The two dimensional thermal aspects are analyzed over a porous permeable surface. The fluidic motion is generated with help of moving wall. Flow is considered steady and incompressible. A variable magnetic field is implanted along vertical direction of heated surface. Thermal aspects are based on Joule heating and viscous dissipation. Slip conditions are used along with variable wall velocity. Figure 1 predicts geometry of generated model. The generated form of PDEs via boundary layer approximations are

$$\frac{\partial {U}_{X}}{\partial x}+\frac{\partial {V}_{Y}}{\partial y}=0,$$ (1)

$${U}_{X}\frac{\partial {U}_{X}}{\partial x}+{V}_{Y}\frac{\partial {U}_{X}}{\partial y}=\frac{{ϵ}^{2}{\mu }_{e}}{\rho }\left(\frac{{\partial }^{2}{U}_{x}}{\partial {y}^{2}}\right)-\frac{{ϵ}^{2}\mu }{\rho {k}^{*}}{U}_{x}+\frac{\sigma }{\rho }{U}_{x}{B}_{0}^{2},$$ (2)

$${U}_{x}\frac{\partial T}{\partial x}+{V}_{y}\frac{\partial T}{\partial y}=\frac{K}{\rho {C}_{p}}\frac{{\partial }^{2}T}{\partial {y}^{2}}+\frac{{ϵ}^{2}}{\rho {C}_{p}}\left[{\mu }_{e}{\left(\frac{\partial {U}_{X}}{\partial y}\right)}^{2}+\frac{\mu {\left({U}_{X}\right)}^{2}}{{k}^{*}}+\frac{\sigma }{\rho {C}_{p}}{B}_{0}^{2}{\left({U}_{X}\right)}^{2}{\text{sin}}^{2}t\right].$$ (3)

Subjected to desired boundary conditions

\({U}_{x}=ax+{\beta }_{*}\frac{\partial {U}_{X}}{\partial U}, {V}_{Y}=-{V}_{0}, T={T}_{s}+{\delta }_{1}\left(\frac{\partial T}{\partial y}\right)\) at wall, (4)

\({U}_{X}\to ax, T\to {T}_{\infty }\) at away from wall.

Variable transformations are

$$\xi =\sqrt{\frac{\alpha }{\nu }}y, u=ax{g}^{\text{'}}, v=-{\left(a\nu \right)}^{\frac{1}{2}}g, \theta =\frac{T-{T}_{\infty }}{{T}_{s}-{T}_{\infty }}.$$ (5)

Dimensionless ODEs are formulated as

\(\gamma {g}^{\text{'}\text{'}\text{'}}-{\left({g}^{\text{'}}\right)}^{2}+g{g}^{\text{'}\text{'}}-{P}_{m}{g}^{\text{'}}+M{\text{sin}}^{2}t{g}^{\text{'}}=0,\) (6)

$$\frac{1}{Pr}{\theta }^{\text{'}\text{'}}-2{g}^{\text{'}}\theta +g{\theta }^{\text{'}}+{E}_{c}\gamma {\left({g}^{\text{'}\text{'}}\right)}^{2}+{E}_{c}{P}_{m}{\left({g}^{\text{'}}\right)}^{2}+{E}_{c}M{\text{sin}}^{2}t{\left(g\text{'}\right)}^{2}=0.$$ (7)

Dimensioless ODEs are modeled as

$$g\left(0\right)=S, {g}^{\text{'}}\left(0\right)=1+\beta {g}^{\text{'}\text{'}}\left(0\right), \theta \left(0\right)=1+\delta {\theta }^{\text{'}}\left(0\right), \theta \left(\infty \right)=0, {g}^{\text{'}}\left(\infty \right)=0.$$& (8)

Skin force and temperature gradient in dimensionless form are developed as

\({Re}^{\frac{1}{2}}{C}_{f}={g}^{\text{'}\text{'}}\left(0\right),\) (9)

$$Nu{Re}^{-1/2}=-{\theta }^{\text{'}}\left(0\right).$$ (10)