Hybrid Nanofluid Flow over a Stretching Curved Surface with Induced Magnetic Field and Homogeneous-Heterogeneous Reactions

Ming Shen¹, Yunhua Zheng¹, Yihong Liu¹, Hui Chen^2*, Mengchen Zhang¹

¹School of Mathematics and Statistics, Fuzhou University, Fuzhou, China.
²School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou, China.

DOI: 10.4236/jamp.2024.1210218                

Abstract

This study explores the 2D stretching flow of a hybrid nanofluid over a curved surface influenced by a magnetic field and reactions. A steady laminar flow model is created with curvilinear coordinates, considering thermal radiation, suction, and magnetic boundary conditions. The nanofluid is made of water with copper and MWCNTs as nanoparticles. The equations are transformed into nonlinear ODEs and solved numerically. The model’s accuracy is confirmed by comparing it with published data. Results show that fluid velocity increases, temperature decreases, and concentration increases with the curvature radius parameter. The hybrid nanofluid is more sensitive to magnetic field changes in velocity, while the nanofluid is more sensitive to magnetic boundary coefficient changes. These insights can optimize heat and mass transfer in industrial processes like chemical reactors and wastewater treatment.

Keywords

Hybrid Nanofluids, Stretching Curved Surface, Induced Magnetic Field, Improved Shooting Method

1. Introduction

The pursuit of more efficient heat transfer technologies has been a key driver for advancements across various industries [1] [2], from energy production to medical applications. Nanofluids [3], which incorporate nanoscale solid particles into base fluids, have emerged as a promising candidate to tackle the challenge of overcoming the thermal limitations of conventional fluids. Nanoparticles demonstrate significant potential to enhance the thermal performance of fluids[4] due to their exceptional thermal conductivity, which paves the way for the development of more efficient heat transfer systems. Consequently, nanofluid technology is widely regarded as a promising and highly potential technical solution. However, existing single nanofluids have limitations in terms of stability and convective heat transfer performance under high-temperature conditions [5]. To overcome these challenges, researchers are actively investigating the potential of hybrid nanofluids, aiming to leverage their combined properties to achieve superior heat transfer performance [6]-[9].

In recent years, an increasing number of researchers have begun to focus on how the dynamic behavior of homogeneous-heterogeneous reactions affects fluid flow and heat transfer characteristics, providing new directions for optimizing reaction processes and improving system efficiency. Chaudhary et al. [10] [11] developed a simplified model of stagnation point boundary layer flow for homogeneous-heterogeneous reactions, analyzing the steady-state behavior of the system when the diffusion coefficients of the reactants and autocatalysts are equal and unequal. Inayat et al. [12] explored the flow of three-dimensional nanofluids over a stretching surface, focusing on the heat transfer caused by melting as well as the effects of homogeneous and heterogeneous reactions. Kotha et al. [13] investigated the flow of thermoradiative viscoelastic fluids with homogeneous and heterogeneous chemical reactions in a helical coil, finding that the regularized L.R. method yielded the best results.

The fluid flow induced by a curved stretching sheet has garnered significant interest from researchers because of its applications in the engineering of stretchable elastic materials. Maria et al. [14] studied homogeneous-heterogeneous reactions in two-dimensional MHD radiative flow caused by a curved stretching surface. Maryiam et al. [15] investigated the influence of homogeneous-heterogeneous reactions on the MHD peristaltic mechanism of Ellis fluid in a curved channel. Sanni et al. [16] investigated the steady MHD viscous flow over a nonlinear quadratic stretching surface. Additionally, they [17] examined the two-dimensional heat transfer of a conductive cross fluid over a power-law stretching surface under the influence of a variable external magnetic field.

In recent years, a new technology called liquid gating has emerged, considered one of the top ten emerging technologies in the chemical field in 2020 by the International Union of Pure and Applied Chemistry [18]. Sheng et al. [19] demonstrated that external magnetic fields can precisely control the collective dynamics of confined magnetic colloids. Additionally, experiments have shown a strong link between the threshold pressure for fluid invasion and magnetic field strength [20]. Chen et al. [21] designed a new boundary condition with the magnetic response to study superior heat and mass transfer of Maxwell ternary nanofluid for a stagnation-point flow.

Upon reviewing the existing literature on fluid flow research, it is evident that the utilization of curvilinear coordinate systems remains relatively scarce. In light of this, our study aims to delve into the impact of homogeneous-heterogeneous reactions and induced magnetic fields on the flow, heat transfer, and mass transfer properties of binary nanofluids within a curvilinear coordinate framework. Additionally, we will examine the effects of permeable surfaces and magnetic-responsive boundaries on these characteristics, thereby contributing to a more comprehensive understanding of fluid dynamics in complex geometries.

2. Mathematical Analysis

2.1. Modeling Assumptions

The curvilinear coordinates (r, s) are employed for our analysis, where r represents the radial distance from the center O of the curved geometry, and s signifies the arc length component aligned with the flow direction. The surface is stretched with velocity (u_w(s) = as) along the s-direction, as displayed in Figure 1. An induced magnetic field M is also taken into account, with M₁ and M₂ denoting the parallel and normal components, respectively.

In the analysis of the homogeneous and heterogeneous reactions, two chemical species, denoted A and B, are taken into account. The homogeneous reaction, following cubic autocatalysis, is expressed as

A + 2B → 3B, rate = τ_cC_{1 $C_2^2$} ,

while the heterogeneous reaction, occurring on the catalyst surface can be described as

A → B, rate = τ_sC₁.

Here ${\tau }_c$ and ${\tau }_s$ are the rate constants, while the concentrations of chemical species A and B are denoted as C₁ and C₂, respectively.

Figure 1. Physical configuration and coordinate system.

2.2. Govern Equations

Under the above assumptions and in the presence of thermal radiation according to Rosseland’s approximation [22], ignoring free charges and the displacement currents, the governing boundary layer equations for the flow, induced magnetic field, energy and concentration equations in curvilinear coordinates can be stated as [23]:

$\begin{array}{c}\frac{\partial }{\partial r}\left\{\left(r+R\right)v\right\}+R\frac{\partial u}{\partial s}=0,\end{array}$ (2.1)

$\begin{array}{c}\frac{u^2}{r+R}=\frac{1}{{\rho }_{hnf}}\frac{\partial p}{\partial r},\end{array}$ (2.2)

$\begin{array}{l}v\frac{\partial u}{\partial r}+\frac{Ru}{r+R}\frac{\partial u}{\partial s}+\frac{uv}{r+R}\\ =-\frac{1}{{\rho }_{hnf}}\frac{R}{r+R}\frac{\partial p}{\partial s}+\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r+R}\frac{\partial u}{\partial r}-\frac{u}{{(r+R)}^2}\right)\\ +\frac{{\mu }_{ef}}{4\pi {\rho }_{hnf}}\left(\frac{R}{r+R}{M}_1\frac{\partial M_1}{\partial s}+\frac{M_1{M}_2}{r+R}+M_2\frac{\partial M_1}{\partial r}\right),\end{array}$ (2.3)

$\frac{\partial }{\partial r}\left\{\left(r+R\right)M_2\right\}+R\frac{\partial M_1}{\partial s}=0,$ (2.4)

$\begin{array}{l}\frac{R}{r+R}u\frac{\partial M_1}{\partial s}+\nu \frac{\partial M_1}{\partial r}+\frac{u{M}_2}{r+R}-\left[\frac{R}{r+R}{M}_1\frac{\partial u}{\partial s}+M_2\frac{\partial u}{\partial r}+\frac{M_1\nu }{r+R}\right]\\ =\zeta \left(\frac{{\partial }^2{M}_1}{\partial r^2}+\frac{1}{r+R}\frac{\partial M_1}{\partial r}-\frac{M_1}{{\left(r+R\right)}^2}\right),\end{array}$ (2.5)

$\begin{array}{c}\left(\frac{R}{r+R}u\frac{\partial T}{\partial s}\right)+\nu \frac{\partial T}{\partial r}=\left(\frac{{\kappa }_{hnf}}{{\left(\rho C_p\right)}_{hnf}}+\frac{16{\sigma }^{*}{T}_{\infty }{}^3}{3K^{*}{\left(\rho C_p\right)}_{hnf}}\right)\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r+R}\frac{\partial T}{\partial r}\right)\\ +\frac{Q}{{\left(\rho C_p\right)}_{hnf}}\left(T-T_{\infty }\right),\end{array}$ (2.6)

$\frac{R}{r+R}u\frac{\partial C_1}{\partial s}+v\frac{\partial C_1}{\partial r}=D_A\left\{\frac{{\partial }^2{C}_1}{\partial r^2}+\frac{1}{r+R}\frac{\partial C_1}{\partial r}\right\}-{\tau }_c{C}_1{C}_2^2$ (2.7)

$\frac{R}{r+R}u\frac{\partial C_2}{\partial s}+v\frac{\partial C_2}{\partial r}=D_B\left\{\frac{{\partial }^2{C}_2}{\partial r^2}+\frac{1}{r+R}\frac{\partial C_2}{\partial r}\right\}+{\tau }_c{C}_1{C}_2^2$ (2.8)

with the following boundary conditions

$u=u_w\left(s\right)=as,$ (2.9)

$v=-s_0-h_0\frac{sgn\left(s_0\right){\mu }_{ef}}{4\pi a^2{s}^2{\rho }_{hnf}}{M}_1{}^2\left(s,0\right),$ (2.10)

$T=T_w,M_2=0,\frac{\partial M_1}{\partial r}=0,$ (2.11)

$D_A\frac{\partial C_1}{\partial r}={\tau }_s{C}_1,D_B\frac{\partial C_2}{\partial r}=-{\tau }_s{C}_1\text{at}r=0,$ (2.12)

$u\to 0,\frac{\partial u}{\partial r}\to 0,T\to T_{\infty },\text{as}r\to \infty$ (2.13)

$M_1\to M_e\left(s\right),C_1\to C_0,C_2\to 0\text{as}r\to \infty .$ (2.14)

Taking into account the profound influence of the magnetic response boundary on the alteration of flow dynamics, the boundary condition (2.10) is designated as the magnetically responsive condition. In the event that h₀ equals zero, this condition reduces to the conventional velocity boundary condition, which encompasses both suction phenomena.

Considering the distinct geometries of the selected nanoparticles, the expressions for thermal-physical properties of hybrid nanofluids are listed in Table 1.

Table 1. Thermal-physical properties of hybrid nanofluids.

2.3. Normalization

To simplify the analysis, we apply the following similarity transformation:

$\eta =\sqrt{\frac{a}{{\vartheta }_f}}r,u=as{f}^{\prime }\left(\eta \right),\nu =-\frac{R}{r+R}\sqrt{a{\vartheta }_f}f\left(\eta \right),$ (2.15)

$p={\rho }_f{a}^2{s}^{2}P\left(\eta \right),M_1=M_{0}s{g}^{\prime }\left(\eta \right),$ (2.16)

$M_2=-M_0\frac{R}{r+R}\sqrt{\frac{{\vartheta }_f}{a}}g\left(\eta \right),\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},$ (2.17)

$C_1=C_{0}G\left(\eta \right),C_2=C_{0}H\left(\eta \right).$ (2.18)

The conservation equations for mass and the magnetic field, namely Equations (2.1) and (2.4), are satisfied identically under the previously mentioned transformation. Consequently, the governing equations are reduced to the following system of nonlinear coupled ordinary differential equations:

$P^{\prime }\left(\eta \right)=A_2\frac{{\left(f^{\prime }\right)}^2}{\eta +k},$ (2.19)

$\begin{array}{l}\frac{1}{A_2}\left(\frac{2k}{\eta +k}\right)P=\frac{A_1}{A_2}\left(f^{‴}+\frac{1}{\eta +k}{f}^{″}-\frac{1}{{\left(\eta +k\right)}^2}{f}^{\prime }\right)\\ -\frac{k}{\eta +k}\left({f^{\prime }}^2-f{f}^{″}-\frac{f{f}^{\prime }}{\eta +k}\right)+\frac{\beta }{A_2}\frac{k}{\eta +k}\left({g^{\prime }}^2-g{g}^{″}-\frac{g{g}^{\prime }}{\eta +k}\right),\end{array}$ (2.20)

$P{r}_M\frac{k}{\eta +k}\left(f^{″}g-f{g}^{″}\right)-P{r}_M\frac{k}{{\left(\eta +k\right)}^2}\left(f^{\prime }g-f{g}^{\prime }\right)g^{‴}+\frac{g^{″}}{\eta +k}-\frac{g^{\prime }}{{\left(\eta +k\right)}^2},$ (2.21)

$\frac{A_4}{Pr{A}_3}\left(1+\frac{Rd}{A_4}\right)\left({\theta }^{″}+\frac{{\theta }^{\prime }}{\eta +k}\right)+\frac{D}{A_3}\theta =\frac{-k}{\eta +k}f{\theta }^{\prime },$ (2.22)

$\frac{1}{Sc}\left(G^{″}+\frac{G^{\prime }}{\eta +k}\right)+\frac{k}{\eta +k}f{G}^{\prime }-{\tau }_{1}G{H}^2=0,$ (2.23)

$\frac{\delta }{Sc}\left(H^{″}+\frac{H^{\prime }}{\eta +k}\right)+\frac{k}{\eta +k}f{H}^{\prime }+{\tau }_{1}G{H}^2=0,$ (2.24)

subject to the corresponding boundary conditions

$f\left(0\right)=S+sgn\left(S\right)h\beta {\left(g^{\prime }\left(0\right)\right)}^2,f^{\prime }\left(0\right)=1,$ (2.25)

$\theta \left(0\right)=1,g\left(0\right)=0,g^{″}\left(0\right)=0,$ (2.26)

$G^{\prime }\left(0\right)={\tau }_{2}G\left(0\right),\delta H^{\prime }\left(0\right)=-{\tau }_{2}G\left(0\right),$ (2.27)

$f^{\prime }\left(\infty \right)\to 0,f^{″}\left(\infty \right)\to 0,\theta \left(\infty \right)\to 0,$ (2.28)

$g^{\prime }\left(\infty \right)\to 1,G\left(\infty \right)\to 1,H\left(\infty \right)\to 0.$ (2.29)

Here, the dimensionless parameters are

$Pr={\vartheta }_f{(\rho c_p)}_f/{\kappa }_f,R_d=16{\sigma }^{*}{T}_{\infty }^3/3K^{*}{\kappa }_f,$

$k=R\sqrt{\frac{a}{{\vartheta }_f}},\beta =\frac{{\mu }_{ef}}{4\pi {\rho }_f}{(\frac{M_0}{a})}^2,P{r}_M=\frac{{\vartheta }_f}{\zeta },D=\frac{Q}{{\left(\rho C_p\right)}_{f}a},Sc=\frac{{\vartheta }_f}{D_A},$

${\tau }_1=\frac{{\tau }_c{C}_0{}^2}{a},{\tau }_2=\frac{{\tau }_s}{D_A}\sqrt{\frac{{\vartheta }_f}{a}},\delta =\frac{D_B}{D_A},S=\frac{s_0}{\sqrt{a{\vartheta }_f}},h=\frac{h_0}{A_2\sqrt{a{\vartheta }_f}}.$

It’s worth noting that the diffusion coefficients D_A and D_B here are equal for both chemical species, i.e. $\delta =1$ and, thus

$G\left(\eta \right)+H\left(\eta \right)=1,$ (2.30)

Then, Equations (2.23) and (2.24) yield

$\frac{1}{Sc}\left(G^{″}+\frac{G^{\prime }}{\eta +k}\right)+\frac{k}{\eta +k}f{G}^{\prime }-{\tau }_{1}G{\left(1-G\right)}^2=0,$ (2.31)

with the boundary conditions

$G^{\prime }\left(0\right)={\tau }_{2}G\left(0\right),G\left(\infty \right)\to 1.$ (2.32)

Furthermore, by differentiating Equation (2.20) and using Equation (2.19) to eliminate pressure from the momentum equation, we ultimately arrive at the task of solving the following system of nonlinear coupled ordinary differential equations:

$\begin{array}{l}{f}^{i\nu }+\frac{2f^{‴}}{\eta +k}-\frac{f^{″}}{{\left(\eta +k\right)}^2}+\frac{f^{\prime }}{{\left(\eta +k\right)}^3}\\ +\frac{A_2}{A_1}\left[\left(\frac{k}{\eta +k}\right)\left(f{f}^{‴}-f^{\prime }{f}^{″}\right)+\left(\frac{k}{{\left(\eta +k\right)}^2}\right)\left(f{f}^{″}-{f^{\prime }}^2\right)-\left(\frac{k}{{\left(\eta +k\right)}^3}\right)f{f}^{\prime }\right]\\ +\frac{\beta }{A_1}\frac{k}{\left(\eta +k\right)}\left(g^{\prime }{g}^{″}-g{g}^{‴}+\frac{g{g}^{\prime }}{{\left(\eta +k\right)}^2}-\frac{g^{\text{'}2}+g{g}^{″}}{\eta +k}\right)=0.\end{array}$ (2.33)

$P{r}_M\frac{k}{\eta +k}\left(f^{″}g-f{g}^{″}\right)-P{r}_M\frac{k}{{\left(\eta +k\right)}^2}\left(f^{\prime }g-f{g}^{\prime }\right)=g^{‴}+\frac{g^{″}}{\eta +k}-\frac{g^{\prime }}{{\left(\eta +k\right)}^2},$ (2.34)

$\frac{A_4}{Pr{A}_3}\left(1+\frac{Rd}{A_4}\right)\left({\theta }^{″}+\frac{{\theta }^{\prime }}{\eta +k}\right)+\frac{D}{A_3}\theta =\frac{-k}{\eta +k}f{\theta }^{\prime },$ (2.35)

$\frac{1}{Sc}\left(G^{″}+\frac{G^{\prime }}{\eta +k}\right)+\frac{k}{\eta +k}f{G}^{\prime }-{\tau }_{1}G{\left(1-G\right)}^2=0,$ (2.36)

incorporating the boundary conditions

$f\left(0\right)=S+sgn\left(S\right)h\beta {\left(g^{\prime }\left(0\right)\right)}^2,f^{\prime }\left(0\right)=1,$ (2.37)

$\theta \left(0\right)=1,g\left(0\right)=0,g^{″}\left(0\right)=0,G^{\prime }\left(0\right)={\tau }_{2}G\left(0\right),$ (2.38)

$f^{\prime }\left(\infty \right)\to 0,f^{″}\left(\infty \right)\to 0,\theta \left(\infty \right)\to 0,$ (2.39)

$g^{\prime }\left(\infty \right)\to 1,G\left(\infty \right)\to 1,H\left(\infty \right)\to 0.$ (2.40)

2.4. The Physical Quantities

The critical measurable parameters of interest, namely the surface shear stress C_f and the Nusselt number Nu, which quantifies the heat flux rate at the surface, are expressed as follows:

$C_f=\frac{{\mu }_{hnf}{\left(\frac{\partial u}{\partial r}-\frac{u}{r+R}\right)}_{r=0}}{{\rho }_{hnf}{\left(as\right)}^2},Nu=\frac{s\left({-{\kappa }_{hnf}\frac{\partial T}{\partial r}|}_{r=0}{-\frac{16{\sigma }^{*}{T}_{\infty }{}^3}{3K^{*}}\frac{\partial T}{\partial r}|}_{r=0}\right)}{{\kappa }_f\left(T_w-T_{\infty }\right)}.$ (2.34)

The expressions given in Equation (2.34), when transformations from Equations (2.15)-(2.17) are applied, can be rewritten as

$C_{f}R{e}_s{}^{1/2}=\frac{A_1}{A_2}\left(f^{″}\left(0\right)-\frac{f^{\prime }\left(0\right)}{k}\right).$ (2.35)

$NuR{e}_s{}^{-1/2}=-\left(A_4+R_d\right){\theta }^{\prime }\left(0\right).$ (2.36)

where $R{e}_s=a{s}^2/{\vartheta }_f$ is the local Reynolds number.

3. Numerical Solution

In this section, we choose water as the base fluid, a substance commonly selected for its heat transfer capabilities. Copper (Cu) and multi-walled carbon nanotubes (MWCNTs) are employed as two different types of magnetic nanoparticles to be added to the base fluid, resulting in the formation of a hybrid nanofluid. The associated thermophysical properties of both the base fluid and the nanoparticles are presented in Table 2.

Employing an improved shooting method, augmented with Runge-Kutta and Newton’s iterative techniques as detailed in [24], we numerically solve the nonlinear governing Equations (2.23) to (2.26), accompanied by the boundary conditions (2.37 - 2.40). The comparison of our results with the published data shows a high level of agreement, as shown in Table 3, which confirms the accuracy of our method.

Table 2. Thermophysical properties of the base fluid and nanoparticles.

Table 3. Comparison of −C_fRe_s^1/2 with different values of curvature parameter. S = 0, β = 0, Φ₁ = Φ₂ = 0.

4. Results and Discussion

For the convenience of analysis, the following parameter values are chosen in the analysis unless otherwise stated: R_d = 10, Sc = 0.1, Φ₁ = Φ₂ = 0.01, Pr_M = 0.1, Pr = 2, h = 10, τ₁ = τ₂ = 0.5, k = 10, D = 0.1, S = 0.5.

4.1. Velocity Profiles

Figure 2 presents the velocity profiles as the curvature radius parameter k varies. It is evident that with the increase of k, both the nanofluid and hybrid nanofluid velocity profiles exhibit an upward trend. This trend is primarily due to the fact that as the parameter k increases, the centrifugal force acting on the fluid during flow also intensifies, thereby promoting an increase in velocity. The influence of parameters β and h on velocity is depicted in Figure 3 and Figure 4. It is observed that in both hybrid nanofluid and nanofluid scenarios, the velocity decreases as β and h increase. Notably, the hybrid nanofluid is more responsive to changes in β, while the nanofluid is more sensitive to variations in h.

4.2. Temperature and Concentration Profiles

Figure 5 displays the temperature profiles as k varies. It is evident that with an increase in k, the temperature profiles for both nanofluid and hybrid nanofluid show a decreasing trend. Notably, the sensitivity of both nanofluids to this parameter is almost identical. Figure 6 delineates the influence of parameter Pr_M on temperature profiles. With the augmentation of Pr_M, the temperature of both nanofluid and hybrid nanofluid increases. Moreover, the hybrid nanofluid demonstrates greater effectiveness.

As the parameter k increases, the trend of concentration for both nanofluids and hybrid nanofluids exhibits an inverse relationship with temperature, as shown in Figure 7. Significantly, a higher value of k corresponds to greater concentrations in these fluids.

Figure 8 shows that higher homogeneous parameter values ${\tau }_1$ lead to lower concentrations due to reactant consumption. Figure 9 mirrors this trend for the heterogeneous parameter ${\tau }_2$ , where increasing ${\tau }_2$ also reduces concentration, aligning with typical reaction dynamics.

Figure 2. Impact of k on velocity $f^{\prime }\left(\eta \right)$ for hybrid nanofluid and nanofluid.

Figure 3. Impact of β on $f^{\prime }\left(\eta \right)$ for hybrid nanofluid and nanofluid.

Figure 4. Impact of h on $f^{\prime }\left(\eta \right)$ for hybrid nanofluid and nanofluid when S = 20.

Figure 5. Impact of k on $\theta \left(\eta \right)$ for hybrid nanofluid and nanofluid.

Figure 6. Impact of Pr_M on $\theta \left(\eta \right)$ for hybrid nanofluid and nanofluid.

Figure 7. Impact of k on $G\left(\eta \right)$ for hybrid nanofluid and nanofluid when D = 10.

Figure 8. Impact of ${\tau }_1$ on $G\left(\eta \right)$ for hybrid nanofluid and nanofluid when k = 50, Sc = 0.013.

Figure 9. Impact of ${\tau }_2$ on $G\left(\eta \right)$ for hybrid nanofluid and nanofluid.

4.3. Effect of Governing Parameters on Skin Friction Coefficient and Nusselt Number

Figure 10 illustrates the effect of the dimensionless curvature radius parameter k on the surface friction coefficient for a hybrid nanofluid. It is evident that increasing the value of k (i.e., reducing the dimensionless curvature) leads to an increase in the magnitude of surface friction on the sheet. Consequently, on the surface of the stretching sheet, more force is required to drag fluid over a flat surface compared to a curved one. It is clear from Figure 11 that the magnitude of the skin friction coefficient increases for hybrid nanofluid with enhanced suction s, which aligns with physical fact. Furthermore, it is found in Figure 10 and Figure 11 that the variation of D and Rd do not affect the surface friction force, which is consistent with Equations (2.33)-(2.36), further confirming the validity of the results.

Figure 12 demonstrates that an increase in the value of k for hybrid nanofluid leads to a slight increase in the Nusselt number. On surfaces with lower curvature, the boundary layer develops more readily because the centrifugal forces acting on the fluid flow are reduced, which in turn enhances heat transfer within the boundary layer. However, an opposite trend is observed where the Nusselt number decreases with the increase of D. Figure 13 reveals that an augment in Rd and s leads to an increase in the Nusselt number for hybrid nanofluid, indicating a significant enhancement in convective heat transfer efficiency.

Figure 10. Impact of k and D on the surface friction coefficient for hybrid nanofluid when S = 3.

Figure 11. Impact of Rd and S on $G\left(\eta \right)$ on the surface friction coefficient for hybrid nanofluid when k = 1.

Figure 12. Impact of k and D on the Nusselt number for hybrid nanofluid when S = 3.

Figure 13. Impact of Rd and S on $G\left(\eta \right)$ onthe Nusselt number for hybrid nanofluid.

5. Conclusions

Conducting a numerical study on the flow of hybrid nanofluid over a stretching curved surface, we have arrived at the following main findings.

1) Fluid velocity increases with the curvature radius parameter k due to stronger centrifugal forces.

2) The increase in β and h decreases fluid velocity, with hybrid nanofluids being more sensitive to β and nanofluids to h.

3) As k increases, fluid temperature decreases while concentration increases.

4) The augmentation of ${\tau }_1$ and ${\tau }_2$ yields a concomitant decrease in concentration attributable to reactant consumption.

5) The surface friction coefficient increases with k, s, indicating greater force is needed to drag the fluid over a stretching surface.

6) The Nusselt number increases with k, Rd and s, indicating an improvement in convective heat transfer efficiency.

In summary, this study provides an understanding of fluid dynamics, heat and mass transfer of hybrid nanofluids under complex flow conditions and offers strategies for optimizing heat transfer processes in industrial applications.

Funding

This research was funded by the Natural Science Foundations of Fujian Province, Grant 2023J01415.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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