.. include:: /images.rst DEM04: Slipping on a rough surface ---------------------------------- This case serves to verify the MFIX-DEM soft-spring collision model through the analysis of the rolling friction model. This test case was originally reported in :cite:Garg2012. .. _description-dem04: Description ~~~~~~~~~~~ A spherical particle of radius, :math:r_{p}, finite translation velocity, :math:u_{0}, and zero angular velocity, :math:\omega_{0}, is placed on a rough surface as illustrated in :numref:dem04fig1. The particle begins to roll while the translational velocity decreases because of rolling friction attributed to slip between the particle and the rough surface at the point of contact (:math:u \neq \omega r_{p}). The rolling friction converts translation velocity to angular velocity until there is no slip at the contact point (:math:u = \omega r_{p}). After the no-slip condition is reached, rolling friction ceases and the particle continues to move with constant translational and rotational velocities. .. _dem04fig1: .. figure:: ../media/dem04-setup.png :align: center A spherical particle with finite translational velocity and zero angular velocity is placed on a rough surface. Forces acting on the particle are indicated. Kinetic friction is the only translational force acting on the particle and is given by .. math:: \frac{\text{du}}{\text{dt}} = \frac{d^{2}x}{dt^{2}} = - \mu g. :label: dem04eq1 where :math:g is the acceleration due to gravity, and :math:\mu is the coefficient of friction. Similarly, the angular velocity is given by .. math:: \frac{\text{dω}}{\text{dt}} = \frac{\text{μgm}r_{p}}{I} :label: dem04eq2 where :math:I = 2mr_{p}^{2}/5 and :math:m are the particle moment of inertia and mass, respectively. Integrating equations (4‑14) and (4‑15) with initial conditions :math:u_{0} and :math:\omega_{0}, an expression for the time when rolling friction ceases (:math:u = \omega r_{p}) is obtained, .. math:: t_{s} = \frac{2u_{0}}{7\mu g} :label: dem04eq3 .. _setup-dem04: Setup ~~~~~ .. _dem04table1: .. csv-table:: DEM-04 Setup, Initial and Boundary Conditions. :widths: auto :header: "Computational/Physical model", " ", " " "1D, Transient", " ", " " "Granular flow (no gas)", " ", " " "Gravity", " ", " " "Thermal energy equation is not solved", " ", " " " ", " ", " " "**Geometry**", " ", " " "Coordinate system", "Cartesian", " ", " " "x-length", "1.0", "$$m$$", " " "y-length", "1.0", "$$m$$", " " "z-length", "1.0", "$$m$$", " " " ", " ", " " "**Solids Properties**", " ", " " "Normal spring coefficient, :math:k_{n}", "10\ :sup:4", "(N·m\ :sup:-1)" "Restitution coefficient, :math:e_{n}", "1.0", "" "Friction coefficient, :math:\mu", "*varied*", "" " ", " ", " " "**Solids 1 Type**", "DEM", " " "Diameter, :math:d_{p}", "0.001", "$$m$$" "Density, :math:\rho_{s}", "10,000", "(kg·m\ :sup:-3)" " ", " ", " " "**Boundary Conditions**", " ", " ", " " "All boundaries", "Solid walls", " " .. _results-dem04: Results ~~~~~~~ Simulations were conducted for nine restitution coefficients, [0.2, 0.3, 0.4, 0.4, 0.6, 0.7, 0.8, 0.9 1.0], using the Adams-Bashforth time-stepping method with the results shown in :numref:dem04fig2. The absolute relative percent error between the MFIX-DEM and analytical value for the non-dimensionalized time when rolling friction ceases, :math:t_{s}/\left( \text{μg}/u_{0} \right), is less than 1% for all reported conditions. Similarly, the absolute relative percent error between the MFIX-DEM and analytical value for the non-dimensionalized tangential and angular velocities is less than 0.1% for all reported conditions. Error between the MFIX-DEM and analytical values can be further reduced (not shown) by increasing the normal spring coefficient, :math:k_{n}, which decreases the DEM solids time-step size. .. _dem04fig2: .. figure:: ../media/image55.png :align: center Comparison between the analytical solution (solid line) and MFIX-DEM simulation (open symbols) of a particle with radius :math:\mathbf{r}_{\mathbf{p}} slipping on a rough surface for various friction coefficients. (left) Dimensionless slip time end and (right) dimensionless equilibrium tangential, :math:\mathbf{u}, and angular, :math:\mathbf{\omega}, velocities.