and boundary conditions (4) become
\[f(0)=0, f'(0)=1, \theta(0)=1, f'(\infty)=0, \theta(\infty)=0\]
Where, \(Pr=\frac{\mu c_p}{k_\infty}\) is the Prandtl number,\(M=\frac{\sigma B_0^2}{c \rho_\infty}\) is the magnetic field parameter, \(\lambda=\frac{c \mu^2}{\rho_\infty k_\infty(T_w-T_\infty)}\) is the viscous dissipation parameter, \(\beta=\frac{\gamma}{2\pi}\frac{\mu_0k(T_w-T_\infty)\rho}{\mu_\infty^2}\)is the ferromagnetic interaction parameter, \(\epsilon=\frac{T_\infty}{T_w-T_\infty}\) is the dimensionless Curie temperature, \(\alpha=\sqrt\frac{c}{\nu_\infty}d\) is the dimensionless distance, \(R_e=\frac{xU_w}{\nu_\infty}\) is the local Reynolds number, \(G_r=\frac{gB^*(T_w-T_\infty)x^3}{\nu_\infty^2}\) is the Grashof number, \(\lambda_1=\frac{G_r}{R_e^2}\) is the mixed convection parameter, \(\theta_r=\frac{T_r-T_\infty}{T_w-T_\infty}=\frac{-1}{\gamma(T_w-T_\infty)}\) is the viscosity variation parameter.
Note that for liquid \(\theta_r<0\) and for gases \(\theta_r>0\) . Also need to consideration that when \(\lambda_1>0\) represent the assist flow and \(\lambda_1<0\) opposes the flow; while \(\lambda_1=0(T_w=T_\infty)\) represents the case when the buoyancy forces are absent.
The most important part of the present study is to skin friction coefficient \(C_f\) and rate of heat transfer \(Nu\) , which are defined as following way:
\[C_f=\frac{\tau_w}{\rho_\infty u_w^2}\ \ and\ \ \ \ Nu=\frac{xq_w}{k(T_w-T_\infty)}\]