Consider the three-dimensional incompressible, steady, and axisymmetric flow of the HNF (MoS₂-Go/water) and NF (Go/water) between the two spinning infinite disks placed along parallel lines (see Figure 1). The notation r , Θ , z has been used for the cylindrical coordinate system (see Figure 1). The flow is exposed to a magnetic field B _o in the z-direction. As a novelty, to develop the model, the Darcy-Forchheimer law is used for porous medium. In addition, the non-uniform heat source/sink, Cattaneo-Christov double diffusion model, and thermal radiation is used to investigate HTR and MTR. Moreover, the last term in the energy equation (Eq. 5) denotes the non-uniform heat source/sink term q ‴ and it is explained later. The lower disk and upper disk are placed at z = 0 and z = H (see Eq. 7). In the boundary conditions, a ₁ and a ₂ denote the shrinkage rates of the lower and upper disks in the radial direction, respectively. Furthermore, b ₁ and b ₂ are taken as the angular velocities of the lower and upper disks, respectively. The outer surfaces of the disks are assumed to be convectively heated by hot liquids having temperatures T _w (lower disk) and T _H (upper disk), with h ₁ (lower disk) and h ₂ (upper disk) as their heat transfer coefficients. In addition, C _w and C _H denote the NPs concentration at the lower and upper disks, respectively.

Based on the aforesaid points, the flow equations are (see Bhattacharyya et al., 2020; Mabood et al., 2021; Yaseen et al., 2022a):

Continuity equation: ∂ u ∂ r + ∂ w ∂ z + u r = 0 (1)

Momentum equations: ρ h n f u ∂ u ∂ r + w ∂ u ∂ z − v 2 r = − ∂ p ∂ r + μ h n f ∂ 2 u ∂ r 2 + ∂ 2 u ∂ z 2 + 1 r ∂ u ∂ r − u r 2 − σ h n f B 0 2 u − μ h n f k f n u − ρ h n f F k f n u 2 (2) ρ h n f u ∂ v ∂ r + w ∂ v ∂ z + u v r = μ h n f ∂ 2 v ∂ r 2 + ∂ 2 v ∂ z 2 + 1 r ∂ v ∂ r − v r 2 − σ h n f B 0 2 v − μ h n f k f n v − ρ h n f F k f n v 2 (3) ρ h n f w ∂ w ∂ z + u ∂ w ∂ r = − ∂ p ∂ z + μ h n f ∂ 2 w ∂ r 2 + 1 r ∂ w ∂ r + ∂ 2 w ∂ z 2 (4)

Energy equation: u ∂ T ∂ r + w ∂ T ∂ z + γ t ∂ T ∂ r w ∂ u ∂ z + u ∂ u ∂ r + ∂ T ∂ z w ∂ w ∂ z + u ∂ w ∂ r + 2 u w ∂ 2 T ∂ z ∂ r + u 2 ∂ 2 T ∂ r 2 + w 2 ∂ 2 T ∂ z 2 = 1 ρ C p h n f k h n f + 16 σ * T H 3 3 k * ∂ 2 T ∂ r 2 + ∂ 2 T ∂ z 2 + 1 r ∂ T ∂ r + q ‴ ρ C p h n f (5)

Concertation equation: u ∂ C ∂ r + w ∂ C ∂ z + γ c ∂ C ∂ r w ∂ u ∂ z + u ∂ u ∂ r + ∂ C ∂ z w ∂ w ∂ z + u ∂ w ∂ r + 2 u w ∂ 2 C ∂ z ∂ r + u 2 ∂ 2 C ∂ r 2 + w 2 ∂ 2 C ∂ z 2 = D 1 r ∂ C ∂ r + ∂ 2 C ∂ r 2 + ∂ 2 C ∂ z 2 (6)

with Boundary conditions (BCs):

u = a 1 r , v = b 1 r , w = 0 , k h n f ∂ T ∂ z = − h 1 T w − T , C = C w a t z = 0 u = a 2 r , v = b 2 r , w = 0 , k h n f ∂ T ∂ z = − h 2 T − T H , C = C H a t z = H (7) where “ u ,   v ,   w are the velocity components along with r ,   Θ ,   z directions,” respectively, “C is nanoparticles concentration,” “T is temperature,” “p is pressure,” “k ^* is the mean absorption coefficient,” “σ^* is the Stefan-Boltzmann constant,” “F is the Forchheimer coefficient and k _fh is the porous medium permeability,” respectively, “D is diffusion coefficient,” “ γ t is the thermal relaxation time,” and “ γ c is the solutal relaxation time.” Furthermore, “k is thermal conductivity, μ is dynamic viscosity, C p is heat capacity, ρ is density, σ is electrical conductivity.” Moreover, subscript hnf is for hybrid nanofluid, nf is for nanofluid, and f, bf is for fluid. In addition, the last term in the energy Eq. 5 i.e., q ‴ denotes the significance of “non-uniform heat source/sink” and it is demarcated as follows (Yaseen et al., 2022b): q ‴ = k h n f u w T w − T H r v h n f A * d f ∂ ξ + B * θ (8) where u w = b 1 r and the heat source/sink corresponding to space coefficients and corresponding to temperature dependence are signified by constants A * and B * , respectively. As a result, the heat source phenomena is characterized by A * > 0 and B * > 0 , whereas the heat sink phenomena is characterized by A * < 0 and B * < 0 .

This study describes the analysis of HNF and NF. The thermal characteristics of HNF and NF are influenced by the base fluid (water) and NPs; as well as by the volume fraction of MoS₂ and Go NPs, as shown in Table 1 (Khan et al., 2020d). The Shape of MoS₂ and Go NPs is taken as spherical. In this paper, the volume fraction of Go NPs and MoS₂ NPs is denoted by ϕ 1 and ϕ 2 , respectively. Furthermore, s ₁ is used for Go NPs, and s ₂ is used for MoS₂ NPs. The MoS₂ belongs to the transition metal class and Go belongs to the oxide class. The combination of MoS₂ and Go NPs is taken because metal nanoparticles make a nanolayer atop oxide nanoparticles, and it produces a thermal interfacial layer between weak boundaries of the working fluid and hybrid NPs, resulting in thermal conductivity augmentation.

The transformations listed below are utilized to transform the governing equations (Yaseen et al., 2022a): u = r b 1 f ′ η , v = r b 1 g η , w = − 2 H b 1 f η , P = b 1 ρ h n f v h n f P η + r 2 ε 2 H 2 , ξ = z H , θ = T − T H T w − T H , ϕ = C − C H C w − C H (15)

Eq. 1 is satisfied, and Eqs 2, 3, 4, 5, 6, 7 reduce to R e ω 2 ω 1 f ′ 2 − 2 f f ″ − g 2 + ω 3 ω 2 M f ′ + F * f ′ 2 + R e λ f ′ = − ε + f ‴ (16) g ″ − R e λ g = R e ω 2 ω 1 ω 3 ω 2 g M + g F * − 2 g ′ f − f ′ g (17) P ′ + 4 ω 2 ω 1 R e f f ′ + 2 f ″ = 0 (18) ω 5 + R d − ω 4 P r R e τ E f 2 θ ″ + R e ω 5 ω 2 ω 1 A * f ′ + B * θ + ω 4 P r R e 2 f θ ′ − τ E f f ′ θ ′ = 0 (19) ϕ ″ = R e τ c S c f f ′ ϕ ′ + f 2 ϕ ″ − 2 R e S c f ϕ ′ (20)

The BCs are obtained as follows: f η = 0 , f ′ η = k 1 , g η = 1 , θ ′ η = − γ 1 ω 5 1 − θ η , P η = 0 , ϕ η = 1   a t η = 0 f η = 0 , f ′ η = k 2 , g η = Ω , θ ′ η = − γ 2 ω 5 θ η , ϕ η = 0   a t η = 1 (21)

In the aforementioned equations, “ R e = b 1 H 2 v f represents Reynold number, F * = F r k f n represents the inertial coefficient, Pr = v f α f is the Prandtl number, λ = v f b 1 k f n is the porosity parameter, M = σ f B 0 2 ρ f b 1 represents the magnetic field parameter, A * and B * represents heat source/sink parameters, R d = 16 σ * T H 3 3 k f k * represents thermal radiation parameter, τ E = 4 γ t b 1 represents the thermal relaxation parameter, τ C = 4 γ c b 1 represents the solutal relaxation parameter, k 1 = a 1 b 1 and k 2 = a 2 b 1 represent shrinking parameters, Ω = b 2 b 1 is the rotation parameter, γ 1 = H h 1 k f and γ 2 = H h 2 k f are Biot numbers, S c = v f D represents Schmidt number and ω 1 = μ f μ h n f , ω 2 = ρ h n f ρ f , ω 3 = σ h n f σ f , ω 4 = ρ C p h n f ρ C p f , ω 5 = k h n f k f are constants”.

Eq. 16 is then differentiated w.r.t η for the elimination of ε (pressure variable), and we get: ω 1 f i v − λ R e f ″ + ω 2 R e 2 f f ‴ + 2 g g ′ − M ω 3 ω 2 f ″ − 2 f ′ f ″ F * = 0 (22)

Eqs 16–21 define the ε (pressure variable) as follows: ε = f ‴ 0 − R e ω 2 ω 1 f ′ 0 2 − 2 f ″ 0 f 0 − g 0 2 + F * f ′ 0 2 + ω 3 ω 2 M f ′ 0 − λ R e f ′ 0 (23)

To solve Eqn. (18) for P, use integration from 0 to η and obtain the following form: P = − 2 f ′ − f ′ 0 + ω 2 ω 1 R e f 2 = 0 (24)

The Nusselt numbers at the lower disk N u r 0 and upper disk N u r 1 are (Mabood et al., 2021) and (Yaseen et al., 2022a): N u r 0 = − ω 4 + R d θ ′ 0   a n d   N u r 1 = − ω 4 + R d θ ′ 1 (24a)

The Sherwood numbers at the lower disk S h r 0 and upper disk S h r 1 are defined as (Khan et al., 2020a): S h r 0 = − ϕ ′ 0   a n d   S h r 1 = − ϕ ′ 1 (25)

Figures 3, 4, 5, 6, 7, 8, 9, 10, 11 show the fluctuation of dimensionless velocity against the similarity variable while altering distinct flow-regulating parameters one at a time while leaving others unchanged. Figures 3, 4, 5 depict the influence of porosity parameter λ on axial velocity distribution f η , radial velocity distribution f ′ η , and tangential velocity distribution g η , respectively. Within the core areas of the boundary layer region (BLR), there is a noticeable decrease in the axial velocity distribution f η and tangential velocity distribution g η with growing values of λ. The radial velocity shows transitioning nature in BLR and the transition point lies near η ∼ 0.5 . f ′ η rises near the lower disk and behavior changes near the upper disk w.r.t to growing values of λ. Physically, the porosity parameter is linked to the friction force. Resistance increases with a rise in the porosity parameter, and therefore the velocity falls. Figures 6, 7, 8 elucidate the effectiveness of magnetic parameter M on f η , f ′ η and g η . The increasing magnitude of parameter M is responsible for the decrease in velocity f η and g η . The radial velocity f ′ η shows transitioning nature in BLR w.r.t parameter M. The decrease in the velocity f η and g η is related to the generation of Lorentz force. The presence of magnetic force generates the Lorentz force opposite to the flow direction, hence it opposes the motion, and hence velocity decreases.

Figures 9, 10, 11 elucidate the effectiveness of volume fraction on velocity distribution. On increasing the volume fraction of each nanoparticle Go φ 1 , and MoS₂ φ 2 in equal quantity φ , the axial velocity f η decreases and tangential velocity g η increases, whereas radial velocity f ′ η shows transitioning nature in BLR, i.e., it first decreases and then increases. The increasing nanoparticles volume fraction in the base fluid causes the hindrance for the motion in the axial direction, hence the velocity f η decreases. On contrary, the tangential velocity g η increases, because the addition of nanoparticles causes the outward movement of fluid. In Figures 3, 4, 5, 6, 7, 8, 9, 10, 11, the authors have presented a comparison in velocity profiles of HNF (MoS₂-Go/water flow) and NF (Go/water flow). HNF is seen to have the higher axial velocity f η and NF has the higher tangential velocity g η . Near the lower disk, HNF has a higher radial velocity, but as fluid approaches the upper disk, NF has a higher radial velocity.

Figures 12, 13, 14, 15, 16, 17 show the fluctuation of dimensionless temperature θ η against the similarity variable while altering distinct flow-regulating parameters one at a time while leaving others unchanged. Figures 12, 13, 14 depict the influence of porosity parameter λ, magnetic parameter M, and radiation parameter R _d on the temperature profile. It is observed that with an increment in the aforesaid parameters, the temperature θ η shows transitioning nature in BLR. The temperature θ η falls near the lower disk and rises near the upper disk. The effect of resistance forces due to the porosity parameter and the effect of generated Lorentz force due to the magnetic field is significant near the upper disk, which opposes the motion of the flow and increases friction. As a result, the temperature rises near the upper disk with increasing porosity parameter λ and magnetic parameter M. Figure depicts that augmenting parameter R _d causes the temperature to decrease first, then after a transition point, it increases. This means that near the upper disk, the heat carried by radiation supplies energy to the particles, increasing their kinetic energy and velocity, and causing an increase in the temperature.

Figure 15 depicts that an increase in the thermal relaxation parameter τ E lead the temperature θ η to rise near the lower disk but for the same, the temperature falls near the upper disk. The non-zero thermal relaxation parameter τ E denotes time lag during heat transfer. For the zero value of τ E , the heat transfer is governed by Fourier’s law. The results depict that near the lower disk the more time lag in heat transfer implies higher temperature but the same condition near the upper disk relates to the lower temperature. Figure 16 elucidates the effectiveness of volume fraction on temperature θ η . On increasing the volume fraction of each nanoparticle Go φ 1 , and MoS₂ φ 2 in equal quantity φ , the temperature falls near the lower disk and surges near the upper one. The rise in the temperature near the upper disk can be attributed to the enhancement in the thermal conduction of the HNF or NF due to an increase in the nanoparticles in the base fluid. Figure 17 depicts that for the case of heat source A * > 0 ,   B * > 0 , the temperature θ η shows transitioning nature in BLR. Temperature first increases then decreases for increasing A * and B * A * > 0 ,   B * > 0 . Whereas, for the case of the heat sink A * < 0 ,   B * < 0 , the temperature increases for increasing magnitude of heat sink parameters. In Figures 12, 13, 14, 15, 16, 17, the authors have presented a comparison of temperature profiles of HNF (MoS₂-Go/water flow) and NF (Go/water flow). Near the lower disk, HNF has a higher thermal profile θ η , but as fluid approaches the upper disk, NF has a higher thermal profile θ η .

Figures 18, 19, 20 show the fluctuation of dimensionless concentration of nanoparticles against the similarity variable while altering distinct flow regulating parameters one at a time while leaving others unchanged. Figures 18, 19 depict the influence of the solutal relaxation parameter τ C and Schmidt number Sc on the concentration ϕ η . Figure 18 depicts that increasing values of the solutal relaxation parameter τ C leads concentration to fall. The non-zero values of the solutal relaxation parameter τ C denote the presence of time lag during mass transfer. The increasing time lag during mass transfer has an adverse effect on concentration. The increasing time lag during mass transfer leads to a fall in the concentration profile. Figure 19 depicts that increasing values of Sc cause concentration to fall. The concentration decreases as Sc increases. The Schmidt number is an important parameter to consider as it physically connects the depths of the hydrodynamic layer and the mass transfer boundary layer. The momentum diffusion of particles in fluid increases as the strength of the Schmidt number increases in fluid flow. As a result, concentration falls. Figure 20 depicts that on increasing the volume fraction of each nanoparticle Go φ 1 , and MoS₂ φ 2 in equal quantity φ , the concentration falls. The reason for this behavior can be attributed to the fact the mixing of more nanoparticles causes the HNF or NF to be more viscous and flow is resisted, hence the concentration of NPs falls. In Figures 18, 19, 20, the authors have presented a comparison in concentration profiles of HNF (MoS₂-Go/water flow) and NF (Go/water flow). It is seen that NF has a higher concentration profile ϕ η as compared to HNF.

Figure 21 visualizes the streamlines pattern of HNF (MoS₂-Go/water flow) and NF (Go/water flow). The streamlines represent the movement of the suspended particles in the stream and which are driven by it. The direction of velocity at each point in the streamline is given by the tangent at that location. The denser streamlines indicate that the fluid velocity is greater than when it is open out. The HNF is seen to have slightly less dense streamlines in the present case which means more nanoparticles cause fluid to be more viscous and flow is resisted.

Table 3 elucidates the effectiveness of pertinent parameters on the Nusselt and Sherwood number. Nusselt number corresponds to the heat transmission rate (HTR) and the Sherwood number corresponds to the mass transmission rate (MTR) in the BLR at both disks. At the higher magnitude of the heat sink parameter, the HTR rises for HNF and NF at the lower disk but contrary behavior is seen at the upper disk. Whereas, the HTR decreases for HNF and NF at both the upper and lower disk. At the higher values of radiation parameter R _d , HTR increases at the lower disk and decreases at the upper disk. An increase in the volume fraction of each nanoparticle Go φ 1 , and MoS₂ φ 2 in equal quantity φ enhances the thermal conductivity of the THNF due to an increase in the nanoparticles in the base fluid. The rising thermal conductivity enhances the MTR and HTR at the upper disk. It is also observed that at the higher values of the thermal relaxation parameter τ E , HTR reduces at both disks. The non-zero values of the thermal relaxation parameter τ E denote the time lag during heat transfer. Thus on augmenting τ E , HTR at both disks reduces. Moreover, on increasing the magnitude of the Schmidt number Sc and solutal relaxation parameter τ c , the MTR reduces at upper disks, whereas, on increasing the magnitude of the same parameters, the MTR rises at the lower disk.

The present study discuss the flow of nanofluid and hybrid nanofluid without the nanoparticle aggregation effect. The study can de extended with proper modeling of fluid flow by utilization of validated thermophysical correlations for nanoparticle aggregation effect.