Fluid flow past a curved stretching sheet (CSS) is a classical fluid mechanics problem with numerous applications in engineering and physics. Investigating nanofluid flow over curved stretched sheets has become an attractive field of study due to its many useful applications, such as cooling in electronic devices, heat exchangers, processing of materials, solar collectors, synthesis of polymers, and microelectronic devices. The behavior of the fluid in this scenario depends on several factors, including the geometry of the surface, the velocity of the stretching motion, and the properties of the fluid itself. Madhukesh et al. (Madhukesh et al., 2021) investigated the Newtonian heating (NH) and non-Fourier heat flux (NFHF) effect on the CSS in the presence of HNF (hybrid nanofluid). Multiple slippages on hydro-magnetic dissipative fluid across a CSS were addressed by Aihem et al. (Duraihem et al., 2023) and discussed their enhanced thermal and mass transmission properties. The impact of Cross dispersion on MHD Casson liquid movement along a CSS was inspected by Lakshmi et al. (Lakshmi et al., 2022). Sakkaravarthi et al. (Sakkaravarthi and Reddy, 2023) made a numerical investigation on entropy formation over a CHNF circulation over CSS. Simulation and theoretical inquiry on CNF over a CSS with the impact of a magnetic field and chemical processes were examined by Kumar et al. (Varun Kumar et al., 2022).

An electromagnetic actuator is a tool used in fluid mechanics to produce an effective liquid motion. A planar surface known as the Riga plate (RP) comprises alternating permanent magnets and electrodes. The magnetic field on the RP is not uniform, producing a Lorentz force that propels the fluid flow. In 1999, Gailitis and Lielausis (Gailitis and Lielausis, 1961) presented the electromagnetic actuator’s basic theory for the first time. In contrast to typical techniques, they showed that employing electrodes and permanent magnets on a Riga plate may considerably increase the fluid flow rate and mixing capabilities. The advantage of employing an electromagnetic actuator to produce liquid flow is that it can do so without the need for mechanical actuators or movable components, which may be costly and prone to failure. Consequently, it is a viable solution for various practical purposes such as improving the exchange of heat, combining, and liquid flow. Asogwa et al. (Asogwa et al., 2022) examined analytical approaches to cross-diffusion and convection effects in the presence of CF over a porous RP. Hussain et al. (Hussain et al., 2022) investigated the impact of the Navier slip on an upward RP with CF displacement. Madhukesh et al. (Madhukesh et al., 2022a) investigated TPD and heat generation of Newtonian NF in an RP. Alshehri as al. (Mohammed Alshehri et al., 2021). Investigated Buoyancy implications in a Micropolar solution over an upward RP.

A nanofluid is a liquid with individual nanoparticles suspended in a solvent. Increased transfer of heat efficiency is a significant benefit of nanofluids over more traditional fluids. Increased thermal conductivity due to nanoparticles in the base fluid makes nanofluids excellent for thermal transfer medium. Nanofluids remain an intriguing field of study because of their revolutionary effects on a wide range of businesses and technology. The use of the change of variables approaches in the hydrothermal investigation of MHD compressing nanofluid circulation in parallel plates was studied by Zabihi et al. (Zabihi et al., 2022). Rizk et al. (Rizk et al., 2022) assessed the influence of the KKL correlation hypothesis on the production of thermal energies in a nanofluid comprising GO and ZnO dissolved in water passing via a permeable vertically spinning substrate. Shah et al. (Shah et al., 2021) researched mesoscopic modelling for magnetized nanofluid movement inside a porous three-dimensional tank. Ullah et al. (Ullah et al., 2022) scrutinized a magnetized 2D nanofluid that included blood, Go, and ZnO nanoparticles and moved via a perforated tube. The computational estimation of mixed convective entropy optimized in Darcy-Forchheimer circulation of Cross nanofluids via an upward plane plate with inconsistent heat source/sink was explored by Hussain et al. (Hussain et al., 2023).

When non-Newtonian behavior and nanofluids are combined, the result is a non-Newtonian nanofluid. Non-Newtonian fluids have a viscosity that varies as a function of the shear or stress rate. The temperature, nanoparticle concentration, and nanoparticle kind may impact this behavior. Khan et al. (Khan et al., 2023) investigated the effects of irregular heat source/sink on the aiding and opposing movements of the Eyring-Powell liquid on wall jet nanoparticles. Alharbi et al. (Alharbi et al., 2022) assessed the influence of viscous dissipation and Coriolis impacts on the mass and heat transmission evaluation of the 3D non-Newtonian flow of liquids. Khan et al. (Khan S. et al., 2021) investigated the study of the movement of a non-Newtonian liquid through a stretching/shrinking permeable material while considering the transmission of heat and mass. Some of the noticeable works on non-Newtonian fluids are found in (Algehyne et al., 2023; Alsulami et al., 2023).

The Casson nanofluid (CNF) idea is built on the assumption that the Casson equation governs liquid circulation and particle motion, a rheological model that explains the momentum behavior of non-Newtonian liquids. The Casson model considers yield stress and plastic solution viscosity, essential factors in various real-world scenarios such as blood circulation, coating layout, and liquid processing. Because of its improved thermal conductivity and specific heat capacity, CNF can considerably improve a fluid’s ability to transmit temperature. Nanoparticles can also affect the fluid’s rheological properties, such as viscosity and yield stress. Madhukesh et al. (Madhukesh et al., 2023) used the Cattaneo–Christov theory to investigate the heat transport of an MHD CMNF (Casson—Maxwell nanofluid) between two porous discs. Mabood et al. (Mabood et al., 2020) studied the free convective movement of time-dependent CNF in a permeable stretched surface. Madhukesh et al. (Madhukesh et al., 2022b) scrutinized the circulation of MHD MCNF in the presence of permeable discs using CCHF and slip impacts. Rasheed et al. (Rasheed et al., 2022) considered the homotopic solutions for the unsteady MHD CNF in a vertical cylinder with viscous dissipation impacts. The exact solution of a CF using Prabhakar-fractional simulations while also experiencing the effects of magnetohydrodynamic and sinusoidal thermal conditions was examined by Raza et al. (Raza et al., 2023).

Because of its temperature, a body emits a specific sort of electromagnetic radiation known as thermal radiation (TR). This radiation is formed by the thermal movement of the molecules and atoms inside the body, and it can go freely into space as there is no requirement for a medium to conduct it. The Stefan-Boltzmann equation describes the relationship between the temperature of a blackbody (an idealized object that absorbs all radiation incident on it) and the intensity of the thermal radiation it emits. Thermal radiation has important practical applications in various fields, including engineering, physics, astronomy, electronics, and energy conversion. Lone et al. (Lone et al., 2022) inspected MHD micropolar nanofluid hybrids circulating across a flat surface exposed to TR and mixed convection. Khan et al. (Khan U. et al., 2021) inspected the nonlinear T-R-influenced entropy production in the presence of NF with mixed convection effects. Naqvi et al. (Raza Shah Naqvi et al., 2022) examined numerical simulations to study the movement of hybrid nanofluids while considering the consequences of TR and entropy formation. Ramesh et al. (Ramesh et al., 2023) scrutinized the hybrid-based CNT movement over a rotating sphere object in the presence of T-R and TPD. Magnetite-based liquid nanofluid three-dimensional layer movement involving non-linear TR and couple stress responses were studied by Ullah et al. (Ullah et al., 2021). The thermal study of slip and magnetohydrodynamic consequences for unstable sheet extending was investigated by Benos et al. (Benos et al., 2019).

The liquid and porous medium’s features affect the rheological behavior of a fluid moving through them. When a non-Newtonian fluid, like a CNF, travels through a porous media, the pores’ porosity, permeability, size, and shape can all impact how the fluid behaves. There has been a rise in interest in CNF flowing through porous media in recent years because of its potential applications in various industries, including increased oil recovery and geothermal energy generation. Understanding Casson nanofluid behavior is crucial for optimizing these processes since the characteristics of the porous medium can significantly impact how they behave. Alrehili et al. (Alrehili et al., 2022) made a numerical investigation of linear radiation and Soret impacts on MHD CNF over a vertical surface with a porous medium. Rallabandi et al. (Rallabandi, 2022) investigated the CNF flow over an inclined permeable stretched surface. Yogeesha et al. (Yogeesha et al., 2022) studied the Dufour and Soret effects to evaluate the dusty TNF circulation across an unstable stretched sheet. Raza et al. (Raza et al., 2022) inspected the activation energy, magnetic field, and binary chemical reaction impact on NF- and HNF through a porous area. Shoaib et al. (Shoaib et al., 2022) made soft computing to investigate the thermal energy’s effects on the MHD CF as it passes over a porous material with an inclined non-linear surface.

The RKF-45, or Runge Kutta Fehlberg 4th 5th order, is a numerical method employed to solve complex systems of differential equations governing fluid flow problems. Many problems arise from simple laminar to complex turbulent flows in fluid mechanics. In mathematics, many of these situations may be modelled using ordinary differential equations, partial differential equations, or a hybrid of the two. Due to its high order accuracy and flexible step size capacity, RKF-45 is a popular numerical approach for modelling fluid dynamics. RKF-45 continuously controls the step size to reach the required level of precision while minimizing computational cost by calculating two estimates of the solution with varying orders of accuracy. The algorithm of the RKF-45 method is in detail given in (Mathews and Fink, 2004), and solving the differential equations using the RKF-45 algorithm was explained in (Abell and Braselton, 2000). Some works that implemented and used the RKF-45 algorithm are provided in (Sarris et al., 2002; Arifeen et al., 2021; Madhukesh et al., 2022b; Yogeesha et al., 2022; Hussain et al., 2023; Madhukesh et al., 2023).

The consider examination originality comes from its emphasis on the as-yet-unstudied subject of Casson-based nanofluid flow over a CSS in the presence of a porous medium and thermal radiation effects. In today’s energy-conscious world, this study has the potential to help create more effective and sustainable thermal energy systems. Overall, studying the fluid flow past a CSS is an important area of research in fluid mechanics, with a significant impact on the development of microscale machines, including microfluidic devices, microscale engines, microsensors, and microscale reactors.

As schematically seen in Figure 1, the flow pattern under study is a two-dimensional, incompressible, non-Newtonian Casson nanofluid flowing over a curved Riga surface. The radius of the curved surface is represented by R 1 , and its curvilinear coordinates are marked by s 1 &   r 1 . The uniform velocity of the Riga surface is u 1 = U w 1 = a s 1 . Let T w 1 &   T ∞ , respectively, stands for the wall and far-field temperatures. Suppose that the Riga surface is being affected by an electromagnetic force, denoted by F m , in order to model the behavior of the fluid flow. A surface-mounted array of electrodes and permanent magnets are used to build the Riga surface. The previously derived governing equations for the fluid flow under these hypotheses are provided in the references (Hayat et al., 2018; Ahmad et al., 2019; Abbas et al., 2020; AdnanZaidi et al., 2020). These equations account for the influence of porous medium, TR, and CNF rheology on fluid flow across the curved Riga surface as follow: R 1 ∂ u 1 ∂ s 1 + ∂ ∂ r 1 v 1 R 1 + r 1 = 0 , (1) u 1 2 R 1 + r 1 = 1 ρ n f ∂ p 1 ∂ r 1 , (2) R 1 R 1 + r 1 u 1 ∂ u 1 ∂ s 1 + v 1 ∂ u 1 ∂ r 1 + u 1 v 1 R 1 + r 1 = − 1 ρ n f R 1 R 1 + r 1 ∂ p 1 ∂ r 1 + exp − π r 1 c 1 π j 0 M 0 8 ρ n f − ν n f k 1 * u 1 + ν n f 1 + 1 β 1 ∂ 2 u 1 ∂ r 1 2 + ∂ u 1 ∂ r 1 1 R 1 + r 1 − u 1 R 1 + r 1 2 , (3) R 1 R 1 + r 1 u 1 ∂ T 1 ∂ s 1 + v 1 ∂ T 1 ∂ r 1 = − k n f ρ C p n f ∂ 2 T 1 ∂ r 1 2 + ∂ T 1 ∂ r 1 1 R 1 + r 1 − 1 ρ C p n f ∂ ∂ r 1 R 1 + r 1 q r 1 R 1 + r 1 (4) in Eq. 4, the term q r is given by: q r = − 16 σ * T ∞ 3 3 k * ∂ T 1 ∂ r 1 , [see (Hayat et al., 2018)].

The respective boundary conditions for the consider model are r 1 = 0 : u 1 = U w 1 , v 1 = 0 , T 1 = T w 1 , r 1 → ∞ : u 1 → 0 , ∂ u 1 ∂ r 1 → 0 , T 1 → T ∞ . (5)

Furthermore, to ease the analysis of the consider investigation, the following similarity variables are introduced as [see (Abbas et al., 2020; AdnanZaidi et al., 2020)]: ζ = U 1 ν f s 1 0.5 r 1 , u 1 = U 1 h ′ ζ , v 1 = − R 1 R 1 + r 1 U 1 ν f s 1 h ζ , p 1 = ρ f U 1 2 P 1 ζ , K 1 = U 1 ν f s 1 0.5 R 1 , θ = T 1 − T ∞ T w 1 − T ∞ . (6)

Therefore, with the help of the above similarity variables stated in Eqs 1, 6 is satisfied and Eqs 2, 3 take the following form as: P 1 ′ = h ′ 2 G 2 ζ + K 1 , (7) 2 K 1 ζ + K 1 P 1 G 2 = 1 + 1 β 1 h ‴ + 1 ζ + K 1 h ″ − h ′ ζ + K 1 2 1 G 1 G 2 + K 1 ζ + K 1 h h ″ − h ′ 2 + K 1 ζ + K 1 2 h ′ h + Q 1 e − z ζ G 2 − λ 1 G 1 G 2 h ′ . (8) here, Q 1 = π j 0 M 0 8 ρ f U 1 a is the modified Hartmann number, λ 1 = ν f k ∗ 1 a is the porous parameter, and z = π 2 ν f a c 1 2 is the parameter related to the width of the magnets and electrodes.

Moreover, to eliminate the pressure terms in Eqs 7, 8, we get 1 + 1 β 1 h ′ ζ + K 1 3 − h ″ ζ + K 1 2 + 2 ζ + K 1 h ‴ + h ⁗ 1 G 1 G 2 + K 1 ζ + K 1 h h ‴ − h ′ h ″ + K 1 ζ + K 1 2 h h ″ − h ′ 2 − K 1 ζ + K 1 3 h ′ h + w 2 Q 1 e − z ζ G 2 − λ 1 G 1 G 2 h ″ + h ′ ζ + K 1 = 0 . (9)

After utilizing the similarity variables, the energy Eq. 4 reduces to the form as: k n f k f + 4 3 N r 1 Pr G 3 θ ″ + 1 ζ + K 1 θ ′ + K 1 ζ + K 1 h θ ′ = 0 , (10) with simplified boundary conditions are ζ = 0 : h ′ ζ = 1 , h ζ = 0 , θ ζ = 1 . ζ → ∞ : h ′ ζ = 0 , h ″ ζ = 0 , θ ζ = 0 . (11) in aforesaid Eqs 9–11, the term w 2 = 1 ζ + K 1 − z is called a dimensionless quantity, Pr ν f ρ C p f k f , & N r = 4 ϕ ∗ T ∞ 3 k ∗ k f refer the Prandtl number, and radiation parameter, respectively.

The important engineering quantities and its reduced form [see (Abbas et al., 2020; AdnanZaidi et al., 2020)]: C f = μ n f ρ f U w 1 2 1 + 1 β 1 ∂ u 1 ∂ r 1 − u 1 R 1 + r 1 r 1 = 0 ⇒ Re C f = 1 G 1 1 + 1 β 1 h ″ 0 − h ′ 0 ζ + K 1 , (12) N u = − s 1 k f T w 1 − T ∞ k n f + 16 ϕ ∗ T ∞ 3 3 k ∗ ∂ T 1 ∂ r 1 r 1 = 0 ⇒ N u Re = k n f k f + 4 3 N r θ ′ 0 . (13)

Hence, Re = U 1 s 1 ν f is the local Reynolds number.

The thermophysical properties of nanofluid are given as follows [see (Khan et al., 2018; Alwawi et al., 2019)].

The effective thermophysical characteristics of nanofluid are given as follows [see (Acharya et al., 2019)] μ n f = μ f G 1 G 1 = 1 − ϕ ∗ 2.5 , (14) ρ n f = ρ f G 2 G 2 = 1 − ϕ ∗ + ϕ ∗ ρ s ρ f , (15) ρ C p n f = ρ C p f G 3 G 3 = 1 − ϕ ∗ + ϕ ∗ ρ s C p s ρ f C p f , (16) k n f k f = k s − 2 ϕ ∗ k f − k s + 2 k f k s + ϕ ∗ k f − k s + 2 k f . (17)

The higher order and two-point boundary conditions in the governing equations for the fluid flow over the curved Riga surface make them challenging to solve analytically. We must transform these into first-order differential equations to achieve a numerical solution. Applying appropriate transformations will allow the higher-order differential equations to be represented as a set of first-order differential equations. Let us take, h , h ′ , h ″ , h ‴ = κ 1 , κ 2 , κ 3 , κ 4 θ , θ ′ = κ 5 , κ 6 , (18) h ⁗ = − G 1 G 2 1 + 1 β 1 1 + 1 β 1 1 G 1 G 2 2 ζ + K 1 κ 4 − κ 3 ζ + K 1 2 + κ 2 ζ + K 1 3 + K 1 ζ + K 1 κ 1 κ 4 − κ 2 κ 3 + K 1 ζ + K 1 2 κ 1 κ 3 − κ 2 2 − K 1 ζ + K 1 3 κ 2 κ 1 + w 2 Q 1 e − z ζ G 2 − λ 1 G 1 G 2 κ 3 + κ 2 ζ + K 1 , (19) θ ″ = − Pr G 3 k n f k f + 4 3 N r K 1 ζ + K 1 κ 1 κ 6 + 1 ζ + K 1 κ 6 1 Pr G 3 k n f k f + 4 3 N r , (20) with the boundary constraints become κ 1 0 = 0 , κ 2 0 = 1 , κ 3 0 = χ 1 , κ 4 0 = χ 2 , κ 5 0 = 1 , κ 4 0 = χ 3 . (21)

The Runge-Kutta Fehlberg 45-order approach was then used to solve the transformed Eq. 19 numerically and (20) as well as the boundary conditions (21). Since the boundary conditions contain unknowns, we employed a shooting technique to find the solution that meets the conditions at infinity. Further, utilised a step size of 0.001 and set the error tolerance to 10^–6 to achieve accurate findings. By substituting appropriate values for the dimensionless variables and using the thermophysical properties of the nanofluid (see Table 1) solutions are obtained. We discovered that our findings were in strong accord with prior work (Sajid et al., 2010), demonstrating the accuracy and dependability of our numerical method (see Table 2).

The purpose of this section is to describe how significant dimensionless parameters affect the temperature and velocity profiles. The RKF-45 method and shooting approach are used to numerically solve the reduced ODEs and boundary conditions acquired in the previous section. The acquired data are shown as graphs to illustrate the impact of various dimensionless parameters on the motion and temperature fields. Also, a discussion of the important technical variables that may have an impact on the system’s flow and thermal transfer characteristics is included in this section. The current study offers useful insights for designing and optimising industrial applications employing Casson-based nanofluid movements over curved surfaces by taking these parameters into account.

Figures 2A, B show the impact of K 1 (curvature constraint) over velocity and temperature profiles, respectively. According to the findings, a rise in the curvature parameter improves the h ′ profile (Figure 2A) but lowers the θ profile (Figure 2B). This is explained by the fact that increasing the radius of the curved surface causes the fluid to move more quickly, which improves the velocity profile by reducing the thickness of both the momentum boundary layer (MBL) and thermal boundary layer (TBL). However, when the fluid moves more quickly, there is less time for temperature distribution, which reduces temperature.

Figure 3A, B display the variation of h ′ and θ profiles in the presence of Q 1 (modified Hartmann number). The improvement in the Q 1 will decreases the velocity profile (see Figure 3A) but improves the temperature profile (see Figure 3B). This is caused by a rise in the Q 1 , which slows the liquid flow and lowers the velocity profile by increasing the magnetic strength and, consequently, the Lorentz force. Yet, this also improves the system’s thermal distribution, leading to a better temperature profile.

The effect of the porosity constraint λ 1 on the h ′ profile is illustrated in Figure 4A. It has been found that a higher λ 1 causes the velocity profile to drop. This is due to the presence of a porous medium, which restricts the movement of fluids by providing a barrier against the motion of the fluids. The h ′ profile is decreased when the resistance rises along with the porous parameter. Figure 4B displayed the influence of the thermal radiation N r parameter on θ profile. The rise in N r will improve the temperature profile. An increase in the value of N r denotes a rise in the system’s thermal radiation output. The energy from the radiation is absorbed by the fluid, raising its temperature, which improves the temperature profile.

The consequence of the Casson parameter β 1 on the h ′ profile is represented in Figure 5A. It is evident that a rise in the values of β 1 causes the velocity profile to fall. This is because a greater β 1 causes the fluid’s yield stress to flow initiation and decrease the h ′ profile. This leads to decline in the overall velocity of the liquid near the boundary as the circulation is impeded by increasing yield stress. Figure 5B displays the variation of θ profile for numerous values of the Casson parameter β 1 . The rise in the values of β 1 will advance the temperature distribution. As explained in Figure 5A, the reduction in the velocity will lead to the liquid’s residence time near the surface. When the β 1 increases, it implies a larger yield stress, meaning that the liquid requires more energy to commence flow. As a result of the higher flow resistance, more energy is released as heat inside the fluid. The temperature profile rises as a result of this phenomena.

Figure 6A represents the effect of skin friction on the porous parameter λ 1 for the rise in the values of solid volume fraction ϕ * . It is observed that surface drag force decreases with improved values of λ 1 and ϕ * . This is due to the fact that raising these parameters generates an increase in the MBL’s thickness, which in turn causes a reduction in the fluid flow at the surface. As a direct consequence of this, the force of surface drag is decreased. Figure 6B shows the variation in Nusselt number for improved values of N r and ϕ * . When these two criteria are improved, the rate at which thermal energy is distributed will increase. However, because nanoparticles are present in the fluid, the thermal conductivity is boosted, which results in an increase in the total heat transfer rate. This offsets the fact that the surface area that is accessible for heat transmission decreases as the percentage rises.

The present study investigates Casson-based nanofluid movement over a curved stretching surface in the presence of porous medium and thermal radiation effects. The ODEs and BCs are obtained by applying suitable similarity constraints to the PDEs. The numerical calculations are done with the aid of RKF-45 and shooting techniques. The outcomes are visualized using a graphical representation. The discussions on important dimensionless constraints are presented. The main conclusions of the study are as follows:

❖ The improvement in the modified Hartman number and porosity factors will decrease the velocity. An increase in these components indicates stronger magnetic impacts and increased permeability. As a result, the velocity of the flow of nanofluid reduces.