The boundary layer is a fundamental concept in understanding fluid transportation over a surface from theoretical fluid mechanics perspectives, pioneered by Prandtl in 1904 [1]. Conventional liquids, such as oil, ethylene glycol (EG), and water, used in various mechanical and technical operations, typically exhibit poor thermal conductivity, limiting the heat transfer efficiency for specific engineering processes. In 1993, Choi et al. [2] introduced a nanotechnology-based fluid aimed at enhancing energy efficiency and heat transfer capacity. As a result, nanofluids became prominent in heat transfer applications, including the biomedical and pharmaceutical sectors, microelectronics, magma solidification, cooling and heating exchangers, drug delivery, and food manufacturing. A mono nanofluid is defined as a single type of solid nanoparticle homogeneously dispersed in an ordinary liquid. Recent attention has shifted toward the introduction of multiple different nanoparticles suspended in that ordinary liquid, referred to as hybrid, ternary, or composite nanofluids. These advanced nanofluids integrate the chemical and physical properties of the suspended nanoparticles within a single phase, yielding diverse effects from the combined elements [3]. Babu et al. [4] found that hybrid nanomaterials display distinct physicochemical characteristics absent in general fluids or mono nanofluids. This discovery spurred further research on various hybrid nanofluids, examining their preparation, synthesis, and characterization stages. It was reported that thermal conductivity in hybrid nanofluids surpassed that of mono nanofluids [5]. In addition, hybrid nanofluids achieved higher heat flux than mono nanofluids in a study of Copper-Alumina/water (Cu-Al₂O₃/H₂O) by Nadeem et al. [6]. Salah et al. [7] reported that using the Al-Mg-TiO₂/water-ethylene glycol ternary hybrid nanofluid substantially increases the heat transfer coefficient for swirl flow within a rotating cone.

Blood is considered an incompressible flow consisting of the boundary layer flow and the potential flow within arteries. Analyzing blood flow over cylindrical surfaces has crucial applications in diagnosing and treating conditions related to plaque deposition and aneurysms in cardiovascular diseases, minimizing post-operative complications, and reducing healthcare costs. This analysis also applies to tumor treatments, blood clot removals, brain aneurysms, and infections. However, selecting appropriate models and approaches to depict blood flow challenges is vital to ensure realistic and effective solutions. According to Akhtar et al. [8], many researchers preferred non-Newtonian boundary layer models to study arterial blood flows, as these models provide a more accurate representation of hemodynamics. Recent studies have also reported on blood nanofluid boundary layer flows. Akhtar et al. [8] simulated blood flow within a symmetrically stenosed artery using the non-Newtonian Casson model, suggesting that their findings are crucial for surgical considerations, including assessing stenosis shape, location, and formation. McCash et al. [9] numerically explored the entropy analysis of the peristaltic flow of a Cu-Ag/water hybrid nanofluid within an elliptical duct with sinusoidal progressing boundaries. Tripathi et al. [10] presented a theoretical and numerical evaluation of unsteady blood flow in a diseased artery featuring irregular stenosis, focusing on drug delivery applications for blood vessels using an Ag-gold/blood hybrid nanofluid boundary layer model. Sharma et al. [11] examined the impact of the Au-Al₂O₃/blood hybrid nanoliquid on the hemodynamic properties of unsteady blood flow in a curved artery with stenosis and aneurysm. They concluded that the chosen hybrid nanomaterials can modulate blood velocity and temperature, enabling surgeons to adjust them as required.

Sutterby liquid is crucial in the polymer industry, making it one of the most frequently discussed non-Newtonian fluids due to its rheological features [12]. This non-Newtonian fluid model characterizes the behavior of pseudoplastic substances. To date, researchers have presented their findings using both analytical and numerical methods across various geometries to evaluate the heat energy efficiency of non-Newtonian Sutterby hybrid nanofluid flows. Waqas et al. [13] studied SiO₂-SWCNT/EG and MoS₂-MWCNT/EG hybrid nanofluid boundary layer flows in the three-dimensional Sutterby model over a stretchy surface affected by thermal convection, radiation, and heat melting. They observed that the temperature and velocity profiles decrease when larger melting parameter values are applied. Al-Mughanam et al. [14] numerically examined the characteristics of mono, duo, and tri-nanoparticles suspended in the Sutterby fluid model using the Finite Element Method (FEM). They noted moderate values of thermal memory effects in the hybrid nanofluid compared to other types of nanofluids under consideration. Bouslimi et al. [15] discussed the heat transport efficiency of Sutterby mono and hybrid nanofluid flows past a slippery hot surface, while Jamshed et al. [16] found in their study that Sutterby nanofluids using hybrid Copper-Sodium-Alginate (Cu-SA) and Gold-Sodium-Alginate (Au-SA) nanoparticles enhance the rate of heat transfer in the Parabolic Trough Solar Collector (PSTC). These studies highlight that non-Newtonian Sutterby fluids have found use in various applications, including lubrication and drilling operations.

The importance of nanoliquids by focusing on the solid nanoparticles’ volume fraction, where the thermophysical properties of both the base fluid and nanoparticles are heterogeneously correlated, was initially examined by Tiwari et al. [17]. Numerous studies on the Tiwari and Das hybrid nanofluid models are now available in the literature. Dinarvand et al. [18] analytically explored the Cu-Ag/water hybrid nanoliquid model developed based on the Tiwari–Das framework near a vertically permeable circular channel. A subsequent study by Dinarvand et al. [19] found that the Tiwari-Das Falkner-Skan model for TiO₂-CuO/water hybrid nanofluids outperforms mono-nanofluids regarding heat flux. Additionally, Ramzan et al. [20] utilized the Tiwari-Das model for SiO₂-TiO₂/water hybrid nanofluid flows through a rotary channel influenced by the Hall current. They indicated that the hybrid nanofluid flow is superior to the performance of mono-nanofluid systems in solar thermal applications. Alwawi et al. [21] developed the Tiwari-Das mathematical model to simulate the behavior of Williamson hybrid nanoliquid flows over a cylinder. They reported that silver-aluminum oxide nanoparticles demonstrate superiority in enhancing the velocity and energy transfer of the base fluid. Furthermore, Saranya et al. [22] examined the thermal behavior of the Blasius-Sakiadis Tiwari-Das flow for water-based ternary hybrid nanofluids, considering the effect of nanoparticle shape.

The attention focused on selecting various types of graphene-based solid nanoparticles for producing single and hybrid nanoliquids offers significant advantages for technological and scientific approaches. Graphene-based nanoparticles provide excellent thermal conductivity and stability and serve as flexible transporters with minimal corrosion and erosion [23]. Moreover, because graphene-based materials exhibit superior electrical conductivity, high chemical stability, and exceptional mechanical behavior, they are efficiently utilized in supercapacitors and other energy storage devices [24]. Graphene-based materials have also demonstrated vast applicability in the medical field for applications such as cancer therapy and diagnosis, sensing and imaging, tissue regeneration, and drug delivery [25]. Mehrali et al. [26] reported that graphene-magnetite hybrid nanoparticles increase the fluid thermal conductivity by approximately 11%. In contrast, Sadeghinezhad et al. [27] observed that graphene nanoparticles offer higher stability and a surface area thousands of times greater than other nanoparticles. Additionally, Purbia et al. [28] reported that the heat flux increases by about 32% at a 0.1% concentration of graphene nanomaterials, attributed to the enhanced thermal conductivity and Reynolds number of the conducting solid nanomaterials. Bouslimi et al. [15] also found that the thermal transmission rate of the Sutterby hybrid Cu-GO/engine oil nanoliquid surpasses that of the mono Cu/engine oil nanoliquid.

Limited studies on the Sutterby liquid model over a cylinder inspired the current research to further investigate the heat flux efficiency of selected hybrid nanofluids. The examination of graphene, graphene oxides (GO), and graphene nanoplatelets (GNP) as potential nanomaterials in fluid mechanics remains infrequent despite their exceptional combination of mechanical and electrical properties. This research is the first to analyze graphene, GO, and GNP nanoparticles dispersed in the radiative Cu/blood mono nanofluid to form various hybrid mixtures around a slanted permeable cylinder in the existing literature. The impacts of thermal radiation and the heat source are also considered in this study. MATLAB’s bvp4c code is utilized to address the transformed boundary value problems derived from the primary set of partial differential equations (PDEs). Comprehensive results are validated and compared, and the effects of specific parameters on the hybrid Sutterby non-Newtonian nanofluids in terms of non-dimensional velocity and temperature distributions, reduced skin friction value, and reduced Nusselt number are thoroughly investigated. The findings are then presented in tables and graphs in the final section of this research.

Accordingly, the contributions of this research are outlined as follows: 1.

This study represents the first exploration of the non-Newtonian Sutterby Tiwari-Das model using blood as the base fluid with hybrid nanoparticles, while previous research focused on other conventional base fluids [15,16].

In the current research, a non-Newtonian Sutterby fluid model over a leaning permeable cylinder is given due consideration. The stress tensor is specified as [15]: (1)T=−Ip+S,where I and p express identity-tensor and pressure, respectively, while S implies an additional stress-tensor, which is defined as follows: (2)S=A 1[sinh−1⁡(γ˙E)γ˙E]χμ0,with E and μ0 are designated as material time-constant and zero-shear rate of viscosity, respectively. The first term of Eq. (1) indicates the element of viscoelasticity. Moreover, the fluid reflects the Newtonian behavior when χ=0, the fluid becomes pseudoplastic (shear-thinning) when χ > 0, and the fluid serves as dilatant (shear-thickening) when χ < 0. Accordingly, the Rivlin-Ericksen tensor of first-order, A1  and the second invariant strain tensor, γ˙ are expressed as follows: (3)A 1=(grad V)+(grad V)T, (4)γ˙=tr(A 1)22.

The viscosity of blood varies with shear rate and is determined by several factors, such as the viscosity of plasma, blood cell distribution, and the mechanical properties of the blood cells. Due to their high concentration and distinct mechanical properties, most non-Newtonian effects originate from red blood cells. As blood exhibits non-Newtonian properties of shear-thinning and viscoelasticity, the present research uses the Sutterby model to represent a steady, incompressible, laminar, non-Newtonian blood fluid flow.

The base blood fluid is initially mixed with copper nanoparticles to form Cu/blood mono nanofluids using the Tiwari-Das hybrid nanofluid model. Subsequently, hybrid nanofluids are fabricated by dispersing three types of selected nanoparticles (graphene, GO, and GNP). Table 1 presents the existing models of mono and hybrid nanofluids’ thermophysical properties [18]. Similarly, quantities from references [12,18,29,30] for the base fluid and selected nanoparticles for this study are provided in Table 2. The velocity of the mainstream flow over the cylindrical coordinates  (x, r) is assumed to be U(x)=U0(x/l). The thermal radiation and source of heating effect are also considered. The thermal radiation and heating effects are considered. The geometry of the hybrid nanofluid flow over a slanted permeable cylinder with radius R is depicted in Fig. 1.

The governed PDEs of continuity, momentum, and energy, as referenced in [15,18,29] with the corresponding boundary conditions (BCs), are given as: (5)∂u∂x(r)+∂v∂r(r)=0, (6)∂u∂x(u)+∂u∂r(v)=νhnf12∂2u∂r2(1−mb22(∂u∂r)2)+σB02ρhnfu+(ρβT)hnfρhnf(T−T∞)gcosΘ, (7)(ρcp)hnf(∂T∂x(u)+∂T∂r(v))=khnf(∂2T∂r2+1r∂T∂r)+16σ∗T∞ 33 k∗∂2T∂r2+Q∗(T−T∞), (8)u=U(x),   v=Hw∗,   T=Tw at r=R, u→0,   T→T∞ as r→∞, },

For these, two cases of suction (Hw∗<0)  and injection (Hw∗>0) are considered. Subsequent similarity transformations are presented as follows: (9)u=U0(xl)f′(η),  v=−Rr(U0vfl)12f(η), η=r2−R22R(U0vfl)12,ψ=(U0vf x2l)12Rf(η), θ(η)=T−T∞Tw −T∞},with ψ as the stream function, is characterized in u=r−1(∂ψ/∂r) and v=−r−1(∂ψ/∂r).

By substituting Eq. (9) into Eqs. (6) and (7), a dimensionless system of nonlinear ODEs is derived as below: (10)2G0G1((1+2γη)f‴−12(1+2γη)2ς f″2f‴)+MG1f′G2G1λcosΘ−f′2+f″f=0, (11)(G4+Rd)(1+2γη)θ″+(2G4+Rd)γθ′+G3Pr f θ′+PrQθ=0,where G0=(1−ϕ1)2.5(1−ϕ2)2.5,G1=(1−(ϕ1+ϕ2))+ϕ1ρs1ρf+ϕ2ρs2ρf,G2=(1−(ϕ1+ϕ2))+ϕ1(ρβT)s1(ρβT)f+ϕ2(ρβT)s2(ρβT)f, (12)G3=(1−(ϕ1+ϕ2))+ϕ1(ρcp)s1(ρcp)f+ϕ2(ρcp)s2(ρcp)f,G4 =[2κf+(ϕ1κs1+ϕ2κs2)ϕ1+ϕ2+2(ϕ1κs1+ϕ2κs2)−2κf(ϕ1+ϕ2)2κf+(ϕ1κs1+ϕ2κs2)ϕ1+ϕ2+(ϕ1κs1+ϕ2κs2)−κf(ϕ1+ϕ2)],

These equations depend on the BCs: (13)f(η)=Hw, f′(η)=1, θ(η)=1 at η=0, f′(η)→0, θ(η)→ 0 as η→∞.},

f and θ are individual functions related to the dimensionless velocity and temperature profiles for the examined hybrid nanoliquids, and primes denote differentiation with respect to η.

The selected parameters used in this problem are defined mathematically as follows: γ=1R(vflU0)12, ς= mb2U03x2vfl3, M=σB02lρfU0, λ=GrxRex2, Pr=(μcp)fκf, (14)Rd=4σ∗T∞3κfk∗, Hw=−rR(lvfU0)12Hw∗ , Q=Q∗U0(ρCp)f, Re=U0lvf.

The skin friction, Cf and Nusselt number, Nux are substantial quantities describing the fluid flow and are defined in [10]. (15)Cf =τwρfU02,  Nux=xqwκf(Tw−T∞),where τw and qw are described as: (16)τw=μhnf(∂u∂r+mb23(∂u∂r)3)r=R, qw=−κhnf(1+16σ∗T∞33k∗νf(ρCp)f∂T∂r)r=R.

When Eq. (9) is substituted into Eq. (14), the resulting reduced skin friction and reduced Nusselt number are obtained: (17)Cf Re 12=1G0(f″(η)+ς3f″(η)3), NuxRe−12=−κhnfκf(1+Rd)θ′η.

In order to address the mathematical model for this problem, a robust solution technique is needed to ensure the accuracy and reliability of the outcomes of the controlling PDEs. Therefore, the numerical procedure for this non-Newtonian Sutterby hybrid nanoliquid flow of Cu-blood with chosen graphene, GO, or GNP over a slanted permeable cylinder is conducted using the bvp4c package in MATLAB. The bvp4c tool, derived from the finite difference technique, uses the collocation process in the Lobatto IIIa [32] formula. Additionally, the package implements a derivative scheme in the form of f′ based on the initial solution estimates and BCs [33]. This approach offers a straightforward algorithm with reduced cost and high computational speed compared to other methods. Furthermore, many researchers have utilized this code, validating it as an effective solution for various mathematical and engineering challenges.

Given the variables: (18)f=F(1), f′=∂F(1)∂η=F(2), f″=∂F(2)∂η=F(3),f‴=∂F(3)∂η,θ=F(4),θ′=∂F(4)∂η=F(5),θ″=∂F(5)∂η,},a shooting approach is employed to reformulate the nonlinear Eqs. (10) and (11) with the BCs (13). The equations are decreased into the first-order DEs as follows: (19)F′(3)=12((1+2γη)−12(1+2γη)2ςF(3)2)(G1G0)[F(2)2−F(3)F(1)−MG1F(2)−G2G1λcosΘ], (20)F′(5)=−1(G4+Rd)(1+2γη)[(2G4+Rd)γF(5)+G3PrF(1)F(5)+PrQF(4)].

Thus, the boundary conditions are defined as follows: (21)F(1)=Hw, F(2)=1, F(4)=1 at η=0, F(2)→0,  F(4)→0 as η→∞.}.

F(3) and F(5) are assumed to have initial values of 0 and the preferred limits for η range from 0 to 10. In addition, the problem is also solved with a residual tolerance of 10⁻⁶.

The range of selected nanoparticles’ volume fraction for ϕ1 and ϕ2 is from 0 to 0.04, which is simulated for the current problem. All calculations consider the specific shape factor of the nanoparticles, as listed in Table 2. Moreover, the Prandtl number, Pr = 19.4049 [12] is utilized to represent the blood fluid. The distributions of velocity, temperature, reduced skin friction value, and reduced Nusselt number for various parameters, including curvature parameter γ, Sutterby fluid parameter ζ, Hartmann number M, mixed convection λ, angle of inclination Θ, thermal radiation Rd, wall permeability parameter Hw and local source of heating parameter Q are presented in figures and tables for clarity. The applied range of parameter values is given in Table 3.

This study examines the effects of thermal radiation and heating sources on non-Newtonian Sutterby hybrid nanofluids consisting of various types of graphene, GO, and GNP with Cu-blood over a slanted permeable cylinder. Initially, the governing PDEs of the fluid model are transformed into nonlinear ordinary DEs using analogous transformational terms. Subsequently, they are addressed with the bvp4c scheme in MATLAB to obtain numerical solutions. The effects of other pertinent parameters on the Sutterby non-Newtonian blood nanofluids are also assessed and presented in figures and tables. The results are as follows:

The velocity distributions increase with increasing values of γ and M, but decrease for higher values of ϕ2, ζ, λ and Hw.

Several potential applications arise from these research findings. For example, guidance for evaluating occupational and public health risks related to radiation and electromagnetic field exposure can be based on blood studies in rats. Moreover, the nanoparticles selected for this study can be particularly effective in medical treatments such as cancer therapy, anti-infection measures, and drug delivery. Hence, they promise to enhance medical systems, equipment, and devices. Nevertheless, further research in this domain must address the study’s limitations and comprehensively meet industrial objectives and practical requirements.