Theoretical Physics

Download PDF (791.1 KB) PP. 1 - 3 Pub. Date: December 30, 2016

DOI: 10.22606/tp.2016.11001

• Ph. Choquard^*
  ITP.EPFL.(CH)

One-dimensional, inviscid, compressible and isentropic fluids under gravity are considered, here, as usefull preambles relevant to the theory of uni-axial meteorological phenomena,[4] .We shaw that there are two new Riemann invariants, [1], incorporating gravity, which are constants of the motion. Expressing the mass densities occuring in these Invariants as product of their initial values times the inverse Jacobian of the characteristics of these fluids with respect to their initial values, we propose, central in this work, first order non-linear PDE’s of Charpit type [2] satisfied by these invariants. Examples of solutions are given and checked to conform with results of, gravity-free, similar PDE’s published in [3].

1-Dimensional, invicid, compressible, isentropic, gravity, Riemann Invriants

[1] B. Riemann, Abhandlung der k?niglichen Gesellschaft der Wissenschaften zu G?ttingen, Vol. 8, 1860

[2] L. Deborath, Nonlinear PDE’s for Scientists and Engineers 3th. edition ch.4 sect,4.2 Birkhauser, 2011

[3] A.D. Polyanin, V.F. Zaitzev, Handbook of Nonlinear PDE’s (2th. edition), ch.2 sect.2.1.2, No 3,p.45, Taylor and Francis, 2012.

[4] Ph. Choquard, M. Vuffray, Physica D, vol. 285, X 2014, pp. 11-22,sect.4, items 3 & 4