## Materials and Methods  
Consider a flow of thin film Williamson fluid on an inclined plate. The plate is oscillating and moving with a uniform velocity \(U\) at time \(t=0^{+}\). The thickness of the liquid film is considered uniform and equal to \(\delta\). The gravity force opposes the fluid motion and tries to make the fluid film drain down the plate. A uniform magnetic field is applied transversely to the translating inclined plate. Furthermore, during fluid motion, constant temperature filed is also considered. The flow is assumed to be unsteady and laminar after a small distance above the film surface. The physical conditions of the unsteady problem are used as:  
\[u(0,t)=U(1+\cos\omega t),\frac{\partial u(\delta,t)}{\partial y}=0, \tag{1}\]  
\[\Theta(0,t)=\Theta_{0},\Theta(\delta,t)=\Theta_{1}, \tag{2}\]  
where \(\omega\) is the frequency of the oscillating plate and \(\delta\) is the thickness of the thin film. The Lorentz force \(\mathbf{J}\,\times\,\mathbf{B}\) for the uniform Magnetic field is defined as:  
\[\mathbf{J}\,\times\,\mathbf{B}=\Big{[}0,-\sigma B_{0}^{2}u(y,t),0\Big{]}, \tag{3}\]  
where \(\sigma\) is the electrical conductivity of the fluid. The governing equations take the form as:  
\[\
abla.\mathbf{u}=0, \tag{4}\]  
\[\rho\frac{d\mathbf{u}}{dt}=-\
abla p+\rho\mathbf{g}\sin(\Theta)+\
abla.\mathbf{ T}+\mathbf{J}\,\times\,\mathbf{B}, \tag{5}\]  
\[\rho C_{p}\frac{D\Theta}{Dt}=k\
abla^{2}\Theta+tr(\mathbf{T}.\mathbf{L}), \tag{6}\]  
where \(\mathbf{u}\) is the velocity vector, \(\rho\) is the fluid density, \(\mathbf{g}\) is the gravity force, \(p\) is the pressure term, \(\Theta\) is the temperature field, \(k\) defines thermal conductivity, \(C_{p}\) is the specific heat and \(\mathbf{L}=\
abla\mathbf{u}\). The extra stress tensor \(\mathbf{T}\) for the Williamson fluid is defined as in [4,8]:  
\[\mathbf{T}=\Big{[}\mu_{\infty}+\big{(}\mu_{0}+\mu_{\infty}\big{)}\big{(}1- \Gamma\dot{\mathbf{\gamma}}\big{)}^{-1}\Big{]}\dot{\mathbf{\gamma}}, \tag{7}\]  
where \(\Gamma\) is the time constant, \(\mu_{0}\) is the zero viscosity, \(\mu_{\infty}\) is the infinite viscosity, and \(\dot{\mathbf{\gamma}}\) is defined as:  
\[\dot{\mathbf{\gamma}}=\sqrt{\frac{1}{2}{\sum\limits_{i}{\sum\limits_{j}{\gamma _{ij}\gamma_{ji}}}}}=\sqrt{\frac{1}{2}\mathbf{\Pi}}, \tag{8}\]  
where \(\mathbf{\Pi}=trace\big{(}A_{1}\big{)}^{2}=2\Big{(}\frac{\partial u}{\partial y }\Big{)}^{2}\) is the second invariant strain tensor derived as in [4,8]. We consider the constitutive Equation (7) in which \(\mu_{\infty}=0\) and \(\Gamma\dot{\mathbf{\gamma}}<1\). The component of the extra stress tensor, therefore, can be written as:  
\[\mathbf{T}=\mu_{0}\Big{[}\big{(}1-\Gamma\dot{\mathbf{\gamma}}\big{)}^{-1} \Big{]}\dot{\mathbf{\gamma}}=\mu_{0}\big{[}\big{(}1+\Gamma\dot{\mathbf{\gamma }}\big{)}\big{]}\dot{\mathbf{\gamma}}, \tag{9}\]  
where \(\dot{\mathbf{\gamma}}=\frac{\partial u}{\partial y}\) is the shear rate and \(\Gamma\) is the time constant. The unsteady unidirectional velocity field and temperature profile is defined as:  
\[\mathbf{u}=[u(y,t),0,0],\ \Theta=\Theta(y,t), \tag{10}\]When (9) and (10) are used in (4)-(6), the continuity equation is identically satisfied and the momentum and energy equations without dynamic pressure are reduced to:  
\[\Theta(0,t)=\Theta_{0},\Theta(\delta,t)=\Theta_{1}, \tag{2}\]0  
\[\Theta(0,t)=\Theta_{0},\Theta(\delta,t)=\Theta_{1}, \tag{2}\]1  
Introduce the following dimensionless parameters:  
\[ \begin{array}{l}\overline{u}=\frac{u}{\partial t},\,\,\overline{y}=\frac{u }{\partial t},\,\,\overline{t}=\frac{\mu t}{\partial\overline{\Theta}},\, \,\overline{\Theta}=\frac{\Theta-\Theta_{0}}{\Theta_{1}-\Theta_{0}},\,\, \overline{\omega}=\frac{\omega\delta^{2}\rho}{\mu},\,\,We=\frac{\Gamma U}{ \delta},\\ p_{r}=\frac{\mu c_{p}}{k},\,\,E_{c}=\frac{\mu U^{2}}{k(\Theta_{1}-\Theta_{0})},\,\,M=\frac{\mu B_{0}^{2}\delta^{2}}{\mu_{0}},\,\,m=\frac{\delta^{2}Pg\sin \Theta}{\mu U}.\end{array}