilde{C}}_{\infty }},\eta =\tilde{\mathrm{z}}{\left(\frac{{\tilde{c}}^{\tilde{n}-2}}{b^{\ast }/{\rho }_f}\right)}^{\frac{1}{\tilde{n}+1}}{\tilde{x}}^{1-\tilde{n}/1+\tilde{n}}.$$

(14)

Equation (7) is now satisfied and equations (8–13) result in the following form

$$Af″\prime +\tilde{n}{(-f″)}^{\tilde{n}-1}f″\prime +\left(\frac{2\tilde{n}}{\tilde{n}+1}\right)ff″-{(-f\prime )}^2+gf″-M^{2}f\prime =0,$$

(15)

$$\begin{array}{c}Ag″\prime +\tilde{n}{(-f″)}^{\tilde{n}-1}g″\prime -(\tilde{n}-1)g″f″\prime {(-f″)}^{\tilde{n}-2}\hfill \\ +\left(\frac{2\tilde{n}}{\tilde{n}+1}\right)fg″-{(g\prime )}^2+gg″-M^{2}g\prime =0,\hfill \end{array}$$

(16)

$$\begin{array}{c}(1+\frac{4}{3}R+\epsilon \theta )\theta ″+\mathrm{Pr}(\left(\frac{2\tilde{n}}{\tilde{n}+1}\right)f\theta \prime +g\prime \theta \prime +\mathrm{Nb}\theta \prime \varphi \prime \hfill \\ +\mathrm{Nt}{(\theta \prime )}^2+M^2\mathrm{Ec}[{(f\prime )}^2+{(g\prime )}^2])=0,\hfill \end{array}$$

(17)

$$(1+{\epsilon }_1\theta )\varphi ″+\mathrm{LePr}(\left(\frac{2\tilde{n}}{\tilde{n}+1}\right)f\prime \varphi \prime +g\varphi \prime )+\left(\frac{\mathrm{Nt}}{\mathrm{Nb}}\right)\theta ″=0,$$

(18)

$$f=g=0,f\prime =1,g\prime =\alpha ,\theta =1,\mathrm{Nb}\varphi \prime +\mathrm{Nt}\theta \prime =0\mathrm{for}\eta =0,$$

(19)

$$f\prime \to 0,g\prime \to 0,\theta \to 0,\varphi \to 0\mathrm{for}\eta \to \infty .$$

(20)

Some parameters are listed as follows:

$$\left\{\begin{array}{c}A=\frac{R{e}_{b^{\ast }}^{\frac{2}{\tilde{n}+1}}}{R{e}_{a^{\ast }}},M^2=\frac{\sigma B_0^2}{{\rho }_f\tilde{c}},\alpha =\frac{d}{\tilde{c}},Pr=\frac{\tilde{x}{\tilde{U}}_{\tilde{\mathrm{w}}}R{e}_{b^{\ast }}^{\frac{-2}{\tilde{n}+1}}}{{\alpha }_{\tilde{m}}},\\ Nb=\frac{\left(p_{\tilde{c}}\right)p{D}_B{\tilde{C}}_{\infty }}{{\left(p_{\tilde{c}}\right)}_f{a}^{\ast }/{\rho }_f},Nt=\frac{\left(p_{\tilde{c}}\right)p{D}_T\left(T_{\tilde{\mathrm{w}}}-T_{\infty }\right)}{{\left(p_{\tilde{c}}\right)}_f{T}_{\infty }{a}^{\ast }/{\rho }_f},Le=\frac{{\alpha }_{\tilde{m}}}{D_B},\\ K\left(T\right)=1+\epsilon \left[\frac{T-T\infty }{T\tilde{\mathrm{w}}-T\infty }\right],D\left(T\right)=1+{\epsilon }_1\left[\frac{T-T\infty }{T\tilde{\mathrm{w}}-T\infty }\right].\end{array}\right\}$$

(21)

Here: The Sisko fluid parameter is represented by  $(A)$ , the magnetic parameter by $(M)$ , the Brownian circulation factor by $(\mathrm{Nb})$ , the thermophoresis factor by $(\mathrm{Nt})$ , the Lewis number by $(\mathrm{Le})$ , the Prandtl number by $(\mathrm{Pr})$ , the variable diffusion coefficient by $D(T)$ , and the temperature-dependent thermal conductivity by $K(T)$ . When an external magnetic field influences the velocity and stability of an electrically conductive fluid, it is represented by the magnetic parameter $(M)$ . When calculating the respective thicknesses of the velocity and thermal boundary layers in a fluid flow, the Prandtl number $(\mathrm{Pr})$ shows the ratio of momentum diffusivity to thermal diffusivity.

The Nusselt numbers and skin friction coefficients are defined by

$$\left\{\begin{array}{c}{\mathrm{Re}}_{b^{*}}^{\frac{1}{\tilde{n}+1}}{\tilde{c}}_{f\tilde{x}}=Af″\left(0\right)-{(-f″\left(0\right))}^{\tilde{n}},\hfill \\ {\mathrm{Re}}_{b^{*}}^{\frac{1}{\tilde{n}+1}}{\tilde{c}}_{f\tilde{x}}=\frac{{\tilde{V}}_{\tilde{\mathrm{w}}}}{{\tilde{U}}_{\tilde{\mathrm{w}}}}(Ag″\left(0\right)+{(-f″\left(0\right))}^{\tilde{n}-1}g″\left(0\right)),\hfill \\ {\mathrm{Re}}_{b^{*}}^{\frac{-1}{\tilde{n}+1}}N{\tilde{u}}_{\tilde{x}}=-\theta \prime \left(0\right).\hfill \end{array}\right\}$$

(22)

This dimensionless mass flow, represented by $S{h}_{\tilde{x}}$ is now identically zero, and $R{e}_{a^{*}}={\tilde{U}}_{\tilde{\mathrm{w}}}\tilde{x}{\rho }_{f/a^{*}}$ and $R{e}_{b^{\ast }}={\tilde{U}}_{\tilde{\mathrm{w}}}^{2-\tilde{n}}{\tilde{x}}^{\tilde{n}}{\rho }_{f/b^{\ast }}$ .

Solutions procedure

Through the use of adjustable parameters, the Optimal Homotopy Analysis Method continuously deforms a homotopy from an initial guess to an exact solution, thereby optimizing convergence and solving ordinary differential equations (ODEs) that arise from the transformation of partial differential equations (PDEs). Control and adaptability over the convergence phase are guaranteed by this approach. When it comes to solving nonlinear differential equations, the Optimal Homotopy Analysis Approach provides improved accuracy and flexibility. Compared to conventional approaches, it enables the convergence-control parameter to be optimally computed, resulting in more precise and dependable solutions. Several modeled equations arising in mathematical physics are nonlinear and the exact solution does not exist. Our current focus is on using the technique of optimal homotopy analysis^42–45 to calculate solutions. The appropriate initial guesses $(f_0,g_0,{\theta }_0,{\varphi }_0)$ and the linear operators $({\overbrace{L}}_f,{\overbrace{L}}_g,{\overbrace{L}}_{\theta },{\overbrace{L}}_{\varphi })$ are outlined below

$$\begin{array}{c}{f}_0\left(\eta \right)=1-\frac{1}{e^{\eta }},g_0\left(\eta \right)=\alpha (1-\frac{1}{e^{\eta }}),{\theta }_0\left(\eta \right)=\frac{1}{e^{\eta }},\hfill \\ {\varphi }_0\left(\eta \right)=-\frac{\mathrm{Nt}}{\mathrm{Nb}}\frac{1}{e^{\eta }},\hfill \end{array}$$

(23)

$${\overbrace{L}}_f=f″\prime -f\prime ,{\overbrace{L}}_g=g″\prime -g\prime ,{\overbrace{L}}_{\theta }=\theta ″-\theta ,{\overbrace{L}}_{\varphi }=\varphi ″-\varphi .$$

(24)

The operators mentioned above meet the criteria listed below

$$\begin{array}{c}\multicolumn{1}{c}{{\overbrace{L}}_f[{\tilde{C}}_1+{\tilde{C}}_2{e}^{\eta }+{\tilde{C}}_3{e}^{-\eta }]=0,{\overbrace{L}}_g[{\tilde{C}}_4+{\tilde{C}}_5{e}^{\eta }+{\tilde{C}}_6{e}^{-\eta }]=0,}\end{array}$$

$${\overbrace{L}}_{\theta }[{\tilde{C}}_7{e}^{\eta }+{\tilde{C}}_8{e}^{-\eta }]=0,{\overbrace{L}}_{\varphi }[{\tilde{C}}_9{e}^{\eta }+{\tilde{C}}_{10}{e}^{-\eta }]=0.$$

(25)

In which ${\tilde{C}}_i$ $(i=1-10)$ the arbitrary constants are denoted by. The problems with zeroth-order deformation are provided by

$$(1-\stackrel{··}{\mathrm{Ϙ}}){\overbrace{L}}_f[\overbrace{f}(\eta ,\stackrel{··}{\mathrm{Ϙ}})-f_0(\eta )]=\stackrel{··}{\mathrm{Ϙ}}{\hslash }_f{\overbrace{N}}_f[\overbrace{f}(\eta ,\stackrel{··}{\mathrm{Ϙ}}),\overbrace{g}(\eta ,\stackrel{··}{\mathrm{Ϙ}})],$$

(26)

$$(1-\stackrel{··}{\mathrm{Ϙ}}){\overbrace{L}}_g[\overbrace{g}(\eta ,\stackrel{··}{\mathrm{Ϙ}})-g_0(\eta )]=\stackrel{··}{\mathrm{Ϙ}}{\hslash }_g{\overbrace{N}}_g[\overbrace{f}(\eta ,\stackrel{··}{\mathrm{Ϙ}}),\overbrace{g}(\eta ,\stackrel{··}{\mathrm{Ϙ}})],$$

(27)

$$\begin{array}{c}(1-\stackrel{··}{\mathrm{Ϙ}}){\overbrace{L}}_{\theta }[\overbrace{\theta }(\eta ,\stackrel{··}{\mathrm{Ϙ}})-{\theta }_0(\eta )]\hfill \\ =\stackrel{··}{\mathrm{Ϙ}}{\hslash }_{\theta }{\overbrace{N}}_{\theta }[\overbrace{f}(\eta ,\stackrel{··}{\mathrm{Ϙ}}),\overbrace{g}(\eta ,\stackrel{··}{\mathrm{Ϙ}}),\overbrace{\theta }(\eta ,\stackrel{··}{\mathrm{Ϙ}}),\varphi (\eta ,\stackrel{··}{\mathrm{Ϙ}})],\hfill \end{array}$$

(28)

$$\begin{array}{c}(1-\stackrel{··}{\mathrm{Ϙ}}){\overbrace{L}}_{\varphi }[\overbrace{\varphi }(\eta ,\stackrel{··}{\mathrm{Ϙ}})-{\varphi }_0(\eta )]\hfill \\ =\stackrel{··}{\mathrm{Ϙ}}{\hslash }_{\varphi }{\overbrace{N}}_{\varphi }[\overbrace{f}(\eta ,\stackrel{··}{\mathrm{Ϙ}}),\overbrace{g}(\eta ,\stackrel{··}{\mathrm{Ϙ}}),\overbrace{\theta }(\eta ,\stackrel{··}{\mathrm{Ϙ}}),\overbrace{\varphi }(\eta ,\stackrel{··}{\mathrm{Ϙ}})],\hfill \end{array}$$

(29)

$$\begin{array}{cc}\multicolumn{1}{c}{\overbrace{f}(0,\stackrel{··}{\mathrm{Ϙ}})}& =0,\overbrace{f}\prime (0,\stackrel{··}{\mathrm{Ϙ}})=1,\overbrace{f}\prime (\infty ,\stackrel{··}{\mathrm{Ϙ}})=0,\overbrace{g}(0,\stackrel{··}{\mathrm{Ϙ}})=0,\hfill \\ \multicolumn{1}{c}{\overbrace{g}\prime (0,\stackrel{··}{\mathrm{Ϙ}})}& =\alpha ,\overbrace{g}\prime (\infty ,\stackrel{··}{\mathrm{Ϙ}})=0,\overbrace{\theta }(0,\stackrel{··}{\mathrm{Ϙ}})=1,\hfill \\ \multicolumn{1}{c}{\overbrace{\theta }(\infty ,\stackrel{··}{\mathrm{Ϙ}})}& =0,\mathrm{Nb}\overbrace{\varphi }\prime (0,\stackrel{··}{\mathrm{Ϙ}})+\mathrm{Nt}\overbrace{\theta }\prime (0,\stackrel{··}{\mathrm{Ϙ}})=0,\overbrace{\varphi }(\infty ,\stackrel{··}{\mathrm{Ϙ}})=0,\hfill \end{array}$$

(30)

$$\begin{array}{c}\multicolumn{1}{c}{{\overbrace{N}}_f[\overbrace{f}(\eta ;\stackrel{··}{\mathrm{Ϙ}}),\overbrace{g}(\eta ;\stackrel{··}{\mathrm{Ϙ}})]=A\frac{{\partial }^3\overbrace{f}}{\partial {\eta }^3}+\tilde{n}{(-\frac{{\partial }^{2}f\prime }{\partial {\eta }^2})}^{\tilde{n}-1}\frac{{\partial }^3\overbrace{f}}{\partial {\eta }^3}}\\ +\left(\frac{2\tilde{n}}{\tilde{n}+1}\right)\overbrace{f}\frac{{\partial }^{2}f\prime }{\partial {\eta }^2}-{\left(\frac{\partial \overbrace{f}}{\partial \eta }\right)}^2+g\prime \frac{{\partial }^{2}f\prime }{\partial {\eta }^2}-M^2\frac{\partial \overbrace{f}}{\partial \eta },\hfill \end{array}$$

(31)

$$\begin{array}{c}\multicolumn{1}{c}{{\overbrace{N}}_g[\overbrace{f}(\eta ;\stackrel{··}{\mathrm{Ϙ}}),\overbrace{g}(\eta ;\stackrel{··}{\mathrm{Ϙ}})]=A\frac{{\partial }^3\overbrace{g}}{\partial {\eta }^3}+{(-\frac{{\partial }^{2}f\prime }{\partial {\eta }^2})}^{\tilde{n}-1}\frac{{\partial }^3\overbrace{g}}{\partial {\eta }^3}}\\ -(\tilde{n}-1)\frac{{\partial }^{2}g\prime }{\partial {\eta }^2}\frac{{\partial }^3\overbrace{f}}{\partial {\eta }^3}{(-\frac{{\partial }^{2}f\prime }{\partial {\eta }^2})}^{\tilde{n}-2}+\left(\frac{2\tilde{n}}{\tilde{n}+1}\right)\overbrace{f}\frac{{\partial }^2\overbrace{g}}{\partial {\eta }^2}\hfill \\ -{\left(\frac{\partial \overbrace{g}}{\partial \eta }\right)}^2+\overbrace{g}\frac{{\partial }^2\overbrace{g}}{\partial {\eta }^2}-M^2\frac{\partial \overbrace{g}}{\partial \eta },\hfill \end{array}$$

(32)

$$\begin{array}{l}{\overbrace{N}}_{\theta }[\overbrace{f}(\eta ;\stackrel{··}{\mathrm{Ϙ}}),\overbrace{g}(\eta ;\stackrel{··}{\mathrm{Ϙ}}),\overbrace{\theta }(\eta ;\stackrel{··}{\mathrm{Ϙ}}),\overbrace{\varphi }(\eta ;\stackrel{··}{\mathrm{Ϙ}})]=\frac{{\partial }^2\overbrace{\theta }}{\partial {\eta }^2}\\ +Pr\left(\frac{2\tilde{n}}{\tilde{n}+1}\right)\overbrace{f}\frac{\partial \overbrace{\theta }}{\partial \eta }+Pr\overbrace{g}\frac{\partial \overbrace{\theta }}{\partial \eta }+\mathrm{NbPr}\frac{\partial \overbrace{\theta }}{\partial \eta }\frac{\partial \overbrace{\varphi }}{\partial \eta }+\mathrm{NtPr}{\left(\frac{\partial \overbrace{\theta }}{\partial \eta }\right)}^2,\end{array}$$

(33)

$$\begin{array}{l}{\overbrace{N}}_{\varphi }[\overbrace{f}(\eta ;\stackrel{··}{\mathrm{Ϙ}}),\overbrace{g}(\eta ;\stackrel{··}{\mathrm{Ϙ}}),\overbrace{\theta }(\eta ;\stackrel{··}{\mathrm{Ϙ}}),\overbrace{\varphi }(\eta ;\stackrel{··}{\mathrm{Ϙ}})]=\frac{{\partial }^2\overbrace{\varphi }}{\partial {\eta }^2}\\ +\mathrm{LePr}\left(\frac{2\tilde{n}}{\tilde{n}+1}\right)\overbrace{f}\frac{\partial \overbrace{\varphi }}{\partial \eta }+\mathrm{LePr}\overbrace{g}\frac{\partial \overbrace{\varphi }}{\partial \eta }+\left(\frac{\mathrm{Nt}}{\mathrm{Nb}}\right)\frac{{\partial }^2\overbrace{\theta }}{\partial {\eta }^2}.\end{array}$$

(34)

In the phrases above, $\stackrel{··}{\mathrm{Ϙ}}\in [0,1]$ represents the embedding parameter, $h_f,h_g,h_{\theta }$ and $h_{\varphi }$ the non-zero auxiliary parameters, and ${\overbrace{N}}_f$ , ${\overbrace{N}}_g$ , ${\overbrace{N}}_{\theta }$ , and ${\overbrace{N}}_{\Phi }$ , the nonlinear operators. Setting $\stackrel{··}{\mathrm{Ϙ}}$ = 0 and $\stackrel{··}{\mathrm{Ϙ}}=$ 1 we have

$$\overbrace{f}(\eta ;0)=f_0\left(\eta \right),\overbrace{f}(\eta ;1)=f\left(\eta \right),$$

(35)

$$\overbrace{g}(\eta ;0)=g_0\left(\eta \right),\overbrace{g}(\eta ;1)=g\left(\eta \right),$$

(36)

$$\overbrace{\theta }(\eta ;0)={\theta }_0\left(\eta \right),\overbrace{\theta }(\eta ;1)=\theta \left(\eta \right),$$

(37)

$$\overbrace{\varphi }(\eta ;0)={\varphi }_0\left(\eta \right),\overbrace{\varphi }(\eta ;1)=\varphi \left(\eta \right),$$

(38)

When $\stackrel{··}{\mathrm{Ϙ}}$ changes as of 0–1 then $\overbrace{f}(\eta ;\stackrel{··}{\mathrm{Ϙ}}),\overbrace{g}(\eta ;\stackrel{··}{\mathrm{Ϙ}}),\overbrace{\theta }(\eta ;\stackrel{··}{\mathrm{Ϙ}})\mathrm{and}\overbrace{\varphi }(\eta ;\stackrel{··}{\mathrm{Ϙ}})$ vary from the preliminary guesses $f_0\left(\eta \right),g_0\left(\eta \right),{\theta }_0\left(\eta \right)$ , and ${\varphi }_0\left(\eta \right)$ to the consequences $f\left(\eta \right)$ , $g\left(\eta \right),\theta \left(\eta \right)\mathrm{and}\varphi \left(\eta \right)$ , respectively. The Taylor series expansion’s implementation provides

$$\overbrace{f}(\eta ;\stackrel{··}{\mathrm{Ϙ}})=f_0\left(\eta \right)+\sum _{\sout{\mathrm{m}}=1}^{\infty }{f}_{\sout{\mathrm{m}}}\left(\eta \right){\stackrel{··}{\mathrm{Ϙ}}}^{\sout{\mathrm{m}}},f_{\sout{\mathrm{m}}}\left(\eta \right)=\frac{1}{\sout{\mathrm{m}}!}\frac{{\partial }^{\sout{\mathrm{m}}}\overbrace{f}(\eta ,\stackrel{··}{\mathrm{Ϙ}})}{\partial {\stackrel{··}{\mathrm{Ϙ}}}^{\sout{\mathrm{m}}}}{|}_{\stackrel{··}{\mathrm{Ϙ}}=0},$$

(39)

$$\overbrace{g}(\eta ;\stackrel{··}{\mathrm{Ϙ}})=g_0\left(\eta \right)+\sum _{\sout{\mathrm{m}}=1}^{\infty }{g}_{\sout{\mathrm{m}}}\left(\eta \right){\stackrel{··}{\mathrm{Ϙ}}}^{\sout{\mathrm{m}}},g_{\sout{\mathrm{m}}}\left(\eta \right)=\frac{1}{\tilde{m}!}\frac{{\partial }^{\sout{\mathrm{m}}}\overbrace{g}(\eta ,\stackrel{··}{\mathrm{Ϙ}})}{\partial {\stackrel{··}{\mathrm{Ϙ}}}^{\sout{\mathrm{m}}}}{|}_{\stackrel{··}{\mathrm{Ϙ}}=0},$$

(40)

$$\theta (\eta ;\stackrel{··}{\mathrm{Ϙ}})={\theta }_0\left(\eta \right)+\sum _{\sout{\mathrm{m}}=1}^{\infty }{\theta }_{\sout{\mathrm{m}}}\left(\eta \right){\stackrel{··}{\mathrm{Ϙ}}}^{\sout{\mathrm{m}}},{\theta }_{\sout{\mathrm{m}}}\left(\eta \right)=\frac{1}{\sout{\mathrm{m}}!}\frac{{\partial }^{\sout{\mathrm{m}}}\theta (\eta ,\stackrel{··}{\mathrm{Ϙ}})}{\partial {\stackrel{··}{\mathrm{Ϙ}}}^{\sout{\mathrm{m}}}}{|}_{\stackrel{··}{\mathrm{Ϙ}}=0},$$

(41)

$$\varphi (\eta ;\stackrel{··}{\mathrm{Ϙ}})={\varphi }_0(\eta )+\sum _{\sout{\mathrm{m}}=1}^{\infty }\varphi \left(\eta \right){\stackrel{··}{\mathrm{Ϙ}}}^{\sout{\mathrm{m}}},{\varphi }_{\sout{\mathrm{m}}}\left(\eta \right)=\frac{1}{\sout{\mathrm{m}}!}\frac{{\partial }^{\sout{\mathrm{m}}}\varphi (\eta ,\stackrel{··}{\mathrm{Ϙ}})}{\partial {\stackrel{··}{\mathrm{Ϙ}}}^{\sout{\mathrm{m}}}}{|}_{\stackrel{··}{\mathrm{Ϙ}}=0}.$$

(42)

The convergence of equations (39–42) strongly is dependent on making the right decisions of ${\hslash }_f,{\hslash }_g,{\hslash }_{\theta }\mathrm{and}$ ${\hslash }_{\varphi }$ . Considering that ${\hslash }_f,{\hslash }_g,{\hslash }_{\theta }\mathrm{and}$ ${\hslash }_{\varphi }$ are selected in a way that equations (39–42) converge at $\stackrel{··}{\mathrm{Ϙ}}$ = 1 then

$$f\left(\eta \right)=f_0\left(\eta \right)+\sum _{\sout{\mathrm{m}}=1}^{\infty }{f}_{\sout{\mathrm{m}}}\left(\eta \right),$$

(43)

$$g\left(\eta \right)=g_0\left(\eta \right)+\sum _{\sout{\mathrm{m}}=1}^{\infty }{g}_{\sout{\mathrm{m}}}\left(\eta \right),$$

(44)

$$\theta \left(\eta \right)={\theta }_0\left(\eta \right)+\sum _{\sout{\mathrm{m}}=1}^{\infty }{\theta }_{\sout{\mathrm{m}}}\left(\eta \right),$$

(45)

$$\varphi \left(\eta \right)={\Phi }_0\left(\eta \right)+\sum _{\sout{\mathrm{m}}=1}^{\infty }{\varphi }_{\sout{\mathrm{m}}}\left(\eta \right).$$

(46)

The $\sout{\mathrm{m}}$ th-order deformation issues are described as follows:

$${\overbrace{L}}_f[f_{\sout{\mathrm{m}}}\left(\eta \right){\tilde{\chi }}_{\sout{\mathrm{m}}}{f}_{\sout{\mathrm{m}}-1}\left(\eta \right)={\hslash }_f{R}_f^{\sout{\mathrm{m}}}\left(\eta \right)],$$

(47)

$${\overbrace{L}}_g[g_{\sout{\mathrm{m}}}\left(\eta \right){\tilde{\chi }}_{\sout{\mathrm{m}}}{g}_{\sout{\mathrm{m}}-1}\left(\eta \right)={\hslash }_g{R}_g^{\sout{\mathrm{m}}}\left(\eta \right)],$$

(48)

$${\overbrace{L}}_{\theta }[{\theta }_{\sout{\mathrm{m}}}\left(\eta \right){\tilde{\chi }}_{\sout{\mathrm{m}}}{\theta }_{\sout{\mathrm{m}}-1}\left(\eta \right)={\hslash }_{\theta }{R}_{\theta }^{\sout{\mathrm{m}}}\left(\eta \right)],$$

(49)

$${\overbrace{L}}_{\varphi }[{\varphi }_{\sout{\mathrm{m}}}\left(\eta \right){\tilde{\chi }}_{\sout{\mathrm{m}}}{\varphi }_{\sout{\mathrm{m}}-1}\left(\eta \right)={\hslash }_{\varphi }{R}_{\varphi }^{\sout{\mathrm{m}}}\left(\eta \right)],$$

(50)

$$f_{\sout{\mathrm{m}}}\left(0\right)=f_{\sout{\mathrm{m}}}\prime \left(0\right)=f_{\sout{\mathrm{m}}}\prime \left(\infty \right),g_{\sout{\mathrm{m}}}\left(0\right)=g_{\sout{\mathrm{m}}}\prime \left(0\right)=g_{\sout{\mathrm{m}}}\prime \left(\infty \right),$$

(51)

$${\theta }_{\sout{\mathrm{m}}}\left(0\right)={\theta }_{\sout{\mathrm{m}}}\prime \left(\infty \right)=0,\mathrm{Nb}{\varphi }_{\sout{\mathrm{m}}}\prime \left(0\right)+\mathrm{Nt}{\theta }_{\sout{\mathrm{m}}}\prime \left(0\right)=0,{\varphi }_{\sout{\mathrm{m}}}\left(\infty \right)=0.$$

(52)

For $\tilde{n}=1,$ the nonlinear operators are provided by

$$\begin{array}{c}\multicolumn{1}{c}{R_f^{\sout{\mathrm{m}}}\left(\eta \right)={\mathrm{Af}}_{\sout{\mathrm{m}}-1}″\prime +f_{\sout{\mathrm{m}}-1}″\prime +\sum _{k=0}^{\sout{\mathrm{m}}-1}(f_{\sout{\mathrm{m}}-1-k}{f}_k″+g_{\sout{\mathrm{m}}-1-k}{f}_k″)}\\ -\sum _{k=0}^{\sout{\mathrm{m}}-1}{f}_{\sout{\mathrm{m}}-1-k}\prime f_k\prime -M^2{f}_{\sout{\mathrm{m}}-1}\prime ,\hfill \end{array}$$

(53)

$$\begin{array}{c}\multicolumn{1}{c}{R_g^{\sout{\mathrm{m}}}\left(\eta \right)={\mathrm{Ag}}_{\sout{\mathrm{m}}-1}″\prime +g_{\sout{\mathrm{m}}-1}″\prime +\sum _{k=0}^{\sout{\mathrm{m}}-1}(f_{\sout{\mathrm{m}}-1-k}{g}_k″+g_{\sout{\mathrm{m}}-1-k}{g}_k″)}\\ -\sum _{k=0}^{\sout{\mathrm{m}}-1}{g}_{\sout{\mathrm{m}}-1-k}\prime g_k\prime -M^2{g}_{\sout{\mathrm{m}}-1}\prime ,\hfill \end{array}$$

(54)

$$\begin{array}{c}\multicolumn{1}{c}{R_{\theta }^{\sout{\mathrm{m}}}\left(\eta \right)={\theta }_{\sout{\mathrm{m}}-1}″+\mathrm{Pr}\sum _{k=0}^{\sout{\mathrm{m}}-1}(f_{\sout{\mathrm{m}}-1-k}{\theta }_k\prime +g_{\sout{\mathrm{m}}-1-k}{\theta }_k\prime )}\\ +\mathrm{PrNb}\sum _{k=0}^{\sout{\mathrm{m}}-1}{\theta }_{\sout{\mathrm{m}}-1-k}\prime {\varphi }_k\prime +\mathrm{PrNt}\sum _{k=0}^{\sout{\mathrm{m}}-1}{\theta }_{\sout{\mathrm{m}}-1-k}\prime {\theta }_k\prime ,\hfill \end{array}$$

(55)

$$R_{\varphi }^{\sout{\mathrm{m}}}\left(\eta \right)={\varphi }_{\sout{\mathrm{m}}-1}″\left(\tilde{n}\right)+\mathrm{LePr}\sum _{k=0}^{\sout{\mathrm{m}}-1}(f_{\sout{\mathrm{m}}-1-k}{\varphi }_k\prime +g_{\sout{\mathrm{m}}-1-k}{\varphi }_k\prime )+\left(\frac{\mathrm{Nt}}{\mathrm{Nb}}\right){\theta }_{\sout{\mathrm{m}}-1}″.$$

(56)

The irregular machinists for $\tilde{n}=2$ are given by

$$\begin{array}{l}{R}_f^{\sout{\mathrm{m}}}\left(\eta \right)=A{f}_{\sout{\mathrm{m}}-1^{\prime \prime \prime }}-2\sum _{k=0}^{\sout{\mathrm{m}}-1}{f^{″}}_{\sout{\mathrm{m}}-1-k}{f}_{k^{‴}}+\frac{4}{3}\sum _{k=0}^{\sout{\mathrm{m}}-1}{f}_{\sout{\mathrm{m}}-1-k}{f}_{k^{″}}\\ -\sum _{k=0}^{\sout{\mathrm{m}}-1}{f}_{\sout{\mathrm{m}}-1-k^{\prime }}{f}_{k^{\prime }}+\sum _{k=0}^{\sout{\mathrm{m}}-1}{g}_{\sout{\mathrm{m}}-1-k^{\prime }}{f}_{k^{″}}-M^2{f}_{\sout{\mathrm{m}}-1^{\prime }},\end{array}$$

(57)

$$\begin{array}{c}\multicolumn{1}{c}{R_g^{\sout{\mathrm{m}}}\left(\eta \right)={\mathrm{Ag}}_{\sout{\mathrm{m}}-1}″\prime -\sum _{k=0}^{\sout{\mathrm{m}}-1}{f″}_{\sout{\mathrm{m}}-1-k}{g}_k″\prime -\sum _{k=0}^{\sout{\mathrm{m}}-1}{g}_{\sout{\mathrm{m}}-1-k}″f_k″\prime }\\ +\frac{4}{3}\sum _{k=0}^{\sout{\mathrm{m}}-1}{f}_{\sout{\mathrm{m}}-1-k}{g}_k″-\sum _{k=0}^{\sout{\mathrm{m}}-1}{g}_{\sout{\mathrm{m}}-1-k}\prime g_k\prime +\sum _{k=0}^{\sout{\mathrm{m}}-1}{g}_{\sout{\mathrm{m}}-1-k}{g}_k″-M^2{g}_{\sout{\mathrm{m}}-1}\prime ,\hfill \end{array}$$

(58)

$$\begin{array}{c}\multicolumn{1}{c}{R_{\theta }^{\sout{\mathrm{m}}}\left(\eta \right)={\theta }_{\sout{\mathrm{m}}-1}″+\frac{4}{3}\mathrm{Pr}\sum _{k=0}^{\sout{\mathrm{m}}-1}{f}_{\sout{\mathrm{m}}-1-k}{\theta }_k\prime +\mathrm{Pr}\sum _{k=0}^{\sout{\mathrm{m}}-1}{g}_{\sout{\mathrm{m}}-1-k}{\theta }_k\prime }\\ +\mathrm{PrNb}\sum _{k=0}^{\sout{\mathrm{m}}-1}{\theta }_{\sout{\mathrm{m}}-1-k}\prime {\varphi }_k\prime +\mathrm{PrNt}\sum _{k=0}^{\sout{\mathrm{m}}-1}{\theta }_{\sout{\mathrm{m}}-1-k}\prime {\theta }_k\prime ,\hfill \end{array}$$

(59)

$$\begin{array}{l}{R}_{\varphi }^{\sout{\mathrm{m}}}\left(\eta \right)={\varphi }_{\sout{\mathrm{m}}-1^{\prime \prime }}\left(\eta \right)+\frac{4}{3}LePr\sum _{k=0}^{\sout{\mathrm{m}}-1}{f}_{\sout{\mathrm{m}}-1-k}{\varphi }_{k^{\prime }}\\ +LePr\sum _{k=0}^{\sout{\mathrm{m}}-1}{g}_{\sout{\mathrm{m}}-1-k}{\varphi }_{k^{\prime }}+\left(\frac{Nt}{Nb}\right){\theta }_{\sout{\mathrm{m}}-1^{\prime \prime }},\end{array}$$

(60)

$${\tilde{\chi }}_{\sout{\mathrm{m}}}=\{\begin{array}{c}0,\sout{\mathrm{m}}\le 1,\hfill \\ \multicolumn{1}{c}{1,\sout{\mathrm{m}}>1.}\end{array}$$

(61)

Likewise for $\tilde{n}$ = 3, 4, etc. The general conclusions $(f_m,f_g,f_{\theta },f_{\varphi })$ to equations (47) to (50) in terms of special conclusions $(f_m^{*},f_g^{*},f_{\theta }^{*},f_{\varphi }^{*})$ are presented in the following:

$$f_{\sout{\mathrm{m}}}\left(\eta \right)=f_{\sout{\mathrm{m}}}^{*}\left(\eta \right)+{\tilde{C}}_1+{\tilde{C}}_2{e}^{\eta }+{\tilde{C}}_3{e}^{-\eta },$$

(62)

$$g_{\sout{\mathrm{m}}}\left(\eta \right)=g_{\sout{\mathrm{m}}}^{*}\left(\eta \right)+{\tilde{C}}_4+{\tilde{C}}_5{e}^{\eta }+{\tilde{C}}_6{e}^{-\eta },$$

(63)

$$f_{\sout{\mathrm{m}}}\left(\eta \right)={\theta }_{\sout{\mathrm{m}}}^{*}\left(\eta \right)+{\tilde{C}}_7{e}^{\eta }+{\tilde{C}}_8{e}^{-\eta },$$

(64)

$${\varphi }_{\sout{\mathrm{m}}}\left(\eta \right)={\varphi }_{\sout{\mathrm{m}}}^{*}\left(\eta \right)+{\tilde{C}}_9{e}^{\eta }+{\tilde{C}}_{10}{e}^{-\eta },$$

(65)

In which the coefficients ${\tilde{C}}_i(i=1-10)$ via boundary situations (51) and (52) are specified as

$${\tilde{C}}_2={\tilde{C}}_5={\tilde{C}}_7={\tilde{C}}_9=0,{\tilde{C}}_3=\frac{\partial f_{\sout{\mathrm{m}}}^{*}\left(\eta \right)}{\partial \eta }{|}_{\eta =0},{\tilde{C}}_1=-{\tilde{C}}_3-f_{\sout{\mathrm{m}}}^{*}\left(0\right),$$

(66)

$${\tilde{C}}_6=\frac{\partial g_{\sout{\mathrm{m}}}^{*}\left(\eta \right)}{\partial \eta }{|}_{\eta =0},{\tilde{C}}_4=-{\tilde{C}}_6-g_{\sout{\mathrm{m}}}^{*}\left(0\right),{\tilde{C}}_8=-{\theta }_{\sout{\mathrm{m}}}^{*}\left(0\right),$$

(67)

$${\tilde{C}}_{10}=\frac{\partial {\varphi }_{\sout{\mathrm{m}}}^{*}\left(\eta \right)}{\partial \eta }{|}_{\eta =0}+\frac{\mathrm{Nt}}{\mathrm{Nb}}(-{\tilde{C}}_8=\frac{\partial {\theta }_{\sout{\mathrm{m}}}^{*}\left(\eta \right)}{\partial \eta }{|}_{\eta =0}).$$

(68)

Optimal convergence control parameters

Observe that the non-zero auxiliary parameters $({\hslash }_f,{\hslash }_g,{\hslash }_{\theta }\mathrm{and}{\hslash }_{\varphi }$ ) are present in the series solutions (46–49), which control the homotopy series solutions rate and convergence region. To find the optimal expressions of $({\hslash }_f,{\hslash }_g,{\hslash }_{\theta }\mathrm{and}{\hslash }_{\varphi }$ ), by utilizing the average squared residual errors, we have used the minimization principle

$${\epsilon }_{\sout{\mathrm{m}}}^f=\frac{1}{k+1}\sum _{j=0}^k{\left[{\overbrace{N}}_f{(\sum _{i=0}^{\sout{\mathrm{m}}}\overbrace{f}\left(\eta \right),\sum _{i=0}^{\sout{\mathrm{m}}}\overbrace{g}\left(\eta \right))}_{\eta =j\delta \eta }\right]}^{2}d\eta ,$$

(69)

$${\epsilon }_{\sout{\mathrm{m}}}^g=\frac{1}{k+1}\sum _{j=0}^k{\left[{\overbrace{N}}_g{(\sum _{i=0}^{\sout{\mathrm{m}}}\overbrace{f}\left(\eta \right),\sum _{i=0}^{\sout{\mathrm{m}}}\overbrace{g}\left(\eta \right))}_{\eta =j\delta \eta }\right]}^{2}d\eta ,$$

(70)

$${\epsilon }_{\sout{\mathrm{m}}}^{\theta }=\frac{1}{k+1}\sum _{j=0}^k{\left[{\overbrace{N}}_{\theta }{(\sum _{i=0}^{\sout{\mathrm{m}}}\overbrace{f}\left(\eta \right),\sum _{i=0}^{\sout{\mathrm{m}}}\overbrace{g}\left(\eta \right),\sum _{i=0}^{\sout{\mathrm{m}}}\theta \left(\eta \right),\sum _{i=0}^{\sout{\mathrm{m}}}\varphi \left(\eta \right))}_{\eta =j\delta \eta }\right]}^{2}d\eta ,$$

(71)

$${\epsilon }_{\sout{\mathrm{m}}}^{\varphi }=\frac{1}{k+1}\sum _{j=0}^k{\left[{\overbrace{N}}_{\varphi }{(\sum _{i=0}^{\sout{\mathrm{m}}}\overbrace{f}\left(\eta \right),\sum _{i=0}^{\sout{\mathrm{m}}}\overbrace{g}\left(\eta \right),\sum _{i=0}^m\theta \left(\eta \right),\sum _{i=0}^{\sout{\mathrm{m}}}\varphi \left(\eta \right))}_{\eta =j\delta \eta }\right]}^{2}d\eta ,$$

(72)

$${\epsilon }_{\sout{\mathrm{m}}}^t={\epsilon }_{\sout{\mathrm{m}}}^f+{\epsilon }_{\sout{\mathrm{m}}}^g+{\epsilon }_{\sout{\mathrm{m}}}^{\theta }+{\epsilon }_{\sout{\mathrm{m}}}^{\varphi }.$$

(73)

Where ${\epsilon }_{\sout{\mathrm{m}}}^t$ is the total squared residual blunder, $\delta \eta$ =0.5, and $k=20$ . The Mathematica package $\mathrm{BVPh}2.0$ is utilized to diminish the total average squared residual blunder. The main idea is to find the associated local optimum convergence control settings while minimizing the overall average squared residuals. For example, a situation has been thought of where $A=0.5,\alpha =0.7,n=2.0,M=0.1=\mathrm{Nt},\mathrm{Nb}=0.3,\mathrm{Le}=0.1,\mathrm{Pr}=1.0,\epsilon =0.2,{\epsilon }_1=0.7,R=0.2$ and $\mathrm{Ec}=0.4$ . At the approximate fourth order, the best values for the convergence control parameters are ${\hslash }_f=-0.2632846314731318,{\hslash }_g=-0.9595099790028533,{\hslash }_{\theta }=-0.8180795452029149$ and ${\hslash }_{\varphi }=-1.323262691215517$ .

Results and discussion

The influences of many physical features are inspected in this segment on the distribution of nanoparticle concentration $\varphi (n)$  and dimensionless temperature $\theta (n)$ .

In this section, we will discuss results and numerical graphs. We observed Sisko fluid with the variable magnetic field, to investigate heat and mass transfer in bi-directional stretched flow. Numerical simulations of our investigation are presented in Figures 2 to 21.

Figure 2.

Influence $A$ on $f\prime$ when $\tilde{n}=2.0,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 3.

Influence of $M$ on $f\prime \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 4.

Influence of $A$ on $g\prime$ when $\tilde{n}=2.0,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 5.

Influence of $M$ on $g\prime$ when $\tilde{n}=2.0,A=0.5,\alpha =0.7,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 6.

Influence of $M$ on $\theta \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 7.

Influence of $\mathrm{Nb}$ on $\theta \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Le}=0.1$ .

Figure 8.

Influence of $\mathrm{Nt}$ on $\theta \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 9.

Influence of $\epsilon$ on $\theta \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 10.

Influence of $\mathrm{Ec}$ on $\theta \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 11.

Influence of $R$ on $\theta \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 12.

Influence of $\alpha$ on $\varphi$ when $\tilde{n}=2.0,A=0.5,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 13.

Influence of $\mathrm{Le}$ on $\varphi$ when $\tilde{n}=2.0,A=0.5,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3.$ .

Figure 14.

Influence of $M$ on $\varphi$ when $\tilde{n}=2.0,A=0.5,\alpha =0.7,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 15.

Influence of $M$ on $\varphi \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 16.

Influence of $\mathrm{Nb}$ on $\varphi \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Le}=0.1$ .

Figure 17.

Influence of $\mathrm{Nb}$ on $\varphi \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Le}=0.1$ .

Figure 18.

Influence of $\mathrm{Nt}$ on $\varphi \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 19.

Influence of $\mathrm{Nt}$ on $\varphi \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 20.

Influence of $\mathrm{Pr}$ on $\varphi \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,M=0.1,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Figure 21.

Influence of ${\epsilon }_1$ on $\varphi \mathrm{when}\tilde{n}=2.0,A=0.5,\alpha =0.7,M=0.1,\mathrm{Pr}=1.0,\mathrm{Nt}=0.1,\mathrm{Nb}=0.3,\mathrm{Le}=0.1$ .

Velocity profiles

Figure 2 represents the impact of the Sisko fluid parameter $(A)$ . We have used different values of $A=0.0,0.5,1.0,1.5,2.0$ . In this case, a rise in the Sisko fluid parameter (A) causes the velocity scenario to expand. Figure 3 shows the impact of the magnetic parameter $(M)$ . We used diverse standards of $M=0.1,0.5,0.9,1.3,1.7$ . In this, the magnetic parameter appeared to decrease the fluid’s $f$ ′ velocity. Due to the MHD effect, a diminishing importance for magnetic parameters is shown. Figure 4 illustrates how the Sisko fluid parameter $(A)$ has an influence. Various values of A have been utilized. When stretching causes the Sisko fluid parameter $(A)$ to rise, the velocity scenario grows. Figure 5 expresses the impression of the magnetic factor $(M)$ . We utilized dissimilar morals of $M=0.1,0.5,0.9,1.3,1.7$ . In this, the magnetic parameter appeared to decrease the fluid’s $g\prime$ velocity. A decreasing role is observed for the magnetic parameter which occurs due to the MHD effect.

Temperature profile

Figure 6 demonstrates how the temperature profile $\theta \left(\eta \right)$ is affected by the magnetic factor $(M)$ . When we raise the values of the magnetic parameter in this case, the temperature increases. The augmentation in magnetic parameter cause to produce friction, which results to enhance temperature field. The behavior of the Brownian motion variable $(\mathrm{Nb})$ for various values is displayed in Figure 7. Figure 8 shows the characteristics of the thermophoresis factor $(\mathrm{Nt})$ in the thermal scenario. It has been noted that when a fluid’s morality increases, its temperature also rises noticeably $(\mathrm{Nt})$ . The suspension of small particles in the fluid causes the temperature difference between the sheet and the liquid to increase. Figure 9 presents the impact of thermal conductivity ( $\epsilon$ ) for different values $\epsilon =0.0,0.6,1.2,1.8,2.4$ . This shows that with an upsurge in $\epsilon$ , the temperature sketch $\theta \left(\eta \right)$ rises close to the stretching surface. An accelerating role is observed for small parameters which occurs due to the variable thermal conductivity model. Figure 10 represents the compartment of Eckert number $(\mathrm{Ec})$ . We used different values of $\mathrm{Ec}=10.0,90.0,170.0,250.0,330.0$ . The Eckert quantity refers to the correlation between motion kinetic energy and the differential in heat enthalpy. Because $(\mathrm{Ec})$ increases kinetic energy, a fluid’s temperature also rises as a result. With an increasing Eckert number, it is seen that the fluid temperature rises. Figure 11 presents the bearing of the radiation parameter $(R)$ for altered values $R=0.1,0.5,0.9,1.3,1.7$ . It exhibits how the thermal draft is exaggerated by the thermal radiation parameter. It is evident from the figure that the fluid temperature increased as $(R)$ values increased.

Concentration profile

Figure 12 presents the bearing of ratio factor $\alpha$ for different values of $\alpha =0.0,0.6,1.2,1.8,2.4$ . The fluid velocity reduces for big values of $\alpha$ because there is less distortion in the stretching of the wall. As was previously stated, heat slowly escapes from thick surfaces into the flow. As a result, raising the value will result in less heat being lost to the flow and lower temperature profiles. In the concentration profile, a similar behavior is seen. Figure 13 presents the compartment of Lewis number $(\mathrm{Le})$ for different values of $\mathrm{Le}=0.6,0.9,1.2,1.5,1.8$ . This illustrates how a weaker nanoparticle concentration field $\varphi \left(\eta \right)$ results from higher values of the Lewis quantity $(\mathrm{Le})$ . Figure 14 shows that a lessening in the nanomaterial spatial field $\varphi \left(\eta \right)$ is caused by an increase in the magnetic parameter . Figure 15 represents the magnetic constraint $(M)$ for different values of $M=8.0,12.0,16.0,20.0,24.0$ . This shows that an escalation in the magnetic constraint ( $M)$ causes the nanoparticle solutal field $\varphi \left(\eta \right)$ to be enhanced. Figure 16 represents the impression of the Brownian motion parameter $(\mathrm{Nb})$ for distinct values of $\mathrm{Nb}=0.3,0.4,0.5,0.6,0.7$ . As a result, the concentration decreases with increasing levels of $(\mathrm{Nb})$ . Figure 17 shows the behavior of the Brownian motion factor $(\mathrm{Nb})$ for different values of $(\mathrm{Nb})$ . Consequently, when the concentration rises, so does the amount of $(\mathrm{Nb})$ . The properties of the thermophoresis parameter $(\mathrm{Nt})$ on the spatial scenarios are represented in Figures 18 and 19. It is observed that a fluid’s solutal sketches considerably rise with growing morals of $(\mathrm{Nt})$ . The escalation in the temperature differential between the liquid and the sheet is caused by the suspension of microscopic particles in the fluid. Figure 20 represents the compartment of the Prandtl amount $(\mathrm{Pr})$ for different values $\mathrm{Pr}=0.7,1.0,1.3,1.6,1.9$ on fluid temperature. This exposes that a decrease in the field of nanomaterial spatial is shown by a rise in the Prandtl $(\mathrm{Pr})$ . Figure 21 represents the behavior of variable thermal conductivity ( ${\epsilon }_1$ ) for different values ${\epsilon }_1$ This shows that with an increase in $({\epsilon }_1)$ , the nanoparticle concentration field $\varphi \left(\eta \right)$ decreases. Small parameters show a diminishing influence, which is brought on by the changing diffusion coefficient model. Dimensionless tresses are tabulated in Table 1 against the Sisko fluid parameter, ratio, and magnetic parameters. Table 1 establishes comparative research for the current solution. Comparative research for the current solution is shown in Table 2. Table 2 shows the temperature transference proportion and the mass transmission ratio is presented in Table 3 against numerous parameters. Table 3 provides a comparison of dimensionless stresses.

Table 1.

For distinct combinations of $A$ , $M$ , and $\alpha$ , mathematical principles of the local skin friction quantities ${\tilde{c}}_{\mathrm{fx}}$ and ${\tilde{c}}_{\mathrm{fy}}$ are provided.

$A$	$M$	$\alpha$	${\tilde{c}}_{f\tilde{x}}$	${\tilde{c}}_{f\tilde{y}}$
0.1	0.5	0.7	−2.00082	−2.94405
0.3	0.5	0.7	−2.15778	−2.99811
0.5	0.5	0.7	−2.30601	−3.02126
0.7	0.5	0.7	−2.44599	−3.18440
0.3	0.1	0.2	−1.99150	−0.68234
0.3	0.3	0.2	−2.02823	−0.70459
0.3	0.5	0.2	−2.10248	−0.74955
0.3	0.7	0.2	−2.21577	−0.81810
0.2	0.4	0.1	−1.97377	−0.34909
0.2	0.4	0.3	−1.99558	−1.11103
0.2	0.4	0.5	−2.01724	−1.95698
0.2	0.4	0.7	−2.03875	−2.88568

Table 2.

Computation for $A,M,\alpha ,\mathrm{Le},\mathrm{Nb},\mathrm{Nt},$ and $\mathrm{Ec}$ are represented numerically by the Nusselt number ( $-\theta \prime \left(0\right)$ ).

$A$	$M$	$\alpha$	$L\mathrm{e}$	$\mathrm{Nb}$	$\mathrm{Nt}$	$\mathrm{Ec}$	$-{\theta }^{\prime }\left(0\right)$
0.1	0.5	0.7	0.4	0.5	0.2	0.2	0.608404
0.3	0.5	0.7	0.4	0.5	0.2	0.2	0.609133
0.5	0.5	0.7	0.4	0.5	0.2	0.2	0.609863
0.7	0.5	0.7	0.4	0.5	0.2	0.2	0.610592
0.3	0.1	0.2	0.4	0.5	0.2	0.2	0.586452
0.3	0.3	0.2	0.4	0.5	0.2	0.2	0.583026
0.3	0.5	0.2	0.4	0.5	0.2	0.2	0.576231
0.3	0.7	0.2	0.4	0.5	0.2	0.2	0.566214
0.2	0.4	0.1	0.4	0.5	0.2	0.2	0.572614
0.2	0.4	0.3	0.4	0.5	0.2	0.2	0.586874
0.2	0.4	0.5	0.4	0.5	0.2	0.2	0.600590
0.2	0.4	0.7	0.4	0.5	0.2	0.2	0.613832
0.3	0.4	0.6	0.1	0.5	0.2	0.2	0.613522
0.3	0.4	0.6	0.3	0.5	0.2	0.2	0.609598
0.3	0.4	0.6	0.5	0.5	0.2	0.2	0.605674
0.3	0.4	0.6	0.7	0.5	0.2	0.2	0.601750
0.7	0.4	0.6	0.8	0.1	0.2	0.2	0.654500
0.7	0.4	0.6	0.8	0.3	0.2	0.2	0.627044
0.7	0.4	0.6	0.8	0.5	0.2	0.2	0.601260
0.7	0.4	0.6	0.8	0.7	0.2	0.2	0.577150
0.3	0.4	0.2	0.5	0.7	0.1	0.2	0.564444
0.3	0.4	0.2	0.5	0.7	0.3	0.2	0.553629
0.3	0.4	0.2	0.5	0.7	0.5	0.2	0.546160
0.3	0.4	0.2	0.5	0.7	0.7	0.2	0.542037
0.7	0.4	0.6	0.8	0.5	0.4	0.1	0.640500
0.7	0.4	0.6	0.8	0.5	0.4	0.3	0.635722
0.7	0.4	0.6	0.8	0.5	0.4	0.5	0.630945
0.7	0.4	0.6	0.8	0.5	0.4	0.7	0.626167

Table 3.

Different values of ${\epsilon }_1$ , $\mathrm{Le},$ and $\mathrm{Nb}$ are represented numerically by the local Nusselt number $-\varphi \prime \left(0\right)$ .

${\epsilon }_1$	$\mathrm{Le}$	$\mathrm{Nb}$	$-\varphi \prime \left(0\right)$
0.1	0.5	0.6	0.627522
0.3	0.5	0.6	0.662896
0.5	0.5	0.6	0.720158
0.7	0.5	0.6	0.799308

Conclusions

Based on the investigation of mass and thermal transportation in the bi-directional stretched movement of Sisko liquid with a variable magnetic field, the following assumptions can be drawn:

• The presence of a magnetic factor $(M)$ significantly impacts the thermal and mass passage characteristics of the bi-directional stretched motion of Sisko liquid. The magnetic field tends to reduce the solutal and thermal sketches near the exterior of the stretched surface, indicating better heat and mass transfer rates.

• Elevated values of the Lewis coefficient $(\mathrm{Le})$ lead to a reduction in the nanoparticle concentration zone.

• The thermal appearance shows enhanced behavior for augmented values of thermal conductivity $(\epsilon )$ .

• Reduction in the range of nanoparticle concentration is noticed with higher values in varying heat conductivity $({\epsilon }_1)$ .

• The Prandtl count $(\mathrm{Pr})$ receives a prominent rise as the concentration of nanoparticles in the atmosphere decreases.

• Temperature profile is positively affected as a result of enhancement in radiated component $(R)$ .

• The amounts of the extending factor strongly influence the flow’s heat and mass transmission characteristics. Developed ethics of the stretching factor lead to better heat and bulk transfer ratios, while lower morals of the stretching factor lead to poorer warmth and mass transfer rates.

• The Sisko fluid exhibits non-Newtonian behavior, and its rheological properties strongly influence the flow’s heat and mass transmission characteristics. Advanced beliefs of the Sisko fluid parameter lead to poorer heat and mass transfer rates, while lower values of the Sisko fluid parameter lead to better heat and mass transport ratios.

• The bi-directional nature of the stretching flow results in the occurrence of two distinct flow regimes, which are characterized by different heat and mass transfer characteristics. The first regime corresponds to the stretching flow along the x-direction, while the second regime corresponds to the stretching flow along the y-direction.

Overall, the investigation reveals that the joint effects of the variable magnetic effect, stretching parameter, and rheological belongings of the Sisko fluid have a substantial impression on the heat and mass transportation characteristics of the bi-directional stretched flow. The findings of this study can be useful in understanding and optimizing heat and mass transfer processes in various industrial applications.

Future directions

The Prandtl count $(\mathrm{Pr})$ increases as the concentration of nanoparticles decreases, indicating a shift in the dynamics of heat and momentum diffusivity. In engineering applications, this affects heat transfer efficiency realistically, but theoretically, it emphasizes the impact of nanoparticles on fluid characteristics. The optimization of nanoparticle concentrations for improved heat management systems may be the subject of future study.