## 2 Description and Formulation of the Governing Equations  
Consider two-dimensional boundary layer flow of a viscous, non-compressible nanofluid over a nonlinearly radiating plate adjacent to a Darcian porous medium. Free stream velocity is defined by \(u_{e}=Kax\). Here \(a\) is constant having dimension 1/s, \(K\) is a constant which relates to fraction of stationary versus moving free stream. The plate surface is exposed to a momentum slip and thermal slip, in addition to a zero mass-flux boundary condition. Figure 1 displays the schematic of the problem under consideration. An exterior magnetic field having uniform strength \(B_{0}\) is applied transverse to the direction of the plate. We ignore the induced magnetic field, due to a sufficiently low value of magnetic Reynolds number. The \(x\)-axis is aligned with the plate, while the \(y\)-axis is perpendicular to the plate. The gravitational acceleration is represented by expression \(\mathbf{g}^{\ast}(t)=g_{0}[1+\varepsilon\cos(\pi\omega t)]\overrightarrow{K}(\text {J})\) ([PERSON] et al. [10]). Here \(g_{0}\) is the time-averaged value  
of the acceleration, due to gravity. \(\mathbf{g}^{*}(t)\) is orientated to the direction of the unit vector \(\overrightarrow{K}\) (concerned within the upward trend) \(\varepsilon\) is a scaling parameter, \(t\) is the time, and \(\omega\) represents oscillation related to frequency ([PERSON] et al. [10]). The forcing can be regarded as \(a\) perturbation of the mean gravity when \(\varepsilon<<1\). Three different boundary layers (velocity, thermal, and concentration) are formed in the vicinity of the surface of the plate. The flow is driven by both the thermal and nanoparticle species buoyancy forces. It is assumed that the velocities are adequately low; hence, the Darcy drag force model is valid. We adopt [PERSON]'s law for an isotropic, homogenous porous medium with constant permeability \(k_{p}\) (m\({}^{2}\)). The Boussinesq approximation is used. Based on the above simplifications, following [PERSON] [40], and using the order of magnitude analysis (OMA), the transient balance equations are presented (in a dimensional form), respectively, as follows ([PERSON] et al. [10]):  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{1}\]  
\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u }{\partial y}=Ku_{\frac{\partial u}{\partial x}}+\frac{\mu}{\partial y}\frac{ \partial u}{\partial y}-\frac{\mu}{k_{p}^{2}}(u-Ku_{\varepsilon})-\frac{ \alpha\beta_{0}^{2}}{\rho}(u-Ku_{\varepsilon}) \tag{2}\]  
\[+[\mathbf{g}*(t)\beta_{T}(T-T_{\infty})+\mathbf{g}*(t)\beta_{C}(C-C_{\infty})] \left(\frac{t}{\tau}\right),\]  
\[\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T }{\partial y}=\alpha\frac{\partial^{2}T}{\partial y^{2}}+\tau_{1}\left[D_{B} \frac{\partial v}{\partial y}\frac{\partial T}{\partial y}+\frac{\eta_{T}}{T _{\infty}}\left(\frac{\partial\tau}{\partial y}\right)^{2}\right]-\frac{1}{p \tau\rho}\frac{\partial v}{\partial y}, \tag{3}\]  
\[\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C }{\partial y}=D_{B}\frac{\partial^{2}C}{\partial y^{2}}+\frac{\eta_{T}}{T_{ \infty}}\frac{\partial^{2}T}{\partial y^{2}}. \tag{4}\]  
Following [PERSON] [33], [PERSON] et al.