Influence of nanoparticles aggregation and Lorentz force on the dynamics of water-titanium dioxide nanoparticles on a rotating surface using finite element simulation

This communication briefings the roles of Lorentz force and nanoparticles aggregation on the characteristics of water subject to Titanium dioxide rotating nanofluid flow toward a stretched surface. Due to upgrade the thermal transportation, the nanoparticles are incorporated, which are play significance role in modern technology, electronics, and heat exchangers. The primary objective of this communication is to observe the significance of nanoparticles aggregation to enhance the host fluid thermal conductivity. In order to model our work and investigate how aggregation characteristics affect the system’s thermal conductivity, aggregation kinetics at the molecular level has been mathematically introduced. A dimensionless system of partial-differential equations is produced when the similarity transform is applied to a elaborated mathematical formulation. Thereafter, the numerical solution is obtained through a well-known computational finite element scheme via MATLAB environment. When the formulation of nanoparticle aggregation is taken into consideration, it is evident that although the magnitude of axial and transverse velocities is lower, the temperature distribution is enhanced by aggregation.

Nanofluids are fluids that are created by the homogenous dispersion of tiny particles of metal or metal oxides at the nanoscale 1 . The host fluid which contains the nanomaterials has a valuable influence on the dynamics of host fluid according to the experimental and numerical investigation 2,3 . The modern era researchers are fascinating due to wide range of nanofluids application in every field of engineering and science 4 . The host fluid temperature enhanced due to mixture nano-sized particles inside the fluid and it is happen because of effective thermal transport of tiny particles which are responsible for enhancing heat transfer rate 5 . The tiny solid particles different shapes influence the host fluid dynamics and significance influence on host fluid temperature 6 . The different particles features inside the nanofluid flows have been investigated numerically, analytically, and experimentally by the research community 7 . Recent, several researchers investigated the different nature of nanoparticles to improve the host fluid thermal enhancement subject to different types of techniques and geometeries [8][9][10] . The inclusion of nano-particles in the base fluid resulted in a rapid rise in temperature, whereas the mono fluid had a smaller impact on temperature than the hybrid nanofluids 11 . The combination of base fluids and small solid fragments can improve the thermal properties of several fluids 12,13 .
In our daily life, the titanium dioxide is key component which is studied in metal oxide surface science as a crystalline oxides because of its capacities as a photocatalyst with sensibly high proficiency for water breakdown 14 .
1. What is influence of magnetic parameter on the characteristics of water-titanium dioxide nanofluid subject to titanium dioxide particles aggregation and non-aggregation? 2. What is impact of Coriolis force on the dynamic of water-titanium dioxide nanofluid subject to titanium dioxide particles aggregation and non-aggregation? 3. What impact do the magnetic and rotating parameters have on Nusselt number and skin friction factors in the presence of titanium dioxide particles aggregation and non-aggregation?

Mathematical formulation
Take into consideration time dependent three-dimensional rotating water-based nanofluid that flows across a stretching subject to magnetic field. On an elongated surface, a three-dimensional, non-transient flow of magnetized nano liquid (TiO2) is considered, as depicted in Fig. 1. By taking into account the corresponding dynamic viscosity and thermal conductivity, the influence of nanoparticle aggregation is evaluated. The movement of nanoparticles suspended in liquid would result in aggregations that significantly impact the base fluids' physical properties. The elaborated problem system is rotating in z direction, which is oriented transverse to xy plane, at an angular velocity of . The fixed origin O(x, y, z) has been chosen, with the x-axis representing the movement of the stretching surface, the y-axis representing the normal of the surface, and the z-axis representing transverse to the xy plane. In the axial (z-direction) direction, a static and uniform magnetic B 0 field is applied 35 . The ambient temperature is represented by T ∞ , and the surface temperature is represented by T w . The components u 1 , u 2 , and u 2 show the velocity components along x, y, and z directions, respectively. Tables 1 and 2 details the aggregation and non-aggregation related physical properties of based fluid and nanoparticles. The governing equations of continuity, momentum, and temperature are given below 36,37 : where µ n f , ρ nf , α n f , are the dynamic viscosity, fluid density, and thermal diffusivity, T represents the fluid temperature. The formulated problem boundary conditions are 38,39 : Similarity transformations (see 36,38 ): t ≥ 0 : u 1 =ãx, u 3 = 0, u 2 = 0, T = T s , as z = 0,   Table 2. Thermophysical properties model of nanofluid 40,41 .

Numerical solution
For resolving partial differential equations, the finite-element method is an effective approach. This approach's underlying concept is dividing the domain into small size, known as finite elements. In modern engineering analysis, this method is such a good numerical method that it can be used to solve integral equations in many different fields, including heat transfer and fluid mechanics 42,43 . The first and second essential steps of this method are to assume the piecewise continuous function in order to obtain the solution and to locate the parameters that correspond to the functions in such a way as to minimize the error in the solution 44,45 . With the boundary conditions (12) for the solution of a system of simultaneous partial differential equations as shown in (9-11), we first assume that: Equations (9)-(12) reduces to: where w f 1 , w f 2 , w f 3 , and w f 4 are the arbitrary test functions. The grid point of computation domain ( e ) is shown in Fig. 2. Equations can be transformed into the finite element model by substituting approximations of the form: � e w f 2 1 � e w f 3 1 j=1� ′ j ϒ j assumed to be the known. The 4 functions are compute at each node. The final system of equations are non-linear forms after assembly, so linearize this system via an iterative algorithm subject to the required precision of 10 −5 .

Results and discussion
Using a graphical representation, the physical effects of the various flow parameters, including the rotating, magnetic, and time dependent parameters, on the dimensionless fluid flow velocity and the temperature are shown. The effects of particle aggregation suspensions with microscopic particles and a fluid that is susceptible to Lorentz and Coriolis forces have been demonstrated by our findings. The each graph show two types of curves for nanoparticles aggregation ( int = 1.0 ) and non aggregation ( int = 1.0 ). Throughout the computation, the others involved parameters default values are: P r = 6.2 , M = 1.0 , = 1.0 , D = 1.8 , = 0.01 , max = 0.650 , and R a /R p = 3.34 . A grid independence study is carried out in order to examine the Galerkin finite element approach's reliability and validity. We finalized all of the results on a grid that is 100 × 100 because the problem input is divided into different mesh densities and there is no more variation after that (see Table 3). In some cases, a validation with previous studies is presented in Tables 3 and 4 to demonstrate that the current results are valid and reliable. As shown, the current results are very similar to the published ones. Table 4 displays the friction factors as well as the primary and secondary directions ( −F ′′ 1 (0) & −F 2 (0)) in response to increasing inputs of ( = 0, 1, 2, 5) at ( Ŵ = 1.0) . The findings are very consistent with those analyzed by Wang et al. 46 and Ali et al. 40 . In addition, the −� ′ (0) finding are compared between Bagh et al. 47 and Adnan et al. 48 , who present finite element results against rising inputs of M & , and discovered that they are in an excellent agreement in Table 5. Therefore, numerical calculations can be validated, and the Matlab created Finite Element Computations have a high rate of convergence.  The response of the variables in the boundary layer to the magnetic field (M), is depicted in Fig. 3a,b. An increase in M results in a significant decrease in the primary velocity F ′ 1 (Ŵ, η) ). With increasing magnetic field, the F ′ 2 (Ŵ, η) also experiences a significant magnitude decrease. The fluid flow in xy plane are both significantly slowed down by the Lorentzian drag forces − M 2 χ 2 F ′ 1 and − M 2 χ 2 F 2 in Eqs. (9) and (10). These magnetic forces are produced in the plane of the sheet, which are transverse to axial direction. The primary velocity never reversal flow because the surface is extend in the x-direction. However, as evidenced by the negative values in Fig. 3b, the secondary fluid flow which is perpendicular over the primary fluid flow, significant experiences backflow. The F ′ 1 (Ŵ, η) profiles indicate that when the magnetic field is at its weakest, M = 1.0 , the Lorentz resistance is approximately the same as the viscous hydrodynamic force in the nanofluid. The primary and second velocity components are all affected by the rotational parameter , as shown in Fig. 4a,b. A growing in the rotational parameter is accompanied by a significant decrease in the primary flow velocity. The magnitude of secondary velocity also decreases when the rotation parameter is increased. As previously stated, the crossflow effects cause the secondary flow to be negative, or backflow. In this case, in the x-direction momentum development is aided by the sheet's stretching direction, while in the y-direction momentum development is counteracts. In Eqs. (9) and (10) respectively, the Coriolis forces are +2 F 2 and −2 F ′ 1 . Even though the body force is positive in first momentum equation, the F ′ 2 (Ŵ, η) is negative, so the primary momentum field is affected negatively overall. As a result, as the rotational parameter escalate, the primary velocity recedes (Fig. 4a). The flow variables' responses to the unsteadiness parameter, ζ , are depicted in Fig. 5a,b. The predominant pattern is that primary velocity decreases as unsteadiness increases (Fig. 4a). Since the unsteadiness effect has been dampened out, the ( F ′ 1 (Ŵ, η) ) is effectively stabilized by a decline in the time dependent parameter. On the other hand, we notice that the regime's temperatures rise significantly with increasing unsteadiness parameter (Fig. 5b). According to Ali et al. 38 , for transient Newtonian convection surface extending flow, this is due to a upgrade in unsteady convection currents, which enhances boundary layer thermal diffusion. In addition, the model and nanoparticle aggregation have a lesser distribution of ( F ′ 1 (Ŵ, η) ) and ( F ′ 2 (Ŵ, η) ), whereas the distribution of F ′ 1 (Ŵ, η) & F ′ 2 (Ŵ, η) is slightly higher than that of non-aggregated nanoparticles case. Physically, the effective viscosity 49 increased as a result of the aggregation of nanoparticles, and the increasing strength of the viscosity slowed the fluid velocity 50 . Table 3. Study of different size of grid independent when Ŵ = 1.0.  Table 4. Comparative of −F ′ 1 (0) and −F ′′ 2 (0) against various strength of at Ŵ = 1.0 subject to no effect of others parameters.
Ali et al. 40 Wang 46 Present  Table 5. Comparison of −θ ′ (0) against various strength of at Ŵ = 1.0 subject to no effect of others parameters.
Adnan et al. 48 Ali et al. 47 Figure 6a demonstrates that for increasing Ŵ(0)rightarrow1 , the (C f x Re x 1/2 ) increases steadily to a fixed rate and does not change noticeably; however, for strengthen M, a significant decrease in the axial friction is observed. Figure 6b shows that the (C f y Re x 1/2 ) decreases steadily as Ŵ(0 → 1 increases until it reaches a constant rate, at which point there is no discernible difference. However, when M is improved, the significance difference near the boundary surface is noted. Figure 7a,b shows that for increasing Ŵ(0 → 1 , the (C f x Re x 1/2 ) is gradually raised until it approach a constant rate, after which no significant  www.nature.com/scientificreports/ change is observed. On the other hand, for increasing , both the (C f x Re x 1/2 ) and the (C f y Re x 1/2 ) are significantly decreased. In addition, the ranges of (C f x Re x 1/2 ) and (C f y Re x 1/2 ) for the model with nanoparticle aggregation are shown to have a distribution that is significantly lower than that of the case with non-aggregated nanoparticles. Figures 8 and 9 depict the distribution of �(Ŵ, η) for various parameters. As can be seen in Fig. 8a, the temperature distribution, �(Ŵ, η) , was enhanced by the magnetic parameter. The temperature profile depicted in Fig. 8a is controlled by a net force known as the Lorentz force subject to internal electric force, and the external magnetic field, whereas the thermal boundary layer thickness recedes while the value increases, as shown in Fig. 8b. Figure 9a   www.nature.com/scientificreports/ The distribution of (Nu x Re x 1/2 ) gradually decreases as M and grow. The (Nu x Re x 1/2 ) decreases significantly in the nanoparticle aggregation model, while the distribution of (Nu x Re x 1/2 ) is slightly higher than in the case of nanoparticles non-aggregation.

Conclusions
A theoretical model for predicting the thermal conductivity of nanofluids has been developed by taking into account the structure of the aggregates and nanoparticles, as well as the physical properties of the base liquid and the nanoparticles. Effective nanofluid thermal conductivity and viscosity for homogeneous models and nanoparticle aggregation were investigated by the authors. It is reasonable to draw the following main findings based on present report analysis: 1. The nanoparticles aggregation causes a decline in magnitude of both primary and secondary velocities, and lesser the strength of C f x Re x 1/2 and C f y Re x 1/2 . 2. The fact that the aggregation model has a temperature profile that is higher than that of typical model and a significant decline in Nu x Re x 1/2 . 3. The magnitudes of axial momentum and transverse momentum decrease when the Coriolis and Lorentz strengths are exceeded, and • improve thermal performance of base fluid and a negative influence on Nu x Re x 1/2 .
• Increase the value of the skin friction factor, Cf x Re 1/2 x .

Data availability
The data used to support the findings of this study are available from the corresponding author upon request.