The components of velocity in \(u\) and v in the direction of \(s-\) and \(r-\) respectively. The \(R\), \(\rho\), \(\nu\),\(\alpha\), \(\tau\), \(c_{p}\), \(h_{w}\), and \(p\) are the radius, density of the fluid, kinematic viscosity, thermal diffusivity of fluid, ratio between heat capacity and base fluid, specific heat, heat transfer coefficient, and pressure respectively. Further, \(\gamma\) takes the form,\(\gamma=\left(\mu+\frac{k^{*}}{2}\right)j=\mu\left(1+\frac{K_{1}}{2}\right)j,\ \ \ j=\frac{\nu}{a}\).
Here \(K_{1}=\frac{k^{*}}{\mu}\) is the micropolar parameter. The above equations \(\left(1\right)-(7)\) are translated into ordinary differential equations by using the following non dimensional and no similar variables [32].