Normal stress differences, their origin and constitutive relations for a sheared granular fluid

Abstract

The rheology of the steady uniform shear flow of smooth inelastic spheres is analysed by choosing the anisotropic/triaxial Gaussian as the single-particle distribution function. An exact solution of the balance equation for the second-moment tensor of velocity fluctuations, truncated at the 'Burnett order' (second order in the shear rate), is derived, leading to analytical expressions for the first and second (N 1 and N 2 ) normal stress differences and other transport coefficients as functions of density (i.e. the volume fraction of particles), restitution coefficient and other control parameters. Moreover, the perturbation solution at fourth order in the shear rate is obtained which helped to assess the range of validity of Burnett-order constitutive relations. Theoretical expressions for both N 1 and N 2 and those for pressure and shear viscosity agree well with particle simulation data for the uniform shear flow of inelastic hard spheres for a large range of volume fractions spanning from the dilute regime to close to the freezing-point density (ν ∼ 0.5). While the first normal stress difference N 1 is found to be positive in the dilute limit and decreases monotonically to zero in the dense limit, the second normal stress difference N 2 is negative and positive in the dilute and dense limits, respectively, and undergoes a sign change at a finite density due to the sign change of its kinetic component. It is shown that the origin of N 1 is tied to the non-coaxiality (φ = 0) between the eigendirections of the second-moment tensor M and those of the shear tensor D. In contrast, the origin of N 2 in the dilute limit is tied to the 'excess' temperature (T ex z = T − T z , where T z and T are the z-component and the average of the granular temperature, respectively) along the mean vorticity (z) direction, whereas its origin in the dense limit is tied to the imposed shear field.

Normal stress differences and beyond-Navier-Stokes hydrodynamics

“…The last hydrodynamic field M(x, t) is required to account for normal stress differences [7,8]. The balance equations for the mass, momentum and second-moment, respectively, can be obtained by taking appropriate moment of the Enskog-Boltzmann equation:…”

“…The balance equations (1-2, 7), along with closure relations for (4), (8) and (9), constitute the Navier-Stokesorder hydrodynamics for which the equation for the deviatoric part of the second moment is satisfied identically.…”

“…is assumed to be zero, leaving only its isotropic part, the heat flux vector (8), to be evaluated as a constitutive relation. Recently we showed [7] that the heat-flux follows a generalized Fourier law of the form…”

“…The balance equations (1-3), along with closure relations for the stress tensor (4), the source of second moment (6) and the heat-flux vector (11), constitute the 10-moment theory for a granular fluid -this represents the minimal continuum model that incorporates (i) normal stress differences as well as (ii) anisotropic heat flux [7,8]. In the remainder of this paper, we will analyse the dense-limit behaviour of the stress tensor and related transport coefficients for uniform shear flow for which the anisotropic Gaussian [2,7,8] serves as the reference distribution function. The latter follows from the maximum entropy principle.…”

“…Such order-one effects must be incorporated in the theoretical modelling of granular fluids so that the theory remains valid from the dilute to the dense limit. Recently we outlined a Grad-like [6] extended hydrodynamic theory for granular fluids [7][8][9], and derived analytical expressions for normal stress differences and other transport coefficients that are valid for (a) a large range of density (encompassing both the dilute and dense limits) and (b) small restitution coefficient (far away from the limit of elastic collisions). After a brief overview of this theory in Sec.…”