# A System of Numerical Solutions of Wall Similarity Temperature Gradient

## Abstract

A system of numerical solutions of wall similarity temperature gradient \(\left( { - \frac{{\partial \theta (\eta, Mc_{x,\infty } )}}{\partial \eta }} \right)_{\eta = 0}\) is obtained and shown for creation of the optimal formalization of the wall similarity temperature gradient. It is seen that with increasing local mixed convection parameter \(Mc_{x,\infty }\), the wall temperature gradient \(\left( { - \frac{{\partial \theta (\eta, Mc_{x,\infty } )}}{\partial \eta }} \right)_{\eta = 0}\) will increase obviously. It demonstrates the coupled effect of Buoyancy force on heat transfer coefficient. Meanwhile, both of the local Prandtl numbers \(Pr_{w}\) and \(Pr_{\infty }\) influence the wall similarity temperature gradient, although the effect of local Prandtl number \(Pr_{\infty }\) is stronger than that of the local Prandtl number \(Pr_{w}\). It reflects the effect of fluid’s variable physical properties on heat transfer of mixed convection. The work of the present chapter has completed the first step for creation of Nusselt number formalization. According to the numerical solutions of the wall similarity temperature gradient \(\left( { - \frac{{\partial \theta (\eta, Mc_{x,\infty } )}}{\partial \eta }} \right)_{\eta = 0} ,\) effect of the local Prandtl number \(Pr_{w}\) was first considered. Then, a system of optimal formalized equations are obtained where the coefficient *a* and exponent *b* are regarded as the functions of local Prandtl number \(Pr_{\infty }\) at local mixed convection parameter \(Mc_{x,\infty }\). In next chapter, the optimal equations of the coefficient *a* and exponent *b* will be obtained with variation of Prandtl number \({ Pr }_{\infty }\) at the mixed convection parameter \(Mc_{x,\infty } .\)