Finite Element Analysis of Thermo-Diffusion and Multi-Slip Effects on MHD Unsteady Flow of Casson Nano-Fluid over a Shrinking/Stretching Sheet with Radiation and Heat Source

In this article, we probe the multiple-slip effects on magnetohydrodynamic unsteady Casson nano-fluid flow over a penetrable stretching sheet, sheet entrenched in a porous medium with thermo-diffusion effect, and injection/suction in the presence of heat source. The flow is engendered due to the unsteady time-dependent stretching sheet retained inside the porous medium. The leading non-linear partial differential equations are transmuted in the system of coupled nonlinear ordinary differential equations by using appropriate transformations, then the transformed equations are solved by using the variational finite element method numerically. The velocity, temperature, solutal concentration, and nano-particles concentration, as well as the rate of heat transfer, the skin friction coefficient, and Sherwood number for solutal concentration, are presented for several physical parameters. Next, the effects of these various physical parameters are conferred with graphs and tables. The exact values of flow velocity, skin friction, and Nusselt number are compared with a numerical solution acquired with the finite element method (FEM), and also with numerical results accessible in literature. In the end, we rationalize the convergence of the finite element numerical solution, and the calculations are carried out by reducing the mesh size.


Introduction
The basic idea of the non-slip condition is concerned with the Navier Stokes theory. To study the temperature and velocity, numerous authors have found both the analytical and numerical solution by the implementation of non-slip boundary conditions. The utmost significance of slip-conditions in nano-channels has stimulated, to a great extent, interest in the study of vibrating values [1]. Now it is not unknown that if fluid contains concentrated suspensions, in that case the slip could be stirring. Soltani and Yilmazer [2] have performed by parallel disk rheometer with prominence on the wall slip phenomena on the rheological characterization of extremely filled suspensions consisting of Newtonian matrix and diverse with two different power sizes of aluminum and two different sizes of bed glass. In the articulations of fluid-like as suspension, emulsion foams, and polymer solution, Abbas et al. [20] deliberated the consequences of radiation in the existence of a uniform magnetic field of nano-fluid on the curved stretching surface by integrating the slip effect. In recent times, Makinde et al. [21] deliberated a numerical study of the effects of radiation on chemically reacting MHD nano-fluid prejudiced by the heat source/sink, and the collective heat and mass transfer analysis aimed at mixed convection flow over the vertical surface, with radiation and chemical reaction explained by Ibrahim et al. [22]. Prasannakumara et al. [23] deliberated the velocity slip effects, temperature jump, solutal slip, and thermal radiation, on steady flow, and the transfer of heat and mass of incompressible Jeffrey nano-fluid over the horizontal stretching surface. Imtiaz et al. [24] scrutinized the unsteady MHD flow of the curved stretching surface.
The literature remains silent whether we have to investigate the impact of multiple slips on MHD unsteady Casson nano-fluid flow, heat, and mass transfer in the presence of heat source with thermo-diffusion effect over a stretching/shrinking sheet. The objective of this article was to prolong the recent work of Fazle Mabood and Standford Shateyi [25]. Appropriate similarities have been used for transformation, and the governing non-linear partial differential equations are rendered in the non-dimensional and non-linear system of ODEs. The resulting system of non-linear ODEs has been solved numerically with an efficient and validated variational finite element method (FEM) with the boundary conditions. We also use a special case for the existing model to compare our outcomes along with previous studies. A parametric study has been performed to inquire into the mass and heat transfer characteristics and also the impact of different parameters of the flow. After that, we have provided a numerical comparison of our results and discussed them with graphs. In future studies, transient flow with slip effects in the presence of mixed convection and chemical reaction at the sheet can be examined.

Mathematical Modeling
The unsteady two-dimensional MHD flow of an incompressible fluid and Casson nano-fluid viscous flow over an electrically conducted stretching/shrinking sheet in the presence of heat source was considered. We choose an xy-ordinate system as the measurement of the sheet will be taken with x-, y-axis, and the y-axis along the vertical direction of the sheet, as given in Figure 1. The sheet is moving with non-uniform velocity U(x, t) = ax/(1 − λt) since a is the stretching rate along the x-axis but λt is positively constant, as by property λt < 1. The transverse magnetic field is supposed be the function of distance from the origin and is defined as where B 0 is the strength of the magnetic field. The induced magnetic field is negligible as compared to the applied magnetic field. The free stream temperature is supposed to be T ∞ , the free mass concentration is C ∞ , and the nano-particle concentration is E ∞ . The governing equations for flow can be put into the form as in ( [25,26]) ∂u ∂x where x and y are the co-ordinates in the x-axis and y-axis; u and v are the velocity components along the x-axis and y-axis respectively; α, ν, σ, ρ are the thermal diffusivity, kinetic viscosity, electrical conductivity, and density of fluid, respectively; g is the acceleration due to gravity; β T is the thermal expansion coefficient; β C is the solutal concentration expansion coefficient; β E is the nano-particle concentration expansion coefficient; T is the temperature; C is the solutal concentration; E is the nano-particle concentration; D M is the molecular diffusivity; D T is the thermal diffusivity; D B is the Brownian diffusivity; σ * is the Stefan-Boltzmann constant; k * is the mean absorption coefficient; T ∞ is the stream temperature; Q is the uniform volumetric heat generation/absorption; C p is the specific heat of fluid; C s is the concentration of susceptibility; and K T is the thermal diffusion ratio. The boundary conditions for the above mathematical model are (see [25]): where x and y are the co-ordinates; u and v are the velocity components along the x-axis and y-axis, respectively; α,ν, σ, ρ are the thermal diffusivity, kinetic viscosity, electrical conductivity and density of fluid, respectively; g is the acceleration due to gravity; β T is thermal expansion coefficient; β C is the solutal concentration expansion coefficient; β E is the nano-particle concentration expansion coefficient; T is the temperature; C is the solutal concentration; E is the nano-particle concentration; D M is the molecular diffusivity; D T is the thermal diffusivity; D B is the Brownian diffusivity; σ * is the Stefan-Boltzmann constant; k * is the mean absorption coefficient; T ∞ is the stream temperature; Q is the uniform volumetric heat generation/absorption; C p is the specific heat of fluid; C s is the concentration of susceptibility; and K T is the thermal diffusion ratio.
Where the injection/suction velocity as is the temperature of sheet and C w (x, t), E w (x, t) are concentrations at surface of the below form (see [25]): where T 0 , C 0 , and E 0 are the reference temperature, reference solutal concentration, and reference nano-particle concentration respectively, such that 0 ≤ T 0 ≤ T w , 0 ≤ C 0 ≤ C w and 0 ≤ E 0 ≤ E w , with these above expressions being valid if (1 − λt) > 0.
Usually, the stream function Ψ is defined as u = ∂Ψ ∂y and v = − ∂Ψ ∂x that the Equation (1) is satisfied. We introduce similarity transformations to solve the above equations (see [25,26]): In view of the above similarity transformations of Equation (7), PDEs from (1)-(5) transform into the following system of nonlinear ODEs: (1 + R) 1 Pr 1 Sc for the above problem the transformed boundary conditions are: where unsteadiness parameter σ = λ/a, M is the magnetic field parameter, Pr is the Prandtl number, Nb is the Brownian motion parameter, Nt is the thermo-phoresis parameter, β is the Deborah number, Sc is the Schmidt number, R is the thermal radiation parameter, λ 1 , λ 2 , and λ 3 are the buoyancy parameters, k p is the permeability parameter, Nd is the Dufour parameter, Sr is the Soret parameter, and f w is the Suction/injection parameter. The primes show the differentiation with respect to η. The parameters used in Equations (8)-(11) are explained as: Also, the local skin friction co-efficient, local Nusselt number, and local Sherwood number are explained as below when Equation (7) is substitited into Equations (17)- (19), the final dimensionless form is obtained; where the local Reynolds number is Re x , reduced skin friction is C f r, reduced Nusselt number is Nur, and the reduced Sherwood number is Shr.

Finite Element Method Solutions
We solve the system of non-linear boundary value problem that is given in Equations (8)-(11) numerically by applying the finite element method (FEM) subjected with the boundary conditions (12)- (16). The FEM is extremely effectual and has been applied to study different problems in fluid mechanics, CFD, mass transfer, heat transfer, solid mechanics, and also in many other fields. The general detail of the finite element method (FEM) can be found in Raddy [26][27][28]. Reddy [29] gives a general detail of the variational finite element method, which also found that the finite element method (FEM) is employed in commercial software like ADINA, ANSYS, MATLAB, and ABAQUS. Swapna et al. and Rana et al. [30,31] explain that the variational finite element method solves the boundary value problem very efficiently and accurately. To solve the non-linear boundary value problem (8)-(11) by using finite element method (FEM), along with boundary conditions (12)- (16), to apply FEM, first we have to consider: The Equations (8)-(11) take the form The corresponding boundary conditions now reduce to the following form:

Finite Element Formulations
The equation of finite element model that is gained from the Equations (27)-(31) by exchanging the FEM approximation in the form: with where the shape function ψ e i are the shape functions for the element (η c , η c + 1) and are taken as Therefore, the finite element model equations are given by where [W mn ] and b m (m,n = 1,2,3,4,5) are the matrices and are given as: and j=1φ j ψ e j are assumed to be known. After the assemblage of element equations, a consequential system of non-linear equations is obtained, after that, it requires an iterative scheme to solve it for an efficient solution. The calculation of f , g, θ, and φ are then conceded out for a higher level, and then proceeding until the required 0.00005 is not obtained. The results in Table 1 show the convergency, as we computed, for the increasing number of elements, n = 300, 500, 1000, 1300, 1500, 1800, 1900, 2100. From the results, it is clear that as the number of elements increases further, no significant difference in the values of f , g, θ, φ, and ξ can be seen as the number of elements increases beyond 1800, so the outcomes at n = 1800 elements are reported.

Results and Discussion
The numerical calculations have been performed for the velocity, temperature, solutal, and Casson nano-fluid volume fraction functions for a different assessment of physical parameters, as magnetic parametre M, Prandtl number Pr, Unsteadiness σ, permeability k p , Brownian motion Nb, thermophoresis Nt, thermal radiation R, Dufour Nd, Schmidt Sc, buoyancy λ 1 , λ 2 , λ 3 , Suction/Injection f w , Soret Sr, Lewis number Le, hydro-dynamic slip S f , thermal slip S θ , solutal slip S φ , and nano-particles concentration slip S ξ .
The detail of the present results and the appraisal of flow velocity is made with the exact solution that is given by Crane [32] as f (η) = 1 − e (−η) under the special case (M = 0, σ = 0, β → ∞, λ 1 = λ 2 = λ 3 = 0, S f = 0, f w = 0, kp = 0). The finite element method's outcomes have decent concurrence with the exact solution, which approves the validity of the finite element method. It can be seen clearly in Table 2, and in Table 3, that the skin friction coefficient attained by the finite element method is equivalent to the numerical outcomes of Gireesha et al. [33] and the exact solution of Mudassar et al. [34] under special case σ = 0, β = 0, λ 1 = λ 2 = λ 3 = 0, S f = 0, f w = 0, kp = 0. To confirm the accuracy of the presented numerical results, the results obtained by the finite element method for skin friction co-efficient for steady and unsteady flow have been compared with the numerical results that have previously been reported in studies and shown in Table 4. Regarding our results, there is admirable conformity among our outcomes and previously available research articles, which approves the cogency and the accuracy of the current results that are obtained by the finite element method (FEM). Table 5 describes the results of the heat transfer rate that are acquired by the finite element method, which are compared with the results of earlier studies and with the accurate solution of Ishak et al. [35] in a special case (Nt = Nb = 0). We observe that our results are in complete agreement and the grid invariance test has performed to sustain accuracy for 4 decimal points. In Table 6, local skin friction coefficient − f (0), the rates of heat transfer −θ (0), and mass transfer −φ (0) that are acquired by FEM, which are also compared with the published research work, show excellent correlation.    Table 6. Comparison of − f (0), −θ (0), and −φ (0) for different values of Pr, and Sc when M = σ =   Figures 3 and 4 exemplify that the velocity boundary layer thickness increases with enhancing the thermal buoyancy parameter λ 1 and the nano-particles concentration buoyancy parameter λ 3 in the non-existence of hydro-dynamic slip and the existence of hydro-dynamic slip. In these two cases, we have to perceive that the momentum boundary layer is prolonged with the increasing values of buoyancy λ 1 and λ 3 . Also in Figures 3 and 4, the velocity profile is increasing in the case of steady and unsteady flow and alike behavior of solutal buoyancy has been observed in Figure 5. It is observed that increasing the value of radiation parameter R is the reason to improve the velocity profile and incentive of instability in parameter k p , and the velocity profile is explained in Figure 6.   Also, we perceive in both cases that as the permeability of k p increases, this causes a decline in the velocity profile with no hydro-dynamic slip and with hydro-dynamic slip. Furthermore, we also perceive in Figure 6 that the suction decreases the momentum boundary layer thickness. Similar behavior of the Deborah number β has been perceived on the velocity profile in Figure 7. Figure 8 indicates the impact of M on the temperature profile with the non-existence of thermal slip and with existence thermal slip. In the figure, it is clear that with an increasing value of M the temperature profile also increases in both cases. From Figure 8 it is also clear that suction f w decreases the thermal boundary layer thickness in both cases and the thickness of the thermal boundary layer can be controlled with the help of suction f w . Figure 9 depicts the similar behavior of M. It is perceived that the temperature increases with the discrepancy in parameter R.  Figure 10 demonstrates the influence of σ and Pr on the temperature profile with and without the existence of thermal slip, with increasing Pr, σ, and S θ causing the thermal boundary layer to decrease. The description of the effect of N b , f w , and S θ with the temperature profile is explained in Figure 11. It is clear that with the increase in N b , the thermal boundary layer of fluid flow increases in both cases. It can also be perceived that f w decreases the thermal boundary layer thickness in both cases. In Figure 12, there is the influence of M and Sc on the solutal profile with the non-existence of solutal slip and with the existence of solutal slip. From the figure, it is clear that with the increasing value of M, the solutal profile also increases in both cases. Furthermore, the effect of S c over the solutal profile declines the profile. The impact of the existence and non-existence of solutal slip and increasing values of S r is depicted in Figure 13. It is seen that increasing the value of the Soret parameter increases the solutal profile in both the steady and unsteady cases. It is also seen that increasing the value of the slip results in decreasing the thickness of the solutal profile boundary.  Figure 14 indicates the impact of N t on the nano-particle volume fraction profile without the existence of nano-particles concentration slip and with the existence of nano-particles concentration slip. It is clear from the figure that in both the steady and unsteady cases, increasing values of N t cause a decline in the nano-particle volume fraction profile. The impacts of σ and S γ over the nano-particle volume fraction profile have been illustrated in Figure 14. It is noticed that the nano-particle volume fraction profile decreases in increments in the values of σ and S γ . Figures 15 and 16 show the impact of Lewis number Le, unsteadiness parameter σ, and suction parameter f w on the nano-particle volume fraction profile, and interpretations about the nano-particle volume fraction profile are made. It is seen that increasing values of Le and f w in the boundary layer of nano-particles causes the volume fraction profile to decline. In addition, it is clear from the figures that the nano-particle volume fraction profile decreases both with and without the existence of nano-particles concentration slip. Figure 17 shows the impact of M, σ, and S f on the skin friction co-efficient and also shows that the skin friction co-efficient decreases with the increase of slip parameter, magnetic and unsteadiness parameter. In Figure 18, the skin friction coefficient increases with the increment in the thermal buoyancy, suction parameters, and solutal buoyancy. Figure 19 represents the increment in the value of Nusselt number with increasing values of magnetic, radiation and thermal slip parameters. As perceive in Figure 20 that when the values of solutal buoyancy and suction intensity increases with the existence and the non-existence absence of thermal buoyancy. Figure 21 demonstrates that the impact of Sc, Sr, and S φ on the reduced Sherwood number. Also from the Figure 22, it clear that the reduced Sherwood number decreases with increment in the values of the unsteadiness parameter σ, magnetic parameter M and the suction parameters f w .  Figure 18. Effects of f w , λ 1 , and λ 2 on the skin friction coefficient.  In Table 7 we analyze the variation of physical parameters M, λ 1 , λ 2 , λ 3 , σ, P r , L e on skin friction co-efficient − f (0), Nusselt number −θ (0), and Sherwood number −ξ (0).

Chamkha et al. [37] FEM(Our Results)
The following results are concluded from Table 7: (i) The skin-friction coefficient is increasing while reducing the local Nusselt and Sherwood numbers through improvement in the Magnetic parameter; (ii) The increment in thermal buoyancy parameters λ 1 , λ 2 , λ 3 causes the skin-friction coefficient to decrease while increasing the local Nusselt number and Sherwood number; (iii) With the increasing unsteadiness parameter σ, the skin-friction coefficient, local Nusselt, and Sherwood numbers are also increasing; (iv) The skin-friction coefficient is increasing with the increment in Prandtl number and the local Nusselt number and Sherwood number also increase; (v) The skin-friction coefficient is increasing with the increasing Lewis number and also increment in the local Nusselt number and Sherwood number. Table 7. Various mathematical values of physical constraints M, λ 1 , λ 2 , λ 3 , σ, P r , L e when R = f w = S r = S t = S f = S p = S g = 0.5, k p = 0. 1, N u

Conclusions
In this paper, we have developed a mathematical model to investigate the unsteady two-dimensional magnetohydrodynamic (MHD) flow and heat transfer of an incompressible electrically conducted fluid. This study was conducted to analyze the multiple slip effects on magneto-hydrodynamic unsteady Casson nano-fluid flow over a non-linear porous shrinking/stretching sheet in the presence of a heat source with Soret effect. Appropriate similarities have been used for transformations and the governing partial differential equations (PDEs) are rendered into a system of ordinary differential equations (ODEs). The resulting system of ordinary differential equations was solved numerically with an efficient and validated finite element method (FEM). We also use a special case of the present model to compare our results with previous studies. A parametric study has been performed to explore the mass and heat transfer characteristics, and also the impact of different parameters, of the flow. The following are the results that can be concluded from the present study.

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The cause of reducing the fluid velocity near the region of the boundary layer is the increment in values of slip, suction, magnetic field, and unsteady parameters.

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The increment in thermal buoyancy parameters λ 1 , λ 2 , λ 3 and slip parameters cause decreasing the skin-friction coefficient while increasing the local Nusselt number and Sherwood number.

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Decreasing the value of the Sherwood number causes an increasing value of the solutal slip parameter, Schmidt number, and Soret number, but the effect is the opposite with increasing values of the unsteadiness, magnetic, and suction parameters.

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With the increasing value of the magnetic parameter M, the slip velocity parameter and suction parameter are found to be reduced in the velocity profile.

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The skin-friction coefficient decreases with the increasing value of slip, magnetic, and unsteadiness parameters, but the effect is the opposite for increasing values of thermal buoyancy, suction parameter, and solutal buoyancy.