Finite Element Study of MHD Impacts on the Rotating Flow of Casson Nanoﬂuid with the Double Diffusion Cattaneo—Christov Heat Flux Model

: A study for MHD (magnetohydrodynamic) impacts on the rotating ﬂow of Casson nanoﬂuids is considered. The concentration and temperature distributions are related along with the double diffusion Cattaneo–Christov model, thermophoresis, and Brownian motion. The governing equations in the 3D form are changed into dimensionless two-dimensional form with the implementation of suitable scaling transformations. The variational ﬁnite element procedure is harnessed and coded in Matlab script to obtain the numerical solution of the coupled nonlinear partial differential problem. The variation patterns of Sherwood number, Nusselt number, skin friction coefﬁcients, velocities, concentration, and temperature functions are computed to reveal the physical nature of this examination. It is seen that higher contributions of the magnetic force, Casson ﬂuid, and rotational ﬂuid parameters cause to raise the temperature like thermophoresis and Brownian motion does but causes slowing the primary as well as secondary velocities. The FEM solutions showing an excellent correlation with published results. The current study has signiﬁcant applications in the biomedical, modern technologies of aerospace systems, and relevance to energy systems.


Introduction
Noteworthy endeavors have been made in recent years to explore nanofluids because of remarkable thermodynamic properties. Nanofluids can be utilized to cool the motors of vehicles, biomedical applications, high-transition gadgets, clothes washers machines, high-power microwaves, diode arrays of heavy-power laser, and various welding frameworks. In addition, significant advances in nano designing have opened up the chance of utilizing nanomaterials to treat various types of human body tumors, pharmacological medicines, artificial organs (lungs, heart surgery), and cancer therapy, etc. Nanofluids, presented by Choi and Eastman [1] in 1995, has gotten impressive consideration in present times. Makinde [2] examined numerically that the boundary layer flow induces in the nanofluid cause of a linear stretching surface with the influence of Brownian motion, and thermophoresis. code is confirmed in the face of previously available data for limiting cases. Furthermore, pictorial representations of some principal findings with a detailed discussion have also been presented.

Statement of the Problem
The time-dependent 3D magnetohydrodynamics of an incompressible Casson nanofluid flow over an extending sheet with a rotating frame of reference are considered as shown in Figure 1. Physically, we consider that the entire system is at rest in the time t < 0; however, for t = 0, the sheet is extended in the x-direction at z = 0 with angular velocity Ω. The mass and heat transfer component is examined through the heat flux model of Cattaneo-Christov double diffusion expressions. The framework is rotating with angular velocity (Ω) along the z-direction. In the z-direction, B o (magnetic field) is applied, the instigated magnetic field is overlooked due to a small magnetic Reynolds number, and Ohmic dissipation and Hall's current impacts are ignored since the field of magnetic is not too much strong [37]. Moreover, we assume that the concentration and temperature at the surface areC w ,T w , respectively, and the ambient concentration and temperature areC ∞ ,T ∞ , respectively. The rheological model for the flow of a Casson fluid can be written as: In Equation (1), π ij , e ij , p y , µ B , π c , and π = e ij e ij are Cauchy stress tensor, deformation rate components (i,j), yield stress of fluid, Casson fluid plastic dynamics viscosity, non-Newtonian based critical values of this product, and product of components of deformation rate with itself, respectively.

Governing Equations
Considering the above suppositions, the consistent mass, momentum, energy, and conservation of nanoparticles volume fraction equations in a Cartesian coordinate system (x, y, z) as follows [35,[38][39][40]: whereT Here, (ũ,ṽ,w) are components of velocity in directions (x, y, z), respectively, ρ n f ,α n f , µ n f , σ n f , λ 1 , and λ 2 are respectively the density, thermal diffusivity, dynamic viscosity, electrical conductivity, relaxation time of heat, and mass fluxes of the nanofluid.T andC are the fluid temperature and nanoparticle volume concentration,D B andD T are the Brownian and thermophoretic diffusion coefficient, respectively, Furthermore, t and C are respectively time and concentration of nanoparticles' volume fraction. The current physical elaborated problem, characterized boundary conditions, are [38,41]: We offer a following set of transformation variables to proceed the analysis (see [38,42,43]): The continuity of Equation (2) is satisfied identically using the similarity transformations above. In light of Equation (11), Equations (3)-(10) reduce into the following nonlinear PDEs in the transformed coordinate system (ξ, η): where θ rT =ff θ +f 2θ , and φ rc =ff φ +f 2φ . The come into view parameters in Equations (12)-(15) are defined as: where β, λ, M, Pr, Le, Nb, Nt, γ T , γ C , and Q s are the Casson fluid parameter, rotating parameter, magnetic parameter, Prandtl number, Lewis number, Brownian motion, thermophoresis, thermal relaxation parameter, concentration relaxation parameter, and heat source, respectively. When τ → ∞, ξ = 1, then the Equations (12)-(15) become: subject to the boundary conditions (16) Skin friction coefficient expressions, local Nusselt number, and Sherwood number are defined as: where the skin friction tensor at wall are τ , the wall heat transfer is q w = −κ T z z=0 , and the mass flux from the sheet is q m = −D B C z z=0 . By the aid of similarity transformation Equation (15), we get:

Finite Element Method Solutions
The transformed set of nonlinear partial differential Equations (12)-(15) is solved numerically utilizing the variational finite element method along with boundary conditions (Equation (16)) because Equations (12)-(15) cannot be solved analytically due to highly nonlinearity. This procedure is a great numerical computational methodology significant for solving the different types of real word problems [44] and problems of engineering [45]-for example, liquids with heat transportation [46], Bio-materials [47], rigid body dynamics [48], and various regions [49,50]. An astounding general description of variational finite elements method outlined by Reddy [51] and Jyothi et al. [52] summed up the basic steps involved in the FEM. Basically, the technique includes a continuous piecewise function for the solution and to get the functions parameters in an efficient way that minimizes the error [53]. The FEM solves boundary value problem adequately, rapidly, and precisely [54]. To reduce the order of nonlinear differential Equations (12)-(16), firstly we consider: The set of Equations (12)-(16) thus reduces to

Variational Formulations
Over a typical rectangular element Ω e , the associated variational form with Equations (23)-(27) is given by Ω e Ω e where w f s (s = 1, 2, 3, 4, 5) are arbitrary weight functions or trial functions.

Finite Element Formulations
Let us divide the rectangular domain (Ω e ) into 4-noded (rectangular element) and (ξ i , η j ) be the domain grid points (see Figure 2). The length of plate and thickness of boundary layer is fixed at ξ max = 2, η max = 5, respectively. The finite model of the element can be obtained from Equations (29)-(33) by replacing the following form of finite element approximations: (34) where Ψ j (j = 1, 2, 3, 4) are the linear interpolation functions for a rectangular element Ω e (see Figure 2) and are given by: The model of finite elements of the equations thus developed is given by: where [W mn ] and [b m ] (m, n = 1, 2, 3, 4) are defined as: where the known values to be considered aref

Validation of Results
The skin friction coefficient C f x Re 1/2 x and C f y Re 1/2 y in xand y-directions are compared with the already published article of Adnan et al. [35] for various values of λ and β when the magnetic field M = 0 and ξ = 1 (final steady state flow) that can be seen an excellent correlation in Table 1. Table 2 shows the values of non-dimensional Nusselt number (−θ (0)) is compared with different values of λ and Pr under special cases like the absence of magnetic field M = 0, Newtonian fluid (β → ∞), no heat source (Q s = 0), pure fluid (Nt = Nb = 0), ξ = 1 (final steady state flow), and Fourier law (γ T = 0). The results are found to be in excellent agreement with Adnan et al. [35]. Table 3 demonstrates that the non-dimensional Nusselt number (−θ (0)) is compared with different values of λ and M when the fluid is pure and Newtonian with no heat source (Q s = 0), ξ = 1 (steady flow), and absence of double diffusion heat flux model (γ T = γ c = 0). It is noted in Table 3 that the comparison of the present results with the existing numerical results of Abbas et al. [38] is in good agreement. A brilliant relationship has been accomplished, which insists the validity of the FEM MATLAB code.

Results and Discussion
This portion gives some noteworthy results of the boundary value problem as finally comprised in Equations (12)- (16). A variational Galerkin method is utilized along with finite element discretization. A broad computational continuing is performed to see the reactions of velocities ( f (ξ, η), h(ξ, η)), temperature θ(ξ, η) and concentration φ(ξ, η) with the differing contributions of influential parameters. In addition, the outcomes for the Nusselt number as well as the coefficients of skin friction are additionally computed. For graphical results, one parameter varies while all the physical parameters are referred to constant values like Nt = 0.2, Nb = Q s = 0.2, β = λ = 1.0, Pr = 5, M = 1.0, Le = 9.0, γ c = 0.1, and γ T = 0.1.
The effect of M (magnetic field) and β (Casson fluid parameter) on f (ξ, η), h(ξ, η), θ(ξ, η) and φ(ξ, η) is depicted in Figure 3a-d. As demonstrated in Figure 3a-d, the advancing contribution of M subsides the primary velocity f (ξ, η) and the magnitude of secondary velocity h(ξ, η). As a matter of fact, the interaction of M delivers an impeding force (Lorentz force) to halt the momentum of the flow in the xy-plane. The perception of Figure 3b shows that h experiences reverse flow because of Lorentz resistive force; therefore, a rise in the h profile is seen close to the sheet and afterward it gets zero. However, as opposed to the velocity, the temperature θ and concentration φ display direct exceeding relation with parameter M (see Figure 3c,d). Similar dealings for progressing input of β is observed (see Figure 3a-d). The velocities show a decline due to the resistance force created by tensile stress because of elasticity.  Figure 4a,b exhibit that both the components of velocity are diminished close to the sheet and afterward experience a variance when λ is incremented. The primary velocity achieves its bigger value for λ = 0. This is to make reference to the extending of the sheet along the x-axis being dependable to increase momentum toward this direction, but the y-direction momentum is denied any supporting element. In this way, Figure 4a,b individually shows that the f is switched meagerly and the h is outstandingly switched. From Figure 4c,d, it is seen that both the θ and φ are incremented with the increasing of λ. The rise in temperature is caused by dissipation produced by receded velocity near the sheet. In addition, the rotational factor directly strengthens the diffusion process, and it gives rise to the species concentration. Furthermore, Figure 4a-d are introduced to clarify the variation of f , h, θ, and φ as affected by τ. It is to clarify that larger τ exemplifies the greater lapse of time after the jerk to the stretching sheet. Thus, Figure 4a reveals that f (ξ, η) is monotonically subsided against increasing τ. Likewise, from Figure 4b, it is seen that the h(ξ, η) is diminished with reverse flow close to the sheet and afterward it changes at some distance away from the sheet. Both of the components of velocity become smoother for larger τ. In opposition to velocity behavior, Figure 4c depicts that θ rises directly with τ. This is because a longer lapse to the stretch for sheet makes the flow smoother and convection currents for heat transportation are established to raise the temperature in the boundary layer regime. The increased values of τ set the φ(ξ, η) into swaying pattern as delineated in Figure 4d. It is seen that the curve of φ(ξ, η) descends close to the sheet, and it ascends away from the sheet. Figure 5a shows the implication of Buongiorno's model parameters on the temperature field. The thermophoretic force causes the moving of the nanoparticles from the hotter area to the cooler area and, subsequently, a greater heat move happens in the boundary layer area. Similarly, the quicker random motion of species particles in nanofluids increased the Brownian forces to boost heat transportation. Thus, the rise of θ(ξ, η) and improvement in the thermal boundary layer is reported in Figure 5a, respectively, when Nt and Nb are dynamically incremented. Figure 5b is presented to reveal the bringing down of temperature θ(ξ, η) when γ T is apportioned higher input values. The greater values of γ T mean lesser thermal diffusion and hence the decline in temperature field occurs. In contradiction to the impact of γ T , the incremented Q s boosts the temperature θ(ξ, η) as displaced in Figure 5b.  As proved in Figure 6a, φ(ξ, η) is lessened when the Lewis number is augmented. Physically, the high Lewis number corresponds to bring down mass diffusivity, and, subsequently, the lesser species concentration in nanoliquid results. Figure 6a is also demonstrated to reveal the bringing down of φ(ξ, η) when γ c is designated higher input values. Figure 6b shows the ramifications of Buongiorno's model parameters on the φ(ξ, η). The thermophoretic force causes the moving of the concentration layer from the lower area to the higher. Thus, the quicker random movement of species particles in nano liquids raised the Brownian forces to boost up the φ(ξ, η).   Nusselt number and reduced Sherwood number. Both of these quantities become uniform against small ξ. Furthermore, Figure 7a,c,d respectively exhibit a similar trend for the Casson fluid parameter β, but opposing behavior is observed for C f y Re 1/2 y (see Figure 7b). The perceptions for Figure 8a portrays that increasing λ created a significant increase in the magnitudes of primary skin friction C f x Re 1/2 x . The diagrams for each estimation of λ get uniform at small estimations of ξ (ξ > 0). Additionally, all the negative estimations of C f x Re 1/2 x show the reversal of primary flow at the surface. In Figure 8b, C f y Re 1/2 y shows negatively raised magnitude with higher λ.     Figures 9a,b and 10a,b respectively draw sketches of reduced Nusselt number and reduced Sherwood number against magnetic field M, rotating parameter λ, and Casson fluid parameter β for varying values of the combine parameters Nb, Nt, and heat source Q s . It is revealed that increments in thermophoresis and Brownian motion parameters recede the wall heat transfer rate, but they boost the wall mass transfer rate. Similar results for wall heat transfer rate and wall mass transfer rate against the Q s are perceived (see Figures 9a,b and 10a,b). Figures 9a,b and 10a,b, in their respective order, demonstrate the meager reducing impacts of M and λ on reduced Nusselt number, and reduced Sherwood number. Furthermore, it is seen that increments in β recede the wall heat transfer rate and wall mass transfer rate. Figure 11a shows the consolidated impact of the Nb (Brownian motion) and Nt (thermophoresis) on the reduced Nusselt number for two cases of Prandtl number, that is, Pr = 1.0 and Pr = 5.0, respectively. It is revealed that increments in thermophoresis and Brownian motion parameters recede the wall heat transfer rate but Pr = 5.0 boost the wall heat transfer rate. Similar results for wall mass transfer rate against the rotational parameter and Prandtl number are perceived (see Figure 11b). The reduced Sherwood number is delineated against different values of Lewis number Le for two cases of Lewis number, that is, Le = 10.0 and Le = 15, respectively. The Lewis number (Le) increases the wall mass transfer rate as disclosed in Figure 11b.

Conclusions
This computational and theoretical work addresses the 3D time-dependent magnetohydrodynamics rotational flow of Casson nanofluids across an extending sheet with double diffusion Cattaneo-Christove and heat source. The transformed 2D partial differential formulation is solved by variational Galerkin procedure. Numerical findings for velocity components, skin friction coefficients, temperature, Nusselt number, nano-particle volume fraction, and Sherwood number are computed for influential parameters. Some of the major outcomes are reported below: • The progressing values of Casson fluid parameter β and magnetic parameter M reduced the magnitude of secondary velocity h(ξ, η) and the primary velocity f (ξ, η) but concentration φ and temperature θ are incremented. • The concentration and temperature are incremented along with rising values of λ.
Both components of velocity diminish near the surface when λ is incremented. • The decline of temperature is noted when thermal relaxation parameter γ T is progressively incremented, but improvement in temperature is reported when Nb, Nt, and Q s are increased.

Conflicts of Interest:
The authors declare no conflict of interest.