# Type of PDE: Hyperbolic or Parabolic or Elliptic?

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 January 17, 2001, 13:15 Type of PDE: Hyperbolic or Parabolic or Elliptic? #1 Guo Guest   Posts: n/a When reading the book, "Computational Fluid Mechanics and Heat Transfer", 2nd adition, by Tannehill, Anderson and Pletcher, on page 624, I read the following: "the unsteady compressible N-S equations are a mixed set of hyperbolic-parabolic equations in time.If the unsteady terms are dropped from these equations, the resulting equations become a mixed set of hyperbolic-elliptic equations, ..." My question is: why are those equations a mixed set of hyperbolic-parabolic or hyperbolic-elliptic? Thank you! flowAlways likes this.

 January 17, 2001, 19:12 Re: Type of PDE: Hyperbolic or Parabolic or Ellipt #2 John C. Chien Guest   Posts: n/a (1). I think, it has something to do with the local flow behavior. (2). If the local flow behavior is something like transient heat conduction, then it is parabolic. (3). If the flow behavior is subsonic, steady state, then it is elliptic. (4). If the flow behavior is supersonic, steady state, then it is hyperbolic. (5). If the flow behavior is transient inviscid, then it is hyperbloic. (6). In other words, flow field solution can exhibit different flow behaviors in different part of the flow field. (thus the word "mixed" is used) (7). There are also standard method to classify the type of a partial differential equations, which I think, is also included in the Anderson' book "Computational Fluid Mechanics and Heat Transfer". (8). The important thing is to understand the local flow behavior based on the local Mach number, Reynolds number, and its transient state. This is important because you can run into a steady-state supersonic flow (hyperbolic)with embedded subsonic pocket or region (elliptic). In that case,solution method and boundary conditions must be changed according to it type. I think, you are in the PhD. domain now.

 January 17, 2001, 19:19 Re: Type of PDE: Hyperbolic or Parabolic or Ellipt #3 John C. Chien Guest   Posts: n/a

 January 17, 2001, 21:52 Re: Type of PDE: Hyperbolic or Parabolic or Ellipt #4 Guo Guest   Posts: n/a Thank you for the answer. I understand the basis of the classification from your words. But it seems to me that, from your answers, the judgement of the types of the N-S equations to be hyperbolic-parabolic or hyperbolic-elliptic is based on the physical characteristics of the solutions which are not known before we get them or if we don't have experience in heat transfer or fluid mechanics. So, like other judgement for rather simple 2nd order PDE's by using b^2-4ac to decide the type which can be carried out by people who have only mathematical knowledge, are there any mathematical methods which can be applied directly to the N-S equations to decide the mixed types? Or, the types can only be judged based on the computational outcome or on some experience?

 January 17, 2001, 22:22 Re: Type of PDE: Hyperbolic or Parabolic or Ellipt #5 John C. Chien Guest   Posts: n/a (1). If you have a nozzle, the flow can be steady and subsonic throughout the nozzle. In that case, the exit condition will affect the rest of the flow field. And the flow will be elliptic. (2). The same nozzle can also produce subsonic-transonic-supersonic flow, or even subsonic-transonic-supersonic-subsonic flow, depending upon the initial and the boundary cinditions. And it is well known that in the supersonic region, the flow can only affect the region in the downstream direction. This portion of the flow is hyperbolic. (3). In both cases, you are using the same governing equations.

 January 18, 2001, 14:02 Re: Type of PDE: Hyperbolic or Parabolic or Ellipt #6 kalyan Guest   Posts: n/a The independent variables need to be identified in order to classify PDEs. If you consider space & time, the N-S equations are parabolic. If you drop viscous terms, they become hyperbolic. However, in space, they are elliptic (for subsonic flows) or hyperbolic (supersonic). As pointed out, hyperbolic equations have characteristics. There are characteristic lines in spac e (like in the method of characteristics used for supersonic nozzle designs) and characteristics in space-time using which unsteady characteristic boundary conditions are designed for flux-split schemes. Therefore, identifying what independent variables do you want to base your classification on is important. flowAlways likes this.