The Riemann problem of planar shock waves is analysed for a dilute granular gas by solving Euler-and Navier-Stokes-order equations numerically. The density and temperature profiles are found to be asymmetric, with the maxima of both density and temperature occurring within the shock layer. The density peak increases with increasing Mach number and inelasticity, and is found to propagate at a steady speed at late times. The granular temperature at the upstream end of the shock decays according to Haff's law (θ(t) ∼ t −2 ), but the downstream temperature decays faster than its upstream counterpart. Haff's law seems to hold inside the shock up to a certain time for weak shocks, but deviations occur for strong shocks. The time at which the maximum temperature deviates from Haff's law follows a power-law scaling with the upstream Mach number and the restitution coefficient. The origin of the continual build-up of density with time is discussed, and it is shown that the granular energy equation must be 'regularized' to arrest the maximum density.

Classical Navier-Stokes equations fail to describe some flows in both the compressible and incompressible configurations. In this article, we propose a new methodology based on transforming the fluid mass velocity vector field to obtain a new class of continuum models. We uncover a class of continuum models which we call the re-casted Navier-Stokes. They naturally exhibit the physics of previously proposed models by different authors to substitute the original Navier-Stokes equations. The new models unlike the conventional Navier-Stokes appear as more complete forms of mass diffusion type continuum flow equations. They also form systematically a class of thermomechanically consistent hydrodynamic equations via the original equations. The plane wave analysis is performed to check their linear stability under small perturbations, which confirms that all re-casted models are spatially and temporally stable like their classical counterpart. We then use the Rayleigh-Brillouin scattering experiments to demonstrate that the re-casted equations may be better suited for explaining some of the experimental data where original Navier-Stokes equations fail.Öttinger [11] also proposed earlier a substitute for the classical Navier-Stokes in his phenomenological GENERIC formalism. In the GENERIC formulation, Öttinger [11] demonstrated that by assuming the row and the column, which are associated with the mass density in the friction matrix to be identically zero, leads to the conventional Navier-Stokes equations. A more general form of friction matrix that includes the basic fact that particles operate diffusive motions, leads to a revised set of transport equations that contain two velocities, which are very similar in nature to the volume and mass velocities. Durst et al [12] based their arguments on that the absence of mass diffusion terms in the continuity equation contradicts constitutive relations for momentum and heat diffusion: when the fluid properties changes spatially in the presence of momentum and heat diffusions, there should also be present the mass diffusion. They later derived the Extended Navier-Stokes equations [12,13,17] based on the mass-diffusion controlled formalism. A late suggestion to substitute the Navier-Stokes is given by Svärd [18].Generally, there are a number of experimental data that standard Navier-Stokes fail to predict. Shock wave structure predictions are a ubiquitous example of Navier-Stokes failure [4,6,19,20]. Experimental data of water flows in carbon nanotubes or confined pores is another research topic showing large deviations from the classical Hagen-Poiseuille equation [21,22]. A convincing theoretical interpretation of this data is still lacking. Leaving aside non-linear configurations, conventional Navier-Stokes equations also fail to describe some of the linear flow problems accurately. For example, it is unsuccessful in describing the actual spectrum shapes in the Rayleigh-Brillouin scattering problem [23][24][25][26][27][28].In the current work, we provide new insights into the ...

Over the past two decades, several researchers have presented experimental data from pressure-driven liquid flows through nanotubes. They quote flow velocities which are four to five orders of magnitude higher than those predicted by the classical theory. Thus far, attempts to explain these enhanced mass flow rates at the nanoscale have focused mainly on introducing wall-slip boundary conditions on the fluid mass velocity. In this paper, we present a different theory. A change of variable on the velocity field within the classical Navier-Stokes equations is adopted to transform the equations into physically different equations. The resulting equations, termed re-casted Navier-Stokes equations, contain additional diffusion terms whose expressions depend upon the driving mechanism. The new equations are then solved for the pressure driven flow in a long nano-channel. Analogous to previous studies of gas flows in micro-and nano-channels, a perturbation expansion in the aspect ratio allows for the construction of a 2D analytical solution. In contrast to slip-flow models, this solution is specified by a no-slip boundary condition at the channel walls. The mass flow rate can be calculated explicitly and compared to available data. We conclude that the new re-casting methodology may provide an alternative theoretical physical explanation of the enhanced mass flow phenomena.

Classical Navier-Stokes equations fail to predict shock wave profiles accurately. In this paper, the Navier-Stokes system is fully transformed using a velocity variable transformation. The transformed equations termed the recast Navier-Stokes equations display physics not initially included in the classical form of the equations. We then analyze the stationary shock structure problem in a monatomic gas by solving both the classical and the recast Navier-Stokes equations numerically using a finite difference global solution (FDGS) scheme. The numerical results are presented for different upstream Mach numbers ranging from supersonic to hypersonic flows. We found that the recast Navier-Stokes equations show better agreement with the experimentally measured density and reciprocal shock thickness profiles.

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