The curvilinear coordinates (r, s) are employed for our analysis, where r represents the radial distance from the center O of the curved geometry, and s signifies the arc length component aligned with the flow direction. The surface is stretched with velocity (u_w(s) = as) along the s-direction, as displayed in Figure 1. An induced magnetic field M is also taken into account, with M₁ and M₂ denoting the parallel and normal components, respectively.

In the analysis of the homogeneous and heterogeneous reactions, two chemical species, denoted A and B, are taken into account. The homogeneous reaction, following cubic autocatalysis, is expressed as

A + 2B → 3B, rate = τ_cC_{1 C 2 2} ,

while the heterogeneous reaction, occurring on the catalyst surface can be described as

A → B, rate = τ_sC₁.

Here τ c and τ s are the rate constants, while the concentrations of chemical species A and B are denoted as C₁ and C₂, respectively.

Under the above assumptions and in the presence of thermal radiation according to Rosseland’s approximation [22], ignoring free charges and the displacement currents, the governing boundary layer equations for the flow, induced magnetic field, energy and concentration equations in curvilinear coordinates can be stated as [23]:

∂ ∂ r { ( r + R ) v } + R ∂ u ∂ s = 0 , (2.1)

u 2 r + R = 1 ρ h n f ∂ p ∂ r , (2.2)

v ∂ u ∂ r + R u r + R ∂ u ∂ s + u v r + R = − 1 ρ h n f R r + R ∂ p ∂ s + μ h n f ρ h n f ( ∂ 2 u ∂ r 2 + 1 r + R ∂ u ∂ r − u ( r + R ) 2 )           + μ e f 4 π ρ h n f ( R r + R M 1 ∂ M 1 ∂ s + M 1 M 2 r + R + M 2 ∂ M 1 ∂ r ) , (2.3)

∂ ∂ r { ( r + R ) M 2 } + R ∂ M 1 ∂ s = 0 , (2.4)

R r + R u ∂ M 1 ∂ s + ν ∂ M 1 ∂ r + u M 2 r + R − [ R r + R M 1 ∂ u ∂ s + M 2 ∂ u ∂ r + M 1 ν r + R ] = ζ ( ∂ 2 M 1 ∂ r 2 + 1 r + R ∂ M 1 ∂ r − M 1 ( r + R ) 2 ) , (2.5)

( R r + R u ∂ T ∂ s ) + ν ∂ T ∂ r = ( κ h n f ( ρ C p ) h n f + 16 σ * T ∞ 3 3 K * ( ρ C p ) h n f ) ( ∂ 2 T ∂ r 2 + 1 r + R ∂ T ∂ r )     + Q ( ρ C p ) h n f ( T − T ∞ ) , (2.6)

R r + R u ∂ C 1 ∂ s + v ∂ C 1 ∂ r = D A { ∂ 2 C 1 ∂ r 2 + 1 r + R ∂ C 1 ∂ r } − τ c C 1 C 2 2 (2.7)

R r + R u ∂ C 2 ∂ s + v ∂ C 2 ∂ r = D B { ∂ 2 C 2 ∂ r 2 + 1 r + R ∂ C 2 ∂ r } + τ c C 1 C 2 2 (2.8)

with the following boundary conditions

u = u w ( s ) = a s , (2.9)

v = − s 0 − h 0 s g n ( s 0 ) μ e f 4 π a 2 s 2 ρ h n f M 1 2 ( s , 0 ) , (2.10)

T = T w ,   M 2 = 0 ,   ∂ M 1 ∂ r = 0 , (2.11)

D A ∂ C 1 ∂ r = τ s C 1 ,   D B ∂ C 2 ∂ r = − τ s C 1     at   r = 0 , (2.12)

u → 0 ,   ∂ u ∂ r → 0 ,   T → T ∞ ,     as   r → ∞ (2.13)

M 1 → M e ( s ) ,   C 1 → C 0 ,   C 2 → 0     as   r → ∞ . (2.14)

Taking into account the profound influence of the magnetic response boundary on the alteration of flow dynamics, the boundary condition (2.10) is designated as the magnetically responsive condition. In the event that h₀ equals zero, this condition reduces to the conventional velocity boundary condition, which encompasses both suction phenomena.

Considering the distinct geometries of the selected nanoparticles, the expressions for thermal-physical properties of hybrid nanofluids are listed in Table 1.

To simplify the analysis, we apply the following similarity transformation:

η = a ϑ f r ,     u = a s f ′ ( η ) ,     ν = − R r + R a ϑ f f ( η ) , (2.15)

p = ρ f a 2 s 2 P ( η ) ,     M 1 = M 0 s g ′ ( η ) , (2.16)

M 2 = − M 0 R r + R ϑ f a g ( η ) ,     θ ( η ) = T − T ∞ T w − T ∞ , (2.17)

C 1 = C 0 G ( η ) ,     C 2 = C 0 H ( η ) . (2.18)

The conservation equations for mass and the magnetic field, namely Equations (2.1) and (2.4), are satisfied identically under the previously mentioned transformation. Consequently, the governing equations are reduced to the following system of nonlinear coupled ordinary differential equations:

P ′ ( η ) = A 2 ( f ′ ) 2 η + k , (2.19)

1 A 2 ( 2 k η + k ) P = A 1 A 2 ( f ‴ + 1 η + k f ″ − 1 ( η + k ) 2 f ′ ) − k η + k ( f ′ 2 − f f ″ − f f ′ η + k ) + β A 2 k η + k ( g ′ 2 − g g ″ − g g ′ η + k ) , (2.20)

P r M k η + k ( f ″ g − f g ″ ) − P r M k ( η + k ) 2 ( f ′ g − f g ′ ) g ‴ + g ″ η + k − g ′ ( η + k ) 2 , (2.21)

A 4 P r A 3 ( 1 + R d A 4 ) ( θ ″ + θ ′ η + k ) + D A 3 θ = − k η + k f θ ′ , (2.22)

1 S c ( G ″ + G ′ η + k ) + k η + k f G ′ − τ 1 G H 2 = 0 , (2.23)

δ S c ( H ″ + H ′ η + k ) + k η + k f H ′ + τ 1 G H 2 = 0 , (2.24)

subject to the corresponding boundary conditions

f ( 0 ) = S + s g n ( S ) h β ( g ′ ( 0 ) ) 2 ,     f ′ ( 0 ) = 1 , (2.25)

θ ( 0 ) = 1 ,     g ( 0 ) = 0 ,     g ″ ( 0 ) = 0 , (2.26)

G ′ ( 0 ) = τ 2 G ( 0 ) ,     δ H ′ ( 0 ) = − τ 2 G ( 0 ) , (2.27)

f ′ ( ∞ ) → 0 ,     f ″ ( ∞ ) → 0 ,     θ ( ∞ ) → 0 , (2.28)

g ′ ( ∞ ) → 1 ,     G ( ∞ ) → 1 ,     H ( ∞ ) → 0. (2.29)

Here, the dimensionless parameters are

P r = ϑ f ( ρ c p ) f / κ f ,     R d = 16 σ * T ∞ 3 / 3 K * κ f ,

k = R a ϑ f ,     β = μ e f 4 π ρ f ( M 0 a ) 2 ,     P r M = ϑ f ζ ,     D = Q ( ρ C p ) f a ,     S c = ϑ f D A ,

τ 1 = τ c C 0 2 a ,     τ 2 = τ s D A ϑ f a ,     δ = D B D A ,     S = s 0 a ϑ f ,     h = h 0 A 2 a ϑ f .

It’s worth noting that the diffusion coefficients D_A and D_B here are equal for both chemical species, i.e. δ = 1 and, thus

G ( η ) + H ( η ) = 1 , (2.30)

Then, Equations (2.23) and (2.24) yield

1 S c ( G ″ + G ′ η + k ) + k η + k f G ′ − τ 1 G ( 1 − G ) 2 = 0 , (2.31)

with the boundary conditions

G ′ ( 0 ) = τ 2 G ( 0 ) ,     G ( ∞ ) → 1. (2.32)

Furthermore, by differentiating Equation (2.20) and using Equation (2.19) to eliminate pressure from the momentum equation, we ultimately arrive at the task of solving the following system of nonlinear coupled ordinary differential equations:

f i ν + 2 f ‴ η + k − f ″ ( η + k ) 2 + f ′ ( η + k ) 3 + A 2 A 1 [ ( k η + k ) ( f f ‴ − f ′ f ″ ) + ( k ( η + k ) 2 ) ( f f ″ − f ′ 2 ) − ( k ( η + k ) 3 ) f f ′ ] + β A 1 k ( η + k ) ( g ′ g ″ − g g ‴ + g g ′ ( η + k ) 2 − g ' 2 + g g ″ η + k ) = 0. (2.33)

P r M k η + k ( f ″ g − f g ″ ) − P r M k ( η + k ) 2 ( f ′ g − f g ′ ) = g ‴ + g ″ η + k − g ′ ( η + k ) 2 , (2.34)

A 4 P r A 3 ( 1 + R d A 4 ) ( θ ″ + θ ′ η + k ) + D A 3 θ = − k η + k f θ ′ , (2.35)

1 S c ( G ″ + G ′ η + k ) + k η + k f G ′ − τ 1 G ( 1 − G ) 2 = 0 , (2.36)

incorporating the boundary conditions

f ( 0 ) = S + s g n ( S ) h β ( g ′ ( 0 ) ) 2 ,     f ′ ( 0 ) = 1 , (2.37)

θ ( 0 ) = 1 ,     g ( 0 ) = 0 ,     g ″ ( 0 ) = 0 ,     G ′ ( 0 ) = τ 2 G ( 0 ) , (2.38)

f ′ ( ∞ ) → 0 ,     f ″ ( ∞ ) → 0 ,     θ ( ∞ ) → 0 , (2.39)

g ′ ( ∞ ) → 1 ,     G ( ∞ ) → 1 ,     H ( ∞ ) → 0. (2.40)

The critical measurable parameters of interest, namely the surface shear stress C_f and the Nusselt number Nu, which quantifies the heat flux rate at the surface, are expressed as follows:

C f = μ h n f ( ∂ u ∂ r − u r + R ) r = 0 ρ h n f ( a s ) 2 ,     N u = s ( − κ h n f ∂ T ∂ r | r = 0 − 16 σ * T ∞ 3 3 K * ∂ T ∂ r | r = 0 ) κ f ( T w − T ∞ ) . (2.34)

The expressions given in Equation (2.34), when transformations from Equations (2.15)-(2.17) are applied, can be rewritten as

C f R e s 1 / 2 = A 1 A 2 ( f ″ ( 0 ) − f ′ ( 0 ) k ) . (2.35)

N u R e s − 1 / 2 = − ( A 4 + R d ) θ ′ ( 0 ) . (2.36)

where R e s = a s 2 / ϑ f is the local Reynolds number.

Figure 2 presents the velocity profiles as the curvature radius parameter k varies. It is evident that with the increase of k, both the nanofluid and hybrid nanofluid velocity profiles exhibit an upward trend. This trend is primarily due to the fact that as the parameter k increases, the centrifugal force acting on the fluid during flow also intensifies, thereby promoting an increase in velocity. The influence of parameters β and h on velocity is depicted in Figure 3 and Figure 4. It is observed that in both hybrid nanofluid and nanofluid scenarios, the velocity decreases as β and h increase. Notably, the hybrid nanofluid is more responsive to changes in β, while the nanofluid is more sensitive to variations in h.

Figure 5 displays the temperature profiles as k varies. It is evident that with an increase in k, the temperature profiles for both nanofluid and hybrid nanofluid show a decreasing trend. Notably, the sensitivity of both nanofluids to this parameter is almost identical. Figure 6 delineates the influence of parameter Pr_M on temperature profiles. With the augmentation of Pr_M, the temperature of both nanofluid and hybrid nanofluid increases. Moreover, the hybrid nanofluid demonstrates greater effectiveness.

As the parameter k increases, the trend of concentration for both nanofluids and hybrid nanofluids exhibits an inverse relationship with temperature, as shown in Figure 7. Significantly, a higher value of k corresponds to greater concentrations in these fluids.

Figure 8 shows that higher homogeneous parameter values τ 1 lead to lower concentrations due to reactant consumption. Figure 9 mirrors this trend for the heterogeneous parameter τ 2 , where increasing τ 2 also reduces concentration, aligning with typical reaction dynamics.

Figure 10 illustrates the effect of the dimensionless curvature radius parameter k on the surface friction coefficient for a hybrid nanofluid. It is evident that increasing the value of k (i.e., reducing the dimensionless curvature) leads to an increase in the magnitude of surface friction on the sheet. Consequently, on the surface of the stretching sheet, more force is required to drag fluid over a flat surface compared to a curved one. It is clear from Figure 11 that the magnitude of the skin friction coefficient increases for hybrid nanofluid with enhanced suction s, which aligns with physical fact. Furthermore, it is found in Figure 10 and Figure 11 that the variation of D and Rd do not affect the surface friction force, which is consistent with Equations (2.33)-(2.36), further confirming the validity of the results.

Figure 12 demonstrates that an increase in the value of k for hybrid nanofluid leads to a slight increase in the Nusselt number. On surfaces with lower curvature, the boundary layer develops more readily because the centrifugal forces acting on the fluid flow are reduced, which in turn enhances heat transfer within the boundary layer. However, an opposite trend is observed where the Nusselt number decreases with the increase of D. Figure 13 reveals that an augment in Rd and s leads to an increase in the Nusselt number for hybrid nanofluid, indicating a significant enhancement in convective heat transfer efficiency.