Elliptic-parabolic/Hyperbolic-parabolic N.S Equations
 

Hi,

I did not get the clear picture of Elliptic-parabolic system (incompressible) or Hyperbolic- parabolic system(compressible) of N.S equations. Let us say for a 2D case, N.S contains 1 continuity and 2 momentum equations. Among these, which will be elliptic/parabolic/hyperbolic? and these hyperbolic/parabolic with respect to time or space?

Please provide me some material/reference to read...

The continuity equation is always hyperbolic (compressible or not flow model), the momentum equation is parabolic for unsteady viscous (compressible or not) flows but becomes hyperbolic for non-viscous flows.

The mathematical classification of systems of PDE is a topic of applied mathematics but you can find also in CFD books some references.

Thank you Filippo Maria Denaro.

I am working on incompressible flow computation, i have read a paper recently which says "elliptic-parabolic nature of Incompressible N.S will become hyperbolic-parabolic when it becomes compressible".

can you please clarify the following.

1) In case of Incompressible unsteady flow, will the continuity equation be hyperbolic?(no time derivative of density)

2) In case of Incompressible steady flow,will the momentum equation be parabolic?

In most cfd books, i could find only the nature of general equations and i could not find anything about N.S system. If u particularly know some books, plz suggest....

1) yes, the continuity equation Div V =0 is hyperbolic, it will be then reorganized in the elliptic pressure equation

2) The Prandtl equations are parabolic in the momentum, while the full NS momentum equation are elliptic in the steady case

Thank you very much.!!!

For system of equation it is very difficult to say its kind. Because all equations are coupled. The best way is to find out the eigen value of the system which says system is elliptic or parabolic.
For NS equation, the eigen values are u+a, u-a, u,u,u. When they are positive definite then it is purely elliptic and when it becomes semi definite the it is parabolic. In case of subsonic flow these eigen values can take zero or positive quantity and so they are elliptic-parabolic system.

Well, u+/-a,u are the eigenvalues for the time-dependent Euler system (hyperbolic both in subsonic and supersonic cases), not for NS ... the mathematical character of the NS depends on steady/unsteady formulations

I understood the following.!!!

1) Unsteady Incompressible N.S will be Elliptic-parabolic type
2) Unsteady Compressible N.S will be Hyperbolic-parabolic type
3) Steady N.S (for Incompressible & Compressible) will be pure Elliptic type