Table 2 Mathematical values of physical constraints at the boundaries for altered values of factors β , R , M , \(\pmb{\gamma_{1}}\) , \(\pmb{\gamma_{2}}\) when \(\pmb{m = 0.5}\) , \(\pmb{\Pr = 1.0}\) , \(\pmb{\mathrm {Sc}= 0.7}\) and \(\pmb{\hbar_{f} = \hbar _{h} = \hbar_{\theta } = \hbar_{\varphi } = - 0.3}\)

β

M

λ

N

\(\boldsymbol{\gamma_{1}}\)

\(\boldsymbol{\gamma_{2}}\)

R

\(\boldsymbol{- ( 1 + \frac{1}{\beta } ) f''(0)}\)

\(\boldsymbol{( 1 + \frac{1}{\beta } ) h'(0)}\)

\(\boldsymbol{- \theta '(0)}\)

\(\boldsymbol{- \varphi '(0)}\)