Inviscid 1 D fluids under gravity

Abstract

Abstract

One-dimensional, inviscid, compressible, isothermal and isentropic fluids under gravity are considered as usefull preambles relevant to the theory of uni-axial meteorological phenomena [4; Sect:4; items 3 & 4]. In Sect.1, the continuity equation, the Euler equations of these fluids, the equations of their charcteristics as well as those of their energies, constants of the motion, are given. Next, the continuy and Euler equations for isentropic fluids are written in a matrix and compact forms; their diagonalized versions are shown to be total derivatives of new constants of the motion, identified as gravitational Riemann invariants, [1]: Then, and in Sect.1, also, the mass densities at time t occuring in the above equations are expressed as product of their initial value times the inverse Jacobian of the characteriistics of the fluids with respect to their initial values, an operation permitting to generate, central in this work, the first order non-linear partial differential equations satisfied by these invariants and, also, those of the other constants of the motion, the energies. In Sect.2, the Charpit scheme, designed to solve non-linear first order PDE's of n variables, in general, [2; ch:4] ; is presented. For systems of two independant variables, the correspding ordinary differential equations are given in  Sect.2.1. The Charpit functions for the Riemannian cases and for the other cases are given in Sect.2.2.Then,in Sect.3, the attention is focussed on the gravitational Riemann invariants only, owing to their originality and also, to the relative simplicity of the numerics implied. Their corresponding ODE's are given in Subsect.3.1 and, in Subsect.3.2, several examples are propposed and some of their explicit solutions, algebraic and graphical are reported. To conclude, due comparaison is made between these results and solutions of equivalent PDE's,  given in [3; No.2:1:2:3; p:45]:

Highlights...

- 1D inviscid, cimpressible, isothermal or isentropic fluids under gravitation.

- Gravitational Riemann Invariants for isentropic systems

- Charpit scheme, EDO's and examples of isentropiic solutions