2. Governing equations

The equation of continuity and momentum of MHD flow passing through porous space is given by [7]

$$\nabla \cdot V = 0,\\ \rho \left(\frac{d\mathbf{V}}{dt}\right) = d\dot{v}\mathbf{r}\mathbf{T} + \mathbf{J} \times \mathbf{B} + \mathbf{R},\tag{1}$$

where velocity is represented by V, density by ρ, Cauchy stress tensor by T, magnetic body force by J � B, current density by J, magnetic field by B, and Darcy's resistance in the porous medium by R.

For an incompressible and unsteady Jeffrey fluid the Cauchy stress tensor is defined as [9]

$$\mathbf{T} = -p\mathbf{I} + \mathbf{S}, \ \mathbf{S} = \frac{\mu}{1+\lambda} \left[ \mathbf{A} + \Theta \left( \frac{\partial \mathbf{A}}{\partial \mathbf{t}} + (\mathbf{V} \cdot \nabla) \mathbf{A} \right) \right],\tag{2}$$

where S and pI represents the extra stress tensor and the indeterminate spherical stress, the dynamic viscosity is denoted by μ, A= L + L<sup>T</sup> is the first Rivlin-Ericksen tensor, L is the velocity gradient, λ is relaxation time and θ is retardation time. The Lorentz force due to magnetic field is

$$\mathbf{J} \times \mathbf{B} = -\sigma \beta\_o^2 \mathbf{V},\tag{3}$$

where σ represents electrical conductivity and β<sup>o</sup> the strength of magnetic field. For the Jeffrey fluid the Darcy's resistance satisfies the following equation

$$\mathbf{R} = -\frac{\mu \phi}{\kappa (1 + \lambda)} \left( 1 + \theta \frac{\partial}{\partial t} \right) \mathbf{V},\tag{4}$$

where κ(>0) and ϕ(0 < ϕ < 1) are the permeability and the porosity of the porous medium.

In the following problem we consider a velocity field and extra stress of the form

$$\mathbf{V} = (0, 0, w(\mathbf{x}, y, t)), \mathbf{S} = \mathbf{S}(\mathbf{x}, y, t) \tag{5}$$

where w is the velocity in the z-direction. The continuity equation for such flows is automatically satisfied. Also, at t = 0, the fluid being at rest is given by

$$\mathbf{S}(\mathbf{x}, y, \mathbf{0}) = \mathbf{0},\tag{6}$$

therefore from Eqs. (2), (5) and (6), it results that Sxx = Syy = Syz = Szz = 0 and the relevant equations

$$
\tau\_1 = \frac{\mu}{(1+\lambda)} \left( 1 + \theta \frac{\partial}{\partial t} \right) \partial\_x w(\mathbf{x}, y, t), \quad \tau\_2 = \frac{\mu}{(1+\lambda)} \left( 1 + \theta \frac{\partial}{\partial t} \right) \partial\_y w(\mathbf{x}, y, t), \tag{7}
$$

where τ<sup>1</sup> = Sxy and τ<sup>2</sup> = Sxz are the tangential stresses. In the absence of pressure gradient in the flow direction, the governing equation leads to

$$\mathbf{u}(1+\lambda)\partial\_t \mathbf{w}(\mathbf{x}, y, t) = \nu \left( 1 + \theta \frac{\partial}{\partial t} \right) \left( \partial\_x^2 + \partial\_y^2 \right) \mathbf{w}(\mathbf{x}, y, t) - \nu \mathbf{K} \left( 1 + \theta \frac{\partial}{\partial t} \right) \mathbf{w}(\mathbf{x}, y, t) - H(1+\lambda)\mathbf{w}(\mathbf{x}, y, t), \tag{8}$$

where <sup>H</sup> <sup>¼</sup> <sup>σ</sup>B<sup>2</sup> 0 <sup>ρ</sup> is the magnetic parameter, <sup>K</sup> <sup>¼</sup> <sup>ϕ</sup> <sup>κ</sup> is the porosity parameter and ν = μ/ρ is the kinematic viscosity.
