.. 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.