The aim of this work was to computationally model and numerically analyse the behaviours of an unsteady magnetised micropolar blood-based HNF containing gold and copper nanoparticles as it flows over a curved surface. The Powell–Eyring fluid model was used in this study, and consideration was given to the viscous dissipation and Joule heating processes. The model objective was to contrast the performances of the HNF models. By investigating the behaviours of Ag and TiO₂ nanoparticles dispersed in blood, our aim was to understand the intricate phenomenon of hybridisation. Artificial bacteria swim in an interstitial nanoliquid that is heated by Joule heating and variable thermal conductivity within a biotic cell. The effects of velocity slip and thermal jump were considered while analysing the curved surface. The basic flow assumptions were considered in the development of a mathematical framework. The numerical technique used to design the model verified previous analyses, demonstrating good agreement for a specific circumstance. The investigations delve into the impacts of various parameters, such as the Prandtl number, volume fraction of the nanoparticles, and blood flow parameters. The outcomes were meticulously elucidated through graphical representations and tabular summaries.

This study is unique and has a novel methodology because of the following:

• The integration of the different elements involved in the flow phenomenon has multiple implications for biomedical and technical applications.

We considered a two-dimensional unsteady boundary layer bioconvection flow of the Eyring–Powell micropolar blood-based magnetised HNF (containing Ag and TiO₂ nanoparticles) over a porous curved stretching surface. The curvilinear coordinate system was chosen, where r , s are the radial components, s is the length of the arc, and stretching velocity in the s direction is u = U w = a s 1 − c t , and r is perpendicular to the tangent. The variable magnetic field intensity B t operates normal to the surface. To precisely describe the dynamics of bacterial density ρ n and nutrient concentration n , the reaction–diffusion equations are utilised.

• Unsteady, incompressible, and 2D boundary layer bioconvection flows of the Eyring–Powell micropolar blood-based magnetised hybrid NF (containing Ag and TiO₂ nanoparticles) over a porous curved stretching surface are considered.

Figure 1 illustrates the coordinate system, velocity field, and relevant elements of the flow problem geometry.

The constitutive equations characterising micropolar behaviour are constructed in vectorial form using new kinematic properties like the gyration tensor and micro-inertia moment tensor, in addition to the body moments, stress moments, and micro-stress. To take into consideration the magnetic effects, these equations are further enhanced, as follows (Ashraf and Wehgal, 2012; Eringen, 1966a): ∇ . V = 0 , (1) ρ f ∂ V ∂ t + V . ∇ V = − ∇ p + μ f + k * ∇ 2 V + k * ∇ × v + J × B (2) ρ f j ∂ v ∂ t + v . ∇ v = α + β + γ ∇ . ∇ v − γ ∇ × ∇ × v + k * ∇ × v − 2 k v , (3)

Here, the velocity vector is represented by V = u , v , w , pressure is indicated by p , microrotation is shown by v , and current density is indicated by J = σ V × B , with σ indicating electrical conductivity and B indicating the magnetic field. Moreover, the fluid density is represented by ρ f , dynamic viscosity is given by μ f , vortex viscosity is given by k * , microinertia density is given by j , and gyroviscosity coefficients are given by α , β , and γ . It is noted that there are restrictions on α , β , γ , μ f , and k * . k * ≥ 0 , 2 μ f + k * ≥ 0 , 3 α + β + γ ≥ 0 , γ ≥ β .

The constitutive equations for the micropolar fluids are described in greater detail in Eringen (1966a). The equations for thermal energy and diffusion of nanoparticles, which are based on the Buongiorno model and obey the laws of Fourier and Fick, are described in Abbas et al. (2019). ρ f c f ∂ T ∂ t + V . ∇ T = k f ∇ 2 T , (4)

where the temperature is represented by T , specific heat of the nanofluid is given by c f , and thermal conductivity is given by k f .

Gyrotactic microorganisms are then taken into consideration according to Uddin et al. (2016). ∂ n ∂ t + ∇ J 1 = 0 , (5)

where J 1 = n V + n V ∼ − D n ∇ n

The flow of microorganisms caused by fluid convection is represented by J 1 , density of the gyrotactic microorganisms is indicated by n , and diffusivity of the microorganisms is given by D n . The cell swimming velocity is represented by the velocity vector V ∼ = 0 , 0 , w ^ , where the velocity component along the z axis is represented by w ^ = b   w c   / ∆ C ∇ C . The maximal cell swimming speed is indicated by w _c , and the chemotaxis constant is represented by b.

Using cylindrical coordinates, the governing Eqs. (1–6) (Hashmi et al., 2012; Ashraf and Wehgal, 2012) are simplified by taking into account a magnetic field of the form B = 0 , 0 , B t , axisymmetric flow, and a microrotation vector normal to the disc surface v = 0 , N , 0 .

The Eyring–Powell model was introduced in 1994 for defining shear in a non-Newtonian flow. Here, the stress–strain relationship is shown using the non-Newtonian Eyring–Powell model by the strain–stress tensor Τ = Ρ Ι + τ , where Ι denotes the identity stress tensor, Ρ signifies the pressure, and τ is the extra stress tensor satisfying the following relation (Rooman et al., 2022): τ = μ ∇ V + 1 β 1 sinh − 1 1 c 1 ∇ V , (6)

where sinh − 1 1 c 1 ∇ V ≈ 1 c 1 ∇ V − 1 6 1 c 1 ∇ V 3 , 1 c 1 ∇ V ≪ 1

The Navier–Stokes flow is a form of fluid motion in which the typical dimension and flow rotation speed, U w , are both very small. Because a small Reynolds number is used to eliminate the inertial term in the Navier–Stokes equation, the Stokes estimate is frequently employed to explain the motion of magnetic microorganisms. Therefore, the fluid speed of the swimming magnetotactic bacteria is determined by the Navier–Stokes and continuity equations. The relevant governing equations to investigate the aforementioned fluid flow are as follows (Abbas et al., 2022; Shi et al., 2021): r ¯ ∂ v ∂ r + r + R ∂ u ∂ s = 0 , (7) u 2 r ¯ − 1 ρ h n f ∂ p ∂ r = 0 , (8) ∂ u ∂ t + ν ∂ u ∂ r + R r ¯ u ∂ u ∂ s + u ν = − 1 ρ h n f R r ¯ ∂ p ∂ s + 1 ρ h n f μ h n f + k * + 1 β 1 c 1 ∂ ∂ r ∂ u ∂ r + u r ¯ − 1 6 β 1 c 1 3 ∂ ∂ r ∂ u ∂ r + u r ¯ 3 − 1 ρ h n f σ h n f B 2 t u − μ h n f k 1 u − k * ∂ N ∂ r ∂ N ∂ t + v ∂ N ∂ r + R u r ¯ ∂ N ∂ s = − γ ρ h n f j ∂ 2 N ∂ r 2 + 1 r ¯ ∂ N ∂ r − k * ρ h n f j 2 N + ∂ u ∂ r + u r ¯ (9)

where N represents the microrotation velocity. Here, γ * = μ + k * 2 = μ 1 + K 1 2 j , where K 1 = k * μ represents the material parameter, j = 2 v L c e s L is the micro inertia per unit mass, and γ * and J indicate the spin gradient and vortex viscosity, respectively. ρ C p h n f ∂ T ∂ t + ν ∂ T ∂ r + R r ¯ u ∂ T ∂ s = 1 r ¯ ∂ ∂ r r ¯ k h n f T ∂ T ∂ r   + σ h n f B 2 t u 2 , (10) ∂ ρ n ∂ t + v ∂ ρ n ∂ r + R r ¯ u ∂ ρ n ∂ s = D n ∂ 2 ρ n ∂ r 2 + 1 r ¯ ∂ ρ n ∂ r + A n , t ρ n . (11)

Subject to the boundary conditions as per Ma et al. (2019), we get u = U w = a s 1 − c t , v = 0 , T = T w , ρ n = ρ n w , N = − m ∂ u ∂ r , a t   r → 0 , u → 0 , ∂ u ∂ r → 0 , T → T ∞ , ρ n → ρ n ∞ , N → 0 , a s   r → ∞ (12)

Here, r ¯ = r + R . In the present discussion, it is assumed that n > K m , a > 0 , and c ≥ 0 with dimension t i m e − 1 .

Consider the following dimensionless similarity transformation (Ma et al., 2019): η = a ν f 1 − c t r , u = a s 1 − c t f ′ η , p = ρ f a s 1 − c t 2 P η , v = R r ¯ a ν f 1 − c t f η , T = T ∞ + T w − T ∞ θ η , N = a s 1 − c t a ν f 1 − c t g η   ρ n = ρ n ∞ + ρ n w − ρ n ∞ χ η , n = n ∞ + n w − n ∞ ω η (13)

The continuity equation is met by the dimensionless quantity in Eq. (13), and upon pressure elimination, the governing Eqs. (7–12) can be represented as follows: μ h n f μ f + K 1 + α 1 f I V + 1 η + K 2 f ‴ −   f ″ η + K + f ′ η + K 2   −   f ″ + f ′ η + K σ h n f σ f   M + μ h n f μ f β 0   − α 2 f ″ 2 + 2 f ′ f ″ η + K + f ′ 2 η + K 2 f I V + f ″ 2 − 3 f ′ f ″ η + K − f ′ 2 η + K 2 f ″ η + K 2 + 3 f ′ 3 η + K 5 + 2 f ″ + f ′ η + K f ‴ 2 + 2 3 f ″ 2 + 2 f ′ f ″ η + K − f ′ 2 η + K 2 f ‴ η + K − K 1 g ″ + g ′ η + K + ρ h n f ρ f   K   f   f ‴ −   f ′   f ″ η + K + K   f   f ″ −   f ′ 2 η + K 2 −   K f   f ′ η + K 3 − γ η + K   η 2   f ″ + f ′ − γ 2   η f ‴ + 3   f ″ = 0 , (14) μ h n f μ f + K 1 2 g ″ + g ′ η + K + K η + K f g ′ − K η + K f ′ g − K 1 2 g + f ″ + f ′ η + K   − γ 2 η g ′ + 3 g = 0 , (15) k h n f k f     1 + β   θ   θ ″ + θ ′ η + K + β   θ ′ 2 + σ h n f σ f M P r E c f ′ 2 + ρ C P h n f ρ C P f   Pr   K f η + K −   γ   η 2   θ ′ = 0 , (16) χ ″ + χ η + K + L b   K f η + K −   γ   η 2   χ ′ + λ   Ω + χ = 0 , (17)

Similarly, pressure can be expressed as P η = μ h n f μ f + K 1 + α 1   f ‴ + f ″ η + K −   f ′ η + K 2 + 1 2   ρ h n f ρ f f   f ″ −   f ′ 2 + f   f ′ η + K − η + K K   γ   η 2   f ″ + f ′ − η + K 2 K   σ h n f σ f M + ρ h n f ρ f β 0   f ′ + α 2 f ″ + f ′ η + K 2 f ‴ + f ″ η + K −   f ′ η + K 2 , (18)

Subject to boundary conditions f ′   0 = θ   0 = χ   0 = ω   0 = 1 , − m 0 f ″   0 = 0 , g 0 = 0 , = 0 , f ′   ∞ = f ″   ∞ = θ   ∞ = χ   ∞ = ω   ∞ = 0 , g ∞ = 0 . (19)

Figure 4A shows the variations between the porosity parameter β 0 and velocity profile f ′ η . The porosity parameter is significant in various fields and disciplines owing to its influence on a wide range of physical, chemical, and engineering processes. When β 0 is enhanced, the velocity profile declines for the HNF; this is because a porous medium with increasing porosity usually has more empty areas. Although this may seem to improve flow at first glance, it also implies that less solid material is available to provide flow stability and order. A decrease in the fluid’s effective flow area therefore causes the velocity profile to diminish. Figure 4B explains the effect of the unsteady parameter γ on the velocity profile f ′ η . By enhancing the parameter γ , we can reduce the velocity profile f ′ η because an increase in γ indicates greater impacts of the time-dependent variables on the fluid flow. Increased fluid inertia property that tends to resist changes in velocity can result from this. The fluid may thus respond more slowly to changes in the external environment, causing the velocity profile to drop. Figure 4C represents the variation between the curvature factor K and velocity profile f ′ η . Increasing the curvature factor K reduces the velocity profile because the pressure gradient along a curved surface tends to get stronger as the curvature increases. A drop in the velocity profile may occur from this increased pressure gradient creating a greater barrier to flow. Because of higher pressure forces oppose the flow, the fluid has a tendency to slow down as it moves around the curved surface. Figure 4D represents the variation between the magnetic parameter M and velocity profile. As M increases, there is a decay in the velocity profile because the strength of the magnetic field operating on the NF increases with increase in M. The fluid’s magnetic nanoparticles prefer to align under this greater magnetic field, which might increase the effective viscosity of the NF. The velocity profile decreases as a result of the resistance of the increasing viscosity to flow.

Figure 5A presents the variation between the porosity parameter β 0 and temperature profile θ η . When β 0 increases, the temperature also increases because there is usually more space for flow within a porous medium when the porosity parameter increases. This may lead to improved convective heat transfer within the medium via increased fluid velocity and circulation. Figure 5B represents the variation between the magnetic parameter M and temperature profile θ η . An increase in M increases the temperature profile of the HNF because the intensity of the external magnetic field acting on the nanoparticles increases with M . The fluid’s nanoparticles may move and align more significantly as a result of this increased magnetic force. Figure 5C represents the variation between the Eckert number E c and temperature profile θ η . As E c increases, the temperature also increases because the kinetic energy contribution is greater than the enthalpy change when the Eckert number increases. This suggests that fluid motion, as opposed to heat transmission, accounts for a greater percentage of the system’s energy. More kinetic energy can improve fluid circulation and mixing, which can help the fluid transfer heat more effectively. Figure 5D presents the variation of the unsteady parameter γ with the temperature profile θ η . As γ increases, the temperature also increases because the flow parameters will vary more quickly over time if the unsteady parameter γ is larger. Higher temperatures may be seen due to the greater heat dispersion and distribution across the fluid domain brought on by the enhanced fluid mixing. Figure 5E presents the variation between the variable thermal conductivity β and temperature profile θ η . When we increase the variable thermal conductivity β , the temperature of the HNF increases because the variable thermal conductivity implies that the fluid’s capacity to transfer heat changes spatially. Better heat transmission throughout the fluid domain is made possible by the fluid’s greater ability to transmit heat when β increases. The fluid’s temperature may increase as a consequence of this increased heat conduction. Figure 5F presents the variation between the curvature factor K and temperature profile θ η . When we increase the curvature factor K , the temperature of the HNF decreases because the surface area per unit volume often increases with increasing curvature; heat is dispersed from the fluid to the surroundings more efficiently owing to this larger surface area. As a result of improved cooling over the curved surface, the temperature of the HNF drops.

Figure 6A presents the variation between the variable thermal conductivity β and micropolar profile g η . When we increase the variable thermal conductivity β , the micropolar profile of the HNF increases because a rise in β suggests that the thermal conductivity of the material is increasingly temperature-dependent. Higher temperatures thus lead to enhanced thermal conduction, which facilitates heat transmission within the fluid. The micropolar profile may increase as a result of this improved thermal conduction, enabling more effective temperature distribution throughout the fluid. Figure 6B presents the variation between the micropolar parameter K 1 and micropolar profile g η . When we increase the micropolar parameter K 1 , the micropolar profile of the HNF increases because the strength of the fluid’s microstructural effects, such as the microrotation and microdeformation of the fluid constituents, is represented by the micropolar parameter K 1 . These microstructural effects intensify as K 1 increases, thereby increasing the fluid’s degree of micropolar activity. The fluid’s total micropolar profile increases as a result of this enhanced micropolar behaviour. Figure 6C presents the variation between the curvature factor K and micropolar profile g η . When we increase the curvature factor K , the micropolar profile of the HNF increases because more complicated microstructural effects arise within the fluid as a result of the curved surface’s flow patterns being impacted by the increase in curvature K . These effects, which are more noticeable in areas with greater curvature, include microrotation and microdeformation of the fluid components. Consequently, the fluid’s micropolar activity intensifies, which increases the micropolar profile.

Figure 7A presents the variation between the maximum growth rate of the bacteria λ and bacterial density field χ η . When we increase the maximum growth rate of the bacteria λ , the bacterial density field of the HNF increases because the bacteria that have a greater maximal growth rate are believed to multiply more quickly. Consequently, the fluid produces more bacteria in a given amount of time, increasing the total bacterial density field. Figure 7B presents the variation between the bacterial difference density parameter Ω and bacterial density field χ η . When we increase the bacterial difference density parameter Ω , the bacterial density field of the HNF increases because the density difference between the bacteria and surrounding fluid is represented by the bacterial differential density parameter Ω. This suggests that the bacteria become more buoyant in relation to the fluid as Ω increases. The bacteria multiply and spread more broadly in the fluid as a result of this enhanced buoyancy, increasing the total bacterial density field. Figure 7C presents the variation between the bioconvection Lewis number L b and bacterial density field χ η . When we increase the bioconvection Lewis number L b , the bacterial density field of the HNF increases because the transport of nutrients to the bacterial cells is more effective when Lb increases. The increase in total bacterial density field inside the HNF is a result of the increased nutrition availability, which encourages bacterial growth and replication.

Figure 8A shows how the skin-friction coefficient is affected by η , β 0 , M , and K . Changes in the porous structure of the medium significantly affect the frictional forces, as demonstrated by the porosity parameter β 0 that has the greatest impact on skin friction. The magnetic parameter M has less of an impact on skin friction than the porosity parameter   β 0 . This suggests that in terms of frictional forces, the magnetic field modifies flow behaviour to a lesser extent. Although not as much as the porosity parameter β 0 , the unsteady parameter γ also increases skin friction. This parameter shows that the frictional forces are influenced by temporal variations in the flow conditions. Out of all the characteristics examined, the curvature factor K has the least incremental impact on skin friction; this implies that the frictional forces are mostly determined by factors other than the surface geometric curvature, such as porosity, magnetic fields, and flow unsteadiness.

The changes in the Nusselt number N u s R e s − 1 / 2 for different values of β 0 , M , and γ are shown in Figure 8B. Stronger magnetic fields appear to inhibit heat transport, as seen by the continuous decline in the Nusselt number as the magnetic parameter M increases. This behavior is consistent with the findings of other studies, which shows that the increased flow mixing brought about by higher magnetic fields can lower the heat transfer rate. The results therefore emphasise the necessity of considering magnetic fields carefully in heat transfer applications, particularly when aiming to improve or regulate the heat transfer rate. Heat transmission is impacted by flow unsteadiness, as seen by the decreasing Nusselt number as the unsteady parameter γ increases. Typically, unsteady flows show variations in temperature gradients and velocity, which may interfere with the formation of boundary layers and lower the effectiveness of heat transmission. As a result, engineering systems functioning in unstable environments could experience reduced heat transfer efficiency, requiring countermeasures to preserve the intended thermal properties. Further investigation is necessary to fully understand the role of porosity in heat transfer processes involving magnetic fields and unsteady flows, as evidenced by the lack of discernible variations in the Nusselt number with increasing values of the porosity parameter β 0 . This finding contrasts with the significant influences of the magnetic and unsteady parameters.

The fluctuations of the nutrient concentration number for various values of L b , Ω , and λ are shown in Figure 8C. There are substantial associations between these variables, as shown by the correlations between the influencing factors and nutrient concentration density provided. A reduction in the density of the nutrient concentration is observed with increases in the bioconvection Lewis number L b , bacterial density difference Ω , and bacterial maximum growth rate λ . These results have significant ramifications for medical cancer therapies that use artificial bacteria and magnetite nanoparticles. One may control the availability of the nutrients in normal cells and decrease tumour cell usage of such nutrients at the same time by adjusting the growth rate and density differential between the bacteria. Because the healthy cells are fed even as the tumour cells are starved, this focused strategy may increase the effectiveness of cancer therapies.

Table 7 illustrates the impacts of varying values of the magnetic parameter M , porosity parameter   β 0 , unsteady parameter γ , and curvature factor K on skin friction. The findings indicate positive correlations between increasing values of these parameters and heightened skin friction. Notably, the porosity parameter   β 0 exhibits the most pronounced enhancement in skin friction compared to the magnetic parameter M , unsteady parameter γ , and curvature factor K . Conversely, the least incremental effect on skin friction is observed with the curvature factor K as compared to the magnetic parameter M , unsteady parameter γ , and porosity parameter   β 0 .

Table 8 illustrates the trend of the Nusselt number for incremental values of the magnetic parameter M , porosity parameter   β 0 , and unsteady parameter γ . The data indicate consistent decreases in the Nusselt number as the values of the magnetic parameter M and unsteady parameter γ increase. Conversely, there is no discernible variation observed with increasing values of the porosity parameter   β 0 . This suggests that the magnetic parameter M and unsteady parameter γ exert significant influences on the Nusselt number, leading to a decline, while the porosity parameter   β 0 remains relatively unaffected. Table 9 presents the correlation between the nutrient concentration density and three influential factors: bioconvection Lewis number L b , bacterial density difference Ω, and maximum growth rate of the bacteria λ . The analysis reveals that increases in the bioconvection Lewis number, bacterial density difference, and maximum growth rate are correlated with decreases in the nutrient concentration density. These mathematical findings suggest careful regulation of the bacterial density difference and bacterial growth rate in medical cancer treatments involving magnetite nanoparticles and artificial bacteria. This regulation may enhance nutrient availability in the normal cells while reducing nutrient utilisation by the tumour cells. In the context of medical treatment, it is advisable to augment the applied magnetic factors while diminishing the ratio of thermal diffusivity to mass diffusivity; this approach aims to optimise the nutrient consumption in normal cells while minimising that in tumour cells. Table 10 shows comparisons for skin friction values with findings from previous work, and it is seen that these results are in good agreement.