Mathematical Modelling of Water-Based Fe3O4 Nanofluid Due to Rotating Disc and Comparison with Similarity Solution


 The current research demonstrates the revolving flow of water-based Fe3O4 nanofluid due to the uniform rotation of the disc. This flow of nanofluid is investigated using CFD Module in COMSOL Multiphysics. However, the similarity solution for this flow is also obtained after transforming the given equation into a non-dimensional form. In the CFD Module, streamlines and surface plots are compared with the similarity solution for the magnitude of the velocity, radial velocity, tangential velocity, and axial velocity. The results from the direct simulation in the CFD Module and the solution of dimensionless equations represent a similar solution of velocity distribution. The derived results show that increasing the volume concentration of nanoparticles and effective magnetic parameters decrease the velocity distribution in the flow. Results in the CFD Module are important for monitoring the real-time particle tracing in the flow and, on the other hand, the dimensionless solution is also significant for the physical interpretation of the problem. Both methods of solution empower each other and present the physical model without sacrificing the relevant physical phenomena.


INTRODUCTION
Ferrofluids are colloidal suspensions of magnetic nanoparticles. These types of fluid are not present in nature. Ferrofluids are synthesised by different methods based on their application. One of the important properties of ferrofluids is that they work in a zero-gravity region. Ferrofluids have numerous applications in the field of engineering and medicine. Heat control loudspeaker and frictionless sealing are important applications of ferrofluids. However, in the field of medicine, ferrofluid is used in the treatment of cancer through magnetohyperthermia.
The rotational flow of viscous fluid over a rotating disc has been examined by researchers using similarity transformations (Cochran, 1934;Benton, 1966;Schlichting and Gersten, 2017). Based on the models of the rotational flow of viscous fluid, researchers have developed the models for magnetohydrodynamic flow due to rotating disc (Attia, 1998(Attia, , 2007Turkyilmazoglu, 2012). Using Rosensweig and Shliomis models, the researchers have demonstrated the flow of ferrofluid due to rotating disc (Ram et al., 2010;2013a,b). The researchers have investigated the flow over a rotating disc to measure the irreversibility of the system and for heat transfer encampment applications (Qayyum et 2021) used Adomain decomposition method to investigate hybrid bio-nanofluid flow in a peristaltic channel. The researchers have been working continuously to improve the numerical solution of a set of non-dimensional coupled differential of flow over a rotating disc problem (Chaturani and Narasimman, 1991;Rahman, 1978;Schultz and Shah, 1979;Hayat et al., 2018). Reddy et al. (2017) investigated the flow of Ag-water and Cuwater nanofluids flow due to rotating disc. Takhar et al. (2003) studied the flow of electrically conducting fluid due to a vertical rotating cone and obtained the similarity solution through the finite difference method. Krishana and Chamkha (2020)  In the present work, the impact of volume concentration and magnetic torque on the swirling flow of water-carrying iron(III) oxide nanofluid due to the rotating disc is investigated. This swirling flow is considered steady and axisymmetric; therefore, cylindrical equations of the flow are directly numerically solved using the two-dimensional CFD Module in COMSOL Multiphysics. Further, the governing equations are reduced into non-dimensional coupled differential equations using a similarity approach and compared with the results obtained through CFD Module. The similarity model is also validated with the previous theoretical models. Fig. 1. Flow of axisymmetric nanofluid due to rotating disc and its two-dimensional CFD domain

MATHEMATICAL MODEL
The schematic diagram of the flow due to a rotating disc in a tank is shown in Fig. 1. The disc rotates uniformly about the z-axis and generates a three-dimensional boundary layer. The flow is axisymmetric; therefore, the three-dimensional flow has been considered over a two-dimensional region in the CFD Module. The velocities of the water-carrying iron(III) oxide nanofluid changes in the radial and axial directions. In the CFD Module, the computational domain is the two-dimensional geometry ( Fig. 1). However, the similarity approach is used on the three-dimensional geometry in Fig. 1. These two approaches of rotational flow for velocity distributions are compared to each other. The flow of magnetic fluid is described by the following equations (Shliomis and Morozov, 1994;Odenbach and Thurm, 2002): The inertial term I Using Eq. (4), Eq. (2) can be written as (Shliomis and Morozov, 1994; Bacri et al., 1995): where −∇p = −∇p + μ 0 M∇H denotes the reduced pressure due to magnetisation force [43].
The equilibrium of the magnetic and viscous torque can be written from Eq. (5) as (Ram and Bhandari, 2013;Bacri et al., 1995): The Eq. (6) can be written for mean magnetic torque as (Bacri et al., 1995): where 1 denotes the effective magnetic number due to the applied magnetic field. Using Eqs (6) and (7), the following relation is obtained (Shliomis and Morozov, 1994; Ram and Bhandari, 2013): In the above equation, 3 2 μ n φ 1 m 1 is the rotational viscosity due to the applied magnetic field. Using the result in Eq. (8), Eq. (6) can be written as: The physical properties of the ferrofluid are as follows (Sheikholeslami and Shehzad, 2018): The density of the nanofluid depends on the density of the base fluid and volume concentration. Enhancing the concentration of iron(III) oxide increases the density of the nanofluid. Similarly, it changes the viscosity of nanofluids.
Since the flow is considered steady and axisymmetric, Eqs (1) and (9) can be written in cylindrical coordinates as (Shliomis and Morozov, 1994; Bhandari, 2020): [ The boundary conditions for the considered flow are as follows: In this case, the disc rotates with the uniform angular velocity ω about the axis perpendicular to its plane. No-slip boundary conditions are considered in the flow. The layer of the nanofluid is at the disc which is carried along with the disc. This layer of nanofluid is driven outwards by the centrifugal force. Then new fluid particles are coming towards the disc in the axial direction and these particles are ejected due to centrifugal force. The centrifugal force and radial pressure gradient have an important role in circulating the nanoparticles at a sufficient distance from the wall.

NUMERICAL SOLUTION
In the present problem, the swirling flow of water-based  Eqs (11)-(15) can also be solved by using similarity transformation. The following similarity transformation has been used to obtain non-dimensional coupled differential equations: The transformed equations are as follows: The boundary conditions are as follows: The reduced pressure can be calculated as: ( ) = 0 + (1 +

RESULTS AND DISCUSSIONS
In this section, the results ofor radial tangential and axial velocity distribution are presented for different values of volume fraction and effective magnetic parameters. 2D surface plots are obtained from the CFD Module in COMSOL Multiphysics, and for the same problem, a system of nonlinear coupled differential equations has been solved numerically. A brief discussion of the results is presented here. Swirling of the disc in the presence of the magnetic field enhances the difference between the rotation of the particles and fluid. This difference enhances the viscosity of water conveying iron(III) oxide nanofluid. Near the rotating equipment, the velocity is higher than that in other places. Increasing these parameters changes the pattern of streamlines. The velocity changes Fe 3 O 4 to become closer than the base fluid. Figs 5(a)-(f) show the tangential velocity distribution for different values of volume concentration and the effective magnetic parameter. The speed of tangential velocity distribution is higher than the radial velocity distribution. When increasing volume concentration and magnetic parameter, the streamlines become closer to each other than in ordinary cases. This situation arises due to the difference in the rotation between the nanoparticles and fluid. The magnetic field increases this misalignment between the rotation of the particles and fluid. Therefore, the viscosity of the nanofluid increases.  Fig. 7 shows the magnitude of the dimensionless velocity distribution for different values of φ 1 . The magnitude for the velocity in dimensionless form is obtained from the expression √E 2 + F 2 + G 2 . The velocity distribution is presented on dimensionless axial distance. It is seen that when increasing values of φ 1 , the magnitude of the velocity decreases. The magnitude of the velocity also decreases for increasing values of m 1 , as shown in Fig. 8. The results in Figs 7 and 8 represent a similar character as presented in Figs 3(a)-(f). Figs 9 and 10 show the radial velocity distribution for different values of φ 1 and m 1 . A case φ 1 = 0 shows the velocity distribution of the base fluid. Adding Fe 3 O 4 nanoparticles with suiTab. surfactants decreases the radial velocity distribution decreases. In the presence of the magnetic field, a decrease in the radial velocity distribution is observed. Figs 11  and 12 show the tangential velocity distribution for different values of φ 1 and m 1 . The impact of the volume concentration and rotational viscosity on the tangential velocity distribution is less as compared to radial and axial velocity distributions. Near the surface of the container, the tangential velocity decreases for in-creasing values of φ 1 and m 1 , and far from the surface, these parameters enhance the tangential velocity distribution. Near the surface, the flow of the nanofluid is influenced by the rotating cylinder. Figs 13 and 14 show the axial velocity distribution for different values of φ 1 and m 1 . In the absence of magnetic field and iron(III) oxide nanoparticles, the problem reduces to the previous theoretical models of rotational flow. The comparative results with previous theoretical models are shown in Tab. 3. Increasing the values of the volume concentration and effective magnetic parameter decreases the axial velocity. It is noticeable that the physical interpretation of the flow dimensionless approach is very useful and aligned with the results obtained through the CFD Module. However, the CFD Module can provide the problem imagination and more regions for the velocity distribution. Kelson and Desseaux [47] 0.510233 -0.615922 Bachok et al. [48] 0.5102 -0.6159 Turkyilmazoglu [49] 0.51023262 -0.61592201 Present Result 0.5102337 -0.6159241

CONCLUSIONS
The swirling flow of water-based Fe 3 O 4 nanofluid with different volume concentrations and rotational viscosity has been studied. The results for the velocity distribution have been obtained through the CFD Module and dimensional analysis. These results indicate that increasing the volume concentration and rotational viscosity decreases the velocity distribution. These results show the comparative results between the CFD Module and dimensionless analysis. Both techniques favour the findings of each other. For the physical interpretation of the results, dimensionless analysis is important. However, for real-time particle tracing CFD Module is important. These results show that both types of studies are relevant to the development of swirling flow analysis. These results might be useful in bringing about an improvement in the sealing of hard disc drives. Vorticity (rad/s) Angular velocity (rad/s) Tangential direction (rad)