Hybrid nanofluid flow within the conical gap between the cone and the surface of a rotating disk

The thermal management of the flow of the hybrid nanofluid within the conical gap between a cone and a disk is analyzed. Four different cases of flow are examined, including (1) stationary cone rotating disk (2) rotating cone stationary disk (3) rotating cone and disk in the same direction and (4) rotating cone and disk in the opposite directions. The magnetic field of strength \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{0}$$\end{document}B0 is added to the modeled problem that is applied along the z-direction. This work actually explores the role of the heat transfer, which performs in a plate-cone viscometer. A special type of hybrid nanoliquid containing copper Cu and magnetic ferrite Fe3O4 nanoparticles are considered. The similarity transformations have been used to alter the modeled from partial differential equations (PDEs) to the ordinary differential equations (ODEs). The modeled problem is analytically treated with the Homotopy analysis method HAM and the numerical ND-solve method has been used for the comparison. The numerical outputs for the temperature gradient are tabulated against physical pertinent variables. In particular, it is concluded that increment in volume fraction of both nanoparticles \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\phi_{{Fe_{3} O_{4} }} ,\phi_{Cu} } \right)$$\end{document}ϕFe3O4,ϕCu effectively enhanced the thermal transmission rate and velocity of base fluid. The desired cooling of disk-cone instruments can be gained for a rotating disk with a fixed cone, while the surface temperature remains constant.

The study exhibits that the cone-disk devices have several practical and technical applications, like in the stability analysis of an Oldroyd-B fluid creeping flow 1 , medical purposes 2 , in the calculation of viscosity of fluid using viscosimetry 3 , for gas turbines in a conical diffuser in the cooling system to compress air 4 . Choi 5 is the pioneer to use the nano sized small particles of the metals, carbides and oxides in the base fluids to enhance the thermal conductivity of the base fluids. The present work also deals with the heat transmission through hybrid nanoliquid passing between the gap of a disk and cone, in which either both are rotating in the same direction or in different with angular velocity, or maybe one remains stationary with respect to another. Such type of studied attract a number of researchers to examine its behavior. Turkilmazoglu 6 used semi-analytical method (HAM) and investigated streamline flow on a spinning cone, and produced well documented outputs successfully. Chamkha et al. 7 numerically computed the time dependent problem with heat and mass transfer from a vertical spinning cone. A series of stability analysis of boundary layer was later studied by Garrett et al. 8,9 , plunging into the convective or absolute behavior of instability due to the revolving cone. The magneto hydrodynamics MHD nanoliquid flow over a spinning cone with the thermophoresis and Brownian motion influence was illustrated in 10 . Similarity solutions of the compressible laminar flows subject to surface mass flux over a group of revolving cones were scrutinized in 11 .
With the immediate development of nanotechnology and modern sciences. The nanomaterials has gained tremendous attention from many of the researchers. The small particles in the nanometer sized are stabile dispersed in the base fluids to perform nanofluids. The metal oxides, carbon materials and so on are used as the nanoparticles. Nanofluids, are mainly used in the thermal engineering, fiber technology and electronic devices. The Scientific Reports | (2021) 11:1180 | https://doi.org/10.1038/s41598-020-80750-y www.nature.com/scientificreports/ 2. The magnetic field is imposed vertically to the flow pattern in the present work while the existing work 47 is without the magnetic field. 3. Four different cases for the flow between a disk and cone (1) stationary disk rotating cone (2) rotating disk stationary cone (3) counter rotating of the disk and cone (4) co-rotating of the disk and cone are examined and discussed and this idea extended to both the velocity and temperature profiles. 4. The HAM technique and BVPh 2.0 package have been used for the solution of the nonlinear problem and this method is compared with the numerical (ND-solve) method. 5. It has been observed that hybrid nanofluids improve the thermal efficiency of the base fluids rapidly as compared to the other fluids.

Mathematical formulation
Consider a disk and cone with an incompressible hybrid nanoliquid under the influence of magnetic field is under consideration. Both tools (disk and cone) are assumed to be either rotating or stationary with angular velocity in the cylindrical coordinate (r, ϕ, z) . The ω and highlight the disk and cone angular velocities respectively. B 0 is the strength of magnetic field that is applied along z-direction, whereas the induced magnetic field is neglected. The flow mechanism is illustrated in Fig. 1. Heat transportation modeling is computed with the addition of viscous dissipation. The phenomenon is successfully applied on the surface of the disk with a radially variable wall temperature T w = T ∞ + cr n , here n and c are kept fixed, where T ∞ is the cone wall 47 . Within the conical gap, p is the pressure depending on both axial z and radial r distances. The governing equations on the basis of above assumption can be stated as 47,48 :  where (u, v, w) are the velocity components along (r, ϕ, z) directions, B 0 is the magnetic strength, p is the fluid pressure. While k hnf , ρ hnf , ν hnf , µ hnf , ρc p hnf and σ hnf is the thermal conductivity, density, dynamic viscosity, heat capacitance and electrical conductivity of hybrid nanoliquid respectively.

Boundary conditions. The obligatory boundary conditions are as
Here γ specified the gap angle between the cone and disk.
Similarity conversion. In order to nondimensionalization, we adopt the following similarity transformation 47 : Here Uw is used as the surface velocity, Pr is the Prandtl number and M is the magnetic field. Now, with the help of these transformations as in Eq. (7), the modeled Eqs. (2)(3)(4)(5) and their boundary conditions modify to the following fashion: The modified conditions are: The volumetric fraction of Thermo-physical properties. The different thermal characteristics of hybrid nanoliquid and water as follows 49 : Here ρ hnf , υ hnf , Cp hnf , k hnf , σ hnf are the density, kinematic viscosity, specific heat, thermal conductivity and electrical conductivity of the hybrid nanofluids. The dimensionless shape of the heat transmission rate from the disk and cone surfaces are defined as: Nu d is the Nusselt number for the disc and Nu c for the cone.  for the 1st time introduced HAM method for the solution of nonlinear differential equations. In the present paper, we also tackled the modeled equations through HAM. The HAM method climbed by Liao addresses all highly nonlinear problems with sufficient choice to select parameters values to permit a convergent series solution. Contrary to numerical schemes, HAM can also tackle the far field boundary value problems. Salient characteristics of the said scheme are, HAM solutions are free from the selection of small/large parameters, unlike the perturbation schemes. The convergence of the series solutions is controlled by the auxiliary parameter instead of the physical parameter. HAM also provides us autonomy for the choice of initial guess estimates by keeping in view the physical system of the problem under consideration. This may be of polynomial, exponential, trigonometric or logarithmic nature. Many studies 53,54 have verified the validity and effectiveness of this method. To reveal the convergence rate, the sum of residual error is calculated through BVP 2.0 package 55,56 . The preliminary approximations are selected in this method which satisfy the initial and boundary conditions. The linear operators are used to find the initial guesses for the model problem. In the HAM technique initial guesses are required to run the Mathematica code. The convergence is totally dependent on the initial guesses (Trial solution).

Problem solution
The initial approximation for velocity F 0 , G 0 , H 0 , temperature 0 are given as The linear operators for the proposed problem are suggested as: The expand form of After applying Liao's idea (BVPh 2.0 package) to Eqs. (9-12) as: The total sum and the square residual are stated as www.nature.com/scientificreports/ The total square residual error is used to calculate the convergence of the proposed problem. The BVPh 2.0 package is also used to obtain the range of the physical parameters.

Result and discussion
The motive behind this section is to investigate the nature of different physical entities of versus velocity and temperature profiles. Figure 1 exhibit the flow mechanism of rotating cone and disk. Features of the parameter M (magnetic number) on the axial velocity F(η) profile are illustrated in Fig. 2. It can be noticed that the rising credit of magnetic parameter declines the fluid velocity F(η) of both copper Cu and magnetic ferrite Fe 3 O 4 hybrid nanoliquid. Physically, the Lorentz force is generated by magnetic number M, which retard the fluid velocity, as a result the velocity decreases. Figure 3 and 4 revealed the impact of volume fraction parameters φ Fe 3 O 4 , φ Cu on the axial velocity F(η) profile. The positive increment in φ Fe 3 O 4 and φ Cu enhancing the boundary layer thickness, which decline the velocity profile. Figures 5, 6 and 7 scrutinized the influence of magnetic parameter M, volume fraction parameter of iron oxide φ Fe 3 O 4 and copper φ Cu on radial velocity G(η) profile respectively. It can be observed that the radial velocity also revealed the same behavior as axial velocity against the nominated parameters.
All four cases regarding to disk, cone angular motion is briefly discussed in Figs. 8, 9, 10 and 11 respectively. Case (1) describes the situation, when the disk is at rest while the cone is rotating. The fluid actually moves between the disk-cone gaps. But the maximum flow intensity is found around the cone, therefore the positive variation in cone velocity Re � = r 2 �/ν enhances the radial profile G(η) . On the other hand, an opposite trend has been found in case (2), when the cone is at rest, while the disk is in motion with angular velocity Re ω = r 2 ω/ν . According to the no-slip condition, the fluid particles at the cone wall produces some resistance to the flow field. So that' why such phenomena have been observed. In case (3) both the disk and cone rotate in the same direction, therefore due to the minimum amount of resistance, the flow field illustrates its dominance against Re � and Re ω respectively. While in Fig. 11. Case (4) highlights that the counter-rotating of disk and cone effectively reduces the fluid velocity, due the maximum amount of resistance.
The nature of temperature distribution �(η) versus magnetic strength M is described via Fig. 12. The Lorentz force retards the fluid to move, as a result, some amount of heat produce, which eventually rises the temperature   Fig. 15. Physically, the high Prandtl fluid has always less thermal diffusivity and vice versa.
To keep in touch with the published work 45 , we strictly fixed Reynolds number 12 and 2463 throughout the computational work. In case of co-rotation, the ratio of Reynolds number is set up to 1.01, and counter rotating case, it is fixed to -1. We confine the values of power index also to n = −1, 0, 2 , only for comparison purpose with the literature 44,45 . It can be varied with the situation, according to the model. Here case 1 means stationary disk with rotating cone, case 2 stationary cone with rotating disk, case 3 co-rotating disk and cone and case 4. When both disk and cone are counter rotating. Figure 16a,b are sketched, in order to discuss the similarity temperature   www.nature.com/scientificreports/ between a stationary disk and rotating cone (case 1), with varying credit of (Re ω ) . In case 1. Temperature field is illustrated within the conical gap from the surface of the cone and disk. It can be seen that the temperature is slightly affected in the increase's manner throughout the thermal layer in normal apex angles. Although not much effect is perceived in case of minor gap angle. Actually, a critical power index n = −1, 0, 1 , is appearing there, so the heat transmission from the surface of the disk become zero, hence the fluid at the disk surface act as an insulator, because no heat transfer phenomena take place.   www.nature.com/scientificreports/ Figure 17a,b are plotted, in order to scrutinize the temperature profile and heat transfer from the cone and the disk surface within the conical gap, according to case 2 (a stationary cone with the rotating disk). It can be seen that case 2 gains high heat transfer rate only for fixed wall temperature (n = 0). However, cooling process increases for a stationary cone and rotating disk for a high range of radially varying disk temperature distribution. Figure 18 summarized case 3 (co-rotating disk and cone) situation. It is concluded that with the co-rotation of the disk and cone, the temperature of the system rapidly reduces. Finally, the case 4 (both disk and cone are      www.nature.com/scientificreports/ error in these tables shows the strong agreement between the HAM and ND-Solve method. Table 6, exhibits the numerical outcomes of the Nusselt number −� ′ (0) at the disc surface for the Fe 3 O 4 and Fe 3 O 4 + Cu respectively. While Table 7, shows the numerical outcomes of the Nusselt number −� ′ (1) at the cone surface for the Fe 3 O 4 and Fe 3 O 4 + Cu respectively. The impact of the important physical parameters is calculated and discussed. The larger amount of the Prandtl number declines the heat transfer rate at both the disc and cone surfaces. The increasing in the Eckert number enhancing the heat transfer rate and it happened due to the viscous dissipation term. The impact of the Prandtl number is more efficient using the hybrid nanofluid Fe 3 O 4 + Cu/H 2 O as shown in the Tables 6 and 7. Similarly, the enrichment in the nanoparticle volume fractions enhancing the heat transfer rate and influential improvement achieved using the hybrid nanofluid Fe The present study is compared in the Table 8 with the existing literature and closed agreement obtained.

Conclusion
In present study real applications are revisited mainly disk-cone apparatus used for the industrial usages. A special type of hybrid nanoliquid containing copper Cu and magnetic ferrite the velocity of fluid and raises its temperature �(η) due to retardation effect known as Lorentz force. • The local Reynolds numbers Re ω = r 2 ω/ν and Re � = r 2 �/ν based on the angular velocity of the disk and cone positively influence the radial velocity profile G(η). • It is concluded that the momentum boundary layer improving with the spinning of the cone and disk in the same direction while the decline in the momentum boundary layer observed in the opposite rotation. • It can be seen that the temperature is slightly affected in the increases manner throughout the thermal layer in normal tip angles. Although not much impact is perceived in case of minor gap angle, due to the appearance of a critical power index n = −1, 0, 2 . So the heat transfer from the disk surface become zero, hence the fluid at the disk surface act as an insulator, because no heat transferring phenomena take place. • The suitable range of the physical constraints for the proposed model is calculated which strengthen the convergence of the problem.            License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.