Aspects of Homogeneous Heterogeneous Reactions for Nanofluid Flow Over a Riga Surface in the Presence of Viscous Dissipation

Abstract

The aim of our study is to delineate the characteristics of fluid flow comprising single-wall and multi-wall carbon nanotubes (SWCNTs and MWCNTs) along the surface of a Riga plate fixed in a porous environment. We carried out in-depth comparative analysis to depict the behavior of SWCNTs and MWCNTs when water and kerosene oil are used as base fluids. Homogeneous–heterogeneous reactions generated a significant impact on flow dynamics; furthermore, we also discuss the impact of viscous dissipation. We assembled numerical solutions for non-dimensionalized ordinary differential equations with the help of the shooting technique; moreover, by employing the same procedure, we report the conduct of dominating parameters on velocity, temperature, and concentration profiles. The results show highly desirable skin friction coefficient and Nusselt number values, which we exhibit in terms of tabular data.

1. Introduction

The use of carbon nanotubes (CNTs) is extensively applied in innumerable fields, such as air and water filtration, field emissions, catalyst support, biomedical applications, thermal materials, fibers, fabrics, etc. Carbon nanotubes are rolled graphene cylindrical sheets 0.7 nm to 50 nm in diameter. Graphene layers in the carbon nanotubes determine its characteristics, and Iijima [1] is credited to be the first who introduced carbon nanotubes. Younes et al. [2] discussed some relevant properties of nanofluids. Among the key features of a carbon nanotube (CNT) is its ability to incorporate with a stretching sheet [3], especially in heat generation/absorption and thermal radiation. Owing to their characteristics, CNTs were used to examine slip conditions and viscous dissipation for Darcy–Forchheimer flow, which revealed promising results for disk rotation [4]. This has highlighted new avenues for the application of CNTs, which has led to several other studies developing mathematical models to depict the influence of buoyancy force and suction on flow over a moving plate [5]. The application of magnetohydrodynamics (MHD) on a hybrid nanofluidic flow subject to convective conditions has various implications, as revealed by Ahmad et al. [6], who observed the influence on flow behavior when the fluid flows over a stretching sheet placed in a porous medium. With clear benefits, carbon nanoparticles have been applied in molecular dynamics [7], improving the efficiency related to the thermal changes of graphene nanotubes. Noranuar et al. [8] applied a magnetic field to a fluid flow due to non-coaxial disk rotation and used carbon nanotubes as nanoparticles in human blood. Their results are obtained via the Laplace transform technique. Several models have proved the application of carbon nanotubes, such as the Cattaneo–Christov model [9], which examined micropolar liquid and the behavior of streamlines for single- and multiple-walled carbon nanotubes. The use of magnetic nanofluid resulted in the resonance of carbon nanotubes, allowing for more useful results. [10]. Further advancements in the applications of carbon nanotubes have grabbed the attention of computational experts, who revolutionized the thermal features of a multi-wall carbon nanotube placed within a curly square enclosure [11], which they then analyzed by the Navier–Stokes equations, subject to specific Rosseland boundary conditions. This has convinced various other researchers to numerically explore the flow of ferro-fluid within carbon nanotubes over a stretched surface while taking into account the effects of Darcy–Forchheimer and mixed convection [12]. The significance of numerical representation further proved helpful in illustrating the thermal properties of nanofluid flow, which involve MWCNTs along with wall conditions [13]. The examination is carried out with the assistance of the classical differential transform method. In a more recent investigation, the optimal homotopy analysis method (OHAM) is applied to a CNT flow model for two stretching disks instead of one. The study also includes the effects of heat generation, joule heating and viscous dissipation [14]. Some of the relevant works are carried out in Refs. [15,16,17,18,19,20].

The importance of the Riga plate for the fluid flow has been explored in a variety of industrial procedures. Gallies and Lilausis [21] found a way to control the fluid flow by generating a Lorentz force acting parallel to the wall, resulting in magnetic and electric fields with the help of the Riga plate. This plate can be used for the radiation of a powerful agent, skin friction, and pressure drag allied with submarines because it abstains from the separation of the boundary layer. This unique practical application has generated interest among scientists examining the behavior of low-electrical-conductive liquids by considering the Lorentz force responsible for aiding or opposing the fluid flows. Classical MHD flows were established for liquids that have enhanced electrical conductivity fields, as they can be easily controlled in the presence of external magnetic fields with a strength of approximate of 1 Tesla; however, the same cannot be done for liquids with weak conductivity. For these liquids, the Riga plate being a strong external agent serves the purpose [22]. Furthermore, Pantokratoras [23] looked for the characteristics of Blasius and Sakiadis flow initiated by a Riga plate. Mallawi et al. [24] presented their research concerning the influence of double stratification and Cattaneo–Christov double flux on second-grade fluid flow due to a Riga plate. In continuation of these research outcomes, much effort was devoted on improving the heat transfer abilities of nano lubricants by considering a stationary/moving Riga plate and numerically analyzing the effects of mixed convection, thermal radiation, and slip boundary conditions [25]. Advancements in this direction revealed that an extended Riga wedge could address the stagnation point and activation energy features of tangent hyperbolic nanofluid flow, thereby introducing beneficial outcomes [26]. The significance of these results influenced further in-depth investigations for a MHD Newtonian fluid flow derived by a stretchy Riga sheet [27]. This analysis was obtained using the homotopy analysis method, which explained detailed aspects pertaining to Cattaneo–Christov flux and activation energy. Owing to this detailed analysis, it was therefore possible to theoretically uncover the mixed convection and radiation effects on Cattaneo–Christov nanofluid (fourth grade) flow towards a Riga surface [28]. Further improvements in understanding the behaviors of nanofluid were possible upon analyzing the impact of Nield and convective conditions, and viscous dissipation. Khan et al. [29] observed a melting phenomenon for third-grade nanofluidic flow along a Riga wall while considering the influence of radiation, a heat source/sink, and a binary chemical reaction for the flow model. Asogwa et al. [30] researched the roles of MHD, heat sinks, and Soret and Dufour effects for a Cassson fluid flow, which was altered due to an inclined Riga surface. Finally, the role of electromagnetic forces on SWCNT was critically analyzed by taking into account heat source/sink and slip effects for a flow towards a Riga plate, and the findings were expressed using RKF-45 method [31].

Chemical reactions, such as homogeneous and heterogeneous reactions, occur in many biochemical systems, and in combustion, catalysis, etc. Homogeneous reactions generally take place in fluid bulk, whereas the reactions occurring on catalytic surfaces are mostly heterogeneous reactions; therefore, the homogeneous reactions taking place in liquids, and the heterogeneous reactions occurring on catalytic surfaces, interact and produce different interactive species with numerous velocities. The changes originate both inside the liquids and the catalytic surfaces. Hence, it became possible to formulate a mathematical model that simultaneously considered the effects of homogeneous–heterogeneous reactions on boundary-layer flow in the vicinity of a stagnation point [32]. This analysis involves isothermal cubic kinetics for the modelling of homogeneous reactions, while the heterogeneous reactions retain first-order kinetics. In addition, Chaudhary and Merkin [33] extended their research and examined results concerning the loss of an auto-catalyst. The solutions close to the leading edge of the flat plate were numerically observed. Merkin [34] analyzed isothermal homogeneous–heterogeneous reactions for the flow over a plane surface. In light of these analyses, the Sutterby fluid flow with a rotating disk of varying thickness was considered and solved through the homotopy analysis method [35]. The outcomes were also influenced by homogeneous–heterogeneous reactions and thermal radiations. Considering the fact that concentration profiles behave differently for homogeneous and heterogeneous reactions, the relevant parameters were observed by Waqas et al. [36] by focusing the homogeneous–heterogeneous reactions and slip conditions for the flow of Walter’s B nanofluid. This led to the formulation of the idea of employing homogeneous–heterogeneous reactions (H-HR) and thermal radiations for Darcy–Forchheimer flow [37]. Eid et al. [38] applied H-HR for non-Newtonian fluid flow in the presence of a surface extending horizontally, and the features of thermal radiation and heat generation/absorptions were also a part of the study. Due to the significant outcomes gained by homogeneous–heterogeneous reactions, further recent studies involved the development of axisymmetric boundary-layer nanofluidic flow owing to a stretchable horizontal cylinder submerged in a non-Darcy porous medium and the homogeneous–heterogeneous reaction effects on the flow regime were taken under consideration [39]. The investigations performed by the Ayub et al. [40] include homogeneous and heterogeneous reactions appearing in a three-dimensional radiative nanofluidic flow surrounded by two rotatory stretching disks, with the additional effects of viscous dissipation along with Joule heating.

Extending the research discussed above, we intend to develop an understanding about the homogeneous–heterogeneous reactions that occur in a flow including carbon nanotubes along a Riga plate set in a porous medium. This has increased the novelty of this analysis, as the study outcome will have innumerable real-world physical applications, including in polymer production and ceramics, and in many other industries related to hydrometallurgy and food processing. In the current study, we also focus on the effects of viscous dissipation during heat transfer. Water and kerosene are accounted for as base fluids in the presence of single- and multi-wall carbon nanotubes. The shooting approach provides the solution for the considered dimensionless governing equations. The graphs display the behavior of admissible parameters, which are responsible for controlling the flow and a numerical calculation is executed for the skin friction coefficient, in addition to the Nusselt number.

2. Formulation

Let us deal with the nanofluidic fluid that is actuated due to the Riga plate (see Figure 1). The flow is passing through both single- and multi-walled carbon nanotubes, and the base fluid is chosen to be kerosene oil and water. The Riga plate is fixed in the direction of the x-coordinate, while the $y$-coordinate remains perpendicular to the plate. The element of viscous dissipation effects the system. For modifications in the mass transfer rates, homogenous–heterogeneous reactions are taken into consideration

Figure 1. Schematic diagram.

For homogeneous reactions, the isothermal cubic autocatalysis is specified as:

$$A+2B\to 3B, \mathrm{Rate}=k_{c}a{b}^2,$$

at the same time, a single first-order isothermal (heterogeneous) reaction occurring on the catalyst surface is recorded in the following way:

$$A\to B, \mathrm{Rate}=k_{s}a,$$

The above expressions relate the homogeneous reactions with the heterogeneous reactions taking place in a boundary-layer flow. In these isothermal reactions, which exist far away from the plate, reactant $A$ possesses a consistent concentration $a_0$ in the absence of auto-catalyst $B$ in the external flow.

To proceed towards the construction of the problem, the system of equations is organized as:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}={\nu }_{nf}\left(\frac{{\partial }^{2}u}{\partial y^2}\right)-{\nu }_{nf}\frac{1}{k^{\prime }}u+\frac{\pi j_0{M}_{0}Exp(-\frac{\pi }{b}y)}{8{\rho }_{nf}},$$

(2)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha }_{nf}\left(1+\frac{16{\sigma }^{\ast }{T}_{\infty }^3}{3k^{\ast }{k}_{nf}}\right) \frac{{\partial }^{2}T}{\partial y^2}+\frac{{\mu }_{nf}}{{(\rho c_p)}_{nf}}{\left(\frac{\partial u}{\partial y}\right)}^2,$$

(3)

$$u\frac{\partial a}{\partial x}+v\frac{\partial a}{\partial y}=D_A\frac{{\partial }^{2}a}{\partial y^2}-k_{c}a{b}^2,$$

(4)

$$u\frac{\partial b}{\partial x}+v\frac{\partial b}{\partial y}=D_B\frac{{\partial }^{2}b}{\partial y^2}+k_{c}a{b}^2,$$

(5)

$$u\left(x, y\right)=U_w\left(x\right)=cx, v(x, y)=0, -k\frac{\partial T}{\partial y}=q_w, D_A\frac{\partial a}{\partial y}=k_{s}a, D_B\frac{\partial b}{\partial y}=-k_{s}a \mathrm{at} y=0,$$

(6)

$$u\left(x,y\right)\to 0, T\to T_{\infty }, a\to a_0, b\to 0 \mathrm{as} y\to \infty .$$

(7)

In the above designed equations, $u$, $v$, and $w$ are noted as the velocity components along the $x$, $y$, and $z$ directions, respectively, ${\nu }_{nf}=\left(\frac{{\mu }_{nf}}{{\rho }_{nf}}\right)$ symbolizes nanofluidic kinematic viscosity, ${\sigma }^{\ast }$ is the Stefan Boltzmann constant, $k^{\ast }$ is the mean absorption constant, $k_{nf}$ is the thermal conductivity of nanofluid, $j_0$ is the current density within electrodes, $M_0$ is the magnetization due to permanent magnets, $b$ indicates the width of the electrodes and magnets, ${\rho }_{nf}$ represents nanofluid density, ${\alpha }_{nf}$ is the thermal diffusivity of nanofluid, ${\mu }_{nf}$ specifies the dynamic viscosity of the nanofluid, ${(\rho c_p)}_{nf}$ stands for the heat capacitance of the nanofluid, $a$ and $b$ are the concentrations of the chemical species recognized as $A$ and $B$, $D_A$ and $D_B$ indicate the respective diffusion coefficients corresponding to the species $A$ and $B$, $k_c$ and $k_s$ represent rate constants, $T$ is temperature of fluid, $T_{\infty }$ denotes ambient temperature and $a_0$ is the uniform concentration.

The earlier models of nanofluid were developed for rotational elliptical or spherical particles with small axial ratios, and these nanofluidic models fail to elaborate the space distribution characteristics regarding carbon nanotubes, as these nanotubes possess large axial ratios. This issue was resolved by introducing theoretical models for CNTs with large axial ratios. The significant properties of CNTs can be indicated through the characteristics of the base fluid and carbon nanotubes; furthermore, the solid carbon nanotube volume friction of the base fluid is displayed as:

$${\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\varnothing \right)}^{2.5}}, {\rho }_{nf}=\left(1-\varnothing \right){\rho }_f+\varnothing {\rho }_{CNT}, {\alpha }_{nf}=\frac{k_{nf}}{{\left(\rho c_p\right)}_{nf}}, v_{nf}=\frac{{\mu }_{nf}}{{\rho }_{nf}},$$

$$\frac{k_{nf}}{k_f}=\frac{\left(1-\varnothing \right)+2\alpha \frac{k_{CNT}}{k_{CNT}-k_f}ln\frac{k_{CNT}+k_f}{2k_f}}{\left(1-\varnothing \right)+2\alpha \frac{k_f}{k_{CNT}-k_f}ln\frac{k_{CNT}+k_f}{2k_f}}, {\left(\rho c_p\right)}_{nf}=\left(1-\varnothing \right){\left(\rho c_p\right)}_f+\varnothing {\left(\rho c_p\right)}_{CNT},$$

(8)

where $\varnothing$ is nanoparticle volume fraction,$k_f$ and $k_{nf}$ denote the thermal conductivities of the fluid and the carbon nanotubes, and ${\rho }_f$ and ${\rho }_{CNT}$ characterize the densities of the fluid and carbon nanotubes, respectively.

The transformations specified for this flow are:

$$\begin{array}{c}\psi =\sqrt{c{\nu }_f}xf(\eta ), \eta =\sqrt{\frac{c}{{\nu }_f}}y, \\ u=\frac{\partial \psi }{\partial y}=cx{f}^{\prime }\left(\eta \right), v=-\frac{\partial \psi }{\partial x}=-\sqrt{c{\nu }_f}f\left(\eta \right),\\ \theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }}, a=a_{0}g\left(\eta \right), b=a_{0}h\left(\eta \right).\end{array}$$

(9)

The continuity equation is identically satisfied while Equations (2)–(5) transform as

$$\frac{1}{{(1-\phi )}^{2.5}(1-\phi +\phi (\frac{{\rho }_{{}_{CNT}}}{{\rho }_f}))}{f}^{‴}+f{f}^{″}-{\left(f^{\prime }\right)}^2-k_1{f}^{\prime }+Q{e}^{-C\eta }=0,$$

(10)

$$\left(\frac{\frac{k_{nf}}{k_f}}{(1-\phi +\phi \frac{{(\rho c_p)}_{CNT}}{{(\rho c_p)}_f})}\right) \left(1+\frac{4k_f}{3k_{nf}}{R}_d\right) {\theta }^{″}+\mathrm{Pr}f{\theta }^{\prime }+\frac{\mathrm{Pr}Ec}{{(1-\phi )}^{2.5}}{f}^{\prime \prime 2}=0,$$

(11)

$$\frac{1}{Sc}{g}^{″}+f{g}^{\prime }-Kg{h}^2=0,$$

(12)

$$\frac{\delta }{Sc}{h}^{″}+f{h}^{\prime }+Kg{h}^2=0,$$

(13)

$$f\left(0\right)=0, f^{\prime }\left(0\right)=1,f^{\prime }\left(\infty \right)\to 0,{\theta }^{\prime }\left(0\right)=-\frac{k_f}{k_{nf}},\theta \left(\infty \right)\to 0,$$

$$g^{\text{'}}\left(0\right)=K_{s}g\left(0\right),g\left(\infty \right)\to 1,\delta h^{\prime }\left(0\right)=-K_{s}g\left(0\right),h\left(\infty \right)\to 0,$$

(14)

where $k_1$ is the porous medium parameter, $Q$ is the modified Hartman number, $C$ is the dimensionless parameter,$\mathrm{Pr}$ is the Prandtl number, $Ec$ is the Eckert number, $R_d$ is the thermal radiation parameter, $\delta$ is the ratio of diffusion coefficients, $Sc$ is the Schmidt number, and $K$ and $K_s$ denote the measure of homogenous and heterogeneous reaction strength, respectively. The non-dimensionalized parameters are determined to be

$$\begin{array}{c}\mathrm{Pr}=\frac{{\mu }_f{\left(c_p\right)}_f}{k}, k_1=\frac{{\nu }_{nf}}{k^{\prime }c}, Sc=\frac{{\nu }_f}{D_A}, K=\frac{k_c{a}_0^2}{c},\\ \delta =\frac{D_B}{D_A}, C=\frac{\pi }{b}\sqrt{\frac{{\nu }_{nf}}{c}}, Q=\frac{\pi j_0{M}_{0}x}{8{\rho }_{nf}{U}_w^2},\\ Ec=\frac{U_w^2}{{(\rho c_p)}_{nf}(T_w-T_{\infty })}, R_d=\frac{4{\sigma }^{\ast }{T}_{\infty }^3}{k^{\ast }{k}_f}.\end{array}$$

(15)

The consideration that both chemical species $A$ and $B$ have diffusion coefficients with comparable sizes implies the equivalence of the diffusion coefficients $D_A$ and $D_B$, suggesting that $\delta =1$. Thus, an important relation between $g\left(\eta \right)$ and $h\left(\eta \right)$ can be written as (Chaudhary and Merkin [32])

$$g\left(\eta \right)+h\left(\eta \right)=1$$

(16)

Accordingly, Equations (12) and (13) reduce to

$$\frac{1}{Sc}{g}^{″}+f{g}^{\prime }-Kg{\left(1-g\right)}^2=0,$$

(17)

which are then subjected to the following boundary conditions

$$g^{\text{'}}\left(0\right)=K_{s}g\left(0\right), g\left(\infty \right)\to 1.$$

(18)

Skin friction coefficient and local Nusselt number are characterized as:

$$C_f=\frac{{\tau }_w}{{\rho }_f{U}_w^2}, N{u}_x=\frac{x{q}_w}{k_f(T_h-T_m)},$$

(19)

with the wall shear stress ${\tau }_w$ and the wall heat flux $q_w$ given as

$${\tau }_w={\mu }_{nf}{\left(\frac{\partial u}{\partial y}\right)}_{y=0}, q_w=-k_{nf}{\left(\frac{\partial T}{\partial y}\right)}_{y=0}.$$

(20)

The dimensionless forms of these quantities are presented to be:

$$C_f\mathrm{R}{\mathrm{e}}_x^{1/2}=\frac{1}{{\left(1-\phi \right)}^{2.5}}{f}^{″}(0), N{u}_x\mathrm{R}{\mathrm{e}}_x^{-1/2}=-\frac{k_{nf}}{k_f}\theta \left(0\right),$$

where $\mathrm{R}{\mathrm{e}}_x=U_{w}x/{\nu }_f$ stands for the local Reynolds number.

3. Solution Methodology

In the present analysis, the solution is achieved by implementing the shooting method approach. This technique assists in solving Equations (10), (11), and (17), along with the associated boundary conditions presented in (14) and (18). Proceeding with this method, the higher order equations in $f(\eta )$, $\theta (\eta )$ and $\phi (\eta )$ are transformed into new ODEs,

$$f=p_1, f^{\prime }=p_2=p_1^{\prime }, f^{″}=p_3=p_2^{\prime },$$

$$f^{‴}=p_3^{\prime }=\left[{\left(1-\phi \right)}^{2.5}\left(1-\phi +\phi \left(\frac{{\rho }_{CNT}}{{\rho }_f}\right)\right)\right] \left(-p_1{p}_3+{\left(p_2\right)}^2+k_{1}p+Q{e}^{-C\eta }\right),$$

$$\theta =p_4, {\theta }^{\prime }=p_5,$$

$${\theta }^{″}=p_5^{\prime }=\frac{1}{\left(\frac{\frac{k_{nf}}{k_f}}{(1-\phi +\phi \frac{{(\rho c_p)}_{CNT}}{{(\rho c_p)}_f})}\right) \left(1+\frac{4k_f}{3k_{nf}}{R}_d\right)}\left(-\mathrm{Pr}{p}_1{p}_5-\frac{\mathrm{Pr}Ec}{{(1-\phi )}^{2.5}}{p}_3^2\right),$$

$$g=p_6, g^{\prime }=p_7,$$

$$g^{″}=p_7^{\prime }=Sc\left[-p_1{p}_7+K{p}_6{\left(1-p_6\right)}^2\right],$$

under the boundary conditions,

$$p_1(0)=0, p_2(0)=1, p_5(0)=-\frac{k_f}{k_{nf}}, p_7(0)=K_s{p}_6(0).$$

Next, initial guesses are made for the boundary conditions for $p_3(0), p_5(0)$, and $p_7(0).$

4. Discussion

The occurrence of stretching surfaces in order to describe the boundary-layer phenomenon is one of the unique aspects that has enabled us to understand the performance of various physical parameters of nanofluids. Results have characteristically provided the evidence that variations in different parameters linked with the assumptions of the problem influence the velocity, temperature, and concentration profiles. Figure 2 reveals the behavior of the modified Hartmann number $Q$ on the velocity profile corresponding to the cases of both SWCNTs and MWCNTs, considering water and kerosene oil as base fluids. An increase of the modified Hartmann number intensifies the velocity profile. This is due to Lorentz force (surface parallel), which supports the flow in the positive x-direction. Additionally, the velocity distribution for carbon–water nanofluid dominates when compared with carbon–kerosene oil for both nanotubes cases. Figure 3 gives the illustration of the nanoparticle volume fraction parameter $\phi$ behavior for the velocity profile. The plot confirms that an increased nanoparticle volume fraction parameter raises the velocity profile. In this fig., it is evident that the velocity profile in the case of carbon–kerosene oil diminishes if compared with carbon–water nanofluid for both carbon nanotubes. Figure 4 depicts the influential behavior of dimensionless parameter $C$ for the velocity profile in the cases of carbon–water and carbon–kerosene oil nanofluids. The dimensionless parameter causes the reduction of the velocity profile. Figure 5 shows the characteristics of the porous medium parameter $k_{1 }$ on the velocity profile for both considered carbon nanofluids. The figure shows that the elevated values of the porous medium parameter correspond to the smaller values of the velocity profile. The porous medium induces resistance to the flow, which eventually causes the decay in velocity profile. The carbon–water nanofluid shows an increasing behavior compared with carbon–kerosene oil for velocity distribution. Figure 6 specifies the action of the dimensionless parameter $C$ on temperature profile. Increased values of the dimensionless parameter are associated with an increase in temperature profile. The temperature distribution for carbon–kerosene oil has a superior enhancing effect than the carbon–water nanofluid (considering both SWCNTs and MWCNTs). Figure 7 displays the behavior of the modified Hartmann number $Q$ on temperature profile. Higher values of the modified Hartmann number are accompanied with a decline of temperature profile, while temperature distribution for carbon–kerosene oil nanofluid dominates for both SWCNTs and MWCNTs compared with carbon–water nanofluids. Figure 8 shows the study of the thermal radiation parameter $R_d$ on the temperature profile for carbon–water, as well as carbon–kerosene oil nanofluids. The amplification of temperature profile is seen when the thermal radiation parameter enhances. Figure 9 exhibits the behavior of the nanoparticle volume fraction parameter $\phi$ corresponding to temperature profile, and the results also take carbon–water and carbon–kerosene oil nanofluids into account. With the increase of the nanoparticle volume fraction parameter, the temperature profile expands. Figure 10 shows the strength of the homogeneous parameter $K$ on concentration profile when carbon–water and carbon–kerosene oil nanofluids are under study. Larger values of the homogeneous parameter are associated with a decay in concentration profile. The concentration distribution of carbon–water nanofluid displays an increasing behavior when compared with carbon–kerosene oil nanofluid for both carbon nanotubes. Figure 11 indicates the strength of heterogeneous parameter $K_s$ for the concentration profile, which rises for advanced values of $K_s.$ Enhanced concentration profiles for carbon–kerosene oil nanofluid are noticed when compared with carbon–water nanofluid. Figure 12 shows the outcome of the Schmidt number $Sc$ on the concentration profile. By increasing the Schmidt number, the increasing behavior of the concentration profile is detected. The Schmidt number is interpreted to be the ratio of momentum to mass diffusivity; therefore an enhanced Schmidt number results in a decrease of mass diffusivity, which in turn triggers the concentration profile.

Figure 2. Behavior of f′(η) for increased values of Q.

Figure 3. Behavior of f′(η) for increased values of ϕ.

Figure 4. Behavior of f′(η) for increased values of C.

Figure 5. Behavior of f′(η) for increased values of $k_1$

Figure 6. Behavior of θ(η) for increased values of C.

Figure 7. Behavior of θ(η) for increased values of Q.

Figure 8. Behavior of θ(η) for increased values of $R_d$

Figure 9. Behavior of θ(η) for increased values of ϕ.

Figure 10. Behavior of g(η) for increased values of K.

Figure 11. Behavior of g(η) for increased values of $K_s$

Figure 12. Behavior of g(η) for increased values of S_c.

Table 1 shows the thermophysical properties (density, specific heat capacity, and thermal conductivity) of base fluid (water and kerosene oil) and carbon nanotubes. Table 2 specifies the impact of physical parameters on the skin friction coefficient for SWCNT–water, SWCNT–kerosene, MWCNT–water, and MWCNT–kerosene. The attained result highlights that the magnitude of the skin friction coefficient increases as $\phi ,$ $Q,$ $C$, and $k_1$ increase for SWCNT–water, SWCNT–kerosene, MWCNT–water, and MWCNT–kerosene. The skin friction coefficient is larger for SWCNTs when compared with MWCNTs. Table 3 explores the trend of notable parameters for the Nusselt number. Larger $\phi$ and $R_d$ result in an enhancement of the magnitude of the Nusselt number for SWCNT–water, SWCNT–kerosene, MWCNT–water, and MWCNT–kerosene and a decrease when the $k_1,$ $Q,$ $C$, and $Ec$ increase. The Nusselt number is higher for SWCNTs for $k_1, Q, C$, and $R_d$.

Table 1. Thermophysical features of base fluid and nanoparticles (SWCNT and MWCNT) [41].

Physical Features	Base Fluids		Nanoparticles
	Water	Kerosene	SWCNT	MWCNT
$\rho$	997.0	783.0	2600.0	1600.0
$c_p$	4179.0	2090.0	425.0	796.0
k	0.613	0.145	6600.0	3000.0

Table 2. Numerical values of various parameters for skin friction coefficient.

				Water		Kerosene
$\mathit{\phi }$	${\mathit{k}}_1$	Q	C	SWCNT	MWCNT	SWCNT	MWCNT
0	0.3	0.1	0.3	−1.16519	−1.36754	−1.16519	−1.16519
0.1				−1.43133	−1.46554	−1.47503	−1.39579
0.2				−1.76819	−1.62734	−1.86196	−1.69034
0.2	0.0	0.1	0.3	−1.56090	−1.43595	−1.64428	−1.49180
	0.1			−1.63285	−1.50239	−1.71980	−1.56072
	0.2			−1.70184	−1.56609	−1.79225	−1.62680
0.2	0.3	0.0	0.3	−1.73254	−1.59575	−1.82354	−1.65695
		0.1		−1.76819	−1.62734	−1.86196	−1.69034
		0.2		−1.80446	−1.65938	−1.90113	−1.72425
0.2	0.3	0.1	0.0	−1.73254	−1.59575	−1.82354	−1.65695
			0.1	−1.74816	−1.60966	−1.84032	−1.67163
			0.2	−1.75955	−1.61974	−1.85260	−1.68228

Table 3. Numerical values of various parameters for Nusselt number.

						Water		Kerosene
$\mathit{\phi }$	${\mathit{k}}_1$	Q	C	${\mathit{R}}_{\mathit{d}}$	Ec	SWCNT	MWCNT	SWCNT	MWCNT
0.0	0.2	0.1	0.3	0.2	0.3	−1.21703	−1.21703	−1.48849	−1.48849
0.1						−4.13555	−3.98796	−5.12215	−5.09484
0.2						−7.94600	−7.84866	−9.48526	−9.80823
0.1	0.0	0.1	0.3	0.2	0.3	−4.43275	−4.26328	−5.61607	−5.57988
	0.2					−4.13555	−3.98796	−5.12215	−5.09484
	0.4					−3.88179	−3.75257	−4.72038	−4.70091
0.1	0.2	0.0	0.3	0.2	0.3	−4.27314	−4.10677	−5.33273	−5.28795
		0.1				−4.13555	−3.98796	−5.12215	−5.09484
		0.2				−3.99108	−3.86425	−4.90919	−4.90054
0.1	0.2	0.2	0.0	0.2	0.3	−4.27314	−4.10677	−5.33273	−5.28795
			0.1			−4.12502	−3.97947	−5.11316	−5.08679
			0.2			−4.03841	−3.90523	−4.98348	−4.96853
0.1	0.2	0.2	0.2	0.0	0.3	−3.77310	−3.63455	−4.58980	−4.55941
				0.1		−3.90738	−3.77162	−4.78703	−4.76444
				0.2		−4.03841	−3.90523	−4.98348	−4.96853
0.1	0.2	0.2	0.2	0.2	0.0	−7.70126	−7.20680	−18.9855	−17.8097
					0.1	−5.91343	−5.62237	−9.80372	−9.56738
					0.2	−4.03841	−4.60906	−6.60796	−6.54046

5. Closing Remarks

This study deals with fluid flow in carbon nanotubes along a stretchable Riga surface with the existence of homogeneous–heterogeneous reactions. Our notable observations are compiled as follows:

• The obtained results for SWCNT and MWCNT signify that flow acceleration depends on larger values of the modified Hartmann number $Q$ and the nanoparticles volume fraction parameter $\phi$;
• We observed a considerable degree of resistance during flow when the values of the porous medium parameter $k_1$ are increased. We noticed the decreasing effect of the dimensionless parameter $C$ on the velocity profile in both types of carbon nanotubes;
• Single-walled carbon nanotubes possess a higher temperature distribution in comparison with multi-wall carbon nanotubes in cases of nanoparticles volume fraction $\phi$, modified Hartmann number $Q$, dimensionless parameter $C$, and thermal radiation parameter $R_d$. This discovery can assist engineers in the thermal management of different devices;
• The concentration distribution experiences a reducing effect when the strength of the homogeneous parameter $K$ increases;
• With the ascent of heterogeneous parameter strength $K_s$ and Schmidt number $Sc$, the concentration profiles increase for SWCNT and MWCNT. The increase in $Sc$ is related to the increase in the fluid’s kinematic viscosity;
• The skin friction coefficient increase causes additional sheer stress for all the elevated values of the significant parameters, while in case of the Nusselt number, an enhancing behavior is presented only for $\phi$ and $R_d$, which are ultimately responsible for increased flow convection rates.