Multi-Gradient Fluids – Properties and Hyperbolicity

International audienceWe generalize the cases we study in [1-3] of gradient models to the most general case of multi-gradient model corresponding to strong non homogeneous fluids: A volume internal energy function of mass density, volume entropy and their gradients at n-order generates the representation of multi-gradient fluids.Thanks to Hamilton's principle, we obtain a thermodynamic form of the motion equationwhich generalizes the case of perfect compressible fluids.A stress tensor due to density gradients is obtained. First integrals and classification of flows can be extended from the perfect compressible fluid case. The equation of motion is writtenfor dissipative viscous case and the equation of energy is obtained. The dissipative processes are compatible with the two laws of thermodynamics. We can replace the set of process equations by a system of quasi-linear equations in divergence form. Near an equilibrium position, we obtain an Hermitian symmetric form for the equations of perturbations [4-5]. Theequations belong to the class of dispersive systems where the system is proved to be stable [6]. The Noether theorem proves that any law of conservation can be represented by an invariance group. For compressible perfect fluids it has been proved that the conservation laws expressed by Kelvin's theorems correspond to group of permutations corresponding to particles of equal entropy. It is clear that this group keeps the equation of motions invariant also for multi-gradient perfect fluids. Consequently, it is natural to forecast that a general perfect fluid can be identified by this invariance group and as a continuous medium whose motions verify Kelvin's theorems.[1] H. Gouin, Properties of thermocapillary fluids and symmetrization of motion equations, International J. Non-Linear Mech., 85, 152–160 (2016);[2] H. Gouin, G. Saccomandi, Travelling waves of density for a fourth-gradient model of fluids, Continuum Mech. Thermodyn., 28, 1511–1523 (2016).[3] H. Gouin, T. Ruggeri, Symmetric form for the hyperbolic-parabolic system of fourth-gradient fluid model, Ricerche mat. 66, 491–508 (2017).[4] S. Godunov, An interesting class of quasilinear systems, Sov. Math. Dokl., 2, 947–949 (1961).[5] K.O. Friedrichs, P.D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. USA, 68, 1686–1688 (1971).[6] G. Boillat, T. Ruggeri, Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions, Arch. Rat. Mech. Anal., 137, 307–320 (1997)