Brownian Motion and Thermophoresis Effects on MHD Three Dimensional Nanofluid Flow with Slip Conditions and Joule Dissipation Due to Porous Rotating Disk

This paper examines the time independent and incompressible flow of magnetohydrodynamic (MHD) nanofluid through a porous rotating disc with velocity slip conditions. The mass and heat transmission with viscous dissipation is scrutinized. The proposed partial differential equations (PDEs) are converted to ordinary differential equation (ODEs) by mean of similarity variables. Analytical and numerical approaches are applied to examine the modeled problem and compared each other, which verify the validation of both approaches. The variation in the nanofluid flow due to physical parameters is revealed through graphs. It is witnessed that the fluid velocities decrease with the escalation in magnetic, velocity slip, and porosity parameters. The fluid temperature escalates with heightening in the Prandtl number, while other parameters have opposite impacts. The fluid concentration augments with the intensification in the thermophoresis parameter. The validity of the proposed model is presented through Tables.


Introduction
Nanofluid is the suspension (mixture) of base fluid (water, gasoline oil, kerosene oil, ethylene glycol) and nanometer-sized particles, which is called nanofluid. Nanofluids are made of different The MHD nanofluid flow subject to velocity slip conditions is considered here. The nanofluid flow is considered as time dependent and incompressible. The flow is studied over a rotating porous disk. The disk rotates along z−axis with angular velocity Ω (see Figure 1). The magnetic field is functional along the z−direction. The electric and Hall current influences are ignored throughout the study. The fluid flow is treated with viscous dissipation impact. The heat and mass transmission characteristics are analyzed in the presence of thermophoresis and Brownian motion impacts. The nanofluid flow is based on the present situations [5,22,29]: uw r + ww z = υ w rr + w r r + w zz , uT r + wT z = α T rr + T r r + T zz uC r + wC z = D B C zz + C r r + C rr + D T T ∞ T zz + T r r + T rr ,

Problem Formulation
The MHD nanofluid flow subject to velocity slip conditions is considered here. The nanofluid flow is considered as time dependent and incompressible. The flow is studied over a rotating porous disk. The disk rotates along z  axis with angular velocity  (see Figure 1). The magnetic field is functional along the z  direction. The electric and Hall current influences are ignored throughout the study. The fluid flow is treated with viscous dissipation impact. The heat and mass transmission characteristics are analyzed in the presence of thermophoresis and Brownian motion impacts. The nanofluid flow is based on the present situations [5,22,29]   The consistent boundary conditions are Molecules 2020, 25, 729 4 of 20 The similarity transformations are defined as Using (8), (1) satisfies, and ((2)- (7)) are reduced as 2g 1 Pr where the dimensionless parameters are defined as: The dimensionless surface quantities are defined as Entirely the overhead factors are defined in nomenclature.

Analytical Solution
Here, the proposed model is elucidated by using HAM [40][41][42][43]. In view of ((9)-(12)) with (13); the primary assumptions are deliberated as: The L f , L g , L θ and L φ are picked as: with the following properties: where m i (i = 1 − 9) are constants. The resultant non-linear operators N f , N g , N θ , and N φ are indicated as: Molecules 2020, 25, 729 5 of 20 The zeroth-order problem is The equivalent boundary conditions are: where τ ∈ [0, 1] is the imbedding parameter and h f , h g , h θ , and h φ are used to regulate the convergence of the solution. When τ = 0 and τ = 1, we have: Expanding f (ξ; τ), g(ξ; τ), θ(ξ; τ) and φ(ξ; τ) by Taylor's series where Molecules 2020, 25, 729 6 of 20 The secondary constraints h f , h g , h θ , and h φ are selected, such that the series (29) converges at τ = 1, changing τ = 1 in (29), we get: The q th -order problem satisfies the following: The equivalent boundary conditions are: Here where

Convergence Solution
HAM guarantees the convergence of the series solution of the modeled problem. The auxiliary parameter h plays an important role in adjusting the region of convergence of the series solution.

Results and Discussion
The aim of this section is to visualize variations in velocities, temperature, concentration, Nusselt number, and skin friction coefficient due to involved parameters, like magnetic field ( M ), porosity (

Results and Discussion
The aim of this section is to visualize variations in velocities, temperature, concentration, Nusselt number, and skin friction coefficient due to involved parameters, like magnetic field (M), porosity (κ), velocity slip (ψ), Eckert number (Ec), heat source/sink (γ), thermophoresis (Nt), Prandtl number The escalating κ declines f (ξ) and g(ξ) is depicted in Figures 6 and 7. The porous media usually performs opposite behavior to the fluid flow. With an increase in the porous media, the fluid particles motion reduces and, thus, the fluid velocity diminishes. Therefore, the growing estimations of κ diminishes f (ξ) and g(ξ). Figures 8 and 9 depict the escalating ψ diminishes f (ξ) and g(ξ). The velocity slip parameter always performs a reverse impact on velocity profiles. The corresponding boundary layer thickness declines by ψ, which deescalates f (ξ) and g(ξ). Figure 10 depicts the impression of M on θ(ξ). It is witnessed that the escalating M escalates θ(ξ). The influence of γ on θ(ξ) is demonstrated in Figure 11. The heat source/sink plays like heat producer. As the parameter estimations intensify, the fluid particles temperature heightens. For that reason θ(ξ) upsurges. Figure 12 portrays the effect of Ec on θ(ξ). It is used for extremely fast compressible flow. The positive Eckert number represents the freezing of wall and, as a result, the convection of heat transmission to the fluid is augmented. Figure 13 shows the consequence of Pr on θ(ξ). Pr makes the association of fluid viscosity with thermal conductivity. The fluids have high thermal conductivity with large Pr, while the impact is reverse for higher Pr. Hence, the escalating estimations of Pr deescalates θ(ξ). Figure 14 illustrates the effect of Nb on θ(ξ). Higher Brownian motion induces the random acceleration of the fluid particles. Extra energy is generated because of this random acceleration. Therefore, the thermal rise is reported. Figure 15 presents the impression of Nt on θ(ξ). In the thermophoresis phenomenon, tiny fluid particles are forced back from those in the warmer to the cold surface. As a result, the fluid particles returned from those in the warmed surface and the thermal curve then increased. The outcome of Nb and Nt on φ(ξ) are shown in Figures 16 and 17. The higher estimations of Nb shows reverse impact on φ(ξ). Figure 17 illustrates the rising impression of Nt on φ(ξ). Figure 18 demonstrates the influence of Le on φ(ξ). Le is the correlation of mass diffusion to fluid thermal conductivity. The increasing Le causes thickness of the concentration layer, which consequently escalates the concentration profile.                                     Tables 1-3 are displayed to examine the surface drag force, heat transfer rate, and mass transfer rate, respectively. Table 1 depicts that the increasing M and κ reduce both C f and C g , while the rising values of ψ reduces C f and increases C g . From Table 2, the escalating M, ψ, Le, Nb and Nt reduces Nu, while the higher Pr escalates the Nu. From Table 3

Conclusions
The steady and incompressible flow of MHD nanofluid over a porous rotating disc with slip conditions is examined. The mass and heat transmission with viscous dissipation impact is also intentional. The problem is solved with the help of analytical and numerical methods. The core points of the current inspection are mentioned beneath: