The first hydrodynamical models

In 1954 R. A. Bagnold [8] experimentally studied a suspension of wax spheres in a mixture of glycerin, water and alcohol, shearing the spheres in a coaxial cylinder rheometer. He found that at reasonably large concentrations and shear rates, the generated stresses depended on the square of the imposed shear rate and obeyed a relation of the form

$$\mathcal{T}_{ij}=\rho_p\sigma^2\gamma^2 \mathcal{G}_{ij}(n)$$ (2.152)

where $\rho_p$ is the density of a particle, $\sigma$ is the radius of the particle, $\gamma$ is the shear rate and $\mathcal{G}$ is a tensor-valued function of the solid fraction (which is the total volume occupied by the particles divided by the available volume in the container and is proportional to the number density $n$). Other experiments and computer simulations have confirmed this relation. This relation comes quite naturally from a dimensional analysis of the problem; moreover, as the granular temperature has units of squared velocity, it must be proportional to $\sigma^2\gamma^2$ and this means that the effective viscosity $\eta$ (which in Eq. (2.159) is proportional to $\gamma$) varies as the square root of the granular temperature. This is not different from the expressions of the viscosity obtained in the framework of the kinetic theory of elastic hard spheres.(e.g. see Eq. (2.39b)). Moreover the experiment of Bagnold addressed the presence of ``dispersive stresses'': the shear work generates granular temperature which in turn generates normal stresses, in exactly the same way as the thermodynamic temperature generates thermodynamic pressure in a gas. These normal stresses tend to cause dilation of the material (i.e. they ``disperse'' the particles).

The analogy between the random motion of the granular particles and the thermal motion of molecules in the kinetic theory of gases was explicitly addressed by Ogawa [171] who coined the term ``granular temperature'' for the mean squared deviation of the velocity from the mean velocity, $T=<\vert\mathbf{c}_i\vert^2>$ where $\mathbf{c}_i=\mathbf{v}_i-\mathbf{u}$.

One of the very first studies based on a complete continuum description of the granular rapid flow in terms of a Navier-Stokes-like equations is due to P. K.Haff [103]. He considered the several problems lying under a continuum description of granular flows and pointed out issues that are still debated in nowadays conferences. He firstly recognized that the granular temperature could be included in the list of macroscopic fields obeying to an energy balance equation analogous to the Eq. (2.146c) including dissipative term, i.e. Eq. (2.154). He also gave heuristic derivations of the transport coefficients $\eta \propto \sigma^2n\sqrt{T}$ and $\kappa\propto\sigma^2\sqrt{T}$ and for the energy sink $\propto (1-r^2)nT^{3/2}$, stressing the assumption of homogeneity in this derivation. After having calculated these coefficients, Haff has shown qualitative behavior of the solutions of hydrodynamics in the most typical situations. He gave the form (2.158) (with a slightly different constant $\beta$) as the solution of the hydrodynamics in absence of macroscopic flows. Then he studied the temperature profile in the steady state with or without gravity, obtaining solutions very similar to the ones obtained with rigorous analysis in the works of the successive twenty years. He realized the subtle problems posed by the boundary conditions: the effect of vibration of a wall driving the motion of the particles in the nearest layer (where the gas is strongly out of equilibrium) and also the behavior of the free surface in the presence of gravity in the most far layer (where the gas is too rarefied and grains are almost ballistic projectiles). He finally outlined the behavior of a granular gas in the steady-state Couette flow (shear in cylinder), the instabilities that could form in such an experimental setup and the problem of supersonic flow.

The attempt to derive expressions of the transport coefficients by means of the formalism of the kinetic theory of non-uniform dense gases, has begun with Savage and Jeffrey [194], who assumed a Maxwell distribution of velocities: they did not include the energy equation and therefore their theory was not complete. However they realized that in a situation of high shear rate it would be possible to observe an anisotropy of the distribution of angles at which collisions would occur (i.e. failure of Molecular Chaos assumption).

Lun et al. [152] studied the Couette flow obtaining closed hydrodynamic equations for arbitrary coefficients of restitution, imposing the anisotropic distribution of collision angles suggested by Savage and Jeffrey. Moreover they obtained, closed hydrodynamic equations for general situations for nearly elastic particles: in this framework they followed the Chapman-Enskog theory with the assumption of a slight correction to the Maxwell distributions.

Jenkins and Richman [119,120] derived closed hydrodynamic equations for the dense gas of identical, rough, inelastic hard disks, by means of the expansion of distribution around the Maxwellian with two temperatures (translational and rotational).

In several computer simulations (see for example [51] and a references in [52]) it has also been addressed the problem of normal-stress differences: in a simple 2D shear flow with the velocity gradient in the $y$-direction, the $\mathcal{T}_{xx}$ normal stress is always larger than $\mathcal{T}_{yy}$, up to a ratio of order $10$ in very rarefied or inelastic situations. This is due to an anisotropy of the mechanisms of momentum transport which relies on streaming transport (dominant at low densities and transporting only momentum in $x$ direction) and collisional transport (dominant at high densities and transporting momentum isotropically).

It could be said that up to these contributions (and many others of 1980s) the approach to a continuum description of granular gases has been pursued on an engineeristic basis, with attention paid almost only to the ``numbers'' (refining and comparing the various expressions of transport coefficients) and not to the qualitative phenomena and to the physical problems. The pioneering contribution of Haff is surely an exception to this and should be read today with the same (or more) interest as many other very recent articles.