.. 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.