## 2 Mathematical Modelling  
Assume the unsteady incompressible flow of an Oldroyd-B nanofluid coupled with heat transfer near an infinite plate saturated in a porous medium. The fluid is supposed to be electrically conducting; hence, an external magnetic field is employed normally to the flow direction. It is assumed that, initially, the system was at rest. However, after a short interval of time, the convection takes place because of the temperature gradient, and the fluid starts motion in the \(x\)-direction as portrayed in Figure 1.  
The basic governing equations of the problem are given by [26]:  
\[\
abla.\mathbf{V}, \tag{1}\]  
\[\rho_{nf}\frac{d\mathbf{V}}{dt}=\
abla.\mathbf{T}_{-}+\mathbf{J}\times\mathbf{B }+\mathbf{R}+\rho\mathbf{g}, \tag{2}\]  
where \(\rho_{nf}\) is the density of nanofluid, \(\mathbf{V}\) is the velocity vector, \(\mathbf{T}\) is the Cauchy stress tensor, \(\mathbf{J}\) is the current density, \(\mathbf{B}\) is the total magnetic field, \(\mathbf{R}\) is the resistive force of porous medium, \(\rho\mathbf{g}\) is the other body forces, and \(d/dt\) is the material time derivative. In the case of Oldroyd-B nanofluid, the stress tensor is defined by [27]:  
\[\mathbf{T}= p\mathbf{I}_{-}+\mathbf{S}, \tag{3}\] \[\mathbf{S}+\lambda\frac{D\mathbf{S}}{Dt}=\mu_{nf}\Big{(}1+\lambda _{r}\frac{D}{Dt}\Big{)}\mathbf{A}_{-}, \tag{4}\]  
where \(\mathbf{T}\) is already defined, \(p\) is the pressure, \(\mathbf{I}\) is the identity tensor, \(\mathbf{S}\) is the extra stress, \(\lambda\) is the time of relaxation, \(D/Dt\) is the convective time derivative, \(\mu_{nf}\) is the dynamic viscosity of nanofluid, \(\lambda_{r}\) is  
Figure 1: Physical sketch and coordinate system. the retardation time, and \(\mathbf{A}_{-1}\) is the first kind of Rivlin-Ericksen tensor. Using the Maxwells' set of equations and Darcy resistance, the terms \(\mathbf{J}\times\mathbf{B}\) and \(\mathbf{R}\) are defined in the following form [28; 29]:  
\[\mathbf{J}\times\mathbf{B}=-\sigma_{nf}B_{0}^{2}\mathbf{V}, \tag{5}\]  
\[\left(1+\lambda\frac{\partial}{\partial t}\right)\mathbf{R}=-\frac{\mu_{nf} \phi}{k}\left(1+\lambda_{r}\frac{\partial}{\partial t}\right)\mathbf{V}, \tag{6}\]  
where \(\sigma_{nf}\) is the electrical conductivity of nanofluid, \(B_{0}\) is the magnetic field, \(\varphi;0<\varphi<1\) is the porosity, and \(k\) is the permeability of the porous medium. In the case of the present problem, the velocity vector \(\mathbf{V}\) and extra stress \(\mathbf{S}\) are chosen in the following form. \[\mathbf{V}=\left[u(y,t),0,0\right],\mathbf{S}=\mathbf{S}(y,t) \tag{7}\]  
After the settlement of Equations (3)-(7) subject to \(\mathbf{S}(y,0)\Rightarrow S_{yy}=S_{yz}=S_{zx}=S_{zz}=0\) and using the Boussinesq approximation the governing equation of Oldroyd nanofluid yield:  
\[ \begin{array}{ll}\rho_{nf}\Big{(}1+\lambda\frac{\partial}{\partial t}\Big{)} \frac{\partial u(y,t)}{\partial t}=&\left(\mu_{nf}+\mu_{nf}\lambda_{r}\frac{ \partial}{\partial t}\right)\frac{\partial^{2}u(y,t)}{\partial y^{2}}-\sigma _{nf}B_{0}^{2}\Big{(}u(y,t)+\lambda\frac{\partial u(y,t)}{\partial t}\Big{)} \\ &-\left\{\frac{\mu_{nf}\phi}{k}u(y,t)+\frac{\mu_{nf}\phi}{k}\lambda_{r}\frac{ \partial u(y,t)}{\partial t}\right\}+g(\rho\mathcal{P})_{nf}(T(y,t)-T_{\infty} ),\end{array}  \tag{8}\]  
\[\left(\rho C_{p}\right)_{nf}\frac{\partial T(y,t)}{\partial t}=k_{nf}\frac{ \partial^{2}T(y,t)}{\partial y^{2}} \tag{9}\]  
where \(g\) is gravitational acceleration, \((\beta_{T})_{nf}\) is the thermal expansion of nanofluid, \(T(y,t)\) is the temperature, \(\left(C_{p}\right)_{nf}\) is the heat capacitance, and \(k_{nf}\) is the thermal conductivity of nanofluid, and \(\rho_{nf}\), \(\lambda\), \(u(y,t)\), \(\mu_{nf}\), \(\sigma_{nf}\), \(B_{0}\) and \(\lambda_{r}\) were previously defined. Equations (8) and (9) are subject to the following initial and ramped wall boundary [30]  
\[u(y,0)=0,\,T(y,0)=T_{\infty},\,\forall y\geq 0, \tag{10}\]  
\[\left.