Numerical analysis of thermal conductive hybrid nanofluid flow over the surface of a wavy spinning disk

A three dimensional (3D) numerical solution of unsteady, Ag-MgO hybrid nanoliquid flow with heat and mass transmission caused by upward/downward moving of wavy spinning disk has been scrutinized. The magnetic field has been also considered. The hybrid nanoliquid has been synthesized in the presence of Ag-MgO nanoparticles. The purpose of the study is to improve the rate of thermal energy transmission for several industrial purposes. The wavy rotating surface increases the heat transmission rate up to 15%, comparatively to the flat surface. The subsequent arrangement of modeled equations is diminished into dimensionless differential equation. The obtained system of equations is further analytically expounded via Homotopy analysis method HAM and the numerical Parametric continuation method (PCM) method has been used for the comparison of the outcomes. The results are graphically presented and discussed. It has been presumed that the geometry of spinning disk positively affects the velocity and thermal energy transmission. The addition of hybrid nanoparticles (silver and magnesium-oxide) significantly improved thermal property of carrier fluid. It uses is more efficacious to overcome low energy transmission. Such as, it provides improvement in thermal performance of carrier fluid, which play important role in power generation, hyperthermia, micro fabrication, air conditioning and metallurgical field.

Scientific Reports | (2020) 10:18776 | https://doi.org/10.1038/s41598-020-75905-w www.nature.com/scientificreports/ positive increment in disk thickness. Asifa et al. 10 highlighted the fine point of CNTs hybrid nanoliquid flow over revolving surface using HAM technique. They noticed that, the growing credit of disk rotation significantly accelerate the heat transmission rate and fluid velocity. The steady magnetic flow of nanoliquid via a porous surface with slip conditions and viscous dissipation by employing HAM technique is discussed by Alreshidi et al. 28 .
The ambition of consideration is to extend the idea of Ref. 29 and to investigate the effect of two different nanoparticle Silver Ag and magnesium oxide MgO/Water hybrid nanoliquids over a wavy rotating disk, with upward/downward movement. To improve the thermal conductivity of the fluid flow, this study is taken under consideration. The modeled equations are solved analytically via HAM and for validation and comparison purpose of the outcomes, the Parametric continuation method (PCM) has been implemented. Both results manifest best consensus with each other (Fig. 1).

Mathematical formulation
This section will explain the physical interpretation of the problem, thermophysical properties and equation of motion.
Physical description of the problem. Let us consider a three-dimensional flow of Silver magnesium oxide hybrid Ag-MgO/Water nanoliquid over upward/downward moving wavy rotating disk. At time t , the disk has a vertical velocity ω = a(t) and is at a vertical distance Z = a(t) . The disk was a(0) = h at t = 0 . The rotating disk has angular velocity �(t) about z − axis , the buoyancy effects are negligible and it has been assumed that the nanoparticle are distributed consistent and be in equilibrium state. The uniform magnetic field of constant magnitude � B = (B r � e r + B θ � e θ ) and B = B 2 r + B 2 θ is applied respectively, where e r and e θ are unit vectors.
Thermophysical properties of nanoliquid. The specific heat capacity and the density of the hybrid nanoliquid can be expounded are as follow 29 : where ρ s1 , ρ s2 are the density, C p s1 , C p s2 are specific heat capacity and ϕ 1 , ϕ 2 are the volume fraction of the silver and magnesium oxide nanoliquid respectively, which are mentioned in Table 1.
The viscosity µ hnf of nanofluid is calculated by curve fitting on real experimental data 29 .
Here, the Prandtl number and the thermal conductivity of nanoliquid are defined as 29 : (1)  Here body forces F r along x and F θ along z direction respectively. It can be expressed as 29 : Here u, v, w is the velocity component of the fluid, while Ha is LB 0 σ µ , in which B 0 is the magnitude and θ is the direction of magnetic field. Boundary condition. The initial and boundary condition for wavy spinning disk are: Karman's approach. In order to transform the Eqs. (5)(6)(7)(8)(9) and (12) to the system of ODEs, we use the following transformation, we follow 30 .
. Since the physical constraint S controlling the up/down movement of the disk (or the contraction/expansion of the disk) is defined as 30 : Sign ω nominate the constant rotation of wavy disk 30 : And disk temperature parameter, which express temperature distribution: The non-dimensional form of Nusselt number and skin friction are expressed as: where, τ wφ and τ w r stand for transverse and radial stress respectively.

Problem solution
The analytical approach HAM, which was presented by Liao [22][23][24] has been used for the solution of nonlinear modeled differential equations. For strong convergence, BVP 2.0 package has been implementing to show sum of square residual error.
The linear operators π f ,π g ,π h and π � are presented as, The expand form of π f ,π g ,π h and π � are , Taylor's series expansion form is used where N x = 0 if ρ ≤ 1 and if ρ > 1.

Result and discussion
The time dependent, 3D hybrid nanoliquid flow over a wavy rotating disk with upward/downward motion has been studied. The numerical results of the system of differential equations has been acquire through Parametric continuation method (PCM), while for comparison and validity of results and to get analytical output, HAM technique has been applied. The effect of physical parameter has been shown in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. For comparative studies of PCM and HAM, Tables 2, 3 and Tables 4 are plotted. Figure 1 displays the hybrid nanofluid flow over a wavy spinning disk under the magnetic effects. Figures 2  and 3 demonstrate the influence of volume fraction parameter φ 1 or φ Ag and φ 2 or φ MgO on axial velocity profile f (η) . It shows that, by increasing the number of silvers Ag and magnesium oxide nanoparticles, the axial velocity of fluid significantly improve. Figure 4 highlights the dominance of unsteadiness parameter S versus axial velocity f (η). The rising credit of S declines the fluid velocity. Figure 5 depicts the out-turn of rotation parameter ω on the radial velocity g(η) . It can be presumed that, the increment in ω increases the kinematic energy of the  www.nature.com/scientificreports/ fluid particles, consequently, the velocity accumulates, which generate some amount of heat. Eventually, the disk surface becomes heated, which also improve the fluid temperature θ(η). Figures 6 and 7 show the upshot of β and controlling parameter S on the azimuthal velocity respectively. As S control the movement of the spinning disk, when we increases the values of S , the rate of upward/downward motion of the disk also increases. So, the inter-molecular forces between the fluid particles become week, and   www.nature.com/scientificreports/ during the upward motion of the disk the fluid molecule loses its energy, which causes the decline of temperature and azimuthal velocity as well. Figure 8 depict the effects of γ on temperature. Parameter γ actually controls the upward and downward velocity of spinning disk. So, from Fig. 8, we can presume that the increasing values of γ will reduced the hybrid nanofluid temperature. Figures 9 and 10 illustrate the influence of φ 1 and φ 2 on temperature profile θ(η)   www.nature.com/scientificreports/ respectively. As the volume fraction of silver and magnesium MgO nanoparticles increase, the heat absorbing ability of fluid also increased, which result in enhancement of fluid temperature θ (η) . Figure 9 shows the decreases of temperature versus increases in Prandtl number. Pr hnf = µC p hnf /k hnf , physically less Prandtl fluid has higher thermal diffusivity. The thermal boundary layer thickness reduces with larger values of Prandtl number as a result in decrease of the temperature. Figures 12, 13, and 14 revealed the h-curves for axial velocity h f , radial velocity h g and temperature h fields respectively. Table 2 illustrates the numerical comparison of PCM method against HAM approach for radial, azimuthal, and axial velocity. From Table 2, it can be observed that fractional model show fast converges than Runge Kutta order 4 method. Table 3 shows comparative effect of volume fraction parameter φ on radial and tangential velocities f ′ (0), g ′ (0) for different nanofluid, keeping the rest physical parameters are constant. From, Table 3 we can  www.nature.com/scientificreports/ examine that, the radial and tangential velocity of MgO nanofluid are greater than the Silver Ag nanofluid, because the density of Silver nanoparticles are heavy than MgO nanoparticles. Therefore, the viscosity of Ag nanofluid is greater than MgO nanofluid. That's the reason that the radial and tangential velocity of MgO nanofluid is greater. Table 4 shows the comparative effect of volume fraction parameter φ on skin fraction and Nusselt number h ′ (0) and θ ′ (0) for different nanofluid respectively while the rest of physical parameters are constant. The sum and square of the total residual for the Ag and MgO are displayed in Tables 5 and 6. Table 7 displays the comparison of present work with the published literature.

Conclusion
In this work, the three-dimensional, unsteady Ag-MgO/water hybrid nanofluid flow, caused by upward/downward movement of a wavy rotating disk, under the magnetic field influence with mass and heat transport has been studied. The following observations have been made on the basis of above computation: • The wavy rotating surface increases the heat transmission rate up to 15%, comparatively to flat surface 32 .
• The rising credit of rotation parameter ω increases the kinematic energy of fluid, which result in the enhancement of velocity and temperature of hybrid nanoliquid. Table 3. It shows the comparative behavior of Ag and MgO on radial and tangential velocities f ′ (0), g ′ (0) for volume fraction parameter. η

Silver (Ag)
Magnesium oxide MgO       Which can be synthesized at 700 °C to 1500 °C and is mostly efficacious for refractory and electrical applications • The strong bonds between water atom (H + + OH − ) and silver ions Ag + effectively improves the thermophysical properties of water. • The upward/downward movement of wavy rotating disk positively affects the fluid temperature and velocity.
• The use of hybrid nanoliquid is more efficacious to overcome low energy transmission. Such as, it provides improvement in thermal performance of carrier fluid, which play important role in power generation, hyperthermia, microfabrication, air conditioning and metallurgical field.