Variable

Description

Variable

Description

A

centroid of DEM particle contact overlap area

$re{s}_i^{(\dots )}$

residual at mesh node i, (p) for pressure, (u), (v) for respective velocities

$A,\cdots ,J$

nodes of a finite element (index)

$A_e$

area of a finite element

S

centroid of DEM particle contact overlap area A

$a_g$

gravitational acceleration

T

transpose operator

b

distance from circle center to wall

t

time

$C_1,C_2$

DEM particle center of mass

$\widehat{t}$

tangential direction unit vector

d

diameter

$U$

velocity vector

dl

line integral

$U_0$

maximum inflow velocity

e

finite element (index)

u

horizontal velocity

$entr{y}_{\dots }$

entries for the Jacobian J

$\begin{array}{l}{u}_{BC},v_{BC},\\ u_{BCTOP},v_{BCTOP}\end{array}$

velocity boundary conditions

$F_{da}$

DEM damping force

$F_{drag}$

fluid force acting on DEM particles

v

vertical velocity

x

horizontal position

$F_{el}$

DEM elastic force

Y

Young’s modulus

$g_A^{(\dots )},\cdots ,g_J^{(\dots )}$

coefficients for the element functions, (p) for pressure, (u), (v) for respective velocities

y

vertical position

$\Gamma$

DEM particle boundary

$g_i^{(\dots )},g_j^{(\dots )}$

coefficient for the element function at node i or j, (p) for pressure, (u), (v) for respective velocities

${\Delta }_g$

step vector for one Newton-Raphson iteration

$\Delta A$

change of A over $\tau$

$g,g_{\iota },g_{\iota -1}$

solution vector (entirety of all $g_i^{(\dots )},g_j^{(\dots )}$ ), for current $\iota$ or next $\iota +1$ Newton-Raphson iteration

${\delta }_{\epsilon \zeta }$

Kronecker delta

$\epsilon ,\zeta$

Cartesian direction (index)

$i,j$

mesh node (index), indexing variables

$\eta$

dynamic viscosity

${\theta }_A,\cdots ,{\theta }_J$

third order FEM shape functions

$J$

Jacobian of the Newton-Raphson, method

$\iota$

Newton-Raphson iteration step (index)

k

aspect ratio r/b

$\lambda$

wall correction factor

$L_{\dots }$

coefficients of the barycentric coordinate functions

$\mu$

friction coefficient

$l_{char}$

characteristic length for DEM particle contact

$\nu$

kinematic viscosity

$l_1,l_2$

vector $S{C}_{\dots }$

$\rho$

density

$m_1,m_2$

DEM particle mass

${\rho }_F,{\eta }_F$

fluid properties (density. dynamic viscosity)

$m_{red}$

DEM particle contact reduced mass

${\rho }_P,d_P$

particle properties (density, diameter)

$\widehat{n}$

normal direction unit vector

$\tau$

timestep length

$n-X$

value from X timesteps before (index)

${\varphi }_A,\cdots ,{\varphi }_F$

second order FEM shape functions

$P_{\dots }$

polynomial order of a finite element

${\varphi }_i,{\varphi }_j$

second order (or bubble) FEM shape function at node i or j

p

normalized pressure

${\varphi }_{P2+}$

second order finite element function with third order bubble

$p_{actual}$

pressure

${\psi }_A,\cdots ,{\psi }_C$

first order FEM shape functions

q

quadrature point (index)

${\psi }_i,{\psi }_j$

first order FEM shape function at node i or j

r

radius

${\psi }_{P1}$

first order finite element function

$res$

residual vector for the Newton-Raphson method

${\omega }_q$

quadrature weight of q

${\omega }_T$

trial function