Impacts of Viscous Dissipation and Brownian Motion on Jeffrey Nanofluid Flow over an Unsteady Stretching Surface with Thermophoresis

: The goal of this investigation is to explore the influence of viscous dissipation and Brownian motion on Jeffrey nanofluid flow over an unsteady moving surface with thermophoresis and mixed convection. Zero mass flux is also addressed at the surface such that the nanoparticles fraction of maintains itself on huge obstruction. An aiding transformation is adopted to renovate the governing equations into a set of partial differential equations which is solved using a new fourth-order finite difference continuation method and various graphical outcomes are discussed in detail with several employed parameters. The spectacular influence of pertinent constraints on velocity and thermal curves are inspected through various plots. Computational data for the heat transfer rate and skin-friction coefficient are also reported graphically. Graphical outcomes indicate that an augmentation in buoyance ratio and thermophoretic parameter leads to diminish the velocity curves and increase the temperature curves. Furthermore, it is inspected that escalating Deborah number exhibits increasing in the skin friction and salient decreasing heat transmission. Increasing magnetic strength leads to a reduction in the skin friction and enhancement in the Nusselt number, whilst a reverse reaction is manifested with mixed convection aspects.


Introduction
In modern times, the novel investigations of nanofluid flow through stretchable surfaces have got valuable attention among study scientists and community because of its abundant practical utilization, in several areas of science and biotechnology. Nanofluids play a momentous role in the mechanism of heat transfer. The typical base fluids, e.g., water, alcohol, ethylene glycol, and oil have a weak capacity to promote the heat transfer rate. However, this complex scenario was resolved by asserting the tiny sized solid fragments in the base fluids. It was basically proposed by Choi [1] that the tendency of the base fluids to embellish the thermal properties can be more effective by adding nanoparticles in these fluids. Nanofluids are spawned by dispersion of nanoparticles along with base fluids and these fluids are the amalgamations of suspended nano-sized pieces (1-100 nm) in base fluids. The constituents of nanoparticles may contain metals, carbides, nitrides, and oxides.

Modeling
Here, the magneto-mixed convection Jeffrey-nanofluid flow is treated cross over an unsteady vertical moving surface. Brownian motion characteristics are addressed through thermophoresis and dissipation effects. Constant magnetic strength, B0, is utilized perpendicular to the surface and zero nanoparticles mass flux is assumed. As proposed, the constant temperature at the surface Tw, whilst the ambient temperature field is T∞, and the concentration distribution is C∞. The flow sketch is exhibited in Figure 1. The flow under these attentions can be evaluated in the ensuing form: (1 ) x y y y x y y t y y y y The corresponding initial and boundary conditions are: Note that u and v characterize the flow velocities in x-and y-trends respectively while υ = μ/ρ, μ, ρ describe kinematic viscosity, dynamic viscosity, and density of base liquid respectively. The symbols g are for gravity g, and λ1 for retardation time; λ2 for the ratio of relaxation to retardation times; σ symbolizes the electrical conductivity; B0 symbolizes the magnetic field; T the temperature; αm = k/(ρC)f, k, (ρC)f, (ρC)p symbolize the thermal diffusivity, thermal conductivity, liquid heat capacity, and nanoparticles effective heat capacity; is the nanofluid heat capacity ratio, DB the Brownian diffusion coefficient, DT for the thermophoretic diffusion coefficient; and T and C for fluid temperature and concentration, respectively. Launching the dimensionless variables as: into Equations (1)-(5) gives us the following dimensionless equations: Nb Nt Ec The transformed initial and boundary conditions become: where ()′ designates differentiation concerning the transformed transverse coordinate, η. The parameters in Equations (7)-(10) are defined as follows: where Nb symbolizes the Brownian motion parameter, Nt symbolizes the thermophoresis parameter, Nr reflects the buoyancy ratio parameter, Sc reflects the Schmidt number, β relates the Deborah number, Ec symbolizes the Eckert number, λ symbolizes the mixed convection parameter, Grx and Rex define the Grashof and Reynolds numbers, respectively, Ha reflects the Hartmann number, and Pr symbolizes the Prandtl number. The quantities of engineering interest in dimensionless form skin friction and Nusselt number may be defined as:

Fourth-Order Finite Difference Continuation Method (FFDCM)
The partial differential equations (PDE) system (7)-(10) has a multi-scale solution behavior over an infinite interval where the solution varies quickly over the boundary layer and away from this layer the solution varies slowly and behaves regularly and that is according to the time constants of the solution components. Moreover, the PDE system (7)-(10) is very sensitive to the initial conditions due to the singularity associated with the highest derivative-term in the system and hence more accurate and efficient adaptive methods are required for solving this class of PDE systems [5,[27][28][29][30][31][32]. System (7)-(10) is converted into a first-order PDE system and discretized using a fourth-order finite difference method in η -orientation and a two-point backward finite difference method [28] in τ -orientation. The solution is obtained through a continuation technique in η -orientation. We will denote the suggested method as fourth-order finite difference continuation method (FFDCM).
where D is the fourth-order finite-difference differentiation matrix defined in [27], ( 1 ) The solution of (17)-(18) is obtained through a continuation technique in η -orientation taking η ∞ as our continuation parameter. Starting with an initial estimation of the parameter η ∞ and the initial guess † Z , an estimate solution † Z  is obtained for system (17) The numerical results are obtained at 0.05 τ Δ = , 0.005 κ Δ = using absolute tolerance 6 10 − and relative tolerance 3 10 − for the nonlinear Matlab solver 'fsolve'. Table 1 shows the good agreement between the numerical results obtained by FFDCM and Ishak et al. [33] approach for different values of λ and Pr for

Discussion
This precise segment is dispensed to analyze the magneto-mixed convection and heat transport characteristics for convection Jeffrey-nanofluid flow towards an unsteady vertical moving surface. The graphical explanation is deliberated and the variation of velocity, temperature, skin friction and Nusselt number for various thermophysical parameters are characterized in θ τ η curves for magnifying the buoyance ratio Nr and thermophoretic Nt is pointed out in Figures 2b and 3a. It is manifested that velocity diminishes with the augmentation of Nr and Nt, and this phenomenon yields an increase in temperature curves ability that boosts the thermal boundary. This is due to the reality that greater values of Nr boost the temperature fluid. Figure 4a,b uncovers the impacts of Nr and Nt on skin friction coefficient Cf(τ,0) and the local Nusselt number Nu(τ,0) with several values of β. It is observed that both of Cf(τ,0) and Nu(τ,0) diminish sufficiently for physical parameters Nr and Nt. This because the thermophoresis strength triggers the nanoparticles to transmit the warm sheet to cold sheet which creates the thermal related boundary layer thickness to magnify. In addition, an enlarge in β corresponds to an enhancement in the Cf(τ,0) and salient reduction in Nu(τ,0). This is because a weaker Deborah number, β, gives a viscous impact compared to the elastic impact, whilst a greater β exhibits in the elastically solid material in nature which leads to a reduction in Nusselt number.
(a) (b)  Figure 5a,b. From these curves, it is concluded that intensifying λ improving the flow velocity and decaying the temperature curves. Because when growing λ, the impact of gravity is decreased, and then velocity within a boundary layer decreases. In addition, it is shown that the sway of magnetic strength reduces the fluid movement. This is realistic because magnetic strength is responsible to inspire Lorentz intensity which resists the fluid motion. Figure 6a,b determines the Cf(τ,0) and Nu(τ,0) against the λ and Ha with several values of Ec. It is noticed that Nu(τ,0) enhances but Cf(τ,0) reduces for the improved values of Ha, while an opposite reaction is found with λ. The reason, as mentioned above, is that Ha has a tendency to magnify the Lorentz intensity. This intensity resists the fluid movement and provides heat ability in the flowing. Furthermore, it is clear that an elevation in Ec results in an enhancement in the internal source of energy which boosts the thermal boundary layer, which results in a reduction in the Nusselt number for greater Ec. Figure 7a,b are plotted to analyze the impacts of the ratio of relaxation to retardation times λ2 and Schmidt number Sc on velocity ( , ) f τ η ′ and temperature ( , ) θ τ η curves. It is apparent that both of velocity and temperature curves decline with the enhancement in λ2 parameter while opposite results are obtained for the intensity of Sc. The consequences of Sc and λ2 on Cf(τ,0) and Nu(τ,0) on along with Nb is visualized in Figure 8a,b. It is observed that Cf(τ,0) reduces and Nu(τ,0) enhances by augmenting λ2. Moreover, both Cf(τ,0) and Nu(τ,0) reduce sufficiently by enlarging Sc. This may occur owing to the cause that Schmidt number is reversely symmetrical to the Brownian diffusion coefficient. This Brownian diffusion coefficient becomes lower corresponding to the greater values of Sc and also Brownian movement happens in the system of the nanofluid due to contact of nanoparticles with the regular fluid.

Conclusions
The effectiveness of Brownian motion and viscous dissipation on magneto-mixed convection flow of Jeffrey nanofluid through an unsteady moving surface is examined with thermophoresis. An aiding transformation is adopted to renovate the governing equations into a set of PDEs which are sensitive to the initial conditions due to the singularity associated with the highest derivative-term and so the numerical solution is gotten with the aid of a new FFDCM and various graphical outcomes are discussed in detail with several employed parameters.

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A comparative investigation among the current outcomes and the former cited investigation are explored to trust our outcomes and a notable agreement is observed. The following observations are structured for the current investigation: • Augmentation in buoyance ratio and thermophoretic parameters leads to diminish the velocity curves and increase the temperature curves ability that boosts the thermal boundary. • A greater Deborah number exhibits increasing skin friction and salient decreasing heat transmission.

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The Nusselt number enhances and skin friction reduces for the improved magnetic strength, while an opposite reaction is found with mixed convection aspects. • Both velocity and temperature curves decline with the enhancement in the ratio of relaxation to retardation times while opposite results are obtained for the intensity of Schmidt number.