.. include:: /images.rst
DEM03: Two stacked, compressed particles
----------------------------------------
This case serves to verify the MFIX-DEM linear spring-dashpot collision model through analysis of a multi-particle, enduring collision. This test case is based on the work of Chen et al. :cite:`Chen2007` and the MFIX-DEM test case was originally reported in Garg et al. :cite:`Garg2012`.
.. _description-dem03:
Description
~~~~~~~~~~~
Two particles of equal radius, :math:`r_{p}`, are stacked between two fixed walls such that the particles are compressed. The lower and upper walls are located at :math:`y_{l} = 0.0` and :math:`y_{w} = 3.6r_{p}` and the particle centers are initially located at :math:`y_{10} = 0.25y_{w}` and :math:`y_{20} = 0.75y_{w}`. This configuration, illustrated in :numref:`dem03fig1`, ensures that the particles remain in contact and compressed.
.. _dem03fig1:
.. figure:: ../media/dem03-setup.png
:align: center
Two smooth spherical particles stacked between two fixed walls so that the system is always under compression. A sketch of the problem mechanics is provided along with force balances for the lower and upper particles.
An expression for the acceleration of the lower particle (*particle 1*) is
.. math::
:label: dem03eq1
\frac{d^{2}y_{1}}{dt^{2}} =& - g - \frac{k_{\text{nw}}}{m_{1}}\left( y_{1} - r_{p} \right) - \frac{\eta_{n1w}}{m_{1}}\frac{dy_{1}}{\text{dt}}\\
& -\frac{k_{n12}}{m_{1}}\left( 2r_{p} - \left( y_{2} - y_{1} \right) \right) - \frac{\eta_{n12}}{m_{1}}\left( \frac{dy_{1}}{\text{dt}} - \frac{dy_{2}}{\text{dt}} \right)
where :math:`y_{1}` and :math:`y_{2}` are the particle center positions measured from the lower wall, :math:`g` is the acceleration due to gravity, :math:`k_{\text{nw}}` and :math:`k_{n12}` are the particle-wall and particle-particle spring coefficients, :math:`\eta_{n1w}` and :math:`\eta_{n12}` are the particle-wall and particle-particle damping coefficients, and :math:`m_{1}` is the mass of particle 1. Similarly, acceleration of the upper particle (*particle 2*) is given by
.. math::
:label: dem03eq2
\frac{d^{2}y_{2}}{dt^{2}} =& - g - \frac{k_{\text{nw}}}{m_{2}}\left( r_{p} - \left( y_{w} - y_{2} \right) \right) - \frac{\eta_{n2w}}{m_{2}}\frac{dy_{2}}{\text{dt}}\\
& +\frac{k_{n12}}{m_{2}}\left( 2r_{p} - \left( y_{2} - y_{1} \right) \right) + \frac{\eta_{n12}}{m_{2}}\left( \frac{dy_{1}}{\text{dt}} - \frac{dy_{2}}{\text{dt}} \right)
where :math:`\eta_{n2w}` is the particle-wall damping coefficient for the upper particle, and :math:`m_{2}` is the mass of the upper particle.
.. _setup-dem03:
Setup
~~~~~
.. _dem03table1:
.. csv-table:: DEM-03 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", "0.0018", "\(m\)", " "
"z-length", "0.0010", "\(m\)", " "
" ", " ", " "
"**Solids Properties**", " ", " "
"Normal spring coefficient, :math:`k_{n}`", "10\ :sup:`3`", "(N·m\ :sup:`-1`)"
"Restitution coefficient, :math:`e_{n}`", "*varied*", "\( \)"
"Friction coefficient, :math:`\mu`", "0.0", "\( \)"
" ", " ", " "
"**Solids 1 Type**", "DEM", " "
"Diameter, :math:`d_{p}`", "0.001", "\(m\)"
"Density, :math:`\rho_{s}`", "20000", "(kg·m\ :sup:`-3`)"
" ", " ", " "
"**Solids 2 Type**", "DEM", " "
"Diameter, :math:`d_{p}`", "0.001", "\(m\)"
"Density, :math:`\rho_{s}`", "10000", "(kg·m\ :sup:`-3`)"
" ", " ", " "
"**Boundary Conditions**", " ", " ", " "
"All boundaries", "Solid walls", " "
.. _results-dem03:
Results
~~~~~~~
Analytical solutions to equations :eq:`dem03eq1` and :eq:`dem03eq2` describing the motion of the particles are readily obtainable for perfectly elastic :math:`\left( \eta_{n12} = \eta_{n1w} = \eta_{n2w} = 1.0 \right)` particles of equal mass :math:`\left( m_{1} = m_{2} \right)`. This is not the case for inelastic particles of different mass, therefore a fourth-order Runge-Kutta method is used to calculate a secondary numerical solution which is considered to be the analytical solution during the analysis.
Simulations were conducted for six friction coefficients, [0.5, 0.6, 0.7, 0.8, 0.9, 1.0], using the Adams-Bashforth time-stepping method. :numref:`dem03fig2` shows the motion of the lower (left) and upper (right) particles as well as the absolute value of the relative error for a restitution coefficient of 1. The percent relative difference in results remains below 0.1% for this case. This is the largest observed difference across all cases with the difference in relative error decreasing with decreasing restitution coefficient.
.. _dem03fig2:
.. figure:: ../media/image54.png
:align: center
Comparison between the fourth-order Runge-Kutta solution (solid line) and MFIX-DEM simulation (open symbols) for the center position of two stacked particles compressed between fixed walls for a restitution coefficient of 1. The absolute percent relative errors are shown as dashed lines.