Atmospheric convection is an essential aspect of atmospheric movement, and it is a source of errors in Climate Models. Being able to generate approximate limit formulas and compare the estimations they produce, could give a way to reduce them. In this article, it is shown that it is enough to assume that the velocity's L 2 -norm is bounded, has locally integrable, L 1 loc , weak partial derivatives up to order two, and a negligible variation of its first velocity's coordinate in direction parallel to the surface, to obtain a 1 arXiv:1811.03237v1 [math.AP] 8 Nov 2018Reynolds' limit formula for a Dorodnitzyn's compressible gaseous Boundary Layer in atmospheric conditions. MSC 2010: 35Q30, 76N15, 76N20.

Two new conformable spatial derivatives are defined and introduced to a classical viscous steady-state Navier–Stokes 1D model. The functions for the conformable derivatives have parameters [Formula: see text] and the fractional parameter [Formula: see text]. Analytical solutions for the velocity profile and flow rate are obtained from the conformable models and a fractional model with Caputo’s derivative. The parameters in the conformable derivatives are optimized to fit a classical Darcy–Brinkman 1D model with constant and variable permeability, showing that the conformable models reproduce quite accurately the flow through a porous medium. The [Formula: see text]-conformable model describes with high accuracy the flow in a porous media with constant permeability, and also it was compared with experimental information for a flow through plates containing an aligned cylindrical fiber preforms. The other conformable model is the best representation for a medium with variable permeability. Both conformable models are better to depict the velocity profile than the fractional model. Additionally, an expression for the permeability, a classical function of the porosity, the tortuosity, and the size distribution, is given as an explicit function of the parameters in the conformable derivative. Finally, a geometrical interpretation is given, the new conformable derivatives have the potential to describe qualitatively a deformed space that seems like a porous medium.

A previous analysis by the author (published previously in this Journal) showed that a limit formula could be deduced from Dorodnitzyn's compressible boundary layer model by the application of Bayada and Chambat's diffeomorphism. This article is the second part of the same research. Now, a limit formula in terms of the shear stress is deduced from Dorodnitzyn's shear stress model.

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Oleinik's no back-flow condition ensures the existence and uniqueness of solutions for the Prandtl equations in a rectangular domain R ⊂ R 2 . It also allowed us to find a limit formula for Dorodnitzyn's stationary compressible boundary layer with constant total energy on a bounded convex domain in the plane R 2 . Under the same assumption, we can give an approximate solution u for the limit formula if |u|< < < 1 such that:that corresponds to an approximate horizontal velocity component when a small parameter ǫ given by the quotient of the maximum height of the domain divided by its length tends to zero. Here, c > 0, δ is the boundary layer's height in Dorodnitzyn's coordinates, U is the free-stream velocity at the upper boundary of the domain, and T 0 is the absolute surface temperature.