MHD Flow and Heat Transfer in Sodium Alginate Fluid with Thermal Radiation and Porosity E ﬀ ects: Fractional Model of Atangana–Baleanu Derivative of Non-Local and Non-Singular Kernel

: Heat transfer analysis in an unsteady magnetohydrodynamic (MHD) ﬂow of generalized Casson ﬂuid over a vertical plate is analyzed. The medium is porous, accepting Darcy’s resistance. The plate is oscillating in its plane with a cosine type of oscillation. Sodium alginate (SA–NaAlg) is taken as a speciﬁc example of Casson ﬂuid. The fractional model of SA–NaAlg ﬂuid using the Atangana–Baleanu fractional derivative (ABFD) of the non-local and non-singular kernel has been examined. The ABFD deﬁnition was based on the Mittag–Le ﬄ er function, and promises an improved description of the dynamics of the system with the memory e ﬀ ects. Exact solutions in the case of ABFD are obtained via the Laplace transform and compared graphically. The inﬂuence of embedded parameters on the velocity ﬁeld is sketched and discussed. A comparison of the Atangana–Baleanu fractional model with an ordinary model is made. It is observed that the velocity and temperature proﬁle for the Atangana–Baleanu fractional model are less than that of the ordinary model. The Atangana–Baleanu fractional model reduced the velocity proﬁle up to 45.76% and temperature proﬁle up to 13.74% compared to an ordinary model.


Introduction
Due to the relevance of non-Newtonian fluids in some of the optimization processing of food items, the heat transfer phenomenon is an important research area. Non-Newtonian fluids are investigated of fractional Newtonian fluid embedded in a porous medium. Electrically conducting viscous fluid is considered in their study.
The flow of generalized second-grade fluid between parallel plates with the Riemann-Liouville fractional derivative model was investigated by Wenchang and Mingyu [24]. They acquired the exact analytical solution using the Laplace transform and Fourier transform. The flow of second-order fluid induced by a plate moving impulsively with fractional anomalous diffusion was investigated by Mingyu and Wenchang [25]. The Rayleigh-Stokes problem for a fractional second-grade fluid was studied by Shen et al. [26]. Fractional Laplace transforms and Fourier sine transform was employed to obtain the exact solution. Exact analytical solution for the unsteady flow of a generalized Maxwell fluid between two circular cylinders was determined via Laplace and Hankel transforms by Mahmood et al. [27]. Recently, Shen et al. [28] studied fractional Maxwell viscoelastic nanofluid for various particle shapes. A Caputo time-fractional derivative was implemented by Zhang et al. [29] to acquire the numerical and analytical solutions for the problem of the 2D flow of Maxwell fluid under a variable pressure gradian gradient. They used the separation of the variables method to acquire the analytical solution, while for numerical solution, the finite difference method was used. Aman et al. [30] studied fractional Maxwell fluid with a second-order slip effect for the exact analytical solution. Jan et al. [31] determined the solution for Brinkman-type nanofluid using an Atangana-Baleanu fractional model. Owolabi and Atangana [32] analyzed the numerical simulation of the Adams-Bash forth scheme using Atangana-Baleanu Caputo fractional derivatives. Saad et al. [33] established the numerical solutions for the fractional Fisher's type equation with Atangana-Baleanu fractional derivatives. They employed the spectral collocation method based on Chebyshev approximations. In this research work, the spectral collocation method was implemented for the first time to solve the non-linear equation with Atangana-Baleanu derivatives. Some plenteous literature regarding Atangana-Baleanu derivatives and analytical solution can be found in Saqib et al. [34], Abro et al. [35], and Hristov [36] and the references therein.
Due to the importance of MHD in fluids, many researchers have taken MHD in their studies. In recent years, Khan and Alqahtan [37] investigated the MHD effects for nanofluids in a permeable channel with porosity. The solution is obtained by using Laplace transformation. Further, in the same year, Asif et al. [38] studied the unsteady flow of fractional fluid between two parallel walls with arbitrary wall shear stress. The influence of MHD slip flow of Casson fluid along a non-linear permeable stretching cylinder saturated in a porous medium with chemical reaction, viscous dissipation, and heat generation or absorption is investigated by Ullah et al. [39]. Khan et al. [40] investigated the MHD with heat transfer. A generalized modeling is carried out for the proposed problem by using the new idea of a fractional derivative, i.e., Atangana-Baleanu and Caputo Fabrizio. After that, this idea is used by Gul et al. [41] for the study of forced convection carbon nanotube nanofluid flow passing over a thin needle. Atangana and Alqahtani [42] studied the Caputo fractional derivative for analysis of the spread of river blindness disease. Furthermore, the idea of fractional derivatives was examined by Gomez and Atangana [43] with the power law and the Mittag-Leffler kernel applied to the non-linear Baggs-Freedman model, while Muhammad and Atangana [44] examined for dynamics of Ebola disease, and Khan et al. [45] studied the analytical solution of the hyperbolic telegraph equation, using the natural transform decomposition method. Some other related references dealing with fluid motion, heat transfer, or fractional derivatives are given in [46][47][48][49][50][51][52][53].
On the basis of the above literature, this work aims to use the Atangana-Baleanu fractional derivative (ABFD) of the non-singular and non-local kernel for SA (Sodium alginate) fluid. The Laplace transform method is used to get the exact solution, which is graphically plotted via Mathcad-15 software and discussed in detail.

Mathematical Framing of the Problem
Let us consider the unsteady flow of Casson fluid over a vertical plate, i.e., the plate is taken perpendicular to the y − axis, with perpendicular employed magnetic field to the flow of the fluid. The plate is oscillating, and the medium is porous. The heat flux is also taken into consideration. Thermal radiation effect is also considered. Initially, both the plate and fluid are at rest. After t > 0, the plate started oscillation in its plane. The fluid is electrically conducting there by the Maxwell equation: By using Ohm's law: The magnetic field B is normal to V. The Reynolds number is so small that the flow is laminar. Hence: Keeping in mind the above assumptions, the governing equation of momentum and energy are given as (Khalid et al. [46): where T and u represent temperature and velocity. ρ, µ, β, σ, B 0 , k 1 , g, c p , k, q r and γ are the density, dynamic viscosity, thermal expansion coefficient, heat source parameter, magnetic parameter, porosity parameter, gravitation acceleration, heat capacity, thermal conductivity, heat flux, and Casson fluid parameter. Following Makinde and Mhone [47] and Cogley et al. [48], the fluid used is thin with a low density and radiative heat flux given by ∂q r ∂y = 4α 2 0 (T − T 0 ). The physical boundary conditions are: For non-dementalization, the following dimensionless variables are introduced.
With the intention of converting the ordinal time derivative to the Atangana-Baleanu fractional time derivative, Equations (5) and (6) reduced to: The Atangana-Baleanu fractional time derivative is defined as: where N(α) is the normalization function, is the Mittag-Leffler function. After Laplace transform, Equation (13) becomes: where q is represent the Laplace transform operator.

Problem Solution, Skin Friction, and Nusselt Number
Taking the Laplace transform of Equations (8) and (9) and by using Equation (7), we get: where: Note that in Equations (15) and (16), the bar on θ and u shows the Laplace transformed function. After taking the Laplace inverse, we get: where: where: ψ(y, t, a, b, c) = L −1 ψ(y, q, a, b, c) Here, ϕ is the Wright function, and is defined as: Then, the velocity profile becomes: where: is the Mittag-Leffler function. Note that for deep understanding of ABFD, on may refer to the excellent articles [35,53]. However, for the detailed solution procedure of the problem, one can refer to research works [15,18,20,23,31].

Skin Friction and Nusselt Number
Expressions for Nusselt number and skin friction are calculated from Equations (14) and (15) by using the relation from Khan et al. [49]:

Discussion
To study the influence of many embedded parameters on temperature and velocity, the graphs are plotted by using the Mathcad-15 software. Figure 1 shows the physical sketch of the problem.     Figure 2 is plotted for the influence of Casson parameter γ on velocity, which shows that if the value of the Casson parameter is increased, the fluid velocity increases. This is because of the circumstance that with a large value of γ, the yield stress falls through, and the boundary layer thickness reduces. Figure 3 investigates the impact of Grashof number Gr on velocity. The greater value Gr leads to an increase in the velocity of the fluid. Physically, the increase of Gr leads to an increase in the bouncy force, and as a result, the velocity of the fluid increases.     The effect of porosity parameter K against the velocity profile is investigated in Figure 4. To increase the value of the porosity parameter K , first we need to decrease the flow of fluid. Physically, the resistance of the porous medium is depressed, which raises the momentum development of the flow regime, and finally accelerates the velocity of the fluid. Figure 5   The effect of porosity parameter K against the velocity profile is investigated in Figure 4. To increase the value of the porosity parameter K, first we need to decrease the flow of fluid. Physically, the resistance of the porous medium is depressed, which raises the momentum development of the flow regime, and finally accelerates the velocity of the fluid. Figure 5 shows the influence of M on flow of the fluid; the rise in M decreases the flow of fluid. It is physically true because the increase of M means to increase the frictional force (Lorentz force), which leads to a decrease in the velocity of the fluid.     The influence of radiation parameter N is highlighted in Figure 6. For the higher value of N , the fluid velocity decreases. Physically, the increase in radiation parameter means the release of heat energy from the flow region, and so the fluid temperature decreases as the thermal boundary layer thickness become thinner. Figure 7 and Figure 8 illustrate the influence of Prandtl number Pr on velocity and temperature respectively, the increase of Pr causing a decrease in the temperature and as a result a decrease in the fluid velocity. The small degree of thermal diffusion causes expanding in The influence of radiation parameter N is highlighted in Figure 6. For the higher value of N, the fluid velocity decreases. Physically, the increase in radiation parameter means the release of heat energy from the flow region, and so the fluid temperature decreases as the thermal boundary layer thickness become thinner. Figures 7 and 8 illustrate the influence of Prandtl number Pr on velocity and temperature respectively, the increase of Pr causing a decrease in the temperature and as a result a decrease in the fluid velocity. The small degree of thermal diffusion causes expanding in velocity boundary layer width. Pr controls the comparative thickness of the momentum and thermal boundary layers in the heat transfer problems. Subsequently, Pr can be applied to develop the percentage of cooling.        Figure 11 investigates the effect of time fraction derivative parameter  on temperature. It is investigated that the time-fractional derivative parameter controls the temperature profile.  Figure 11 investigates the effect of time fraction derivative parameter α on temperature. It is investigated that the time-fractional derivative parameter controls the temperature profile.   Figure 10 illustrate the outcome of time t on temperature and velocity profiles. Figure 11 investigates the effect of time fraction derivative parameter  on temperature. It is investigated that the time-fractional derivative parameter controls the temperature profile.    A comparison of the Atangana-Baleanu fractional model with an ordinary model is investigated in Figure 12 and Figure 13 for velocity and temperature, respectively. For both cases, it is detected that the temperature and velocity profile for the Atangana-Baleanu fractional model is less than that of the ordinary model.
Note that all these graphs of velocity are plotted for the phase angle t  equal to 90 degrees, and this value cosine is zero; therefore, all the graphs of velocity (Figures 2-7, 9, 12) have a unique pattern of velocity. That is, at the plate surface y = 0, the fluid is at rest or there is no motion in the fluid, and for large values of the independent variable y, the fluid velocity decays, and as y approaches infinity (further bigger values of y), the fluid motion disappears and velocity tends to zero. This physical pattern of the graphs of velocity agrees with the imposed condition on velocity given in Equation (6). Similarly, the unique style of all the graphs of temperature, that is temperature at y = 0, is 1, and far away from the plate surface; that is, for larger values of y, the temperature decays and tends to zero as y tends to infinity. A comparison of the Atangana-Baleanu fractional model with an ordinary model is investigated in Figures 12 and 13 for velocity and temperature, respectively. For both cases, it is detected that the temperature and velocity profile for the Atangana-Baleanu fractional model is less than that of the ordinary model.
Note that all these graphs of velocity are plotted for the phase angle ωt equal to 90 degrees, and this value cosine is zero; therefore, all the graphs of velocity (Figures 2-7, Figure 9, Figure 12) have a unique pattern of velocity. That is, at the plate surface y = 0, the fluid is at rest or there is no motion in the fluid, and for large values of the independent variable y, the fluid velocity decays, and as y approaches infinity (further bigger values of y), the fluid motion disappears and velocity tends to zero. This physical pattern of the graphs of velocity agrees with the imposed condition on velocity given in Equation (6). Similarly, the unique style of all the graphs of temperature, that is temperature at y = 0, is 1, and far away from the plate surface; that is, for larger values of y, the temperature decays and tends to zero as y tends to infinity.     Table 1 clarified that the Nusselt number is increased when Pr, and N and are increased, while an increase in fractional derivative parameter  and t decreased the Nusselt number. The behavior of the present results is identical with the published results of Khan et al. [49] and Ali et al. [50]. Table 2 shows the impact of the deferent parameter on skin fraction. It is observed that ,, t Gr    Table 1 clarified that the Nusselt number is increased when Pr, and N and are increased, while an increase in fractional derivative parameter α and t decreased the Nusselt number. The behavior of the present results is identical with the published results of Khan et al. [49] and Ali et al. [50]. Table 2 shows the impact of the deferent parameter on skin fraction. It is observed that t, α, Gr, and Pr have a positive (increasing) impact on the skin fraction, while M, N, γ, and K and show a negative (decreasing) impact on the skin fraction. This behavior of skin fraction against different parameters is identical with the published results of Mackolil and Mahanthesh [51].

Conclusions
In this attempt, the exact solution for the heat transfer analysis in unsteady MHD flow of fractional SA fluid past a vertical flat plate is obtained. Generalized Casson fluid model is obtained using the Atangana-Baleanu fractional derivative (ABFD) of the non-local and non-singular kernel. The exact solution for velocity and temperature are obtained by applying the Laplace transform method, and then, the influence of various embedded parameters are presented graphically. Some important outcomes are: Velocity rises for a large value of Gr, γ , and t. Velocity reduces for a large value of M, Pr, N, and K. Temperature is increased by increasing t and α, while decreasing with the increase of Pr. The temperature and velocity of the fractional fluid model converge faster compared to an ordinary fluid model. The Atangana-Baleanu fractional model reduced the velocity profile up to 45.76% and temperature profile up to 13.74% compared to an ordinary model.

Suggestions for future research work.
The researchers extend this work for different kind of nanofluids.