Dilute Granular Flow as a Micropolar Fluid

We show that a micropolar fluid model successfully describes granular flows on a slope. Micropolar fluid is the fluid with internal structures, and the coupling between the rotation of each particle and the macroscopic velocity field is taken into account; it is a hydrodynamical framework suitable for granular systems which consists of particles with macroscopic size. It is demonstrated that the model equations can quantitatively reproduce the velocity and angular velocity profiles obtained from the numerical simulation of the dilute surface flow using the parameters consistent with our simple estimate. 45.70.Mg, 47.50.+d, 47.85.-g Typeset using REVTEX 1 Granular material consists of dissipative particles with macroscopic size. The surprisingly rich phenomena found in granular systems has been attracting the interest of physicists [1]. Even a simple situation like the granular material on a inclined plane shows non-trivial behavior: When the inclination is smaller than the angle of repose, they remain at rest. In a finite range of the angle, however, the stationary state may be metastable and sufficiently large perturbation may trigger the avalanche [2]. If the plane is inclined further, they begins to flow like a fluid. The interaction between the particles are dominated either the friction or the collision, depending on a situation. It is also found that they show spontaneous wave formation in some situation [3]. In spite of the long history of research on the granular flows, the theoretical framework for them has not been established yet. One of the reasons which make analytical treatment difficult is that the separation of the length scale is not easy; the size of each particle is often comparable with the scale of the macroscopic collective motion. Therefore, there are many situations that simple hydrodynamic approaches cannot be used to characterize granular flows [4]. Even when we consider the rapid granular flows [5] in which the density is low enough and kinetic theory seems to be valid, the coupling between the rotation of each particle and macroscopic velocity field is not negligible. Thus, the behavior of flow in general deviates from that of Newtonian fluid. Micropolar fluid model is a continuum model to describe a fluid which consists of particles with internal structure [6]. The model equations include the asymmetric stress tensor and the couple stress tensor to describe the microrotation of constituents. Therefore, the model can be suitable framework to describe the granular flows. Although some researches on the application of the micropolar fluid model to granular flows have been carried out [7], most of them considered the dense granular flow, and the constitution relations adopted for stress and couple stress were very complicated; hence it was difficult to interpret the results physically, or to apply the model equation to description of different situations. In this paper, we apply a micropolar fluid model to the dilute granular flow. The adopted 2 constitution relations are simple and natural extension of those for Newtonian fluid. We calculate the velocity and angular velocity profiles of the uniform steady flow on a slope, and demonstrate that the micropolar fluid model reproduces the result of numerical simulation. It is easy to derive the following equations for the system consists of identical particles with mass m and moment of inertia I from the conservation laws of mass, momentum, and angular momentum [6]: Dtn = −n∂kvk, (1) mnDtvi + vi = mnfi + ∂jSji, (2) nIDtωi = ∂jCji + s (a) i , (3) where vi and ωi are the ith component of the velocity and the microrotation fields, respectively. The summation convention applies to suffixes occurring twice. ∂i represents the partial differentiation with respect to the ith coordinate and Dt ≡ ∂/∂t + vk∂k is the Lagrange’s derivative. n is the number density, Sij is the stress tensor, fi is the body force, and Cij is the couple stress tensor. s (a) i represents the torque due to the asymmetric part of the stress tensor defined as s (a) i = ǫijkSjk, (4) where ǫijk is Eddington’s epsilon. For the constitution relations of the stress tensor S and couple stress tensor C, we adopt following equations [6]: Sij = (−p+ λ∂kvk)δij + μ(∂ivj + ∂jvi) +μr [(∂ivj − ∂jvi)− 2ǫijkωk] , (5) Cij = c0∂kωkδij + μB + μA 2 (∂iωj + ∂jωi) + μB − μA 2 (∂iωj − ∂jωi), (6) with Kronecker’s delta δij . These constitutive equations are natural extension of those for classical hydrodynamics; When μr = 0, eq. (5) recovers the symmetric stress tensor for Newtonian fluid. 3 It is debatable whether such straightforward extension of Newtonian constitution relations can be applied to granular flow because the hydrostatic term in granular material may have different form. For fully developed flow, however, we expect such an effect may not be important. Therefore, here we concentrate on the dilute granular flow in the collisional flow region. Coefficients of viscosity which appear in eqs. (5) and (6) have been derived based on the kinetic theory of the three dimensional spheres with rough surface [9,10]. Here, for later convenience, and also to make the physical meaning of the model clear, we briefly summarize the rough estimation of the coefficients of viscosity for two dimensional disks using the elementary kinetic theory. Let us consider the two dimensional flow which consists of identical disks with diameter σ and is flowing uniformly in x direction, namely n = n(y), v = (u(y), 0, 0), and ω = (0, 0, ω(y)). Then we get Syx = μu ′(y) + μr[u ′(y) + 2ω(y)], (7) Cyz = μBω ′(y), (8) where the prime represents the derivative by its argument. Here, Syx (Cyz) is the x (z) component of the force (torque) acting on the plane perpendicular to y axis per unit area. μ in eq. (7) corresponds to the kinetic viscosity in dilute gas, and we can find the estimation by kinetic theory in textbooks of statistical physics, e.g. ref. [11]: It is given by