OHAM analysis of quadratic radiative heat flux and chemical reactions in hybrid nanofluid flow over variable thickness stretching surface

Abstract

High-temperature heat transfer mechanisms seen in many systems are significantly influenced by thermal radiation. When there are significant temperature fluctuations within the thermal boundary layer, radiative heat transport is often inadequately captured by conventional linear radiation models. To overcome this constraint and provide a more accurate depiction of radiative heat transport, the quadratic thermal radiation model is used. Given these appealing features, the present study investigates the aqueous-based hybrid nanofluid over an extending surface with variable thickness influenced by quadratic thermal radiation. The hybrid nanofluid comprises carbide nanoparticles (SiC, TiC) suspended in water. The novelty of the proposed model lies in considering both homogeneous and heterogeneous reactions under convective boundary conditions at the surface. Analytical solutions are derived for the anticipated model using the optimal homotopy asymptotic method (OHAM). The trends of velocity and temperature distributions are analyzed via graphs against distinct parameters. The outcomes show that fluid velocity declines by approximately 52% as the volume fraction of carbide nanoparticles increases from 0.01 to 0.07. Also, the temperature increases by approximately 57% as quadratic thermal radiation increases from 0.1 to 3.1. Tabular results indicate that the carbide nanoparticle volume fraction ranges from 0.01 to 0.05, enhancing the heat transfer rate by approximately 37.5%. These results demonstrate that nonlinear radiation modeling and carbide-based hybrid nanofluids can greatly enhance heat transport. As a result, this study’s findings could offer valuable theoretical guidance for the development and improvement of advanced thermal systems across fields such as fluid mechanics, thermal management, and chemical processing technologies. The model’s trustworthiness is confirmed by comparing it with existing published work.

Introduction

The transmission of energy in the form of electromagnetic radiation from objects whose temperature is above absolute zero is called thermal radiation. The subject phenomenon is fundamental to numerous aspects of everyday life, including nuclear power plants, aircraft propulsion systems, and gas turbines. It is pertinent to mention that thermal radiation (Rosseland thermal radiation) is reported in both linear and nonlinear forms. When the heat transfer rate is linearly related to the temperature, it is known as Rosseland linear thermal radiation^{1, 2, 3}; otherwise, it is nonlinear or quadratic Rosseland thermal radiation^{4, 5, 6}. Nonetheless, the supremacy of the quadratic thermal radiation form is obvious owing to its applicability with precision in numerous applications. This idea was pitched by Mahanthesh⁷ and Thriveni and Mahanthesh⁸, following the notion of Bataller⁹, by truncating the series after the quadratic term rather than including a linear radiative term. Researchers are motivated to discuss varied aspects of quadratic thermal radiation. Kouz et al.¹⁰ analyzed quadratic thermal radiation and convection in the flow of viscoelastic fluid with heat sources. This is followed by studies^{11, 12} that highlighted the importance of quadratic radiation in the fluid flow, as described by the Buongiorno and Tiwari-Das models. Mandal and Pal¹³ investigated the influence of quadratic radiation on the hybrid nanofluid flow over an extended surface, with an irreversibility analysis. Recently, Vinutha et al.¹⁴ examined the flow of a tri-hybrid nanofluid with quadratic radiation using a sensitivity analysis method. The flow of the Carreau fluid with the effects of melting heat and quadratic radiation over a cylinder is analytically investigated by Jiann et al.¹⁵.

The role of intrinsic cooling mechanisms is vital in electronic devices and numerous industrial processes, and can be accomplished by augmenting the heat transfer capabilities of the coolants. It can be performed by adopting active and passive modes. Active mode involves external power for heat transfer enhancement; however, no external power is used in the passive approach. The latter is more dependable and economical owing to the non-availability of moving parts. Lately, substantial consideration has been given to nanofluids owing to their enhanced heat transfer capabilities. Nanofluids serve diverse purposes spanning heat conveyance, energy applications, and biomedical engineering. A nanofluid comprises a fluid with minute quantities of suspended nanoparticles, typically smaller than 100 nanometers, submerged in a base fluid such as water or oil. Introducing nanoparticles into conventional liquids can enhance thermal conductivity, viscosity, and the coefficient of heat transmission. Carbides such as SiC and TiC nanoparticles are important in nanofluids because they enhance heat transfer efficiency, provide thermal stability, and enable a range of advanced thermal and industrial processes. SiC and TiC are more resistant to oxidation and have lower densities than metals such as Cu or Al, which helps maintain stable suspensions. They are less expensive and last longer in high-temperature situations. This is the reason the current study focuses on these nanoparticles. The seminal work on nanofluids was conducted by Choi and Eastman¹⁶ in 1995 and is considered a game changer in enhancing the thermal conductivity and heat transfer of fluid flows. Significant studies of the evolution of the nanofluids have been conducted by many other researchers^{17, 18, 19, 20}.

The notion of “hybrid nanofluids” refers to a blend of two or more distinct nanoparticles with a customary liquid. A careful selection of a mixture of nanoparticles with varied sizes, shapes, compositions, and chemical properties, combined with liquids, results in excellent improvements in thermal and electrical conductivities^{21, 22, 23, 24, 25}. Generally, research on hybrid nanofluids is a dynamic field that can enhance the efficacy and functionality of various engineering systems, as discussed by Uddin et al.²⁶. Muhammad et al.²⁷ found that, for carbon nanotubes (CNT), the rate of heat transfer increases with increasing nanoparticle volume fraction. A performance-based analysis of hybrid nanofluids was conducted by Waqas et al.²⁸, who concluded that the thermal distribution performs well across a range of values of the thermal radiation effect parameter. In a multiple linear regression analysis, Neethu et al.²⁹ reported that the enhanced volume fraction of the hybrid nanoparticles improves the temperature profile. In another study by Mohana et al.³⁰, the impact of shape effects on hybrid nanofluid flow is assessed. The results showed that the thermal performance of the hybrid nanofluid is significantly higher than that of the simple nanofluid combination at the same nanoparticle volume fraction. It was also revealed that blade-shaped nanoparticles outperform brick-shaped nanoparticles at the same set of volume fractions for both hybrid nanofluid flow combinations. Rafique et al.³¹ examined hybrid nanofluid flow with variable viscosity, rather than constant viscosity, and a slip condition to better reflect reality. The fluid temperature and velocity show opposing trends under the slip effect, an interesting outcome. In a recent study by Pattnaik et al.³², it is reported that increasing estimates of solid particle volume concentration leads to improved fluid thermal characteristics, particularly when greater levels of thermal radiation are incorporated. Considering the impact of inertial drag, Sharma et al.³³ discussed the Mintsa and Gherasim thermal-conductivity models for heat transfer analysis and concluded that fluid velocity is affected by inertial drag. Yasmin³⁴ reported that fluid velocity is influenced by the rotation parameter in hybrid nanofluid flow with microorganisms. With the impact of the Thompson and Troian slip, Gupta et al.³⁵ evaluated the hybrid nanofluid flow considering the Reynolds viscosity with carbon nanotubes of both types in kerosene oil. The outcomes revealed that the MWCNTs mixture performs better than the SWCNTs solution. A ballooned increment of 30.11% if the ratio of volume fraction is enhanced from 1% to 10%. Panda et al.³⁶ discussed heat transfer and magnetic field effects in a hybrid CNT nanofluid past a penetrable, convectively curved, stretched surface with thermal radiation. The strong thermal conductivity of SiC and the superior stability and dispersion properties of TiC led to the selection of the SiC–TiC hybrid nanoparticle combination. Compared with single-nanoparticle nanofluids, their combined presence creates efficient thermal transport channels and enhances suspension stability, thereby improving heat transfer performance.

The proposed mathematical model is handled using OHAM, a well-known series solution technique. While researchers typically favor numerical methods for these problems, analytical schemes are recognized as an alternative for obvious reasons³⁷. Among these methods, perturbation techniques³⁸ are widely applied to various engineering problems. However, their heavy reliance on small or large parameters limits their applicability to only weaker nonlinear equations. To address this shortcoming, non-perturbation schemes, namely the variation iteration scheme³⁹, the δ−expansion approach⁴⁰, Lyapunov’s artificial small parameter method⁴¹, and Adomian decomposition technique⁴² have been developed. However, their applicability to only weaker nonlinear equations and convergence issues raises questions about their reliability. Initially, the idea of homotopy was merged with perturbation schemes. He and Liao did the initial spadework; later, Liao^{43, 44} suggested the HAM, and He pioneered the Homotopy Analysis Method (HPM). Marinca et al.⁴⁵ floated the notion of OHAM, a mixture of HAM and HPM, to find the approximate solution of nonlinear models. This technique has the following characteristics:

• Compared with other analytical approaches, OHAM is a straightforward and powerful technique.

• The results from OHAM are more reliable, faster, and better in agreement with numerical schemes.

• OHAM provides a straightforward approach to controlling the convergence of the solution inherited from the HAM technique, with ample flexibility. It is free to opt for small or large parameters.

Boundary layer flows have attracted interest owing to their increasing applications in industry and engineering^{46, 47}, including glass fiber production, metal extraction, and glass blowing. A large number of studies have discussed the boundary-layer flow over a flat, stretched surface. However, in real scenarios, the sheet thickness may vary owing to its to-and-fro movement from the pivotal point⁴⁸, which depends on the power index value, and a stretched surface with variable thickness may better reflect reality. In industrial processes, including polymer extrusion, metal rolling, fiber drawing, coating technologies, and electronic cooling systems, hybrid nanofluid flow across a stretching surface with variable thickness has significant applications. Materials are continuously stretched during these operations, and their surface thickness changes. Researchers are motivated to discuss varied avenues discussing surfaces with variable thicknesses. Fang et al.⁴⁹ conducted a pioneering study of fluid flow over a surface with variable thickness. Kumar et al.⁵⁰ deliberated the heat transfer of ferrous nanoliquid flow along a sheet with variable thickness. They concluded that, as magnetic parameter counts increase, the concentration distribution broadens. Sushma et al.⁵¹ numerically addressed the flow of a time-independent hybrid nanoliquid with mixed convection effect on the variable thickness sheet. A variable-thickness, extending sheet is computationally explored by Dawar et al.⁵² to examine the flow of a micropolar nanoliquid containing aluminum alloy nanoparticles. The findings demonstrated that the velocity profile of the AA7075–methanol nanofluid reduces as the volume percentage of AA7075 nanoparticles increases. Ramesh et al.⁵³ focused on chemical reactions, including endothermic and exothermic reactions, in the flow of an electrically conducting ternary-hybrid nanoliquid over a variable-thickness surface. The findings imply that the exothermic process and the wall thickness parameter affect the heat transfer rate. The influence of varied thickness surfaces on nanofluid flow is investigated^{54, 55, 56, 57, 58, 59, 60}.

Considering the aforementioned studies and the available literature, there have been limited attempts to discuss hybrid nanofluid flows over stretching surfaces with variable thickness. Nonetheless, there is a scarcity of investigations on aqueous-based hybrid nanofluids over an extended surface with variable thickness, accounting for quadratic thermal radiation. However, no study has so far attempted to deliberate the impacts of quadratic thermal radiation on the flow of a hybrid nanofluid, with novel aspects of homogeneous and heterogeneous reactions and varying surface convective conditions. The anticipated model is handled with a renowned analytical technique, OHAM. The plots and tables represent the salient outcomes. The purpose of the presented study is to answer the following questions:

• What is the effect of assumed carbide nanoparticles on the surface drag coefficient and velocity profile?

• How variable thickness of the assumed surface affect the hybrid nanofluid velocity?

• What is the influence of quadratic thermal radiation on the hybrid nanofluid temperature?

• How does nanoparticle volume fraction influence the heat transmission rate?

• What is the impact of the heterogeneous chemical reaction parameter on the fluid concentration?

Table 1 highlights the unique contributions of the current work by comparing it with previously published findings.

Table 1 A distinction of the current investigation from related published research efforts.

In contrast to Bashir et al.⁵⁸, this study considers a hybrid nanofluid comprising a duo of Carbide nanoparticles mixed in water, amalgamated with quadratic thermal radiation instead of five nanoparticles in a nanofluid flow.

Mathematical formulation

This mathematical model relies on these core assumptions:

• i. Here, an incompressible, steady, and viscous hybrid nanofluid flow over a stretched surface with variable thickness is considered.

• ii. A hybrid nanofluid comprising carbide-based nanoparticles (TiC and SiC) is suspended in water as the base fluid.

• iii. The heat mechanism is elaborated by considering a quadratic thermal radiation in the temperature equation. Also, the convective boundary condition is incorporated on the assumed surface, whose expression is given by $- {k_{HNF}}\left( {\frac{{\partial T}}{{\partial Y}}} \right)={h_n}({T_f} - {T_\infty })$.

• iv. This current investigation is based on assuming the homogeneous and heterogeneous reactions to enhance the energy process.

• v. The stretching of the sheet is along $X -$direction with ${U_s}=a{\left( {X+b} \right)^m}$as an extending velocity, and the flow is considered along $Y -$direction.

• vi. The liquid temperature at the ambient is taken as ${T_\infty }$. In Fig. 1, the flow configuration is plotted.

Fig. 1

Flow diagram.

Chemical reactions^{12, 55, 56, 57} are considered:

$$\bar{A} + 2\bar{B}\xrightarrow{{}}3\bar{B},rate = k_{r} \bar{A}\bar{B}^{2} .$$

(1)

Temperature remains constant during chemical reactions. The isothermal chemical process described below is anticipated as:

$$\bar {A}\xrightarrow{{}}3\bar {B},\,\,\,\,\,\,rate={k_s}\bar {A}.$$

(2)

Following the above assumptions, the model equations are formulated as^{12, 13, 26, 27, 54, 55, 56, 57}:

$$\frac{{\partial U}}{{\partial X}} + \frac{{\partial V}}{{\partial Y}} = 0,$$

(3)

$$U\frac{{\partial U}}{{\partial X}} + V\frac{{\partial U}}{{\partial Y}} = \left( {\frac{{\mu _{{HNF}} }}{{\rho _{{HNF}} }}} \right)\frac{{\partial ^{2} U}}{{\partial Y^{2} }},$$

(4)

$$\left( {\rho C_{p} } \right)_{{HNF}} \left( {U\frac{{\partial T}}{{\partial X}} + V\frac{{\partial T}}{{\partial Y}}} \right) = \left( {k_{{HNF}} } \right)\left( {\frac{{\partial ^{2} T}}{{\partial Y^{2} }}} \right) - \frac{{\partial q_{r} }}{{\partial Y}},$$

(5)

$$U\left( {\frac{{\partial \Omega _{1} }}{{\partial X}}} \right) + V\left( {\frac{{\partial \Omega _{1} }}{{\partial Y}}} \right) = D_{1} \left( {\frac{{\partial \Omega _{1}^{2} }}{{\partial Y^{2} }}} \right) - k_{r} \Omega _{1} \Omega _{2}^{2} ,$$

(6)

$$U\left( {\frac{{\partial \Omega _{2} }}{{\partial X}}} \right) + V\left( {\frac{{\partial \Omega _{2} }}{{\partial Y}}} \right) = D_{2} \left( {\frac{{\partial \Omega _{2}^{2} }}{{\partial Y^{2} }}} \right) + k_{r} \Omega _{1} \Omega _{2}^{2} ,$$

(7)

where in energy Eq. (5), the quadratic thermal radiation heat flux using Rosseland approximation for radioactive heat transfer is given by ${q_r}= - \left( {\frac{4}{{3{k^*}}}} \right)\nabla \left( {{e_b}} \right),\,\,\,\,\,\,\,\,{e_b}={\sigma ^*}{T^4},$ where ${T^4} \approx 3T_{\infty }^{4} - 8T_{\infty }^{3}T+6T_{\infty }^{2}{T^2}$. Higher-order temperature changes are considered in the quadratic radiation approximation. The quadratic model accounts for stronger temperature gradients within the thermal boundary layer, unlike the traditional linear and nonlinear models, which are only valid for modest temperature differences. This method preserves the mathematical tractability of the governing equations while increasing the accuracy of radiative heat transfer estimates.

Taking the derivative of ${q_r}$, we get the following expression:

$${q_r}=\left( {\frac{{4{\sigma ^ * }}}{{3{k^ * }}}} \right)\frac{{\partial {T^4}}}{{\partial y}}=\left( {\frac{{4{\sigma ^ * }}}{{3{k^ * }}}} \right)\frac{\partial }{{\partial y}}\left( {3T_{\infty }^{4} - 8T_{\infty }^{3}T+6T_{\infty }^{2}{T^2}} \right)=\left( { - \frac{{32{\sigma ^ * }T_{\infty }^{3}}}{{3{k^ * }}}} \right)\left( {\frac{{\partial T}}{{\partial Y}}} \right)+\left( {\frac{{24{\sigma ^ * }T_{\infty }^{2}}}{{3{k^ * }}}} \right)\frac{\partial }{{\partial Y}}\left( {{T^2}} \right).$$

(8)

Using Eq. (8) in Eq. (5) we get

$${\left( {\rho {C_p}} \right)_{HNF}}\left( {U\frac{{\partial T}}{{\partial X}}+V\frac{{\partial T}}{{\partial Y}}} \right)=\left( {{k_{HNF}}+\frac{{32{\sigma ^ * }T_{\infty }^{3}}}{{3{k^ * }}}} \right)\left( {\frac{{{\partial ^2}T}}{{\partial {Y^2}}}} \right) - \left( {\frac{{24{\sigma ^ * }T_{\infty }^{2}}}{{3{k^ * }}}} \right)\frac{{{\partial ^2}}}{{\partial {Y^2}}}\left( {{T^2}} \right).$$

(9)

The above system of governing equations is supported by the ensuing boundary constraints^{26, 54, 55, 56, 57}:

$$\begin{gathered} U={U_s}=a{\left( {X+b} \right)^m},\,\,V=0,\,\, - {k_{HNF}}\left( {\frac{{\partial T}}{{\partial Y}}} \right)={h_n}({T_f} - {T_\infty }),\,\, \hfill \\ {D_1}\left( {\frac{{\partial {\Omega _1}}}{{\partial Y}}} \right)={k_s}{\Omega _1},\,\,{D_2}\left( {\frac{{\partial {\Omega _2}}}{{\partial Y}}} \right)= - {k_s}{\Omega _1},\,\,\,\,at\,\,\,Y=\lambda {\left( {X+b} \right)^{\frac{{1 - m}}{2}}}, \hfill \\ U \to 0,\,\,T \to {T_\infty },\,\,{\Omega _1} \to {\Omega _o},\,\,{\Omega _2} \to 0\,,\,\,\,as\,\,Y \to \infty . \hfill \\ \end{gathered}$$

(10)

In the above Eq. (10), the 1st boundary condition is stretching of geometry, 2nd boundary condition says that the stretching surface is impermeable, 3rd condition depicts the convective heat constraint, where the non-linear heat transfer coefficient⁵⁷ is ${h_n}=h{(X+b)^{\frac{{m - 1}}{2}}}$. Next, the 4th and 5th conditions depict the heterogeneous reactions at the surface. Far away from the surface, the temperature is ambient, concentration of species A is uniform.

Considering the thermophysical models, we get^{13, 27}:

$$\begin{gathered} \mu _{{HNF}} = \mu _{F} \left( {1 - \psi _{1} } \right)^{{ - 2.5}} \left( {1 - \psi _{2} } \right)^{{ - 2.5}} , \hfill \\ \hfill \\ \end{gathered}$$

(11)

$$\frac{{\rho _{{HNF}} }}{{\rho _{F} }} = \left( {1 - \psi _{2} } \right)\left( {\left( {1 - \psi _{1} } \right) + \psi _{1} \frac{{\rho _{1} }}{{\rho _{F} }}} \right) + \psi _{2} \frac{{\rho _{2} }}{{\rho _{F} }},$$

(12)

$$\frac{{\left( {\rho C_{p} } \right)_{{HNF}} }}{{\left( {\rho C_{p} } \right)_{F} }} = \left( {1 - \psi _{2} } \right)\left( {\left( {1 - \psi _{1} } \right) + \psi _{1} \frac{{\left( {\rho C_{p} } \right)_{1} }}{{\left( {\rho C_{p} } \right)_{F} }}} \right) + \psi _{2} \frac{{\left( {\rho C_{p} } \right)_{2} }}{{\left( {\rho C_{p} } \right)_{F} }},$$

(13)

$$\frac{{k_{{HNF}} }}{{k_{{NF}} }} = \frac{{k_{2} + 2k_{{NF}} - 2\psi _{2} \left( {k_{{NF}} - k_{2} } \right)}}{{k_{2} + k_{{NF}} + \psi _{2} \left( {k_{{NF}} - k_{2} } \right)}},$$

(14)

$$\frac{{k_{{NF}} }}{{k_{F} }} = \frac{{k_{1} + 2k_{F} - 2\psi _{1} \left( {k_{F} - k_{1} } \right)}}{{k_{1} + k_{F} + \psi _{1} \left( {k_{F} - k_{1} } \right)}},$$

(15)

where ${\psi _1}$ and $\,{\psi _2}$are the nanoparticles’ volume fraction. Applying transformation^{55, 56}:

$$\begin{gathered} U = a(X + b)^{m} \bar{f^{\prime}}(\zeta ),\;\;V = \sqrt {\left( {\frac{{m + 1}}{2}} \right)\nu _{F} a(X + b)^{{m - 1}} } \left[ {\bar{f}(\zeta ) + \zeta \left( {\frac{{m - 1}}{{m + 1}}} \right)\bar{f^{\prime}}(\zeta )} \right], \hfill \\ \zeta = Y\sqrt {\left( {\frac{{m + 1}}{2}} \right)\frac{a}{{\nu _{F} }}(X + b)^{{m - 1}} } ,\;\;\bar{g}(\zeta ) = \left( {\frac{{\Omega _{1} }}{{\Omega _{o} }}} \right),\;\;\bar{h}(\zeta ) = \left( {\frac{{\Omega _{2} }}{{\Omega _{o} }}} \right),\;\;T = \bar{\theta }(\zeta )(T_{f} - T_{\infty } ) + T_{\infty } . \hfill \\ \end{gathered}$$

(16)

Utilizing Eq. (16), we obtain the subsequent system of equations:

$$\left( {\frac{{\left( {1 - \psi _{1} } \right)^{{ - 2.5}} \left( {1 - \psi _{2} } \right)^{{ - 2.5}} }}{{\left( {1 - \psi _{2} } \right)\left( {\left( {1 - \psi _{1} } \right) + \psi _{1} \frac{{\rho _{1} }}{{\rho _{F} }}} \right) + \psi _{2} \frac{{\rho _{2} }}{{\rho _{F} }}}}} \right)\bar{f^{\prime\prime\prime}} - \left( {\frac{{2m}}{{m + 1}}} \right)\bar{f^{\prime}}^{2} + \bar{f}\,\bar{f}^{\prime\prime}\, = 0,$$

(17)

$$\left( {k_{{HNF}} - \left[ {\frac{{8N_{r} }}{3}} \right]} \right)\bar{\theta ^{\prime\prime}} + \left[ {\frac{{6N_{r} }}{{3\left( {\theta _{r} - 1} \right)}}} \right]\left\{ {\left( {1 + \left( {\theta _{r} - 1} \right)\bar{\theta }} \right)^{2} } \right\}^{{\prime \prime }} - \left( {\rho C_{p} } \right)_{{HNF}} \Pr \bar{f}\bar{\theta ^{\prime}} = 0,$$

(18)

$$\bar{g^{\prime\prime}} + S_{c} \bar{g^{\prime}}\bar{f} - \left( {\frac{{2K_{r} S_{c} }}{{m + 1}}} \right)\bar{g}\bar{h}^{2} = 0,$$

(19)

$$\bar{h^{\prime\prime}} + \frac{{S_{c} }}{\Upsilon }\bar{f}\bar{h^{\prime}} + \left( {\frac{{2K_{r} S_{c} }}{{\Upsilon \left( {m + 1} \right)}}} \right)\bar{g}\bar{h}^{2} = 0.$$

(20)

With transformed boundary constraints

$$\begin{gathered} \bar {f}^{\prime}(\zeta )=1,\,\,\bar {f}(\zeta )=0,\,\,\bar {\theta ^{\prime}}(\zeta )= - \frac{{{k_F}}}{{{k_{HNF}}}}{{\text{B}}_{\text{i}}}\left( {1 - \bar {\theta }(\zeta )} \right),\,\,\bar {g^{\prime}}(\zeta )=\sqrt {\frac{2}{{m+1}}} {K_s}\bar {g}(\zeta ), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\bar {h^{\prime}}(\zeta )= - \sqrt {\frac{2}{{m+1}}} \frac{{{K_s}}}{\Upsilon }\bar {g}(\zeta ),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\zeta =\varpi , \hfill \\ \bar {f}^{\prime}(\zeta ) \to 0,\,\,\bar {\theta }(\zeta ) \to 0,\,\,\bar {g}(\zeta ) \to 1,\,\,\bar {h}(\zeta ) \to 0.\,\,\,\,\,\,\,\,\,\,\,\zeta \to \infty . \hfill \\ \end{gathered}$$

(21)

Here, $\varpi =$ $\lambda {\left( {\frac{{a\left( {m+1} \right)}}{{2{\nu _F}}}} \right)^{1/2}},$ and selecting suitable transformation as $\bar {f}(\zeta )=$$f(\zeta - \varpi )=f(\eta )$, $\bar {\theta }(\zeta )=$$\theta (\zeta - \varpi )=\theta (\eta )$, $\bar {g}(\zeta )=$$g(\zeta - \varpi )=g(\eta )$ and $\bar {h}(\zeta )=$$h(\zeta - \varpi )=h(\eta )$ transmuted the aforementioned flow equations as:

$$\left( {\frac{{\left( {1 - \psi _{1} } \right)^{{ - 2.5}} \left( {1 - \psi _{2} } \right)^{{ - 2.5}} }}{{\left( {1 - \psi _{2} } \right)\left( {\left( {1 - \psi _{1} } \right) + \psi _{1} \frac{{\rho _{1} }}{{\rho _{F} }}} \right) + \psi _{2} \frac{{\rho _{2} }}{{\rho _{F} }}}}} \right)f^{{\prime \prime \prime }} - \left( {\frac{{2m}}{{m + 1}}} \right)f^{{\prime 2}} + ff^{{\prime \prime }} = 0,$$

(22)

$$\left( {k_{{HNF}} - \left[ {\frac{{8N_{r} }}{3}} \right]} \right)\theta ^{\prime\prime} + \left[ {\frac{{6N_{r} }}{{3\left( {\theta _{r} - 1} \right)}}} \right]\left\{ {\left( {1 + \left( {\theta _{r} - 1} \right)\theta } \right)^{2} } \right\}^{{\prime \prime }} - \left( {\rho C_{p} } \right)_{{HNF}} \Pr f\theta ^{\prime} = 0,$$

(23)

$$g^{\prime\prime} + S_{c} g^{\prime}f - \left( {\frac{{2K_{r} S_{c} }}{{m + 1}}} \right)gh^{2} = 0,$$

(24)

$$h^{\prime\prime} + \frac{{S_{c} }}{\Upsilon }fh^{\prime} + \left( {\frac{{2K_{r} S_{c} }}{{\Upsilon \left( {m + 1} \right)}}} \right)gh^{2} = 0,$$

(25)

with transformed boundary constraints

$$\begin{gathered} f^{\prime}(\eta )=1,\,\,f(\eta )= - \varpi \left( {\frac{{m - 1}}{{m+1}}} \right)0,\,\,\theta ^{\prime}(\eta )= - {k_{HNF}}{{\text{B}}_{\text{i}}}\left( {1 - \theta } \right),\,\,g^{\prime}(\eta )=\sqrt {\frac{2}{{m+1}}} {K_s}g(\eta ), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,h^{\prime}(\eta )= - \sqrt {\frac{2}{{m+1}}} \frac{{{K_s}}}{\Upsilon }h(\eta ),\,\,\,\,\,at\,\,\,\,\,\,\eta =0, \hfill \\ f^{\prime}(\eta ) \to 0,\,\,\theta (\eta ) \to 0,\,\,g(\eta ) \to 1,\,\,h(\eta ) \to 0,\,\,\,\,\,\,\,as\,\,\,\,\,\,\eta \to \infty . \hfill \\ \end{gathered}$$

(26)

Furthermore, we assume that diffusion is in the same medium so, ${D_1}={D_2},$ or $\Upsilon =1,$

$$g(\eta )+h(\eta )=1,$$

(27)

which eventually transformed the Eqs. (24) and (25) with (27) as:

$$(1 - g)^{\prime\prime}+{S_c}\,f(1 - g)^{\prime} - \frac{{2{K_r}\,{S_c}}}{{m+1}}\,g{\left( {1 - g} \right)^2}=0,$$

$$g^{\prime\prime}+{S_c}\,fg^{\prime} - \frac{{2{K_r}\,{S_c}}}{{m+1}}\,g{\left( {1 - g} \right)^2}=0,$$

(28)

along with boundary constraints

$$g^{\prime}(0)= - \sqrt {\frac{2}{{m+1}}} {K_s}g(0),\,\,\,\,\,\,and\,\,\,\,\,\,\,g(\infty ) \to 1.$$

(29)

The dimensionless parameters are given as:

$$\begin{gathered} \theta _{r} = \frac{{T_{f} }}{{T_{\infty } }},\Pr = \frac{{(\mu c_{p} )_{F} }}{{k_{F} }},S_{c} = \frac{{\nu _{F} }}{{D_{1} }},\Upsilon = \frac{{D_{2} }}{{D_{1} }},N_{r} = \frac{{16\sigma ^{ * } T_{\infty } ^{3} }}{{3k_{F} k^{ * } }}, \hfill \\ \varpi = \lambda \sqrt {\frac{{m + 1}}{2}\frac{a}{{\nu _{F} }}} ,K_{r} = \frac{{k_{r} \Omega _{^\circ } ^{2} }}{a},K_{s} = \frac{{k_{s} \sqrt {\nu _{F} } }}{{D_{1} \sqrt a }},B_{i} = \sqrt {\frac{2}{{m + 1}}\frac{{\nu _{F} }}{a}} \frac{{h_{n} }}{{k_{F} }}. \hfill \\ \end{gathered}$$

(30)

Engineering quantities

The skin friction coefficient and the Nusselt number are critical for the development and optimization of real-world systems involving fluid flow and heat transfer. They give a mathematical framework for enhancing performance, energy efficiency, and safety in a variety of engineering processes.

Surface drag coefficient and surface heat transmission rate is computed as^{12, 13, 55, 56, 57}:

$$\,{C_F}=\frac{{2{\tau _s}}}{{{\rho _F}{U_w}^{2}}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,{N_U}=\frac{{(X+b){q_n}}}{{{k_F}({T_f} - {T_\infty })}},$$

(31)

In Eq. (31), ${\tau _s}$ and represent the wall stress and thermal flux, and are specified as^{12, 13, 56, 57}:

$${\tau _s}={\mu _{HNF}}{\left. {\left( {\frac{{\partial U}}{{\partial Y}}} \right)} \right|_{Y=\lambda {{(X+b)}^{{\raise0.7ex\hbox{${1 - m}$} \!\mathord{\left/ {\vphantom {{1 - m} 2}}\right.\kern-0pt}\!\lower0.7ex\hbox{$2$}}}}}},\,\,\,\,\,{q_n}=\left[ { - {k_{HNF}}\left( {\frac{{\partial T}}{{\partial Y}}} \right)+{{\left. {\frac{{16{\sigma ^ * }{T_\infty }^{3}}}{{3{k^ * }}}\left( {2\frac{{\partial T}}{{\partial Y}} - \frac{{3T}}{{{T_\infty }}}\frac{{\partial T}}{{\partial Y}}} \right)} \right|}_{_{{Y=\lambda {{(X+b)}^{{\raise0.7ex\hbox{${1 - m}$} \!\mathord{\left/ {\vphantom {{1 - m} 2}}\right.\kern-0pt}\!\lower0.7ex\hbox{$2$}}}}}}}}} \right].$$

(32)

The non-dimensional form of the above equations is appended as under:

$$\begin{gathered} \,{C_f}{\operatorname{R} _e}_{X}^{{1/2}}=\sqrt {\frac{{m+1}}{2}} {(1 - {\psi _1})^{ - 2.5}}{(1 - {\psi _2})^{ - 2.5}}f^{\prime\prime}(0), \hfill \\ {N_{UX}}{\operatorname{R} _e}_{X}^{{1/2}}= - \sqrt {\frac{{m+1}}{2}} \left[ {\frac{{{k_{nf}}}}{{{k_f}}} - 8{N_r}+12{N_r}\left( {\theta (0)({\theta _r} - 1)+1} \right)} \right]\theta ^{\prime}(0). \hfill \\ \end{gathered}$$

(33)

Here, local Reynold number is ${\operatorname{R} _e}_{X}=\frac{{{U_w}(X+b)}}{{{\nu _F}}}.$

Optimal homotopic solution

The nonlinear governing equations are approximated analytically using the Optimal Homotopy Asymptotic Method (OHAM). OHAM offers clear analytical expressions and enables direct examination of the impact of physical parameters on the flow characteristics. Furthermore, the method incorporates convergence-control parameters that enable rapid solution convergence without requiring linearization or discretization of the nonlinear system. Also, it makes it easier to understand physical parameters. The transformed coupled nonlinear ODEs are evaluated using the semi-analytical optimal series solution method. First, we suppose the appropriate initial guesses as:

$$\begin{gathered} f_{^\circ } (\eta ) = \varpi \left( {\frac{{1 - m}}{{1 + m}}} \right) + 1 - \exp \left( { - \eta } \right),\;\;\theta ^{\prime}_{ \circ } (\eta ) = - B_{i} \left( {1 - \theta \left( \eta \right)} \right), \hfill \\ g_{^\circ } (\eta ) = 1 - \frac{1}{2}\exp \left( { - \sqrt {\frac{2}{{1 + m}}} k_{s} \eta } \right), \hfill \\ \end{gathered}$$

(34)

with the underlying linear differential operators.

$${L_f}=f^{\prime\prime\prime} - f^{\prime},\,\,\,\,{L_\theta }=\theta ^{\prime\prime} - \theta ,\,\,\,\,{L_g}=g^{\prime\prime} - g.$$

(35)

Every operator is set to zero after being applied to a trial solution or a Homotopy approximation, which are usually exponential functions. The features of the operators are as follows:

$$\begin{gathered} {L_f}\left[ {{\Lambda _1}+{\Lambda _2}{e^\eta }+{\Lambda _3}{e^{ - \eta }}} \right]=0, \hfill \\ {L_\theta }\left[ {{\Lambda _4}{e^\eta }+{\Lambda _5}{e^{ - \eta }}} \right]=0, \hfill \\ {L_g}\left[ {{\Lambda _6}{e^\eta }+{\Lambda _7}{e^{ - \eta }}} \right]=0, \hfill \\ \end{gathered}$$

(36)

where, ${\Lambda _n}(n=1 - 7)$ illustrate the constants, which is dependent on the number of boundary conditions.

Optimal convergence control variables

This section discusses the use of non-zero auxiliary parameters ${\hbar _{\tilde {f}}}\,,\,{\hbar _{\tilde {\theta }}}\,{\text{and}}\,{\hbar _{\tilde {g}}}$ in the Homotopy Analysis Method to manage convergence rate and range. These parameters are appropriately selected using an error-minimization approach.

The M^th order averaged square residual errors of ${\hbar _{\tilde {f}}}\,,\,{\hbar _{\tilde {\theta }}}\,{\text{and}}\,{\hbar _{\tilde {g}}}$, are considered as:

$$\tilde{e}_{M} ^{f} = \frac{1}{{k + 1}}\sum\limits_{{l = 0}}^{k} {\left[ {\mathbb{N}_{f} \left( {\sum\limits_{{i = 0}}^{M} {f(\eta ),} \sum\limits_{{i = 0}}^{M} {\theta (\eta )} } \right)_{{\eta = l\delta \eta }} } \right]^{2} } ,$$

(37)

$$\tilde{e}_{M} ^{\theta } = \frac{1}{{k + 1}}\sum\limits_{{l = 0}}^{k} {\left[ {\mathbb{N}_{\theta } \left( {\sum\limits_{{i = 0}}^{M} {f(\eta ),} \sum\limits_{{i = 0}}^{M} {\theta (\eta )} } \right)_{{\eta = l\delta \eta }} } \right]^{2} } ,$$

(38)

$$\tilde{e}_{M} ^{g} = \frac{1}{{k + 1}}\sum\limits_{{l = 0}}^{k} {\left[ {\mathbb{N}_{g} \left( {\sum\limits_{{i = 0}}^{M} {f(\eta ),} \sum\limits_{{i = 0}}^{M} {\theta (\eta )} ,\sum\limits_{{i = 0}}^{M} {g(\eta )} } \right)_{{\eta = l\delta \eta }} } \right]^{2} } ,$$

(39)

where k signifies the number of distinct points of the dimensionless coordinate axis.

Following

$${\tilde {e}_M}^{t}={\tilde {e}_M}^{f}+{\tilde {e}_M}^{\theta }+{\tilde {e}_M}^{g},$$

(40)

where ${\tilde {e}_M}^{t}$is total squared residual error, $\delta \eta =0.5\,\,\,{\text{and}}\,\,\,k=20.$ The suitable values of the parameters for the convergence control are ${\hbar _f}= - \,{\text{0}}{\text{.338352}}\,,\,{\hbar _\theta }= - \,{\text{0}}{\text{.0306896}}\,\,{\text{and}}\,\,{\hbar _g}= - \,{\text{1}}{\text{.90375}}$. Table 2 demonstrates the attributes of working liquid and carbide nanofluids at room temperature. Averaged squared residual errors for carbide nanofluid are displayed in Table 3. A decrease is noticed for large orders of approximation. Total residual error curves for $({S_i}C - {T_i}C/water)$, nanofluids are drawn in Fig. 2.

Table 2 Attributes of working fluid and carbide nanoparticles^{57, 58}:

Table 3 Average squared residual errors for hybrid nanofluid.

Fig. 2

Errors for $({S_i}C - {T_i}C/water)$.

Graphical discussion

This section presents graphical analyses of various physical parameters against velocity, temperature, and concentrationprofiles^{11, 12, 13, 26, 55, 56, 57}. The ranges of all the parameters are $\begin{gathered} 0.01 \le \psi _{2} \le 0.07,0.1 \le \varpi \le 0.7,0.01 \le \psi _{1} \le 0.03,0.1 \le m \le 1.0,0.1 \le N_{r} \le 3.1,0.22 \le {\text{B}}_{{\text{i}}} \le 0.55, \hfill \\ 0.1 \le K_{r} \le 0.4,0.2 \le K_{s} \le 0.5. \hfill \\ \end{gathered}$

Velocity profile

In Fig. 3, the velocity profile $f^{\prime}(\eta )$is observed for incremental values of the volume fraction of nanoparticles ${\psi _2}$. Computations are carried out for a fixed value of ${\psi _1}=0.03$. A decrease in $f^{\prime}(\eta )$ is examined for the higher estimates of the nanoparticles volume fraction $({\psi _2}=0.01,0.03,0.05,0.07)$. Resistance in nanoparticles increases with snowballing, and therefore the velocity decreases. As the volume fraction of nanoparticles in the nanofluid increases, its effective viscosity generally increases due to the added resistance from the solid nanoparticles. The effective viscosity and particle–fluid interactions within the nanofluid are improved by raising the nanoparticle parameter. The increased resistance to momentum conveyance, due to higher viscous forces, slows fluid motion and lowers the velocity profile. As a result, as the nanoparticle parameter increases, the velocity lowers, and the momentum barrier layer thickens. This increase in viscosity typically leads to a decline in the fluid’s velocity.

Fig. 3

$f^{\prime}(\eta )$ versus ${\psi _2}$. Taking $\varpi =0.3$, ${\psi _1}=0.03$, $m=0.1$, ${S_c}=1.2$, ${N_r}=1.1$, ${{\text{B}}_{\text{i}}}=0.22$, ${K_r}=0.1$, ${K_s}=0.2.$

The consequences of variable wall thickness $\varpi$ on the velocity distribution $f^{\prime}(\eta )$ are presented in Fig. 4. It elucidates that increasing the wall thickness parameter $(\varpi =0.1,0.3,0.5,0.7)$ decreases the fluid velocity. Increased flow resistance from a thicker surface wall section can lead to greater reductions in velocity within specific areas. Increasing the wall thickness parameter reduces fluid velocity by increasing boundary resistance and drag. It also reduces momentum transfer near the wall and increases boundary-layer effects, which prevent the fluid from moving freely.

Fig. 4

$f^{\prime}(\eta )$ versus $\varpi$. Taking ${\psi _1}=0.03$, ${\psi _2}=0.01$, $m=0.1$, ${S_c}=1.2$, ${N_r}=1.1$, ${{\text{B}}_{\text{i}}}=0.22$, ${K_r}=0.1$, ${K_s}=0.2.$

The effects of the power law index $f^{\prime}(\eta )$ on the $f^{\prime}(\eta )$ are noticed in Fig. 5. An increasing arrangement of velocity is witnessed near the surface of the mounting $(m=0.1,0.4,0.7,1.0).$For the higher values of $m,$the effective viscosity may drop near the surface owing to high shear rates. This fact allows the hybrid nanofluid to move more freely, thereby enhancing fluid velocity. Increasing the power-law index can increase velocity by altering the shear-viscosity relationship, lowering near-wall resistance, and improving momentum transmission, particularly in flows caused by stretching or moving surfaces.

Fig. 5

$f^{\prime}(\eta )$ versus m. Taking ${\psi _1}=0.03$, ${\psi _2}=0.01$, $\varpi =0.3$, ${S_c}=1.2$, ${N_r}=1.1$, ${{\text{B}}_{\text{i}}}=0.22$, ${K_r}=0.1$, ${K_s}=0.2.$

Temperature profile

The effects of the Radiation parameter ${N_r},$ and Biot number ${{\text{B}}_{\text{i}}}$ on the temperature profile $\theta (\eta )$ are captured in this segment. Here, Fig. 6 expresses an association of radiation parameter${N_r}$ with the temperature profile. It is observed that the temperature profile improves for rising values of the$({N_r}=0.1,1.1,2.1,3.1)$. In a hybrid nanofluid flow, increasing the radiation parameter raises the fluid’s internal temperature distribution. Increasing the radiation parameter efficiently raises the hybrid nanofluid’s thermal conductivity by increasing radiative heat flux within the boundary layer. A thicker thermal boundary layer and a higher temperature distribution result from enhanced energy transfer from the heated surface to the fluid.

Similarly, in Fig. 7, $\theta (\eta )$ grows for the incremental values of the Biot number $({{\text{B}}_{\text{i}}}=0.22,0.33,0.44,0.55)$. The ratio of the conductive heat resistance within a body to the convective heat transfer resistance at its boundary is the Biot number, a dimensionless quantity. Convective heat transfer at the fluid-solid interface becomes more pronounced as the Biot number increases. This may result in heat transfer from the solid border which is more efficient. From the standpoint of heat transfer, more heat can enter the fluid region when the Biot number is larger, since it improves the thermal energy exchange at the border. As a result, the thermal boundary layer’s temperature rises, and its thickness grows.

Fig. 6

$\theta (\eta )$ versus ${N_r}$. Taking ${\psi _1}=0.03$, ${\psi _2}=0.01$, $\varpi =0.3$, ${S_c}=1.2$, $m=0.1$, ${{\text{B}}_{\text{i}}}=0.22$, ${K_r}=0.1$, ${K_s}=0.2.$

Fig. 7

$\theta (\eta )$ versus ${{\text{B}}_{\text{i}}}$. Taking ${\psi _1}=0.03$, ${\psi _2}=0.01$, $\varpi =0.3$, ${S_c}=1.2$, $m=0.1$, ${N_r}=1.1$, ${K_r}=0.1$, ${K_s}=0.2.$

Concentration profile

Figure 8 illustrates how the homogeneous reaction parameter${K_r}$ affects the concentration profile $g(\eta ).$ Diminishing trends of concentration of hybrid nanofluid are noted with increasing $({K_r}=0.1,0.2,0.3,0.4).$ The concentration profile of hybrid nanofluids is greatly influenced by the homogeneous reaction parameter. Steeper concentration gradients and faster reactant species consumption are caused by higher values of the homogeneous reaction parameter. Raising the homogeneous reaction parameter lowers the reactant concentration in a hybrid nanofluid. It is because the boundary layer thickness increases for large ${K_r}$. The rate at which chemical reactions take place consistently throughout the fluid domain is represented by the homogeneous reaction parameter from the standpoint of mass transfer. Reactant species in the fluid are consumed more quickly as this parameter increases, because the chemical reaction rate increases.

The effect of ${K_s}$( heterogeneous reaction parameter) on the concentration profile is plotted in Fig. 9. Decreasing trends are noticed. By accelerating the rate of reaction at the surfaces in comparison to the transport processes, raising the heterogeneous reaction parameter $({K_s}=0.2,0.3,0.4,0.5)$ lowers the reactant concentration in a hybrid nanofluid. Rapid consumption of reactive species at the surface results from strengthening the chemical reaction at the fluid-solid interface by increasing the heterogeneous reaction parameter. This improves the concentration gradient and mass diffusion toward the wall, but the dominant response rate lowers the total concentration within the boundary layer, leading to a narrower concentration boundary layer and a declining concentration profile.

Fig. 8

$g(\eta )$ versus ${K_r}$. Taking ${\psi _1}=0.03$, ${\psi _2}=0.01$, $\varpi =0.3$, ${S_c}=1.2$, $m=0.1$, ${N_r}=1.1$, ${{\text{B}}_{\text{i}}}=0.22$, ${K_s}=0.2.$

Fig. 9

$g(\eta )$ versus ${K_s}$. Taking ${\psi _1}=0.03$, ${\psi _2}=0.01$, $\varpi =0.3$, ${S_c}=1.2$, $m=0.1$, ${N_r}=1.1$, ${{\text{B}}_{\text{i}}}=0.22$,${K_r}=0.1$.

Heat transfer rate and surface drag coefficient

Table 4 shows the impact of ${\psi _1}$, ${\psi _2}$, $\varpi$and m on the surface drag coefficient ${C_f}{R_{eX}}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt}\!\lower0.7ex\hbox{$2$}}}}$ and explains how heat transfer rates ${N_{UX}}{R_{eX}}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt}\!\lower0.7ex\hbox{$2$}}}}$ are affected by carbide nanoparticle inclusion. The table indicates contrasting trends of both the surfaces drag coefficient and the Nusselt number against the concentration of carbide nanoparticles in the hybrid nanofluid, the power law index m and $\varpi$. The heat transfer rate increases from to $\:1.5363$, as the volume fraction of nanoparticle increases from 0.01 to 0.04. This is because of the high thermal conductivity of Carbide nanoparticles that boosts the thermal conductivity, enabling a higher heat transfer rate. Whereas the skin friction coefficient decreases from − 1.1757 to -1.3712 as volume fraction of nanoparticle increases from 0.01 to 0.04. This is due to the enhanced viscosity of the liquid that thickens the boundary layer and creates a hurdle to the velocity gradient and wall and eventually to the surface drag.

Table 5 presents a comparison between the current study and the previously published work¹³. A comparison of $- \theta ^{\prime}(0)$ with available published work by Mandal et al.¹³, Hamad and Ferdows⁶¹, and Khan and Pop⁶² by suppressing the additional parameters and selecting ${B_i} \to \infty ,\,\,{\text{and}}\,\,Nr,\,{\theta _r},\,\varpi ,\,m \to 0.$An excellent concurrence is observed. Table 6 highlights a comparison of current results with Fang et al.⁶³ for the Skin friction coefficient $f^{\prime\prime}(0)$considering ${\psi _1}={\psi _2}=0.0,$ by varying m and taking $\varpi =0.5$. Figure 10 shows the comparison plot of velocity curve with Fang et al.⁶³ for a fixed value of power law index $m=0.5$, which further explains the accuracy of the present results

Table 4 Numerical estimation of ${C_f}{R_{eX}}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt}\!\lower0.7ex\hbox{$2$}}}}$ and ${N_{UX}}{R_{eX}}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt}\!\lower0.7ex\hbox{$2$}}}}$ for $(Sic - Tic/water)$ hybrid nanoparticles volume fraction.

Table 5 Comparison of $- \theta ^{\prime}(0)$ with already published work by suppressing the additional parameters. Selecting ${B_i} \to \infty ,\,\,{\text{and}}\,\,Nr,\,{\theta _r},\,\varpi ,\,m \to 0$.

Table 6 Comparison of outcomes with Fang et al.⁶³ for Skin friction coefficient $f^{\prime\prime}(0)$considering ${\psi _1}={\psi _2}=0.0,$ by varying m and taking$\varpi =0.5$.

Fig. 10

Comparison plot of velocity curve with Fang et al.⁶³ for a fixed value of power law index $m=0.5$.

Concluding remarks

We have examined the trend of carbide nanoparticles in hybrid nanofluid flow across an extended sheet of varying thickness in this investigation, which is influenced by a quadratic thermal radiative heat flux. Homogeneous and heterogeneous reactions are also explored in the fluid flow. The OHAM approach yields an analytical solution. The following is a summary of the current paper’s findings:

• For higher carbide nanoparticle volume fractions, the velocity profile and surface drag coefficient decrease. As the volume fraction of nanoparticles in the nanofluid increases from 0.01 to 0.07, its effective viscosity generally increases due to the added resistance from the solid nanoparticles. This increase in viscosity tends to lower the fluid’s velocity by approximately 52%.

• Variable wall thickness exhibits a declining trend in the hybrid nanofluid velocity distribution. Thicker sections from 0.1 to 0.3 exhibit lower velocity owing to inflated resistance and friction effects.

• For greater estimations of radiation parameters from 0.1 to 3.1, the similarity parameter $\eta =2,$ the temperature distribution increases by approximately 57%. Higher temperatures and steeper temperature gradients near the heated surface result from the increased contribution of thermal radiation to overall heat transfer. Consequently, this leads to a significantly elevated fluid temperature.

• The heat transmission rate at a variable wall-thickness surface increases by approximately 37.5% when the estimated volume fraction of carbide nanoparticles is increased from 0.01 to 0.03. This is due to the high thermal conductivity of the carbide nanoparticles, which enhances overall thermal conductivity and enables a higher heat transfer rate.

• The concentration distribution diminishes with increasing counts of heterogeneous parameters, from 0.2 to 0.5. Increasing the heterogeneous reaction parameter speeds up surface reactions relative to transport processes. This reduces the reactant concentration in a hybrid nanofluid, creating steeper concentration gradients and accelerating reactant consumption near the surfaces.

Advantages and limitations of the study

• Advantages: By combining homogeneous–heterogeneous chemical reaction effects with quadratic thermal radiation, the current study offers a thorough theoretical framework for examining hybrid nanofluid flow over a stretched surface of variable thickness. Compared to traditional single-particle nanofluids, the use of hybrid nanoparticles, such as silicon carbide and titanium carbide, dispersed in water enables improved heat transfer analysis. Furthermore, in the presence of large temperature fluctuations, the quadratic radiation formulation provides a more accurate representation of radiative heat transfer. The Optimal Homotopy Asymptotic Method, which provides semi-analytical solutions and makes it easier to understand how parameters affect the velocity, temperature, and concentration fields, is used to solve the governing nonlinear equations.

• Disadvantages/limitations: Despite these benefits, the model’s generalizability is constrained by several simplifying assumptions. The analysis ignores potential three-dimensional and transient effects and is limited to two-dimensional, steady boundary-layer flow. Furthermore, the hybrid nanofluid is treated as a homogeneous single-phase fluid, with thermophoretic transport, Brownian motion, and nanoparticle aggregation ignored. Moreover, strong external flow interactions or regions distant from the stretching surface may make the boundary-layer approximation less accurate.

Future work directions

• Future research could expand the current model to include three-dimensional flow configurations, multiphase nanofluid dynamics, and transient behavior.

• This current work can also be extended by considering magnetic dipole effects.

• Extension can also be achieved by introducing an exponentially stretched sheet and including microorganisms.