Hyperbolic incompressibility

It is clear that sound, meaning pressure waves, travels at finite speed. Thus some of the hyperbolic—wavelike-characteristics associated with pressure are in accord with everyday experience. As a fluid becomes more incompressible (e.g., water relative to air), the sound speed increases. In a truly incompressible fluid, pressure travels at infinite speed. When the wave speed is infinite, the pressure effects become parabolic or elliptic, rather than hyperbolic. The pressure terms in the Navier-Stokes equations do not change in the transition from hyperbolic to elliptic. Instead, the equation of state changes. That is, the relationship between pressure and density change and the time derivative is lost from the continuity equation. Therefore the situation does not permit a simple characterization by inspection of first and second derivatives. 

The partial differential equations system for steady flows of Maxwell type (i.e., with = 0) is of composite type, neither elliptic, nor hyperbolic. This is not surprising, the same being true for instance for the stationary system of ideal incompressible fluids. The new feature, discovered in [8], is that some change of type may occur. In fact an easy but tedious calculation shows that three types of characteristics axe present ... 

It is noted that the given pressure-correction method can be applied for arbitrary Mach number flows. At low Mach numbers (almost incompressible flow), the Laplacian term dominates and we recover the Poisson equation. On the other hand, at high Mach number (highly compressible flow), the convective term dominates, reflecting the hyperbolic nature of the flow. Solving the pressure-correction equation is then equivalent to solving the continuity equation for density. Thus, the pressure correction method automatically adjusts... 

The governing equations (1) and (2) are of a mixed parabolic-elliptic nature. A key feature of incompressible flow is that that the time derivative of pressure vanishes from the equations. Hence the equations do not transmit any pressure history directly, and it is as if a new pressure field is established at each step. This situation does not arise for compressible flow where, owing to the presence of the time derivative of the pressure term in the continuity equation, one can solve the coupled hyperbolic system by advancing in time. In the absence of such a term, the algebraic system of equations becomes singular. This is also why attempts to solve the incompressible flow problem as a low Mach-number, compressible-flow problem lead to ill-conditioned algebraic systems with poor algorithmic efficiency and accuracy. For a detailed discussion of these issues, see Ref. 74, p. 642. 

Biot and Darcy theory shortcomings have been largely overcome by development of a coupled diffusion-dynamic formalism (de la Cruz et al. 1993, Spanos 2001, Spanos et al. 2(X)3). Porosity is treated as an explicit thermodynamic variable, so that dnumerical model development. Nevertheless, if they are solved subject to the assumption of the incompressibility of a liquid saturant, the existence of a slow wave is predicted. It is called the porosity dilation (PD) wave it is not a strain wav, it is a coupled liquid-solid displacement wave, and it has some interesting properties. 

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