Nomenclature

A Avogadro’s number

c Velocity space, m/s

c_s Instantaneous particle velocity in i direction, m/s

C_s Peculiar particle velocity, m/s

d_ip = (d_t + d_p)/2 Average diameter of particles i and p, m

d_{s and}, d_p Diameter of particle, m

E Internal energy, J

e Restitution coefficient

e_sw Particle-wall restitution coefficient frequency distribution of

velocities of particles

F_Dip Momentum source (drag between solid phases), N

f Maxwellian velocity distribution function

f Single velocity distribution function

fp_p Pair velocity distribution function

g Gravity acceleration, m/s²

g₀ Radial distribution function

g_ip Radial distribution function at contact between particles i and p

g_ss Radial distribution function at contact between particles of the

same phase

H Gap between two plates, m

I Identity tensor

k_B The Boltzmann constant (1.3805 x 10~²³ J/K)

k=k- Unit vector connecting the centers of the two particles

⁼ ip

22

Kf Turbulent kinetic energy of fluid, m /s

k_s Granular conductivity, kg/m s

M Molecular weight, kg/kmol

m_s Mass of particle of phase s, kg

m₀ = m_i + m_p Total mass of two colliding particles, kg

N Number of moles

n Compact factor

n_s Number density of phase s

N_ip Energy dissipation

P_c Collisional pressure tensor, N/m²

P_k Kinetic pressure tensor, N/m²

p_c Critical state pressure, N/m²

pf Frictional pressure, N/m²

Q Momentum flux, N/m²

q_c Collisional flux, J/s m²

q_k Kinetic flux, J/s-m²

Rj Rate of heterogeneous reaction between phases, mol/m² s

Re_p Particle Reynolds number

(S) Average strain rate

r Position

S Deviatoric stress tensor

T Average of random kinetic energy

v Hydrodynamic velocity, m/s

v Center of mass mean velocity, m/s

v_s Mean velocity of phase s, m/s

V₀ Velocity of plates, m/s

v_s,_para Particle slip velocity parallel to wall, m/s

v_ts Solid particle terminal velocity, m/s

y_i Species mass fraction

Subscripts

• 1 Large
• 2 Small

cr Critical

fr Frictional

g Gas phase

col Collisional

kin Kinetic

max Maximum

T Total

s Solid phase

w Wall

Greek Symbols

X_ip Collisional flux, J/s m²

e_s Solid volume fraction of phase s

e_T Total solid volume fraction

(p_s Perturbation to Maxwellian distribution function

Y_ip Collisional source of granular temperature, J/m² s

p Viscosity, kg/m. s

p_ss Granular viscosity of phase s, kg/m. s

i_ip Mixture granular viscosity, kg/m. s

p_s Solid density of phase s, kg/m³

d_s Fluctuating granular energy m²/s²

d_m Mixture fluctuating granular energy, m²/s²

t_n Mixture normal stress, N/m²

t_s Solid-phase shear stress, N/m²

T_t Mixture shear stress, N/m²

y_s Property of particle

в Granular temperature (random kinetic energy per unit mass), m²/s²

ф Specularity coefficient Ф Angle of internal friction

gs Granular energy exchange between phases

p Bulk density, kg/m³

e_s Volume fraction of solids

p_s Solid density, kg/m³

11_2D Second invariant of the deviatoric stress tensor

Y Shear rate, 1/s

в Interface momentum exchange coefficient

Y Collisional energy dissipation, J/m³ s

? Solid-phase bulk viscosity, kg/m-s