Mathematical analysis of heat and mass transfer on unsteady stagnation point flow of Riga plate with binary chemical reaction and thermal radiation effects

Abstract

To aid in the prevention of reaction explosions, chemical engineers and scientists must analyze the Arrhenius kinetics and activation energies of chemical reactions involving binary chemical mixtures. Nanofluids with an Arrhenius kinetic are crucial for a broad variety of uses in the industrial sector, involving the manufacture of chemicals, thermoelectric sciences, biomedical devices, polymer extrusion, and the enhancement of thermal systems via technology. The goal of this study is to determine how the presence of thermal radiation influences heat and mass transfer during free convective unsteady stagnation point flow across extending/shrinking vertical Riga plate in the presence of a binary chemical reaction where the activation energy of the reaction is known in advance. For the purpose of obtaining numerical solutions to the mathematical model of the present issue the Runge-Kutta (RK-IV) with shooting technique in Mathematica was used. Heat and mass transfer processes, as well as interrupted flow phenomena, are characterized and explained by diagrams in the suggested suction variables along boundary surface in the stagnation point flow approaching a permeable stretching/shrinking Riga Plate. Graphs illustrated the effects of many other factors on temperature, velocity, concentration, Sherwood and Nusselt number as well as skin friction in detail. Velocity profile increased with $Z,\lambda$ and $S$ and decreased with $\epsilon \text{.}$ Increasing values of $\epsilon ,\lambda$ and $S$ decline the temperature profile. The concentration profile boosts up with $Z,\alpha$ and slow down with $\epsilon ,Sc,\beta ,\delta$ and $n1$ parameters. Skin friction profile increased with $Z$ and $S$ and decreased with $\epsilon \text{.}$ Nusselt number profile increased with $S,Z\text{,}$$\epsilon$ and radiation. Sherwood number profile shows upsurges with $\epsilon ,Z,\alpha ,Sc,\beta ,S$ and $n1$ whereas slow down with $\delta$. So that the verdicts could be confirmed, a study was done to compare the most recent research with the results that had already been published for a certain case. The outcomes demonstrated strong concordance between the two sets of results.

Highlights

• •Chemical Binary reaction and thermal radiation impact on thermal enhancement of Riga Plate is analyzed.
• •Heat and Mass transfer across a permeable stretched Riga Plate with distinct physical phenomena are explored.
• •How does the Modified Hartmann number and unsteady parameter affect the flow's characteristics?
• •Would increasing the suction parameter prevent the boundary layer from splitting and improving heat and mass transfer?

Nomenclature

$a,c$

Constants $\left(s^{-1}\right)$

$\mathrm{C}$

Species concentrations

$C_w$

Concentrations at the wall

$C_{\infty }$

Ambient concentrations

$C_f$

Local skin friction coefficients (non-dimensional)

$D$

Molecular diffusivity

$d$

Width of electrodes and magnets parameter

$j_0$

Current density at electrode

${Nu}_x$

“Nusselt number (non-dimensional)

${Sh}_x$

Sherwood number (non-dimensional)”

$q_r$

Radiative heat flow

$T$

“Temperature of fluid $\left(K\right)$

$T_w$

Temperature at the wall $\left(K\right)$”

$T_{\infty }$

Temperature in the surrounding area $\left(K\right)$

$x,y$

Cartesian co-ordinates

$u,v$

Velocity components in $x$ & $y-$ directions $\left(m{s}^{-1}\right)$

$u_w$

Stretching/shrinking velocity $\left(m{s}^{-1}\right)$

$u_e$

Velocity of ambient fluid $\left(m{s}^{-1}\right)$

$v_w\left(x\right)$

Velocity of the mass flow $\left(m{s}^{-1}\right)$

$t$

Time $\left(s\right)$

$M_o$

Magnetization of magnets

$p$

Width of magnets and electrodes

$k^{*}$

Constant of mean absorption

$k$

Thermal conductivity $\left(W{m}^{-1}{K}^{-1}\right)$

$S$

Suction parameter (non-dimensional)

$R$

Radiation parameter (non-dimensional)

$Sc$

Schmidt number parameter (non-dimensional)

$\mathit{Pr}$

Prandtl number parameter

$Z$

Modified Hartmann number (non-dimensional)

${\tau }_w$

Surface shear stress

$Re$

Reynold number (non-dimensionless)

$q_m$

Mass flow

Greeks Letter

${\sigma }^{*}$

Stephen-Boltzmann constant

$\eta$

Similarity variable

$\epsilon$

Unsteadiness parameter (non-dimensional)

$\mu$

Dynamic viscosity $\left(kg{m}^{-1}{s}^{-1}\right)$

$\rho$

Density ($kg{m}^{-3})$

$\lambda$

Stretching/shrinking parameter (non-dimensional)

$\tau$

Heat capacitive ratio

$\nu$

Kinematic viscosity $\left(m^2{s}^{-1}\right)$

$\beta$

Reaction rate parameter (non-dimensional)

$\alpha$

Temperature difference parameter (−)

$\delta$

Activation energy parameter (−)

$\theta \left(\eta \right)$

Temperature profile

$g\left(\eta \right)$

Concentration profile

MHD

Magnetohydrodynamic

Superscripts

$\prime$

“Differentiation with respect to ${\eta }^{″}$

Subscripts

$w$

“Conditions at the wall

$\infty$

The outer edge of the boundary layer”

1. Introduction

An electrostatic operator, often referred to as a Riga plate,” may be made by installing interchanging resistors and a circumferential arrangement of magnetic material on a plate, as seen in Fig. 1. The direction of the magnet's north and south poles is indicated by the characters $N$ and $S^{\prime }\text{,}$ which, taken individually, stand for the terms north and south.

Fig. 1.

Riga plate.

This results in the creation of an actuator with electromagnetic properties. As a direct consequence of this, an electromagnetic actuator will be produced [1]. It has been proven that the Riga plate may be utilized to decrease the boundary layer separation, this lowers the amount of pressure drag and skin friction that a submarine must deal with. This was shown by Farooq et al. [2]. The analysis of the Blasius flow that was done by utilizing the Grinberg [3] demonstration directed at a Riga plate was pioneered by Ref. [4]. The wall parallel Lorentz force was determined to be a factor that was revealed to be a component that effects the stability of the Blasius flow as a result of their studies. After some time had passed, researchers [5] looked into the mixed convection Blasius flow that occurred beyond the Riga plates [2]. They came to the conclusion that higher velocity dispersion was responsible for the rise in the modified Hartmann number, and they published their findings about this development. Researchers from Ref. [6] conducted an investigation into the stagnation point flow in a viscous fluid while using permeable Riga plate as the experimental setup. On the other hand, researcher [1] explored the boundary layer flow features of nanofluid by modifying the varying stiffness of a Riga plate. This allowed them to study the flow properties of the nanofluid. Both groups of researchers were interested in the same thing: how a nanofluid behaves when flowing over a permeable Riga plate. There has been a recent uptick in the number of researchers that are looking into the heat transfer performance of combining different hits toward the Riga plate in different fluids [[7], [8], [9], [10]].

Extrusion operations, the transportation of biofluids, crystal manufacturing, high-temperature flattening, the conditioning of brass substrate, elastic strips, the effectiveness of lubricating and pigments, deep portrayal, liquefying, fabrication of polymers, the extruding of monomers, and supersonic plastic sheet ejection, etc., are all examples of engineering and technological procedures that need study into flow on a stretched sheet. This kind of research has gotten a significant amount of attention over the last several decades. As part of their research, several researchers [[11], [12], [13], [14], [15], [16], [17], [18]] are looking at how fluid moves over the stretched surface.

There are many different applications that can be found in the domains of engineering and science that make use of the influence that heat radiation has on convective fluid movements. Some examples of these applications include gas-cooled nuclear reactors, steam turbines, rocket engines, supersonic flight airfares, space craft, engineering for renewable energy, nuclear power plants, and a great many other commercial fields and applications that are now being researched. Thermal radiation has captured the interest of a number of researchers [19,20].

In modern mechanical and physiological changes, the investigation of the MHD motion of an electrically conducting fluid is absolutely of utmost importance. This occurs as a result of the fact that the magnetic field has an effect on the performance of a variety of systems that use fluids that are electrically conductive, including the modification of boundary layer flow control. As a result of its use in a diverse range of different technical problems, including combustion analyses, hydroelectric power separations, MHD engines, nuclear reactor safety, and furnace construction, this particular form of flow has captured the interest of a number of academics [21]. With a magnetic field and hydromagnetic methods, molten metals may be cleansed of contamination caused by non-metallic additions. In nuclear reactors, aerospace engineering, evaporation practices, cosmological hydrodynamics, thermal exchangers, geothermal, analytical chemistry, construction techniques, photovoltaic physics, solar cells, and oil recovery, the study of heat, in addition to the mass transport of fluids across an exponentially expanding surface with consequences for chemical reactions, plays a vital role. This study is also important for understanding the implications of chemical reactions. A large number of researchers [22,23] are analyzing chemical processes that take place across a stretched sheet.

Researchers used similarity solutions for steady [24] and unsteady [25] flows in order to analyze the flow and heat transfer that occurred through a stretching and contracting sheet. A plot of the dual solutions was created against either the stretching or shrinking characteristic in the situation where there was a continuous flow. The plot may have been either positive or negative. When it came to the scenario including unsteady flow, the responses were provided in the context of the unsteadiness constraint. An examination of stability was carried out so that we could establish the degree to which the solutions could withstand an unsteady flow. In each case, it was shown that there were two solutions in a bigger area with a higher suction parameter value. The investigation on nanofluid flow and heat transfer that was conducted numerically was included in the publication [26], which is now accessible for review. The document also contains an extensive analysis of the study. From a different point of view, various experimental investigations employing a different method of analysis and theoretical modelling have been described by Refs. [27,28]. These studies relate to micro-nano fluidic systems, and they were conducted to investigate these systems. These articles contain fascinating findings as well as findings contextualized within the context of real-world application issues.

In addition to the many applications that can be found in momentum and heat transfer, the technological revolution also makes use of mass transfer in a number of different ways. In the course of this specific line of investigation, it has been taken into consideration that the mass transfer of a chemical reactant takes place in the boundary layer glows as it travels over a shrinking sheet. Anderson et al. [29] inspected the comportment of a reactive fluid's dispersal while passing it over a stretched sheet. Afify [30] looked at the flow of the boundary layer in addition to the mass and thermal transportation of a reactive species over a continuously stretched surface. The purpose of this research was to better understand how physical and chemical properties and polymer manufacturing can be applied in industrial settings. The study that was carried out by Ref. [31] had the primary objective of determining how different chemical reactions affect the flow field, concentration, and heat of fluid flow via a permeable wedge. Viscoelastic reacting fluids were studied by Cortell [32] in light of the different technical uses of boundary layer flow of interacting compounds across a stretchable permeable sheet. Bhattacharyya and Layek [33] investigated the mass transfer of reactive intermediates over a stretched sheet by considering the effects of a magnetic field as well as blowing and suctioning. Their findings were presented in the form of a mathematical model. Bhattacharyya [34] gave dual solutions for stagnation point flow and mass transfer of a chemical across a stretching or shrinking sheet. These solutions were presented in the context of stagnation point flow. In addition, Bhattacharyya [34] identified further uses of stretching sheets, together with the aeronautical ejection of tarpaulins, the manufacturing of fiber reinforcement, and the chilling and rinsing of pulp and fabrics. Researchers [35] explored how the flow of a Maxwell fluid through a permeable stretched sheet is affected by the manifestation of chemical changes and magnetic fields. This study looked at how these factors influence mass and heat transfer. Takhar et al. [36] investigated the mass and heat transmission of a reacting chemical over a stretched sheet, finding conclusions that differ from those of previous versions by focusing on the features of the boundary layer. This was done in order to determine how the chemical transferred heat and mass. The papers cited in Refs. [[37], [38], [39]]. Provide a synthesis of the relevant literature.

Bestman [40] looked at binary chemical reactions that were dependent on the activation energy as part of his study of the boundary layer qualities of a permeable plate that was entrenched in a porous medium. This study was conducted as part of an assessment of the properties of the boundary layer. The problem was theoretically explored by employing asymptotic analysis for massive suction, and the flow parameters were shown by utilizing variations in activation energy. Zaib et al. [41] inspected the influences that binary chemical reactions have on the flow of Casson nanofluid through a wedge because of the importance of binary chemical reactions in bioengineering, the evaporation of nuclear reactions, power generation in underground aquifers, and the healing process of heat sources. This was done because of the significance that binary chemical reactions have in these areas.

The modelling of non-integer-order issues that are associated with science and technology is one of the most challenging and competitive disciplines within the scientific community. In order to provide a more accurate description of nonlinear phenomena, the fractional models seem to be more accurate than the classical models. This is due to the fact that the classical models do not fully capture all of the constraints that are imposed by nonlinear issues. In addition, fractional models are more suitable than other models for calculating the real impact that the constraints have inside the constrained space. This is due to the fact that the parametric impact on chronic ailments at short intervals is essential for determining the range and constraints of the parameters. The idea of the non-integer-order derivatives was first presented by Caputo in Ref. [42] 1967. This was done so that issues concerning a fractional differential equation with beginning conditions may be discussed. Researchers [[42], [43], [44], [45]] showed the scientific community the many non-integer order operators and used them to deal with more realistic physical problems. Higher-order problems in science, the corona virus with optimal control, wave equations, q-integrodifferential equations, fraction hybrid differential operators, a solution to the fractional Allen–Cahn equations and solution to the fractional Burgers equation, are all examples of this type of problem. Similarity transformations were developed so that model PDEs may be converted into nonlinear classical-order ODEs. This was necessary since managing nonlinear issues in the form of PDEs is not a simple task.

It is evident, after conducting an analysis of the pertinent literature, that it is important to take into account the phenomena of mass transfer and heat transfer in the context of the manifestation of a binary chemical reaction accompanied by thermal radiation over a permeability-changing Riga plate. These phenomena are observed in a variety of engineering and industrial settings, so they are significant. The aforementioned literature served as both inspiration and motivation for the current effort, which is one of a kind since it explores the properties of an unsteady stagnation point flow with regard to heat and mass transfer across a permeable stretching/shrinking Riga Plate exposed to a binary chemical reaction changing with activation energy and thermal radiation effect. In this study, unsteady stagnation point flow that happens over a stretching/shrinking Riga Plate under the effects of mass suction and thermal radiations is looked into. In addition to this, the influence of a binary chemical reaction that is characterized by a finite Arrhenius activation energy is also considered. In order to solve a system of nonlinear ordinary differential equations, transformations are used as a means toward accomplishing the goal of effectively completing the process of altering similarity. The shooting approach is employed to figure out how to solve the resulting equations. When talking about the results, the focus is on how the flow changes as a result of changes in a number of physical constraints. According to the authors' best acquaintance, this dilemma with the flow has not been investigated in any of the studies that have been done. Since this is the case, the following is worthy of some consideration.

• I.What kind of impact does the presence of an unsteady parameter has on the mass and heat transfer rate as well as the skin friction coefficient?
• II.Does the existence of suction upshot in an expansion or decrease in the temperature $\theta \left(\eta \right)$ and velocity distributions $f^{\prime }\left(\eta \right)$ of the flow?
• III.How do the temperature, concentration, and velocity distributions of the flow change depend on the other governing parameters?

2. Description of the problem

Consider an unsteady, 2D, incompressible and laminar heat-and-mass-transfer flow that moves over a semi-infinite vertical Riga plate at the stagnation point under the impact of thermal radiations and binary chemical reaction. The selection of the coordinate system needs to be carried out to such an extent as to ensure that the $x-$ axis is aligned in a direction that is parallel to the Riga plate and that the $y-$ axis is aligned in a direction that is perpendicular to the Riga plate. The velocity of the ambient fluid is presumed to be ${\tilde{u}}_e=\frac{cx}{1-at}$ and the stretching/shrinking velocity is ${\tilde{u}}_w=\frac{cx}{1-at}$. Here $a>0$, $c$ and $a$ are enduring real values. If $c$ is positive, the Riga plate will decrease in size over time (also known as contracting), but if $c$ is negative, the Riga plate will increase in size over time (e.g., expanding). This information comes from Ref. [9]. In addition, let us assume that the velocity of the mass flow is denoted by the symbol ${\tilde{v}}_w\left(x\right)$ where ${\tilde{v}}_w\left(x\right)<0$ for suction and ${\tilde{v}}_w\left(x\right)>0$ for injection, correspondingly.

In the meanwhile, we will assume that the temperature at the wall, denoted by $T_w$ and the concentration at the wall, denoted by $C_w$ remain unchanged. The temperature of the fluid in the surrounding area, denoted by $T_{\infty }$ and $C_{\infty }$ its ambient concentration, will remain unchanged (see Fig. 2). We also took into account a binary chemical reaction, which was characterized by activation energy and expressed in terms of the Arrhenius reaction rate. Throughout the course of this research, the Grinberg term $\left(\frac{\pi j_{0M_0}}{8{\rho }_{nf}}{e}^{\raisebox{1ex}{$-\pi y$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}\right)$ has been used to denote the Riga plate.

Fig. 2.

For stretching and shrinking flow, the geometry of the governing model.

On the basis of these presumptions, the continuity, momentum, energy, and concentration equations that describe the boundary layer of the fluid can be written as follows (see Refs. [9,46]):

$$\frac{\partial \tilde{u}}{\partial x}+\frac{\partial \tilde{v}}{\partial y}=0\text{,}$$ (1)

$$\frac{\partial \tilde{u}}{\partial t}+\tilde{u}\frac{\partial \tilde{u}}{\partial x}+\tilde{v}\frac{\partial \tilde{u}}{\partial y}=\frac{\partial {\tilde{u}}_e}{\partial t}+{\tilde{u}}_e\frac{\partial {\tilde{u}}_e}{\partial x}+\nu \frac{{\partial }^2\tilde{u}}{\partial y^2}+\frac{\pi j_{o{M}_o}}{8\rho }{e}^{-\pi y/p}\text{,}$$ (2)

$$\frac{\partial T}{\partial t}+\tilde{u}\frac{\partial T}{\partial x}+\tilde{v}\frac{\partial T}{\partial y}=\tau \frac{{\partial }^{2}T}{\partial y^2}-\frac{1}{\rho C_p}\left(\frac{\partial q_r}{\partial y}\right)\text{,}$$ (3)

$$\frac{\partial C}{\partial t}+\tilde{u}\frac{\partial C}{\partial x}+\tilde{v}\frac{\partial C}{\partial y}=D\frac{{\partial }^{2}C}{\partial y^2}-k_r^2{\left(\frac{T}{T_{\infty }}\right)}^{n1}\left(C-C_{\infty }\right)e^{\frac{-E}{R^{\prime }T}}\text{,}$$ (4)

$\tilde{u},\tilde{v}$ designates the velocity components along $x$-axis and $y-$ axis correspondingly. $T$ (fluidic temperature), $C$ is the species concentration, ${\tilde{v}}_w\left(x\right)$ (wall mass transfer velocity via permeable surface), $M_o$ (magnetization of magnets), and $p$ (the width of the magnets and electrodes), stretching velocity of the wall is represented by ${\tilde{u}}_w\left(x\right)$. $j_0$ represents the current density at the electrodes. The radiative heat flow $q_r$ is obtained by the following expression using the Roseland approximation (see Ref. [9]):

$$q_r=-\frac{4{\sigma }^{*}}{3k^{*}}\frac{\partial T^4}{\partial y}\text{,}$$ (5)

$k^{*}$ is the constant of mean absorption, with ${\sigma }^{*}$ indicating the Stephen-Boltzmann incessant. We assume that there isn't much of a difference in temperature inside the flow. A Taylor series expansion of the equation of ($T-T_{\infty }$) yields the expression $T^4$, excluding higher-order terms.

$$T^4\approx 4T_{\infty }^{3}T-3T_{\infty }^3\text{.}$$

The following is a generalization of the energy equation (see Ref. [9]):

$$\frac{\partial T}{\partial \mathrm{t}}+\tilde{u}\frac{\partial \mathrm{T}}{\partial \mathrm{x}}+\tilde{v}\frac{\partial \mathrm{T}}{\partial \mathrm{y}}=\tau \frac{{\partial }^{2}T}{\partial y^2}+\frac{1}{\rho C_p}\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}}\frac{{\partial }^{2}T}{\partial y^2}\text{,}$$ (6)

The compatible boundary conditions are (see Refs. [9,46]):

$$\tilde{u}={\tilde{u}}_w\left(x\right),\tilde{v}={\tilde{v}}_w\left(x\right),T=T_w\left(x\right),C=C_w\left(x\right)\mathrm{a}\mathrm{t}y=0\text{,}$$

$$\tilde{u}\to {\tilde{u}}_e\left(x\right),T\to T_{\infty },C\to C_{\infty }\mathrm{a}\mathrm{s}y\to \infty \text{.}$$ (7)

Introducing the similarity transformation (see Ref. [9])

$$\tilde{u}=\frac{cx}{1-\alpha t}{f}^{\prime }\left(\eta \right),\tilde{v}=-\sqrt{\frac{a\nu }{1-\alpha t}}f\left(\eta \right),\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},g\left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\eta =\sqrt{\frac{a}{\nu \left(1-\alpha t\right)}}y\text{.}$$ (8)

Where prime indicates differentiation in relation to $\eta$. While ${\tilde{v}}_w=-\sqrt{\frac{a\nu }{1-\alpha t}}S$. Using Eq. (8) we obtained the following non-dimensional unsteady equations for similarity solutions.

$$f^{‴}+f{f}^{″}-f^{\prime 2}+1+\epsilon -\epsilon \left(f^{\prime }+\frac{\eta }{2}{f}^{″}\right)+Z{e}^{-d\eta }=0\text{,}$$ (9)

$${\left(1+\frac{4}{3}R\right)\mathrm{\theta }}^{″}+\mathit{Pr}\left(f{\mathrm{\theta }}^{\prime }-\frac{\epsilon }{2}\eta {\mathrm{\theta }}^{\prime }\right)=0,$$ (10)

$$\frac{1}{Sc}{g}^{″}-\frac{\epsilon }{2}\eta g^{\prime }+f{g}^{\prime }-\beta {\left(1+\alpha \theta \right)}^{n1}g{e}^{-\frac{\delta }{1+\alpha \theta }}=0\text{,}$$ (11)

$$f\left(\eta \right)=S,f^{\prime }\left(\eta \right)=\lambda ,\theta \left(\eta \right)=1,g\left(\eta \right),\mathrm{a}\mathrm{t}\eta =0$$

$$f^{\prime }\left(\eta \right)\to 1,\theta \left(\eta \right)\to 0,g\left(\eta \right)\to 0,as\eta \to \infty \text{.}$$ (12)

Equations (9), (10), (11), (12) include the following variables:

$Z=\pi j_0{M}_0/8a^2\rho$, is modified Hartmann number. The dimensionless parameter $d=\frac{\pi }{p}\sqrt{\frac{\nu }{a}}$ refers to the width of the electrodes and the magnets. Prandtl number $\mathit{Pr}=\nu /a$, where $S>0$ signifies the suction parameter of mass transpiration. $R=\frac{4\sigma T_{\infty }^3}{K}$, represents thermal radiation parameter. $\lambda =\frac{c}{a}$ is said to stretch or shrink the surface, when $\lambda >0$ denotes the stretching surface. $\beta =k_r^2/\nu /a^2$ is non-dimensional reaction rate. $Sc=\frac{\nu }{D}$ is the Schmidt number, $\alpha =T_w-T_{\infty }/T_{\infty }$ is the temperature difference constraint, $\delta =E/R^{\prime }{T}_{\infty }$ is the dimensionless activation energy.

3. Physical quantities of interest

The following are the measures that are of relevance from a physical perspective:

Local skin friction coefficient $C_f=\frac{{\tau }_w}{{\rho \tilde{u}}_e^2}$, Nusselt number ${Nu}_x=\frac{x{q}_w}{k\left(T_w\left(x\right)-T_{\infty }\right)}$ and Sherwood number ${Sh}_x=\frac{x{q}_m}{k\left(C_w\left(x\right)-C_{\infty }\right)}\text{.}$.

The surface shear stress ${\tau }_w$, heat flow $q_w$ and mass flow $q_m$ have the form of

$${\tau }_w={\left(\frac{\partial \tilde{u}}{\partial y}\right)}_{y=0},q_w=-k{\left(\frac{\partial T}{\partial y}\right)}_{y=0}+{\left(q_r\right)}_{y=0},q_m=-D{\left(\frac{\partial C}{\partial y}\right)}_{y=0}\text{,}$$

The lowered skin friction coefficient, local heat and mass transfer rates are given by:

$${Re}_x^{1/2}{C}_f=f^{″}\left(0\right),{Re}_x^{-1/2}{Nu}_x=-\left(1+\frac{4}{3}R\right){\theta }^{\prime }\left(0\right),{Re}_x^{-1/2}{Sh}_x=-g^{\prime }\left(0\right)\text{.}$$

4. Numerical procedure for solution

Nonlinear differential equations (9), (10), (11) with the appropriate boundary conditions (12) may be unraveled mathematically by manipulating the Runge-Kutta (RK) fourth order approach in combination with the shooting method. This allows for the equations to be solved in an exact manner. The higher order non-linear differential equations (9), (10), (11) are transformed into synchronous differential equations of the first order at the very first step of the process. After that, the shooting approach is used to convert them into an initial value issue. The Runge-Kutta fourth order approach may then be used to solve the issue using the starting value. The solution to the ordinary differential equations (9), (10), (11) is to reduce them to a set of seven systems of equations of first order with seven unknown factors. In the current work, the convergence criteria are used by comparing the quantities of the dependent variables in both the current and previous rounds of the process. This comparison is made based on the difference between these two values. Once the absolute quantities of the difference approach ${10}^{-6}$ which shows that the solution has converged to the level of correctness that was sought after, the iteration method is completed. This confirms that the solution has achieved the degree of precision that was sought for. The non-linear ordinary differential equations that regulate the situation may be compacted into a set of instantaneous first-order differential equations as shown in the following scenario (see Refs. [47,48]):

$${\mathrm{X}}_1=f,{\mathrm{X}}_2=f^{\prime },{\mathrm{X}}_3=f^{″},{\mathrm{X}}_4=\theta ,{\mathrm{X}}_5={\theta }^{\prime },{\mathrm{X}}_6=g,{\mathrm{X}}_7=g^{\prime }\text{,}$$ (13)

using equation (13) (see Refs. [47,48]):

$${\mathrm{X}}_1^{\prime }=f^{\prime },{\mathrm{X}}_2^{\prime }=f^{″},{\mathrm{X}}_3^{\prime }=X{\mathrm{X}}_1=f^{‴},{\mathrm{X}}_4^{\prime }={\theta }^{\prime },{\mathrm{X}}_5^{\prime }=X{\mathrm{X}}_5={\theta }^{″},{\mathrm{X}}_6^{\prime }=g^{\prime },{\mathrm{X}}_7^{\prime }=={\mathrm{X}\mathrm{X}}_7=g^{″}\text{,}$$ (14)

By rearranging equations (9), (10), (11), (12)) in the form of bellows, we can get the values $f^{‴}$, ${\theta }^{″}$ and $g^{″}$ that are reflected in equation (14) (see Refs. [47,48]):

$$X{\mathrm{X}}_1=-\left[{\mathrm{X}}_1{\mathrm{X}}_3-{{\mathrm{X}}_2}^2+1+\epsilon -\epsilon \left({\mathrm{X}}_2+\frac{\eta }{2}{\mathrm{X}}_3\right)+Z{e}^{-d\eta }\right]\text{,}$$ (15)

$$X{\mathrm{X}}_5=-\frac{1}{\left(1+\frac{4}{3}R\right)}\left[\mathrm{Pr}\left({\mathrm{X}}_1{\mathrm{X}}_5-\frac{\epsilon }{2}\eta {\mathrm{X}}_5\right)\right]\text{,}$$ (16)

$${\mathrm{X}\mathrm{X}}_7=-Sc\left[{\mathrm{X}}_1{\mathrm{X}}_7-\frac{\epsilon }{2}\eta {\mathrm{X}}_7-\beta {\left(1+\alpha \theta \right)}^{n1}{\mathrm{X}}_6{e}^{-\frac{\delta }{1+\alpha \theta }}\right]\text{,}$$ (17)

$${{\mathrm{{\rm X}}}_1}_a=S,{{\mathrm{{\rm X}}}_2}_a=\lambda ,{{\mathrm{{\rm X}}}_4}_a=1,{{\mathrm{{\rm X}}}_6}_a=1\text{,}$$

$${{\mathrm{{\rm X}}}_2}_b=1,{{\mathrm{{\rm X}}}_5}_b=0,{{\mathrm{{\rm X}}}_7}_b=0\text{,}$$ (18)

The locations indicated by the subscripts $\text{'}a\text{'}$ and $\text{'}b\text{'}$ on the sheet at $\eta =0$ and the distance from the sheet for a given value of are denoted by those values, respectively. In this particular investigation, the value of $\eta =15$ is chosen for this position. Mathematica is a very effective piece of computer software, and we use it for simulations.

5. Discussion of results

In order to solve numerically the scheme of highly nonlinear differential equations (9), (10), (11) that are sensitive to the boundary conditions (12), the Runge-Kutta-Fehlberg numerical method with the shooting procedure is used. On the Riga plate seen in Fig. 1, there is a physical model that may be used to simulate stretching and shrinking flow. Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12 show the results for velocity $f^{\prime }\left(\eta \right)$, temperature $\theta \left(\eta \right)$, concentration $g\left(\eta \right)$, skin friction $\left({Re}_x^{1/2}{C}_f\right)$, heat transfer $\left({Re}_x^{-1/2}{Nu}_x\right)$ and mass transfer $\left({Re}_x^{-1/2}{Sh}_x\right)$. Together, these figures provide a full picture of the current situation in the scheme. The Prandtl number was believed to be $\mathit{Pr}=5\text{,}$ apart from comparisons with the previous case. According to the tables and charts, the control parameters' estimates are adjusted. The far-field boundary conditions (12) guide the selection of these values. Further physical parameters are chosen as $\epsilon \in \left[0.0,-1.0\right]$ (unsteadiness), $\lambda \in \left[-1.0,1.0\right]$ (stretching/shrinking), $Z\in \left[-0.5,0.5\right]$ (Modified Hartmann Number), $d=0.5$ (electrode width fixed), $\beta \in \left[1.0,3.0\right]$ (reaction rate), $\alpha \in \left[0.25,2.0\right]$ (temperature difference), $\delta \in \left[2.0,5.0\right]$ (activation energy), $n1\in \left[-1.0,1.0\right]$ (fitted rate), $Sc\in \left[0.0,1.0\right]$ (Schmidt number) and $R\in \left[0.0,1.0\right]$ (radiation parameter) are selected in accordance with the most important references and solutions that may be found in the opposing flow.

Fig. 3.

(a) Significance of $\epsilon$ on $f^{\prime }\left(\eta \right)\text{.}$(b) Significance of $\epsilon$ on $\theta \left(\eta \right)\text{.}$(c) Impact of $\epsilon$ on dimensionless $g\left(\eta \right)\text{.}$.

Fig. 4.

(a) Stimulus of $Z$ on $f^{\prime }\left(\eta \right)\text{.}$(b) Stimulus of $Z$ on $\mathrm{\theta }\left(\mathrm{\eta }\right)\text{.}$(c) Stimulus of $Z$ on $g\left(\eta \right)\text{.}$.

Fig. 5.

(a) Significance of $\lambda$ on $f^{\prime }\left(\eta \right)\text{.}$(b) significance of $\lambda$ on $\theta \left(\eta \right)\text{.}$.

Fig. 6.

(a) Stimulus of $S$ on $f^{\prime }\left(\eta \right)\text{.}$(b) Stimulus of $S$ on $\mathrm{\theta }\left(\mathrm{\eta }\right)\text{.}$(c) consequence of $R$ on $\theta \left(\eta \right)$.

Fig. 7.

(a) Stimulus of $Sc$ on $g\left(\mathrm{\eta }\right)\text{.}$(b) influence of $\alpha$ on $g\left(\mathrm{\eta }\right)\text{.}$(c) consequence of $\beta$ on $g\left(\eta \right)\text{.}$(d) consequence of $\delta$ on $g\left(\eta \right)\text{.}$(e) Stimulus of $n1$ on $g\left(\mathrm{\eta }\right)\text{.}$.

Fig. 8.

(a) Skin friction of $Z$ against $\lambda \text{.}$(b) Nusselt Number of $Z$ against $\lambda \text{.}$(c) Sherwood Number of $Z$ against $\lambda \text{.}$.

Fig. 9.

(a) Skin Friction of $S$ against $\lambda \text{.}$(b) Nusselt number of $S$ against $\lambda \text{.}$.

Fig. 10.

(a) Skin friction of $\epsilon$ against $\lambda \text{.}$(b) Nusselt Number of $\epsilon$ against $\lambda \text{.}$(c) Sherwood Number of $\epsilon$ against $\lambda$.

Fig. 11.

Nusselt number of $R$ against $\lambda$.

Fig. 12.

(a) Sherwood number of $Sc$ against $\lambda \text{.}$(b) Sherwood number of $\alpha$ against $\lambda \text{.}$. (c) Sherwood number of $\beta$ against $\lambda \text{.}$(d) Sherwood number of $\delta$ against $\lambda$. (e) Sherwood number of $n1$ against $\lambda$.

In Table 1, it is shown that the average amount of central processing unit (CPU) time needed for calculating each result is around 0.65 s. The solutions were found by coming up with different starting estimates that must eventually meet the requirements of the far-field boundary (12). The validation aspect of the model is also explained in Table 1, which compares $f^{″}\left(0\right)$ to our model with those of Zainal et al. [9], for $\gamma =\epsilon =d=R=Z=S=\beta =\alpha =n1=\delta =Sc=0$ and $\mathit{Pr}=1.0$ (flat plate without Modified Hartmann number). The bvp4c solver in MATLAB was used by Ref. [9] whereas in present investigations used RK-IV with shooting in MATHEMATICA. In Table 1, according to the results of [9], the present findings are in excellent accord. Based on these results, researchers are sure that they can use this method and model.

Table 1.

$f^{″}\left(0\right)\text{,}$$\gamma =\epsilon =d=R=Z=S=\beta =\alpha =n1=\delta =Sc=0$ and $\mathit{Pr}=1.0$ for variety of estimates of $\lambda >0$ (for stretching case).

$\lambda$	Present Results		Zainal et al. [9]
	$f^{″}\left(0\right)$	Time, $t\left(s\right)$
$0$	$1.232589$	$0.27$	$1.232587$
$0.2$	$1.051130$	$0.03$	$1.051129$
$0.5$	$0.713295$	$0.05$	$0.713294$
$1.0$	$0.00000$	$0.13$	$0.00000$
$2$	$-1.887306$	$0.28$	$-1.887306$
$5$	$-10.264749$	$1.63$	$-10.264749$
$8$	$-21.684800$	$2.16$

5.1. Velocity ${\mathit{f}}^{\prime }\left(\mathit{\eta }\right)$ with temperature $\mathbf{\theta }\left(\mathbf{\eta }\right)$ and concentration $\mathit{g}\left(\mathbf{\eta }\right)$ curves

On non-dimensional velocity, temperature, and concentration curves, effects of many important physical constraints are studied. These include the unsteadiness parameter $\epsilon$, Modified Hartmann number parameter $Z\text{,}$ stretching/shrinking constraint $\lambda$, the suction constraint $S\text{,}$ the thermal radiation constraint $R\text{,}$ the Schmidt number $Sc\text{,}$ the temperature difference parameter $\alpha$, chemical reaction parameter $\beta$, activation energy parameter $\delta$, and the fitted rate parameter $n1$. The fact that the solutions that are currently available can meet the requirements of far field boundary limits shows its correctness.

In addition, effect of $\epsilon$ on velocity $f^{\prime }\left(\eta \right)$, temperature $\theta \left(\eta \right)$ and concentration $g\left(\eta \right)$ profiles are shown in Fig. 3a-c. The decrease in velocity profile all around the plate coincided with a rise in the value of $\left|\epsilon \right|$. (see Fig. 3a). As can be seen in Fig. 10a, the boundary layer became thicker, which led to a drop in the value of ${Re}_x^{1/2}{C}_f$. Additionally, this led to a reduction in the velocity gradient $f^{″}\left(0\right)$. It was interesting to see that the viscous fluid's velocity profile, which was positioned a considerable distance from the plate, had a tendency that increased with $\left|\epsilon \right|$. On other hand, an increment in $\left|\epsilon \right|$ brought about declination in temperature and concentration profiles, as well as a decrease in the thermal and concentration boundary layers (see Fig. 3b and c). Following this, there was an increment in both temperature and concentration gradient at surface of the Riga plate, which led to a rise in both the ${Re}_x^{-1/2}{Nu}_x$ and ${Re}_x^{-1/2}{Sh}_x\text{.}$.

A decrease in the unsteadiness parameter $\epsilon =-0.1,-0.3,-0.5,-0.7$, outcomes in growth in thickness of boundary layer, as can be seen in Fig. 3 (a), which, in turn, eliminates the velocity outflow on the stretching or shrinking Riga plate. The impact of the $\epsilon$ against $\eta$ on the temperature curve of the stretching and contracting Riga plate is seen in Fig. 3 (b). Because there is a rise in the dimensionless boundary layer thickness when value of $\epsilon$ against $\eta$ is made smaller, temperature gradient of the flow will likewise become smaller as a result. Therefore, rate of heat transmission inside Riga plate rises when it either stretches or shrinks. The impact of the $\epsilon$ against $\eta$ on the concentration curve of the stretching and shrinking Riga plate is seen in Fig. 3 (c). It has been discovered that when the estimates of $\epsilon$ drops, the concentration curve of the flow likewise drops. This is because there is a growth in the dimensionless thickness of boundary layer. Consequently, rate of mass transfer inside liquid that is stretching or shrinking rises. In general, when a value of $\eta =5$ is used, these profiles asymptotically meet the far-field boundary criteria (12).

As $Z$ changes, Fig. 4a-c demonstrates the distributions of velocity, temperature, and concentration along stretching plate when ($\lambda =0.5$). When $Z$ was made more intense, the solution's velocity profile $f^{\prime }\left(\eta \right)$ showed an increase in width of momentum boundary layer, as seen in Fig. 4 (a). Fig. 4 (b) and 4(c) look at the effect that $Z$ has on the thermal profile denoted by $\theta \left(\eta \right)$ and the concentration profile, denoted by $g\left(\eta \right)$ respectively. In a physical sense, the rise in $Z$ was connected to an expansion of the electrical field that is cause for generation of Lorentz force. As result of this force, boundary layer was slowed down, which led to an increment in velocity profile (as shown in Fig. 4a). The overshoot of velocity profiles was subsequently enhanced by further increases in $Z$ (shown in Fig. 4a), which suggest that velocity of flow near Riga plate exceeded free stream velocity despite the presence of strong Lorentz forces. While this was going on, thermal and concentration boundary layers began to contract as $Z$ continued to rise. Fig. 4(b) and show that both the temperature and concentration profiles (c) decreased after that point. In general, when ${\eta }_{\infty }=5$ was carried out, all three profiles $f^{\prime }\left(\eta \right)\text{,}$ $\theta \left(\eta \right)$ and $g\left(\eta \right)$, asymptotically met far-field boundary conditions (12).

Fig. 5 (a) and 5(b) exemplify the upshot that the stretching parameter has on velocity and temperature of fluid, respectively. At greater levels of $\left|\lambda \right|\text{,}$ there is a proportional upswing in the velocity, but there is a drop in the temperature. Because the surface is contracting at a faster rate, the flow velocity of the nanofluid is increasing, which results in a drop in temperature.

Fig. 6a and (b) illustrates consequence of suction constraint $S$, on velocity $f^{\prime }\left(\eta \right)$ and temperature profiles of the viscous flow for stretching Riga Plate case. It is possible to see in this image an increase in velocity along the profile as a result of an increase in the quantity of the suction parameter. This is a consequence of the higher density brought about by the incorporation of more particles into the mixture. As a consequence of the suction, the thickness of the momentum boundary layer is reduced, which ultimately leads to an increase in the flow of the stretched plate. The behavior of $\theta \left(\eta \right)$ in response to the suction parameter $S$ is seen in Fig. 6b. By having a higher cumulative value of $S$, the temperature of the nanofluid may be lowered. There is a one-to-one relationship between the value of $S$ at the surface of a permeable stretching plate and the amount of suction that is experienced by the permeable stretching plate. The augmentation of the $S$ relied on delaying the molecules while they were in the fluid regime in hopes of enhancing the sluggish flow of the fluid on the expanding sheet. This was done in order to increase the viscosity of the fluid.

The response of $R$ is shown in Fig. 6 (c), which may be found here. Fig. 6 (c) illustrates how the variable $R$ affects the function $\theta \left(\eta \right)$. The temperature of the region comprising the boundary layer may be made higher by increasing $R$. It can be seen in Fig. 6 (c) that the estimate of $\theta \left(\eta \right)$ goes up as time goes on when there is a greater amount of $R$. As a result of an increased parameter $R$, there is a release of heat energy in the direction of the flow; as a result, the fluid temperature is raised. The graphical depiction shows that the temperature $\theta \left(\eta \right)$ of the thermal radiation increases as a result.

Fig. 7(a–e) demonstrates the effects of $Sc,\alpha ,\beta ,\delta$ and $n1$ on dimensionless concentration profile $g\left(\eta \right)$. The impact that different values of Schmidt number, $Sc$, have had on distribution of the concentration is seen in Fig. 7a. The Schmidt number has a relationship that is inversely proportional to the molecular (species) diffusivity. Rate of momentum diffusion is greater than the rate of species diffusion whenever $Sc$ is greater than 1. On other hand, $Sc<1$ exhibited opposite behavior. If $Sc$ is equal to 1, both the concentration layer (which contains species) and the momentum layer will have the same thickness and diffusivity rates. It has been discovered that the concentration distribution shown in Fig. 7a brings about a lessening of the concentration for increasing values of $Sc$. Because of the related drop in mass diffusivity, there is now a smaller amount of forcible mass transfer, which not only brings the concentration levels down but also brings thickness of concentration boundary layer down. This is due to the fact that mass transfer is dependent on the interaction between species profile and the velocity field in materials, both of which may be altered by adjusting the Schmidt number.

On display in Fig. 7 (b) is an investigation into how the temperature difference parameter $\alpha$ affects the mass fraction field. When there is a greater temperature differential between the air at the surface and the air around it, thickness of concentration boundary layer diminishes. As consequence, this leads to a boost in the mass fraction field. Fig. 7 (c) depicts the effect that the reaction rate constant $\beta$ has on mass fraction field. It has been discovered that an intensification in response rate constant leads to an improvement in the factor $\beta {\left(1+\alpha \theta \right)}^{n1}{e}^{-\frac{\delta }{1+\alpha \theta }}$. As a consequence of this, a destructive chemical reaction takes place, and in addition to this, there is a reduction in the mass fraction field.

On display in Fig. 7 (d) are the effects that the activation energy $\delta$ has on the mass fraction field. The term “activation energy” refers to the least expanse of energy that ought to be present formerly a chemical reaction may begin. It has been discovered that a combination of a low temperature and a high activation energy result in a boost in reaction rate constant. This, speeds up chemical reaction, and additionally, results in decrease in mass fraction field. In a manner analogous, the concentration profile in Fig. 7 (e) drops as the value of the fitted rate constant $n1$ increases.

5.2. Skin friction with heat and mass transfer

In manufacturing procedures, some characteristics, such as skin friction $\left({Re}_x^{1/2}{C}_f\right)$ and local rate coefficients of heat transfer $\left({Re}_x^{-1/2}{Nu}_x\right)$ and mass transfer $\left({Re}_x^{-1/2}{Sh}_x\right)$, are necessary. The relevance of these figures cannot be brought into doubt because of their widespread usage in marketing.

The progression of $\left({Re}_x^{1/2}{C}_f\right)$, $\left({Re}_x^{-1/2}{Nu}_x\right)$, and the $\left({Re}_x^{-1/2}{Sh}_x\right)$, respectively, for raise in modified Hartmann number $Z$, is depicted in Fig. 8(a–c), respectively. According to the findings, higher values of $Z$ contribute to a growth in the rate of $\left({Re}_x^{1/2}{C}_f\right)$ in a corresponding manner. There is a link between a growth in $Z$ and an enhancement in velocity of nanofluid. This is because existence of Lorentz drag force causes there to be such a correlation. When the value of $Z$ is raised, a pattern that is discernible before may be seen in the performance of the $\left({Re}_x^{-1/2}{Nu}_x\right)$, as shown in Fig. 8 (b). When $Z$ is enhanced, there is a significant increase in the amount of $\left({Re}_x^{-1/2}{Nu}_x\right)$. The $\left({Re}_x^{-1/2}{Nu}_x\right)$ upsurges to its full potential with the adoption of modified Hartman number; this represents significant step forward in performance. In essence, transverse magnetic field generates resistive Lorentz force that acts in opposite direction as the field enhances the degree of friction that occurs between the layers of fluid, this causes temperature of fluid to rise as direct result. This cycle continues until the fluid reaches the desired temperature. Because of this force, the temperature of the nanofluid may also spread from the surface into the flow as it moves. This might happen because of the flow. Despite the fact that Fig. 8 (c) displays a trend in the opposite direction, we are able to draw the conclusion that adoption of modified Hartmann number is responsible for a major boost of thermal efficiency of this certain study as a result of what has been said before.

The suction parameter S is one of factor affected in this research, and it also drives pace at which heat is transferred from one place to another. Fig. 9(a) and (b) reveals effects of the suction parameter on $\left({Re}_x^{1/2}{C}_f\right)$ and $\left({Re}_x^{-1/2}{Nu}_x\right)$, respectively. It is important that when suction parameter is accessible, values of $\left({Re}_x^{1/2}{C}_f\right)$ increase. Fig. 9a interprets these changes. Suction effect at boundary slows down movement of the nanofluid and magnifies velocity differential along permeable Riga Plate as it stretches and shrinks. A velocity gradient that was not anticipated is produced as a result of the suction emergence. This induces the heated fluid gestures that are addressing the wall to begin, and as a consequence, the buoyant strengths that are stimulated by the repercussions of the robust viscosity are slowed down. Suction emergence creates an unexpected velocity gradient. Next, the values of the $S$ are shown in Fig. 9b, which shows that there was a rise in the $\left({Re}_x^{-1/2}{Nu}_x\right)$ when $S$ was accessible along the surface of the stretching and shrinking Riga. In a broad sense, the magnitude of $S$ has an effect that corresponds to an improvement in the ${Re}_x^{-1/2}{Nu}_x$. When the suction parameter value went up, thermal boundary layer got thinner, which made temperature difference at surface go up.

The impact of unsteadiness constraint $\epsilon$ on variable quantity of stretching and shrinking plate is revealed in Fig. 10(a–c) correspondingly. Fig. 10 (a) depicts behavior of the fluid flow in relation to the variable $\left({Re}_{x^{1/2}}{C}_f\right)$ when was altered in the direction of the stretching/shrinking plate. As seen in Fig. 10 (a), the answer has become less restrictive in terms of $\left({Re}_{x^{1/2}}{C}_f\right)$ as has been smaller. When decreases, As the depth of the boundary layer grows, the velocity gradient also decreases, which is a positive feedback loop and results in a smaller $\left({Re}_{x^{1/2}}{C}_f\right)$ value. On the other hand, as shown in Fig. 10 (b and c), when is decreased, the obtained findings of Nusselt number and Sherwood number improve in solution. It has been shown that there is significant increase in the heat and mass transfer rate in boundary layer flow when unsteadiness parameter leads. The authors may reach the conclusion that the unsteadiness parameter considerably enhances heat and mass transmission based on the research that has been conducted recently as well as in the past. In spite of this, the authors are of the opinion that the results may be different depending on which regulatory elements are taken into consideration.

The impact of thermal radiation $R$ on $\left({Re}_x^{-1/2}{Nu}_x\right)$ can be seen in Fig. 11, where the Nusselt number $\left({Re}_x^{-1/2}{Nu}_x\right)$ increases as $R$ becomes more intense. By raising $R$, the heat transfer rate can be realistically enhanced, that helps in transferring more heat into fluid because $R$ is directly linked to thermal radiation parameter. This study is noteworthy because it demonstrates that greater quantities of thermal radiation may increase the efficiency with which heat is transferred. As consequence of thermal radiation effect, a greater amount of heat is generated and transported into flow, resulting in an increase in the thermal performance. In order to avoid turbulent flow, it is advised to utilize the thermal radiation control parameter with a suitable value. It is crucial to note that decreasing this control parameter also aids in slowing the process of boundary layer separation. Despite the fact that it is advised to apply a thermal radiation parameter with a suitable value to avoid turbulent flow, this nevertheless occurs. The authors conclude by saying that creating unique adaptable nanofluids will be necessary to meet the world's energy needs in the future. The creation of novel hybrid, ternary hybrid, and tetra hybrid nanofluids is now hampered by compatibility concerns, which are a hurdle to their development. Because hybrid and ternary nanocomposites often comprise more than two or three distinct kinds of nanoparticles, it is challenging to find and create a nanocomposite that is compatible.

Impacts of $Sc,\alpha ,\beta ,\delta$ and $n1$ towards $\lambda$ on mass transfer are presented in Fig. 12(a–e). Fig. 12 (b) illustrates the influence that the temperature differential parameter $\alpha$ has on the Sherwood number $\left({Re}_x^{-1/2}{Sh}_x\right)$. A higher value of $\alpha$ will result in a higher value for $\left({Re}_x^{-1/2}{Sh}_x\right)$. When considering values of dimensionless activation energy $\delta$ that are lower, the $\left({Re}_x^{-1/2}{Sh}_x\right)$ and $Sc$ are found to be appreciably greater when the value of α is increased. However, when bigger values of $\delta$ are used, the $\left({Re}_x^{-1/2}{Sh}_x\right)$ shows a little rise while $Sc$ nearly completely retains its previous value. In this particular scenario, we choose $n1=0.9$ since a higher number of $\alpha$ results in a faster response rate. As a result, it has been discovered that the Sherwood number rises with $\alpha \text{.}$.

The impact that dimensionless reaction rate $\beta$ has on Sherwood number is broken out in detail in Fig. 12 (c), which can be found here. The extent of the $\left({Re}_x^{-1/2}{Sh}_x\right)$ emerges as the value of $\beta$ rises, and a considerable influence may be seen when the dimensionless activation energy is made smaller. In addition, linear dependency of reaction rate $\beta$ on causes domain in which it is possible for a solution to exist to become more constrained as the value of increases $\beta$. Fig. 12 (d) depicts the influence that the dimensionless activation energy, denoted by the symbol $\delta$, has on the $\left({Re}_x^{-1/2}{Sh}_x\right)$. A very tiny value of $\delta$ may have a significant impact on the mass transfer. When the dimensionless activation energy $\delta$ is raised, there is a proportional reduction in the amount of mass that is transferred. The rate of mass transfer and impact of $\delta$ are both reduced to a modest degree when bigger values of $\delta$ are present. This is because the amount of energy required to turn a reactant into a product increase with rising.

The effect that the fitted rate constant $n1$ has had on Sherwood number is seen in Fig. 12 (e). It is possible for the fitted rate constant to have a completely distinct influence on the $\left({Re}_x^{-1/2}{Sh}_x\right)$, and this is all dependent on the dimensionless activation energy $\delta$. When $\delta$ has small values, the Sherwood number is found to be strongly influenced by $n1\text{,}$ however when $\delta$ has greater values, this impact is discovered to have less of an effect. According to Eq. (11), the reaction rate rises when $n1$ is raised from $-1.0$ to $1.0$ because this causes the Sherwood number to grow. As a direct consequence of this, a greater proportion of reactants are transformed into products, which causes the Sherwood number to rise.

6. Conclusion

Over a shrinking/stretching Riga plate that was also vulnerable to suction through the surface, heat and mass transport of chemically reacting unsteady stagnation point flow were examined. Additionally taken into consideration is the effect of heat radiation. Using the RK-IV with shooting technique in MATHEMATICA program, the lowered nonlinear ordinary differential equations (ODEs) combined with associated boundary conditions were mathematically resolved and calculated. The influence of several different operating parameters, for instance unsteadiness $\left(\epsilon \right)$, stretching/shrinking $\left(\lambda \right)$, modified Hartmann number $\left(Z\right)$, electrode width $\left(d\right)$, reaction rate ($\beta )$, temperature difference $\left(\alpha \right)$, activation energy $\left(\delta \right)$, fitted rate $\left(n1\right)$, Schmidt number $\left(Sc\right)$ and radiation parameter $\left(R\right)$ were scrutinized. The following may be said as the result of this research:

• •The velocity profile increased with $Z$, stretching $\lambda$ and suction $S$ parameters and decreased with unsteadiness $\epsilon$ parameter.
• •The temperature profile diminished with growing amounts of unsteadiness $\epsilon$, stretching $\lambda$, and suction $S$ parameters.
• •The concentration profile slow down with $Z,\alpha ,\epsilon ,Sc,\beta ,\delta$ and $n1$ parameters.
• •Skin friction profile increased with $Z$ and $S$ and decreased with $\epsilon \text{.}$.
• •Nusselt number profile increased with $S$ and $\epsilon$.
• •In addition, the relatively recent modification of the Hartmann number $Z$ had contributed to an increase in the $\left({Re}_x^{-1/2}{Nu}_x\right)$ as well as a reduction in temperature distribution profile. Because of this, thermal boundary layer is pushed below, which causes an improvement in the performance of the $\left({Re}_x^{-1/2}{Nu}_x\right)$. On the other hand, effect of thermal radiation results in an increase in $\left({Re}_x^{-1/2}{Nu}_x\right)$, it ultimately results in a boost up in the heat transfer of the material.
• •Sherwood number profile shows upsurges with $Z,\epsilon ,Sc,\beta ,\alpha$ and $n1$ whereas slow down with $\delta$.
• •If these discoveries are confirmed, they might have a substantial effect on the growth of technology that makes use of the conjugate effects of various fluids and chemical reactions.

The model formulation and the conclusions of present study are important and may be utilized as a guide, particularly for the researchers who are working in the field of nanofluids. For a more thorough analysis, additional features including viscosity dissipation, melting, and Joule heating may be taken into account in future studies. It is recommended to use a variety of base fluids and magnetic nanoparticles. Even though this study is based on theory, it is strongly suggested that an experimental study of this flow issue also be done to add to the research that is already being done.

Author contribution statement

Umar Khan: Conceived and designed the experiments.

Zafar Mahmood: Conceived and designed the experiments; Wrote the paper.

Sayed M Eldin: Performed the experiments; Wrote the paper.

Basim M. Makhdoum: Performed the experiments; Contributed reagents, materials, analysis tools or data.

Bandar M. Fadhl: Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data.

Ahmed Alshehri: Analyzed and interpreted the data.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement

Data will be made available on request.

Declaration of interest's statement

The authors declare no conflict of interest.