**1. 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 much more, but Casson fluids among them have been investigated by a small number of researchers.

This model was introduced by Casson in 1959 [1]. This fluid has grea<sup>t</sup> importance in many fields of industries and medicines; see for examples, Qasim and Ahmad [2], Shehzad et al. [3], and Qasim and Noreen [4]. The Casson fluid model plays a vital role in many materials such as chocolate and blood, according to Mukhopadhyay and Mandal [5]. The metachronal coordination between the beating cilia in a cylindrical tube is one of the factors under which inspiration, and some mathematical formulations, is made to analyze Casson fluid flow, according to Siddiqui et al. [6]. A system of equations for non-linear flow problems are modeled with the help of axisymmetric cylindrical coordinates. Simplification is taken place by inducting long wavelength and low Reynolds number assumptions.

The mathematical model for Casson fluid for mixed convection taking a stretching sheet was studied by Hayat et al. [7]. The velocity and temperature fields are calculated by solving partial di fferential equations. The homotopy analysis method is incorporated. The Atangana–Baleanu fractional derivative (ABFD) definition was based on the Mittag–Le ffler function and promises an improved description of the dynamics of the system with the memory e ffects by Asjad et al. [8]. The mathematical model for some geological formation is often called an aquifer, which is commonly used to solve subsurface motion by a fractional-order derivative, as in Atangana and Baleanu [9]. It was an alternative to Caputo-Fabrizio [10]. A relation between the solution obtained by the Atangana–Baleanu fractional derivative and through experimental methods is highlighted. Exact solutions were obtained by integral transforms.

Aliyu et al. [11] studied the model of transmission of vector-borne diseases and cure using the Atangana–Baleanu fractional operator in a Caputo sense. They also discussed the existence and uniqueness via numerical simulations. This model was taken into consideration by the Atangana–Baleanu fractional operator in the Caputo sense (ABC) with non-singular and non-local kernels. The Picard–Lindel method is adopted. The same method is used by Koca [12].

The e ffectiveness of this model is seen by a huge reduction in the disease growth rate. A comparison of heat transfer enhancement in fractional nanofluids with ordinary nanofluids is made by Azhar et al. [13]. Water is taken as a base fluid mixed with fractional nanofluids. A closed form of solution for motion and temperature is calculated and graphically underlined with a higher heat enhancement rate than ordinary fluids. The influence of memory on nanofluid behavior is di fficult to elaborate by classical nanofluids. This can be easy to describe by fractional nanofluids with Caputo time derivatives. Such investigations are made by Fetecau et al. [14]. Closed-form results are obtained and presented in a Wright function. Heat enhancement was found to be lower for fractional nanofluids.

The Caputo–Fabrizio and Atangana–Baleanu derivatives are those fractional derivatives that transformed the classical model to a generalized model. Comparison investigations were made by Sheikh et al. [15]. Karaagac [16] applied two step Adams Bashforth method for time fractional tricomi equation with non-local and non-singular Kernel. Furthermore, Ali et al. [17] investigated MHD flow for free convection generalized Walter's-B fluid. The exact solution is gained by the Laplace technique. The flow is over a static vertical plate. After that, Saqib et al. [18] studied the application of Atangana–Baleanu fractional derivative to MHD channel flow. The fluid is taken through a porous medium. Abro et al. [19] investigated the MHD convection flow for Maxwell fluid as the application of the Atangana–Baleanu fractional derivative. The fluid is taken over a vertical plate in a porous medium. The same author in [20] investigated the exact solution for MHD flow of nanofluids via Atangana–Baleanu and Caputo–Fabrizio fractional derivatives. The general analytical solutions are expressed in the layout of Mittage Le ffler function and generalized Mittage Le ffler function. The comparison of new fractional derivative operators involving exponential and Mittag–Le ffler kernels is investigated by Yavuz and Ozdemir [21]. The comparison was investigated between Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. Furthermore, Imran et al. [22] investigated the comprehensive report on the MHD convective flow of Atangana–Baleanu and Caputo–Fabrizio fractional derivatives. The viscous fluid is subject to generalized boundary conditions. Abro and Khan [23] examined the study of Atangana–Baleanu fractional derivatives for the MHD flow 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 di ffusion 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 di fference method was used. Aman et al. [30] studied fractional Maxwell fluid with a second-order slip e ffect 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 e ffects 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–Le ffler 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 [46–53].

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 ge<sup>t</sup> the exact solution, which is graphically plotted via Mathcad-15 software and discussed in detail.

#### **2. 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:

$$\text{div}\mathbf{B} = \mathbf{0}, \text{Curl}\mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \text{Curl}\mathbf{B} = \mu\_{\mathcal{E}}\mathbf{I}.\tag{1}$$

By using Ohm's law:

$$\mathbf{J} = \sigma(\mathbf{E} + \mathbf{V} \times \mathbf{B}),\tag{2}$$

The magnetic field **B** is normal to **V**. The Reynolds number is so small that the flow is laminar. Hence:

$$\frac{1}{\rho}\mathbf{J}\times\mathbf{B}=\frac{\sigma}{\rho}[(\mathbf{V}\times\mathbf{B}\_0)\times\mathbf{B}\_0]=-\frac{\sigma B\_0^2\mathbf{V}}{\rho}.\tag{3}$$

Keeping in mind the above assumptions, the governing equation of momentum and energy are given as (Khalid et al. [46):

$$\rho \frac{\partial u(y,t)}{\partial t} = \mu \left( 1 + \frac{1}{\gamma} \right) \frac{\partial^2 u(y,t)}{\partial y^2} - \left( \sigma B\_0^2 + \left( 1 + \frac{1}{\gamma} \right) \frac{\mu}{k\_1} \right) u(y,t) + \rho \beta g(T - T\_{\infty}) \tag{4}$$

$$
\rho c\_p \frac{\partial T(y,t)}{\partial t} = k \frac{\partial^2 T(y,t)}{\partial y^2} - \frac{\partial q\_r}{\partial y},\tag{5}
$$

where T and u represent temperature and velocity. ρ, μ, β, σ, *B*0, *k*1, *g*, *cp*, *k*, *qr* 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 ∂*qr* ∂*y* = <sup>4</sup>α20(*<sup>T</sup>* − *<sup>T</sup>*0).

The physical boundary conditions are:

$$\begin{array}{ll} T(y,0) = T\_{\infty \prime} \ T(0,t) = T\_{\text{w} \prime} \ T(\infty,t) = T\_{\infty} \\ u\left(y,0\right) = 0, \quad u\left(0,t\right) = H(t) \cos(\omega t), \text{ } u\left(\infty,t\right) = 0. \end{array} \tag{6}$$

For non-dementalization, the following dimensionless variables are introduced.

$$u^\* = \frac{u}{\mathcal{U}\_0}, \quad y^\* = \frac{\mathcal{U}\_0}{\nu} y, \quad t^\* = \frac{\mathcal{U}\_0^2}{\nu} t, \quad \theta = \frac{T - T\_{\text{co}}}{T\_w - T\_{\text{co}}}.\tag{7}$$

By using Equation (4), the dimensionless form of Equations (1)–(3) are: (asterisk \* is omitted for convenience):

$$\frac{\partial u(y,t)}{\partial t} = \left(1 + \frac{1}{\gamma}\right) \frac{\partial^2 u(y,t)}{\partial y^2} - Hu(y,t) + Gr\theta(y,t),\tag{8}$$

$$\Pr \frac{\partial \theta(y,t)}{\partial t} = \frac{\partial^2 \theta(y,t)}{\partial y^2} + N^2 \theta(y,t),\tag{9}$$

$$\begin{aligned} \theta(y,0) &= 0, \; \theta(0,t) = 1, \; \theta(\infty,t) = 0 \\ u\left(y,0\right) &= 0, \; u\left(0,t\right) = H(t)\cos(\omega t), \; u\left(\infty,t\right) = 0, \end{aligned} \tag{10}$$

where:

$$Gr = \frac{\varrho \beta \nu (T\_w - T\_\infty)}{l l\_0^3}, \ M = \frac{\rho B\_0^2 V}{\rho l l\_0^2}, \ K = \frac{\nu^2}{k\_1 l l\_0^2}, \ \frac{1}{a\_1} = 1 + \frac{1}{\gamma},$$

$$H = \frac{K}{a\_1} + M, \ \Pr = \frac{\mu c\_p}{k}, \ N^2 = \frac{4 \alpha\_1^2 \nu^2}{k l l\_0^2},$$

where *M*, *K*, Pr, *Gr*, α1 and *N* represent the magnetic, porosity, Prandtl number, Grashof number, mean radiation absorption, and radiation parameter, respectively.

With the intention of converting the ordinal time derivative to the Atangana–Baleanu fractional time derivative, Equations (5) and (6) reduced to:

$$D\_t^\alpha u(y, t) = \left(1 + \frac{1}{\gamma}\right) \frac{\partial^2 u(y, t)}{\partial y^2} - H u(y, t) + G r \theta(y, t), \tag{11}$$

$$\Pr D\_t^a \theta(y, t) = \frac{\partial^2 \theta(y, t)}{\partial y^2} + N^2 \theta(y, t). \tag{12}$$

The Atangana–Baleanu fractional time derivative is defined as:

$$D\_t^\alpha f(y, t) = \frac{N(\alpha)}{1 - \alpha} \int\_0^t E\_\alpha \left( -\alpha \frac{\left(t - \tau\right)^\alpha}{1 - \alpha} \right) f'\left(y, t\right) d\tau,\tag{13}$$

where *<sup>N</sup>*(α) is the normalization function, *N*(1) = *N*(0) = 1 and α ∈ (0, <sup>1</sup>). where *E*α = ∞ *<sup>n</sup>*=0 *z<sup>n</sup>* <sup>Γ</sup>(α*n*+*b*) is the Mittag–Leffler function. After Laplace transform, Equation (13) becomes:

$$L\left\{D\_t^\alpha f(t)\right\} = \frac{q^\alpha L\{f(t)\} - q^{\alpha - 1}f(0)}{q^\alpha (1 - \alpha) + \alpha} \tag{14}$$

where *q* is represent the Laplace transform operator.

#### **3. Problem Solution, Skin Friction, and Nusselt Number**

Taking the Laplace transform of Equations (8) and (9) and by using Equation (7), we get:

$$\overline{\partial}(y,q) = \left(\frac{1}{q^{1-\alpha}}\right) \frac{1}{q^{\alpha}} e^{-y\sqrt{q\_5}\sqrt{\frac{q^{\alpha}+q\_6}{q^{\alpha}+s\_4}}} \tag{15}$$

$$\overline{u}(y,q) = \left(\frac{q}{q^2 + a^2} - \frac{Gr}{q}\frac{q^a + a\_4}{a\gamma q^a + a\_8}\right)e^{-y\sqrt{a\_1a\_2}}\sqrt{\frac{q^a + a\_5}{q^a + a\_4}} + \frac{Gr}{q}\frac{q^a + a\_4}{a\gamma q^a + a\_8}e^{-y\sqrt{a\_5}\sqrt{\frac{q^a + a\_6}{q^a + a\_4}}}\tag{16}$$

where:

 $a\_2 = \frac{1 + H(1 - \alpha)}{1 - \alpha}$ ,  $a\_3 = \frac{H\alpha}{1 + H(1 - \alpha)}$ ,  $a\_4 = \frac{\alpha}{1 - \alpha}$ ,  $a\_5 = \frac{\Pr + N^2(1 - \alpha)}{1 - \alpha}$ ,  $a\_6 = \frac{N^2\alpha}{\Pr + N^2(1 - \alpha)}$ ,  $a\_7 = a\_2 - \frac{a\_5}{a\_1}$ ,  $a\_8 = a\_2a\_3 - \frac{a\_5a\_6}{a\_1}$ .

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:

$$\Theta(y,t) = \bigcup\_{0}^{t} h(t - q, \alpha) \psi(y \sqrt{a\_5}, t, 0, a\_6, a\_4) dq\_\prime \tag{17}$$

where:

$$\overline{\psi}(y,q,a,b,c) = \frac{1}{q^a+a} e^{-y\sqrt{\frac{q^a+b}{q^b+c}}}, \\ \psi(y,t,a,b,c) = \int\_0^\infty \psi(y,t,a,b,c)g(u,t)du$$

where:

$$
\psi(y, t, a, b, c) = L^{-1} \{ \overline{\psi}(y, q, a, b, c) \},
$$

$$\psi(y, t, a, b, c) = e^{-at - y} - \frac{y\sqrt{b - c}}{2\sqrt{\pi}} \int\_0^\infty \int\_0^t \frac{e^{-at}}{\sqrt{t}} \times \exp(at - ct - \frac{y^2}{4u} - u) I\_1(2\sqrt{(b - c)ut}) dt du,$$

$$L^{-1}(\frac{1}{q^{1 - \alpha}}) = h(t, a) = \frac{1}{t^\alpha \Gamma(1 - \alpha)}, \; \text{g}(u, t) = L^{-1} \{ \exp[-uq^{-\alpha}] \} = t^{-1} \varphi(0, -\alpha, -ut^{-\alpha}),$$

Here, ϕ is the Wright function, and is defined as:

$$\varphi(k\_{2\prime} - \alpha, \tau) = \sum\_{n=0}^{\infty} \frac{\tau^n}{n! \Gamma(k\_2 - n\alpha)}$$

Then, the velocity profile becomes:

$$\overline{\psi\_1}(y, a, b, c, d, q) = \exp\left(-y\sqrt{\frac{aq^a + b}{cq^a + d}}\right), \\ \overline{\psi\_2}(a, b, c, q) = \frac{aq^a + b}{q(q^a + c)}$$

$$= \cos\omega t \circ \psi\_1(y\sqrt{a\_1 a\_2}, 1, 1, a\_3, a\_4, t) - \frac{Gr}{d\mathcal{T}} \psi\_2(1, a\_4, \frac{a\_8}{d\mathcal{T}}, t) \circ \left[ \begin{array}{c} \psi\_1(y\sqrt{a\_5}, 1, 1, a\_6, a\_4, t) \\ -\psi\_1(y\sqrt{a\_1 a\_2}, 1, 1, a\_3, a\_4, t) \end{array} \right] \tag{18}$$

where:

*<sup>u</sup>*(*y*, *t*)

$$\begin{split} \psi\_{1}(y,a,b,c,d,t) &= t^{-1} \varphi(0, -\alpha, \mu t^{-\alpha}) \ast \psi\_{3}(y,a,b,c,t), \\ \psi\_{3}(y,a,b,c,t) &= \frac{a}{\varepsilon} e^{-\frac{t}{\varepsilon}} \int\_{0}^{\infty} \varepsilon r f c \left(\frac{t}{2\overline{z}}\right) - e^{-\frac{b}{\varepsilon}} I\_{0} \Big(2\sqrt{(a-cb)t}\overline{z}\Big) d\overline{z}d \\ &+ \frac{b}{\varepsilon} \int\_{0}^{t} \varepsilon r f c \left(\frac{t}{2\overline{z}}\right) - e^{-\frac{b\varepsilon + a}{\varepsilon}} I\_{0} \Big(2\sqrt{(a-cb)t}\overline{z}\Big) d\varsigma d\overline{z}, \\ \psi\_{2}(a,b,c,t) &= aE\_{\alpha}(-ct) - b[a - E\_{\alpha}(-ct)] \end{split}$$

*E*α = ∞ *<sup>n</sup>*=0 *z<sup>n</sup>* <sup>Γ</sup>(α*n*+*b*) 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].

#### **4. 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]:

$$Nu = -\frac{\partial T(y, t)}{\partial y}\bigg|\_{y=0} \tag{19}$$

$$\mathcal{C}\_f = \mu \left( 1 + \frac{1}{\gamma'} \right) \frac{\partial u(y, t)}{\partial y} \bigg|\_{y=0} \tag{20}$$
