**2. Problem Formulation**

#### *2.1. Boundary Layer Governing Equations*

The MHD flow of two-dimensional incompressible viscous fluid over a continuously unsteady shrinking surface is considered. The velocity of mass transfer and the shrinking surface are assumed to be *vw* (*x*, *t*) and *uw* (*x*, *t*), respectively, where *t* is the time and *x* is the coordinate measured with the shrinking surface. Under these assumptions with viscous dissipation, the governing Navier–Stokes (NS) equations of this problem are given by:

$$
\frac{\partial \mu}{\partial x} + \frac{\partial v}{\partial y} = 0 \tag{1}
$$

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{1}{\rho} \frac{\partial P}{\partial x} + \theta \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) - \frac{\sigma^\* B^2 u}{\rho} \tag{2}$$

$$\frac{\partial v}{\partial t} + \mu \frac{\partial v}{\partial \mathbf{x}} + v \frac{\partial v}{\partial y} = -\frac{1}{\rho} \frac{\partial P}{\partial \mathbf{x}} + 8 \left( \frac{\partial^2 v}{\partial \mathbf{x}^2} + \frac{\partial^2 v}{\partial y^2} \right) \tag{3}$$

$$\frac{\partial T}{\partial t} + \mu \frac{\partial T}{\partial \mathbf{x}} + \upsilon \frac{\partial T}{\partial y} = a \left( \frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial y^2} \right) + \frac{\mu}{\rho c\_p} \left[ \left( \frac{\partial u}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial u}{\partial y} \right)^2 \right] \tag{4}$$

Subject to the following boundary conditions:

$$t < 0: \ u = v = 0, \ T = T\_{\text{co}} \text{ for all } \mathbf{x}; \text{ } y$$

$$u \ge 0: \ u = u\_{\text{w}}(\mathbf{x}, t) = -\frac{c\mathbf{x}}{1 - \gamma t}, \ v = v\_{\text{w}}(\mathbf{x}, t) = -\sqrt{\frac{\mathfrak{d}\_f \mathfrak{c}}{1 - \gamma t}} f(0) = S, \ T = T\_{\text{w}}(\mathbf{x}, t) = T\_{\text{co}} + \frac{b\mathbf{x}^m}{1 - \gamma t} \text{ at } \ y = 0$$

$$u \to 0; \ T \to T\_{\text{co}} \text{ as } y \to \infty \tag{5}$$

where the pressure of fluid is denoted by *P*, velocity components along the *x* and *y* directions are represented by *u* and *v*, respectively, temperature of fluid is *T*, kinematic viscosity of the fluid is ϑ, density of the fluid is ρ, thermal diffusivity of the fluid is α, *B* = *<sup>B</sup>*<sup>0</sup> (1−γ*t*) 1/2 is the transverse magnetic field of strength which is applied with the normal surface direction, and *b*, *c*, and *m* are all positive constants. It is worth mentioning that *m* = 1 and *m* = 0 indicate linear and constant variation with *x* of the wall of temperature, respectively.

Now, we introduce the similarity variables for Equations (1)–(5) as follows:

$$u = \frac{c\chi}{(1 - \gamma t)} f'(\eta); \; v = -\sqrt{\frac{c\upsilon}{(1 - \gamma t)}} \; f(\eta), \; \theta(\eta) = \frac{T - T\_{\text{os}}}{T\_w - T\_{\text{os}}} \tag{6}$$

Substituting (6) into Equations (2)–(4) yields the following system of ordinary differential equations

$$f'''' + ff'' - (f')^2 - A\left(\frac{\eta}{2}f'' + f'\right) - Mf' = 0\tag{7}$$

$$\frac{1}{Pr}\Theta'' + f\Theta' - mf'\Theta - A\left(\frac{\eta}{2}\Theta' + \Theta\right) + Ec(f'')^2 = 0\tag{8}$$

with reduced boundary conditions

$$f(0) = \text{S}; \; f'(0) = -1; \; \theta(0) = 1$$

$$f'(\eta) \to 0; \; \theta(\eta) \to 0 \text{ as } \eta \to \infty \tag{9}$$

where *A* = <sup>γ</sup> *<sup>c</sup>* is the unsteadiness parameter, *<sup>M</sup>* <sup>=</sup> <sup>σ</sup>(*B*0) 2 <sup>ρ</sup>*<sup>c</sup>* is the magnetic parameter, *Ec* <sup>=</sup> *<sup>U</sup>*<sup>2</sup> *w Cp*(*Tw*−*T*∞) is the Eckert number, and *Pr* = <sup>ϑ</sup> <sup>α</sup> is the Prandtl number. In our problem, we consider a decelerating shrinking surface (*A* < 0) as assumed in [35,36].
