Fig. Schematic diagram of two particles (“micro polar”) just before collision
We consider a mono disperse system of rough, inelastic spheres of size d, mass m, and the moment of inertia I , interacting via hard-sphere potential. The pre-collisional translational and angular velocities of particle i are denoted by ci and ωi (see Fig) , respectively, and the corresponding post-collisional velocities are denoted by the primed symbols, c′i and ω′i . Let kij = zj− zi = k be the unit vector directed from the center of the i-th particle to that of j -th particle. The total pre-collisional relative velocity at contact, gij , between particle i and j is given by gij = cij -1/2 dk×(ωi + ωj ) where cij = ci − cj is the translational velocity of particle i relative to j, We have performed molecular dynamics (MD) simulations of the uniform shear flow of a rough dilute granular gas. Using particle simulations of the uniform shear flow of a rough dilute granular gas, we show that the translational and rotational velocities are strongly correlated in direction, but there is no orientational correlation-induced singularity at perfectly smooth (β=-1) and rough (β = 1) limits for elastic collisions (e =1); both the translational and rotational velocity distribution functions remain close to a Gaussian for these two limiting cases.
Fig. (a) Variationof Λ(t) =cos2Ψ with time for different values of e with β=0,φ=0.01 and N=8000. Left and right insets show the distributions of cos2Ψ and cosΨ, respectively. (b) Variation of <cos2Ψ> with β for different e. Larger symbols (triangulated-cycle) at β=0 for each e correspond to simulations with N=4000 and 16000, respectively.
In Fig. 1(a) we have plotted the temporal variation of Λ(t) (main panel) for two values of normal restitution coefficients (e = 1, 0.5), with the tangential restitution coefficient being set to β=0 0; the corresponding probability distribution of cosΨ is shown in the left inset. [The probability distribution of cosΨ, P(cosΨ), is symmetric about its zero mean for all e, but its width becomes narrower with decreasing e, see the right inset.] From the main panel and the left inset, we find that even for e = 1 the mean value of Λ is different from 1/3 (for a molecular gas), signaling the presence of orientational or directional correlation; decreasing the value of e to 0.5 decreases its value to <Λ(t)>= 0.26, thus enhancing orientational correlation significantly. The variation of the temporal average of Λ(t) with particle roughness, β, is shown in Fig. (b); the dot-dashed line represents the limiting value of 1/3 for a molecular gas. Note that the data points for e =1 (thick blue dashed line) and e = 0.99 almost overlap with each other. For any e, the orientational correlation is maximum at β =0 and it decreases monotonically as we approach the perfectly smooth (β = -1) and perfectly rough ( β=1) limits. This latter observation is in contrast to that in a freely cooling dilute granular gas for which h.i varies nonmonotonically with . for .1<β<0 and 0<β<1. Another difference with freely cooling gas is that the magnitude of h.i is much larger in shear flow. It must be noted that even though the translation and rotation are decoupled at .. .1 (independent of the value of e), the smooth limit is singular for any e = 1 in shear flow. However, there is no orientational correlation-induced singularity at both the perfectly smooth (β =-1) and rough (β = 1) limits for the limiting case of e = 1.
B. Gayen & M. Alam (2008) Orientational Correlation and Velocity Distributions in Uniform Shear Flow of a Dilute Granular Gas. Phys. Rev. Lett., 100, p. 068002.