My main research interest is aerodynamics and what I discuss here is mainly focussed on the dynamics of (almost) ideal gases. The extension to real gases is possible by invoking real gas laws etc, that account for the storage of energy within the molecules as well as accounting for their non-infinitesimal sizes. However, the model in its current form, does not account for strong inter-molecular forces, that at the macroscopic scale would be manifested as a viscous force in addition to diffusion that is dominant for ideal gases.
Short story:
The inability to prove well-posedness, or even existence of solutions, for the Navier-Stokes equations may well be a consequence of how a compressible fluid is modelled. Hence, I have studied the basic assumptions upon which the Navier-Stokes equations are derived. This has led me to propose a model of compressible viscous flows that is entirely derived in an Eulerian (fixed in space), rather than Lagrangian frame (moving mass element), with diffusion as the main physical principle rather than viscous stresses. Extensive validation indicate that the new model is as accurate as the standard Navier-Stokes model.
For a longer explanation, click here
Its name…
I have used several different names for the model: Eulerian model, alternative Navier-Stokes model and modified Navier-Stokes. Currently, I call it diffusive compressible Euler model.
The original name, Eulerian model, was to emphasise its origins and differences to the Navier-Stokes model. However, Eulerian is a quite common name for models and it has also been misunderstood as an inviscid Euler model.
«Alternative/modified Navier-Stokes» highlighted its intended use. However, both names suggest that it is another version of the Navier-Stokes equations, which it is not.
The name «diffusive compressible Euler» highlights its Eulerian roots while emphasising that it is not modelling an inviscid gas.
Journal publications:
Validation,NACA0012 (with Vít Dolejší)
Validation, sound attenuation (with Karl Munthe)
Validation and under-resolved flows.
Rebuttal of criticism of the model
By other research groups:
Validation, aerodynamics (by group at KAUST)
Validation, incompressible limit (by researchers at the Polish Academy of Sciences)
Numerical tests by E. Padway (NASA)