1 Introduction

Thrust of emerging technology in the world paws a way for developing new technology. One such concept gaining traction in the scientific community is nanotechnology, often referred to as engineered fluids within the realm of fluid dynamics and heat transfer. The thermal conductivity of water and ethylene glycol (conventional fluid) have low values compared with thermal conductivity of nanoparticle. To attain this property, researchers have explored the concept of nanofluids—mixtures of base fluids and nanoparticles—which offer enhanced thermal conductivity. Adding nanoparticles to the base fluid improves its thermal properties. Choosing the right type of nanoparticles can significantly influence the thermal conductivity, viscosity, and other properties of the nanofluid. Foremost person who gave a different opinion to the scientific community and industrial area to enhance their development through nanoparticles was introduce Choi [1]. Hybrid nanofluids (HNFs), which combine traditional base fluids with nanoparticles, have emerged as a promising pathway for improving heat transfer rates over conventional single-component nanofluids. Researchers and engineers have delved into experimental studies to optimize HNF properties for specific applications, ranging from heat exchangers to electronics cooling systems and solar collectors.

Wong and Leon [2] provided insights into various applications of nanofluids, covering both current and future technologies. Subsequent studies by Yu and Xie [3] focused on nanofluid preparation, stability, and applications, while Shoaib et al. [4] investigated 3D radiative HNF motion via a rotational disk. Ratha et al. [5] explored the 2D unsteady motion of incompressible Casson nanofluids over a shrinking horizontal sheet.

The term ‘hybrid material’ refers to substances that blend the chemical and physical properties of multiple materials, offering unique qualities absent in their individual components. Two-step and single-step methods are employed to synthesize nanoparticles into HNFs. Various factors such as nanoparticle dispersion, purity levels, and synthesis methods can impact HNF efficiency. Mustafa [6] conducted an investigation using linear stability theory to analyze nanoparticles in a single-phase model.

Understanding the behavior of viscous fluids across different conditions requires a basic understanding of viscous dissipation principles, which are fundamental to fluid dynamics. It is the mechanism by which a fluid transfers mechanical energy into heat while flowing because of its natural viscosity. This phenomenon occurs due to internal friction between adjacent fluid layers, leading to a gradual loss of kinetic energy and a corresponding increase in temperature. Viscous dissipation is a crucial aspect of liquid motion analysis. Which influences various engineering and scientific applications (like, the cooling of electronic components, etc.). Ohmic viscous dissipation is an interesting mechanism that converts electrical energy in conductive fluids into heat due to electrical resistance. The effects of viscous dissipation on heat transfer in a cold liquid with a coupled stress mechanism were examined by Masthanaiah et al. [7]. Mahesh et al. [8] presented the impact of MHD coupled stress HNF through a porous surface with dissipation effect. Moreover, Sharma et al. [9] investigated the NF motion via rotating disk in moving up/down wards with viscous dissipation phenomena. Additionally, Rashad et al. [10] analyzed the variations of heat transfer by using Williamson HNF over a curved porous medium.

Internal heat generation and absorption is indeed crucial for analyzing, designing, and predicting temperature distributions in various systems, from power plants to refrigeration units. Internal heat generation refers to heat production within a system due to internal processes, while internal heat absorption involves heat transfer from the surroundings or within the system itself. This phenomenon is often encountered in systems such as power plants, electronic devices, chemical reactors, and even living organisms. Yaseen et al. [11] investigated the effect of shape on the flow of radiative hybrid nanofluids over a permeable sheet, observing an increase in heat transfer.

Absolutely, thermal radiation is a fundamental aspect of engineering, particularly in fields like HVAC (Heating, Ventilation, and Air Conditioning), materials science, and thermal engineering. Understanding how materials emit, absorb, and transmit thermal radiation is crucial for designing efficient heating and cooling systems, as well as for selecting appropriate materials for various applications. The homogeneous and heterogeneous chemical reactions on Bioconvection HNF with radiation effect was developed by Puneeth et al. [12]. subsequently, the heat transfer rate has been determined by Nayak et al. [13] by examining thermal radiation and heat dissipation effect on nanofluid motion.

Indeed, Magnetohydrodynamics (MHD) is a fascinating interdisciplinary field that merges principles of fluid dynamics and electromagnetism. It’s concerned with the behavior of electrically conducting fluids, such as plasmas, liquid metals, and electrolytes, in the presence of magnetic fields. Mustafa et al. [14] examine the mixed convective flow of magneto-nanofluid bounded by a vertical stretchable surface. Krishna et al. [15] explore unsteady magnetohydrodynamic (MHD) motion of a non-Newtonian Casson HNF over a vertically moving porous surface under slip velocity in a rotating frame.

The concept of a porous medium is fundamental across various fields due to its widespread occurrence and diverse applications. Here’s how porous media impact different domains: Geosciences and Hydrology, Engineering and Civil Infrastructure, Chemical Engineering and Catalysis, Biomedical Applications, Environmental Science, Material Science and Manufacturing. Nandi et al. [16] investigate the flow of magneto-convective and chemically reactive Casson nanofluid along an inclined permeable stretching with saturated medium. The HNF magnetized between two coaxially rotated porous disks was investigated by Qureshi et al. [17]. Abad et al. [18] analyze the heat and mass transfer in HNF motion on a cylindrical bluff-body with effect of chemical reactions.

Entropy generation is a crucial topic in thermodynamics and engineering to understand the efficiency and irreversibility of energy and heat transmission systems. It plays an essential role in understanding energy transmission phenomena and directionalities of the physical process. The entropy can take on any value of negative, positive, or zero-but this one is always either zero or positive. The generation of entropy is possible but its destruction is impossible. According to 2nd law of thermodynamics, the rate of local entropy generation per unit volume can be estimated as \(S_{gen}^{\prime \prime \prime } = \frac{k}{{T_{0}^{2} }}\left( {\nabla T} \right)^{2} + \frac{\mu }{{T_{0} }}\phi + \frac{{q^{\prime \prime \prime } }}{{T_{0} }}\). The entropy generation mechanism finds application in various sectors such as air conditioners, heat pumps, and freezers. Recently, Bejan [19, 20] explained the entropy generation caused by heat transfer in convective mode across a stretching sheet. Baag et al. [21] investigated the production of entropy generation with convective heat and mass transfer effects. Some of the researchers [28, 29] published the HNF with nanoparticles GO and MoS2 with different geometry, like cylinders and disks and also with different base fluids with different boundary conditions.

The present study aims to investigate the behaviour of hybrid nanofluids over a stretching sheet incorporating thermal radiation, heat source/sink, MHD, ohmic heating and viscous dissipation effects. By numerically solving using the BVP4C MATLAB solver, we aim to build upon the work of Pal et al. [17] and explore irreversibility mechanisms. This study holds implications for various fields, including medicine, where nanocomposites can serve as coolants in electronic devices and aid in cancer treatment through hyperthermia therapy. By optimizing energy efficiency and minimizing entropy generation, nanocomposites can contribute to more effective cancer treatments.

2 Mathematical model of the problem

We consider the steady, incompressible, two-dimensional magnetohydrodynamic (MHD) hybrid nanofluid (HNF) with two different types of nanoparticles namely Graphene-Oxide and Molybdenum-Disulphide over a stretching surface. The effects of heat source/sink, thermal radiation and Ohmic-viscous dissipation are taken into account. Additionally, Tw and T∞ represents the wall temperature and ambient temperature, respectively. An external magnetic field Bo is applied in the normal direction to the liquid flow surface. The plane’s surface is considered in the x-direction and normal plane is taken in y-direction as shown in Fig. 1. The governing non-linear partial differential equation for boundary layer with the Darcian flow of HNF through a porous medium as follows (ref. [22, 27]).

$$\frac{\partial u}{{\partial x}} + \frac{\partial v}{{\partial y}} = 0$$
(1)
$$\frac{1}{{\varepsilon^{2} }}\left( {u\frac{\partial u}{{\partial x}} + v\frac{\partial u}{{\partial y}}} \right) = \frac{1}{\varepsilon }\frac{{\mu_{hnf} }}{{\rho_{hnf} }}\frac{{\partial^{2} u}}{{\partial y^{2} }} - \frac{{\sigma_{hnf} }}{{\rho_{hnf} }}B_{0}^{2} u - \frac{{\mu_{hnf} }}{{\rho_{hnf} k_{p} }}u$$
(2)
$$\left( {u\frac{\partial T}{{\partial x}} + v\frac{\partial T}{{\partial y}}} \right) = \frac{{K_{hnf} }}{{\left( {\rho c_{p} } \right)_{hnf} }}\frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{\sigma_{hnf} }}{{\left( {\rho c_{p} } \right)_{hnf} }}B_{0}^{2} u^{2} + \frac{{\mu_{hnf} }}{{\left( {\rho c_{p} } \right)_{hnf} }}\left( {\frac{\partial u}{{\partial y}}} \right)^{2} - \frac{1}{{\left( {\rho c_{p} } \right)_{hnf} }}\frac{{\partial q_{r} }}{\partial y} + \frac{{Q_{0} }}{{\left( {\rho c_{p} } \right)_{hnf} }}\left( {T - T_{\infty } } \right)$$
(3)
Fig. 1
figure 1

Physical diagram of the problem

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Equation 3 represents the heat transported by a hybrid nanofluid through convection and conduction, wherein the magnetic field induces joule heating, viscous dissipation terms, heat generation and absorption, and radiation terms.

The boundary conditions in dimensional form are provided below [22, 27]:

$$\begin{aligned} & u = 0,v = v_{w} ,T = T_{w} \;at\;y = 0 \\ & u \to 0, T \to T_{\infty } \;as\;y \to \infty \\ \end{aligned}$$
(4)

Here, the u value is set to zero, which represents the velocity of fluid particle vanishing at the solid boundary and it gradually increasing away for the surfaces to attain free stream velocity. Similarly, the fluid particles near to surface are at rest, heat is conducted from a solid surface into the fluid particle, causing a temperature gradient for them to reach the wall temperature.

Here, Radiative heat flux \(q_{r} \left( { = - \frac{{4\sigma^{*} }}{{3k^{*} }}\frac{{\partial T^{4} }}{\partial y}} \right)\) is defined using Rosseland approximation.

The T4 is expanded in terms of T∞, by using the Taylor series is,

$$T^{4} \approx T_{\infty }^{4} - 4T_{\infty }^{3} \left( {T - T_{\infty }^{{}} } \right) + 6T_{\infty }^{2} \left( {T - T_{\infty }^{{}} } \right)^{2} + \ldots$$
(5)

Here we omit the higher order of temperature difference,

$$T^{4} \approx 4TT_{\infty }^{3} - 3T_{\infty }^{4} ,$$

Transformations for the proposed model is referred from Pal et al. [22] and its listed below,

$$\left. \begin{gathered} u = \left( {g\beta_{T} A} \right)^{\frac{1}{2}} xf^{\prime } ,v = - \left( {g\beta_{T} A\upsilon_{f}^{2} } \right)^{\frac{1}{4}} f,\eta = \left( {\frac{{g\beta_{T} A}}{{\upsilon_{f}^{2} }}} \right)^{\frac{1}{4}} y \hfill \\ f = \frac{\psi }{{\left( {g\beta_{T} A\upsilon_{f}^{2} } \right)^{\frac{1}{4}} x}},\theta = \frac{{\left( {T - T_{\infty } } \right)}}{{\left( {T_{w} - T_{\infty } } \right)}} \hfill \\ \end{gathered} \right\}$$
(6)

Using the above transformation (5), Eq. (2) and (3) are changed as follows, and Eq. (1) is satisfied trivially,

$$\frac{1}{\varepsilon }\frac{{\mu_{hnf} }}{{\mu_{f} }}f^{\prime\prime\prime} + \frac{1}{{\varepsilon^{2} }}\frac{{\rho_{hnf} }}{{\rho_{f} }}\left( {ff^{\prime\prime} - f^{{\prime}{2}} } \right) - \left( {\frac{{\sigma_{hnf} }}{{\sigma_{f} }}M + \frac{{\mu_{hnf} }}{{\mu_{f} }}Kp} \right)f^{\prime} = 0$$
(7)
$$\left( {\frac{{K_{hnf} }}{{K_{f} }} + Rd} \right)\theta^{\prime\prime} + \frac{{\left( {\rho c_{p} } \right)_{hnf} }}{{\left( {\rho c_{p} } \right)_{f} }}\Pr f\theta^{\prime} + \Pr Ec\left( {\frac{{\sigma_{hnf} }}{{\sigma_{f} }}Mf^{{\prime}{2}} + \frac{{\mu_{hnf} }}{{\mu_{f} }}f^{{\prime\prime}{2}} } \right) + \Pr Q\theta = 0$$
(8)

With non-dimensional boundary conditions are given as follows,

$$\left. \begin{gathered} f^{\prime} = 0,f = 1,\theta = 1 ,\,\,at\,\,\eta = 0 \hfill \\ f^{\prime} \to 0,\theta \to 0\,\,as\,\,\eta \to \infty \hfill \\ \end{gathered} \right\}$$
(9)

The physical variables play a vital role in revealing the explosive interaction between fluid structures at a rigid surface, where the viscous frictional force is quantified as the skin friction coefficient, also referred to as the surface force.

Here, the dimensionless forms of the local Nusselt number and skin friction are provided.

$$C_{f} {\text{Re}}_{x}^{\frac{1}{2}} = \frac{{f^{\prime\prime}\left( 0 \right)}}{{\left( {1 - \phi_{1} } \right)^{2.5} \left( {1 - \phi_{2} } \right)^{2.5} }},Nu{\text{Re}}_{x}^{{ - \frac{1}{2}}} = - \left( {\frac{{K_{hnf} }}{{K_{f} }} + Rd} \right)\theta^{\prime}\left( 0 \right)$$
(10)

and Rex \(\left( { = \frac{{\left( {g\beta_{T} A} \right)^{\frac{1}{2}} x}}{{\upsilon_{f} }}} \right)\) is referred as Reynolds number.

3 Mathematical formulation of entropy generation

In the system, entropy generation dissipates the available energy. Therefore, understanding the rate of entropy generation is crucial for enhancing the energy efficiency and overall system performance. Entropy generation for this problem [21] is formulated considering various factors: heat transfer, viscous dissipation, joule heating, and porosity. The first term in Eq. (10) represents irreversibility due to heat transfer, the second term represents viscous dissipation, the third term accounts for the magnetic field effect on local entropy formation, and the fourth term represents the effect of porosity.

$$S_{G}^{\prime \prime \prime } = \frac{1}{{T_{\infty }^{2} }}\left[ {\frac{{K_{hnf} }}{{K_{f} }} + \frac{{16\sigma^{*} T_{\infty }^{3} }}{{3K^{*} K_{f} }}} \right]\left( {\frac{\partial T}{{\partial y}}} \right)^{2} + \frac{{\mu_{hnf} }}{{T_{\infty } }}\left( {\frac{\partial u}{{\partial y}}} \right)^{2} + \frac{{\sigma_{hnf} }}{{T_{\infty } }}B_{0}^{2} u^{2} + \frac{{\mu_{hnf} }}{{T_{\infty } K_{p} }}u^{2}$$
(11)

The characteristics entropy generation rate is \(E_{{G_{o} }} = \frac{{T_{w} - T_{0} }}{{T_{\infty } }}\frac{{\left( {g\beta_{T} A} \right)^{\frac{1}{2}} }}{{\upsilon_{f} }}K_{nf}\).

Using Eq. (6), the entropy generation in non-dimensional form is expressed as follows,

$$\begin{aligned} N_{s} & = \frac{{S_{G}^{\prime \prime \prime } }}{{E_{{G_{o} }} }} = N_{ST} + N_{SFF} \\ & = \left\{ {\left( {\frac{{K_{hnf} }}{{K_{f} }} + Rd} \right)\alpha_{1} \theta \prime^{2} } \right\} + \frac{{\rho_{f} }}{{\rho_{hnf} }}\left\{ {\frac{{\mu_{hnf} }}{{\mu_{f} }}Brf^{\prime \prime 2} + \frac{{\sigma_{hnf} }}{{\sigma_{f} }}MBrf^{\prime 2} + \frac{{\mu_{hnf} }}{{\mu_{f} }}KpBrf^{\prime 2} } \right\} \\ \end{aligned}$$
(12)

In the above equation, Ns denotes the total entropy of the system, NST denotes the entropy number due to the thermal effect including thermal radiation and NSFF denotes entropy number due to the fluid friction, magnetic field and porous term.\(\alpha_{1} \left( { = \frac{{T_{w} - T_{\infty } }}{{T_{\infty } }}} \right)\) denote the temperature difference in the form of non-dimensional, \(Br\left( { = \frac{{\rho_{hnf} }}{{k_{nf} }}\frac{{\left( {g\beta_{T} A} \right)x^{2} }}{{\left( {T_{w} - T_{\infty } } \right)}}\upsilon_{f} } \right)\) and denote the Brinkman number used to correlate the heat produced by dissipation and the computed heat.

The Bejan number, defined as the ratio of irreversibility of heat transfer to the total irreversibility due to fluid friction and heat transfer, plays a vital role [24]. Its primary function is to evaluate the irreversibility of heat transfer and fluid flow processes, defined locally at every point of the boundary layer.

$$\begin{aligned} Be & = \frac{{N_{ST} }}{{N_{ST} + N_{SFF} }} \\ & = \frac{{{\text{The}}\;{\text{entropy}}\;{\text{generation}}\;{\text{due}}\;{\text{to}}\;{\text{the }}\;{\text{thermal}}\;{\text{irreversibility}}}}{{{\text{The}}\;{\text{total}}\;{\text{entropy}}\;{\text{generation}}}} \\ \end{aligned}$$
(13)

4 Flow chart for numerical method

4.1 Method of solution

Solving the problem with non-linear ordinary differential equations (ODEs) exactly is challenging. Therefore, the system of governing equations with boundary conditions is solved numerically using BVP4C solver, known for its robustness and accuracy as confirmed by various heat transfer papers. Equation 4 provides the asymptotic boundary condition, with the value of the similarity variable ηmax is set as 3. The non-linear coupled boundary value problem (BVP) of 3rd order in \(f\) and the 2nd order in \(\theta\) is then reformulated into a system of first-order simultaneous equations with five unknowns, which we consider \(y_{1} = f,y_{2} = f^{\prime},y_{3} = f^{\prime\prime},y_{4} = \theta ,y_{5} = \theta ^{\prime}\). The deduced system of first-order equations is as follows:

$$y_{1}^{\prime } = y_{2}$$
(14)
$$y_{2}^{\prime } = y_{3}$$
(15)
$$y_{3}^{\prime } = \varepsilon \frac{{\mu_{f} }}{{\mu_{hnf} }}\left( {\left( {\frac{{\sigma_{hnf} }}{{\sigma_{f} }}M + \frac{{\mu_{hnf} }}{{\mu_{f} }}Kp} \right)y_{2} - \frac{1}{\varepsilon }\frac{{\rho_{hnf} }}{{\rho_{f} }}\left( {y_{1} y_{3} - y_{1}^{2} } \right)} \right)$$
(16)
$$y_{4}^{\prime } = y_{5}$$
(17)
$$y_{5}^{\prime } = - \frac{1}{{\left( {\frac{{k_{hnf} }}{{k_{f} }} + Rd} \right)}}\left( {\frac{{\left( {\rho c_{p} } \right)_{hnf} }}{{\left( {\rho c_{p} } \right)_{f} }}\Pr y_{1} y_{5} + \Pr Ec\left( {\frac{{\sigma_{hnf} }}{{\sigma_{f} }}My_{2}^{2} - \frac{{\mu_{hnf} }}{{\mu_{f} }}y_{3}^{2} } \right) + \Pr Qy_{5} } \right)$$
(18)

and the corresponding boundary conditions are,

$$y_{2} (0) = 1, \, y_{1} (0) = \delta_{1} , \, y_{4} (0) = 1, \, y_{2} (\infty ) = \delta_{2} , \, y_{4} (\infty ) = \delta_{3}$$
(19)

Utilizing the shooting method, initial values are guessed, then the resulting 5th order differential equations were solved by BVP4C with a step size of h = 0.01. The entire procedure is iteratively repeated until the desired result is obtained with a degree of accuracy of 10–5. To ensure accuracy and reliability, the following convergence criteria are implemented: the absolute tolerance ‘AbsTol’ is set to 1e−5 to control the absolute error and ensure solution precision, while the relative tolerance is specified as 1e−4 to manage variations in the solution and achieve convergence. These convergence criteria collectively govern the termination condition for the iterative solution process. The chosen tolerance and iteration limits are determined based on the problem, aiming to achieve a satisfactory balance between solution accuracy and computational efficiency.

5 Results and discussions

In the analysis, the nanoparticle volume fraction (ϕ) varied between 1 and 4%, while the radiation parameter (Rd) ranging from 1 to 3 was observed to significantly affect the temperature distribution within the system. The Brinkman number (Br), magnetic parameter (M), heat source/sink parameter (Q), Eckert number (Ec), and porosity term (ε) were investigated within the ranges of 0–2, 1–3, 0.1–0.3, 0.1–0.3, and 1–3, respectively.

Furthermore, the physical and thermophysical properties of nanoparticles are detailed in Tables 1 and 2, respectively. Table 3 provides a comparative analysis of skin friction values obtained from different magnetic parameter (M) values, compared to earlier research. Notably, Table 4 highlights a significant finding: the skin friction for the hybrid nanofluid is observed to be lower than that of the mono-fluid, while simultaneously, the heat transmission rate of the hybrid nanofluid exceeds that of the mono-fluid. This observation is crucial in understanding the enhanced heat transfer capabilities of the hybrid nanofluid system. Furthermore, Table 5 demonstrates that adjusting the emerging parameters can effectively reduce entropy generation, leading to a notable increase in the Bejan number for the hybrid nanofluid. Additionally, Table 6 presents data confirming the hybrid nanofluid’s higher heat transfer rate compared to the mono-fluid. These results are graphically depicted, with the hybrid nanofluid represented by dashed lines, Graphene Oxide (GO) by dotted lines, and Molybdenum Disulphide (MoS2) by solid lines. The parameters are set as follows: ε = 1.2, M = α1 = Ec = Kp = 0.1, Br = 0.5, and Q = 0.01, ensuring consistent and controlled conditions for analysis.

Table 1 The properties of hybrid nanoparticles and nanomaterials are taken from Fazle et al. [23] and Hussain et al. [30], and are listed in this table (with the shape of the nanoparticles considered here as 3.7)
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Table 2 Physical properties value is taken from the Fazle et al. [23] work
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Table 3 The skin friction across different values of M when Rd = Q = Ec = 0
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Table 4 Variation against M on Skin friction and Nusselt number for Hybrid nanofluid and nanofluid with the values of ϕ1 = 0.01, ϕ2 = 0.005, m = 3.7, Pr = 6.2, Rd = Ec = 0.1 and Q = 0.01
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Table 5 Variation against M on Entropy and Bejan number for Hybrid nanofluid and nanofluid with the values of ϕ1 = 0.01, ϕ2 = 0.005, m = 3.7, Pr = 6.2, Rd = Ec = 0.1 and Q = 0.01
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Table 6 Variation against various parameters on Nusselt number for Hybrid nanofluid and nanofluid with the values of ϕ1 = 0.01, ϕ2 = 0.005, m = 3.7, Pr = 6.2, Rd = Ec = 0.1 and Q = 0.01
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The study further reveals variations in entropy generation and Bejan number across different values of emerging characteristics, including the magnetic parameter (M), radiation parameter (Rd), heat source variable (Q), porosity variable (ε), Brinkman number (Br), as well as velocity and temperature profiles. In Fig. 2, it is evident that the temperature of the hybrid nanofluid exceeds that of the mono-fluid. This finding emphasizes the potential of hybrid nanofluids in enhancing heat transfer efficiency. Researchers can utilize this insight to optimize heat transfer processes.

Fig. 2
figure 2

Temperature profile for the nanofluid and hybrid nanofluid

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Thermal radiation as one of the mode of heat transmission in the form of electromagnetic waves, influences the heat transfer within fluid particles, there by increasing it. From these physical aspects, higher values of Rd, as depicted in Fig. 3a and b, lead to increase temperature and Nusselt number profiles for the hybrid nanofluid compared to the mono-fluid. With an increase in the radiation parameter value, there is a corresponding rise in entropy. To reduce this increase in entropy, control over the Rd parameter becomes essential, as demonstrated in Fig. 3c. The enhancement of the radiation parameter strengthens the growth in the Bejan profile, as shown in Fig. 3d. The rise in the Bejan profile is caused by the dominance of fluid friction irreversibility resulting from heightened thermal radiation.

Fig. 3
figure 3

a–d Impact of radiation parameter with Nusselt, Temperature, Entropy and Bejan profiles

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The Brinkman number measures the significance of viscous heat relative to conductive heat transfer by comparing the heat generated through viscous dissipation where in fluid motion converts energy into internal energy to the heat transferred via molecular conduction. As illustrated in Fig. 4a, viscous dissipation plays a primary role during the heating process, resulting in increased entropy generation in hybrid nanofluids compared to nanofluids alone. Thus, managing Br is essential to minimize entropy growth. From Fig. 4b, the value of Br decreases with the increasing Bejan number for hybrid nanofluid than the nanofluid, because fluid friction (that is viscous dissipation) is dominated so that Bejan profile gets reduced.

Fig. 4
figure 4

a, b Impact of Br with entropy and Bejan profiles

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The temperature differences, depicted by parameter α1 in Fig. 5a, grow along with the thermal energy. This increase corresponds to a simultaneous rise in entropy generation. Therefore, it is essential to regulate the temperature difference factors in order to decrease the entropy. The Bejan number in Fig. 5b represents the ratio between thermal irreversibility and total irreversibility. In the current context, an increase in the temperature difference parameter predominantly affects the numerator of the Bejan number. Consequently, for higher values of α1, the Bejan profile strengthens.

Fig. 5
figure 5

a, b Impact of α1with entropy and Bejan profiles

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The porous medium is made up of pores or voids that contain the fluids. More holes in a medium will cause the fluid to flow slower because they restrict the flow of fluid as it passes through them. Meanwhile, thermal energy is induced by a decrease in velocity. More pores lead to better thermal efficiency from the perspective of physical properties. Figure 6a, b provide a solid explanation of the above-mentioned concept, demonstrating that the hybrid nanofluid exhibits higher heat transfer rate compared to the mono-fluid.

Fig. 6
figure 6

a, b Impact of ε with velocity and temperature profiles

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The heat source parameter determines the heat transfer rate. Figure 7a, b demonstrate that, related to the mono-fluid, the temperature and Nusselt profile of the hybrid nanofluid are greater for larger values of Q (the heat source parameter) because an increase in Q also increases the heat transfer rate.

Fig. 7
figure 7

a, b Impact of Q with Temperature and Nusselt profiles

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Magnetic parameter cause (resistive force) like a drag force because it acts against the direction of the motion of the fluid magnetic field refer exerts a force on a particle, is also named as Lorenz force. At that point, the fluid becomes subjected to increased magnetic influences, which cause frictional forces to remain constant. When friction increases, thermal energy also increases. Figure 8a, b illustrate that, in comparison to mono-fluids, the heat transfer rate for hybrid nanofluids increases as the magnetic parameter value increases. Entropy increases for the higher values of M, but entropy is minimized when it reaches the surface for the higher values of M. Because the magnetic field applied to the moving fluid flow produces a resistance for the moving fluid particle which creates some disorder in the flow. The mono-particle also has better control over entropy than the hybrid nanoparticle. However, the Bejan profile displays the inverse trend because a rise in magnetic force causes an increase in fluid friction, which in turn causes a reduction in pressure due to resistance in the flow of fluid. In order to maximize system performance and minimize energy usage, pressure drop minimizing is essential. Illustrations Fig. 8c, d make the above concepts crystal evident.

Fig. 8
figure 8

a–d Impact of Magnetic parameter for the Temperature, Nusselt, Entropy and Bejan profiles

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The Eckert number tells us how much the flow’s kinetic energy fluctuates from the variation in enthalpy across the thermal boundary layer. Viscosity loss is a key component of understanding how heat spreads through high-speed flows. It’s used to characterise the phenomenon of heat dissipation in high-speed flows, where in the impact of viscosity important. The rate of heat transmission can be enhanced by increasing Eckert number, which not only accelerates the velocity but also simulates the self-heating-up process. As clearly depicted in Figs. 9a, b, the heat transfer rate is higher for the hybrid nanofluid than the mono-fluid for the higher Eckert number. Because an increase in the Eckert number causes the fluid to flow faster, the skin friction with the fluid decreases as the number increases. When a fluid comes into contact with a solid surface, one traditional technique to measure the resistance to motion is skin friction. Compared to mono-fluid skin friction, hybrid nanofluid skin friction is reduced in this case. And may see that in Fig. 9c. Here, the various shapes of the nanoparticles were considered to determine which shape of the particle gave the best performance. And the values considered for m are 8.26, 3.2, 4.82, 5.72, 3.00 for blade, brick, cylinder, platelets and sphere respectively.

Fig. 9
figure 9

a–c Impact of Eckert number with Temperature, Nusselt and Skin friction profiles

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Figure 10a demonstrate the Nusselt profile with various shapes of the hybrid nanoparticles. It displays the flow of heat transmission rate. Also, it provides a measurement of the convective heat transfer that takes place at the surface level. In physical terms, the Nusselt number is the ratio of convective heat transmission to conductive heat transfer. So, the blade-shaped nanoparticle has a higher rate of heat transmission than the others is described in Fig. 10b. Figure 10c depicts the entropy production for the various shapes of the nanoparticles. Minimizing entropy generation increases thermal efficiency. This picture clearly says that the sphere shape of the nanoparticle reduces entropy generation compared to the other shapes of the nanoparticle. From a physical perspective, a low Bejan number enhances the heat transfer rate since it determines a proportion of heat transmission irreversibility compared to the total proportion of heat transfer and fluid friction-induced irreversibility. Figure 10d shows that the sphere-shaped nanoparticle has greater thermal efficiency than the other shape of the nanoparticle.

Fig. 10
figure 10

a–d Impact of various shapes with Nusselt, Temperature,Entropy and Bejan profiles

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6 Conclusion

This study investigates entropy generation in hybrid nanofluid flow incorporating magnetohydrodynamic, thermal radiation, and ohmic viscous dissipation phenomena over a stretching sheet. Graphene Oxide (GO) and Molybdenum Disulphide (MoS2) are employed as hybrid nanoparticles dispersed in water, serving as the base liquid. Entropy generation analysis helps in determining the irreversibility and inefficiencies present in a system. The system of governing equations is solved using MATLAB bvp4c solver. The behavior of the various emerging parameters has been explained and analyzed graphically. The primary finding obtained from this study are summarized as follows:

  1. a.

    The heat transmission rate is observed to be higher for the hybrid nanofluid compared to the mono-fluid.

  2. b.

    Effective control of the temperature difference parameter, Rd (Radiation parameter), and Br (Brinkman number) is essential for minimizing the entropy generation rate.

  3. c.

    Blade-shaped nanoparticles demonstrate superior heat transfer rates, whereas sphere-shaped nanoparticles contribute to reducing entropy production.

The results obtained from this study are significant for various biomedical fields and coolant mechanism research.