The overall objectives to support analytically the mathematical background of hydraulics, linking the Navier-Stokes with hydraulic formulas, which origin is experimental but have wide and varied application. This, lea...The overall objectives to support analytically the mathematical background of hydraulics, linking the Navier-Stokes with hydraulic formulas, which origin is experimental but have wide and varied application. This, leads us study the inverse problem of the coefficients of differential equations, such as equations of the porous medium, Saint-Venant, and Reynolds, and accordingly with the order of derivatives. The research led us to see that the classic version suffers from a parameter that reflects the fractal and non-local character of the viscous interaction. Motivated by the concept of spatial occupancy rate, the authors set forth Navier-Stokes＇s fractional equation and the authors obtain the fractional Saint-Venant. In particular, the hydraulic gradient, or friction, is conceived as a fractional derivative of velocity. The friction factor is described as a linear operator acting on speed, so that the information it contains is transferred to the order of the derivative, so that the same is linearly related to the exponent of the friction factor. It states Darcy＇s non-linear law. The authors take a previous result that describes the nonlinear flow law with a leading term that contains a hyper-geometric function, whose parameters depend on the exponent of the friction factor and the exponent of the hydraulic radius. It searches the various laws of flow according to the best known laws of hydraulic resistance, such as Chezy and Manning.展开更多
文摘The overall objectives to support analytically the mathematical background of hydraulics, linking the Navier-Stokes with hydraulic formulas, which origin is experimental but have wide and varied application. This, leads us study the inverse problem of the coefficients of differential equations, such as equations of the porous medium, Saint-Venant, and Reynolds, and accordingly with the order of derivatives. The research led us to see that the classic version suffers from a parameter that reflects the fractal and non-local character of the viscous interaction. Motivated by the concept of spatial occupancy rate, the authors set forth Navier-Stokes＇s fractional equation and the authors obtain the fractional Saint-Venant. In particular, the hydraulic gradient, or friction, is conceived as a fractional derivative of velocity. The friction factor is described as a linear operator acting on speed, so that the information it contains is transferred to the order of the derivative, so that the same is linearly related to the exponent of the friction factor. It states Darcy＇s non-linear law. The authors take a previous result that describes the nonlinear flow law with a leading term that contains a hyper-geometric function, whose parameters depend on the exponent of the friction factor and the exponent of the hydraulic radius. It searches the various laws of flow according to the best known laws of hydraulic resistance, such as Chezy and Manning.
