governing Equations

Consider a steady, incompressible two-dimensional stagnation-point flow of an electrically conducting Casson fluid towards a vertically stretching sheet at \(y=0\) with the flow confined in the region\(y>0\). Along the stretching surface in the \(x\)-axis, two equal and opposite forces are being applied with a uniform magnetic field strength\(B_{0}\) applied perpendicular to the surface. The induced magnetic field is neglected and the ambient fluid is moved with a velocity\(U_{\infty}\left(x\right)=ax\). The rheological equation of state for an isotropic and incompressible flow of a Casson fluid\(\mathbf{[18,26]}\) is given by
\begin{equation} \tau_{\text{ij}}=\left\{\begin{matrix}\left(\mu_{B}+\frac{P_{y}}{\sqrt{2\pi}}\right)2e_{\text{ij}}\ ,\ \ \ \ \pi>\pi_{c}\\ \left(\mu_{B}+\frac{P_{y}}{\sqrt{2\pi_{c}}}\right)2e_{\text{ij}}\ ,\ \ \ \ \pi>\pi_{c}\\ \end{matrix}\right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\nonumber \\ \end{equation}
where, \(\pi\) is the product of the component of deformation rate with itself, \(\pi=e_{\text{ij}}e_{\text{ij}}\), \(e_{\text{ij}}\) are the\(\left(i,\ j\right)^{\text{th}}\) component of the deformation rate and \(\pi_{c}\) is a critical value of this product based on the non-Newtonian model, \(\mu_{B}\) is plastic dynamic velocity of the non-Newtonian fluid and \(P_{y}\) is the yield stress of the fluid.
The MHD boundary layer equations for the steady stagnation-point flow are given by