Slow viscous flows in micropolar fluids

Rotational Particle Separation in Solutions: Micropolar Fluid Theory Approach

We develop a new mathematical model for rotational sedimentation of particles for steady flows of a viscoplastic granular fluid in a concentric-cylinder Couette geometry when rotation of the Couette cell inner cylinder is prescribed. We treat the suspension as a micro-polar fluid. The model is validated by comparison with known data of measurement. Within the proposed theory, we prove that sedimentation occurs due to particles' rotation and rotational diffusion.

Micromagnetorotation of MHD Micropolar Flows

The studies dealing with micropolar magnetohydrodynamic (MHD) flows usually ignore the micromagnetorotation (MMR) effect, by assuming that magnetization and magnetic field vectors are parallel. The main objective of the present investigation is to measure the effect of MMR and the possible differences encountered by ignoring it. The MHD planar Couette micropolar flow is solved analytically considering and by ignoring the MMR effect. Subsequently, the influence of MMR on the velocity and microrotation fields as well as skin friction coefficient, is evaluated for various micropolar size and electric effect parameters and Hartmann numbers. It is concluded that depending on the parameters’ combination, as MMR varies, the fluid flow may accelerate, decelerate, or even excite a mixed pattern along the channel height. Thus, the MMR term is a side mechanism, other than the Lorentz force, that transfers or dissipates magnetic energy in the flow direct through microrotation. Acceleration or deceleration of the velocity from 4% to even up to 45% and almost 15% deviation of the skin friction were measured when MMR was considered. The crucial effect of the micromagnetorotation term, which is usually ignored, should be considered for the future design of industrial and bioengineering applications.

Steady electro-osmotic flow of a micropolar fluid in a microchannel

We have formulated and solved the boundary-value problem of steady, symmetric and one-dimensional electro-osmotic flow of a micropolar fluid in a uniform rectangular microchannel, under the action of a uniform applied electric field. The Helmholtz–Smoluchowski equation and velocity for micropolar fluids have also been formulated. Numerical solutions turn out to be virtually identical to the analytic solutions obtained after using the Debye–Hückel approximation, when the microchannel height exceeds the Debye length, provided that the zeta potential is sufficiently small in magnitude. For a fixed Debye length, the mid-channel fluid speed is linearly proportional to the microchannel height when the fluid is micropolar, but not when the fluid is simple Newtonian. The stress and the microrotation are dominant at and in the vicinity of the microchannel walls, regardless of the microchannel height. The mid-channel couple stress decreases, but the couple stress at the walls intensifies, as the microchannel height increases and the flow tends towards turbulence.

Rheology of a granular gas under a plane shear

The rheology of a two-dimensional granular gas under a plane shear is investigated. From the comparison among the discrete element method, the simulation of a set of hydrodynamic equation, and the analytic solution of the steady hydrodynamic equations, it is confirmed that the fluid equations derived from the kinetic theory give us accurate results even in relatively high density cases.

Numerical tests of constitutive laws for dense granular flows

We numerically and theoretically study the macroscopic properties of dense, sheared granular materials. In this process we first consider an invariance in Newton's equations, explain how it leads to Bagnold's scaling, and discuss how it relates to the dynamics of granular temperature. Next we implement numerical simulations of granular materials in two different geometries--simple shear and flow down an incline--and show that measurements can be extrapolated from one geometry to the other. Then we observe nonaffine rearrangements of clusters of grains in response to shear strain and show that fundamental observations, which served as a basis for the shear transformation zone (STZ) theory of amorphous solids [M. L. Falk and J. S. Langer, Phys. Rev. E. 57, 7192 (1998); M.R.S. Bull 25, 40 (2000)], can be reproduced in granular materials. Finally we present constitutive equations for granular materials as proposed by Lemaître [Phys. Rev. Lett. 89, 064303 (2002)], based on the dynamics of granular temperature and STZ theory, and show that they match remarkably well with our numerical data from both geometries.

Collisional granular flow as a micropolar fluid

We show that a micropolar fluid model successfully describes collisional granular flows on a slope. A micropolar fluid is the fluid with internal structures in which coupling between the spin 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. We demonstrate that the model equations can quantitatively reproduce the velocity and the angular velocity profiles obtained from the numerical simulation of the collisional granular flow on a slope using a simple estimate for the parameters in the theory.

Condensation of hard rods under gravity: exact results in one dimension

We present exact results for the density profile of a one dimensional array of N hard rods of diameter D and mass m under gravity g. For a strictly one dimensional system, the liquid-solid transition occurs at zero temperature, because the close-packed density straight phi(c) is 1. However, if we relax this condition slightly such that straight phi(c)=1-delta, we find a series of critical temperatures Tc(i)=mgD(N+1-i)/mu0 with mu0=1/delta-1, at which the ith particle undergoes the liquid-solid transition. The functional form of the onset temperature, Tc(1)=mgDN/mu0, is consistent with the previous result [Physica A 271, 192 (1999)] obtained by the Enskog equation. We also show that the increase in the center of mass is linear in T before the transition, but it becomes quadratic in T after the transition because of the formation of solid near the bottom.