A Shear Stress Reynolds' Limit Formula

Abstract: Historically, meteorological and climate studies have been prompted by the need for understanding precipitation to have better logistics in food production. Despite all efforts, nonlinearity in atmosphere dynamics is still a source of uncertainty. On the other hand, aeronautical science studies the boundary layer separation through the \emph{shear stress}. In this work, a mathematical interpretation of methods in classical aerodynamics theory in terms of successive layers of \emph{diffeomorphisms} over \emph{Lipschitz domains} allows us to estimate the boundary layer's \emph{shear stress}, $\tau^{*}_d$ and $\tau^{*}_m$, in dry and humid atmospheric conditions without assuming that there is not a convective derivative term in the conservation of momentum equation or that the gaseous boundary layer is incompressible: [ \tau^{*}_d = \frac{U}{h}\ \left(1-\frac{U^2}{2c_{pd}\ T_0}\right)^{19/25}, \hspace{7pt} \tau^{*}_m = \frac{U}{h}\ \left(1-\frac{U^2}{2c_{pm}\ T_0}\right)^{19/25},] where $h$ is the boundary layer's height, $T_0$ is the surface temperature, $U$ is the \emph{free stream velocity}; $c_{pd}$ is the \emph{specific heat at constant pressure for dry air} and $c_{pm}$ is the \emph{specific heat at constant pressure for moist air}. Furthermore, if $\hat{R}m$ is a \emph{gas constant for moist air} and $p_0$ is the pressure at the surface, the density $\rho \hspace{2pt} \cong \hspace{2pt} p_0 \hspace{2pt} T_0^{\frac{2b}{b-1}-1} \hspace{2pt} \hat{R}{m}^{-1} \hspace{2pt} \left[1-\left(U^2/2c_{ph}T_0\right)\right]^{\frac{b}{(b-1)}-1}$ for $b=1.405$. Moreover, this opens the possibility of finding a different deterministic family of atmosphere natural convection models.