Bi	the Biot number
c	specific heat, J K−1 kg−1
d	diameter, m
g	gravitational acceleration, m s−2
H	cavity height, m
h	specific enthalpy, J kg−1
k	thermal conductivity, W K−1 m−1
L	cavity length, m
Lm	fusion energy or latent heat of melting, J kg−1
Os	Ostrogradsky number
p	pressure, Pa
Pr	Prandtl number
Q	heat transfer rate per unit of volume, W m−3
Ra	Rayleigh number
Ste	Stefan number
t	time, s
T	temperature, K
Tm	melting temperature
u, v	velocity components in Cartesian coordinate along x and y, m s−1
U, V	dimensionless velocity components
x, y	Cartesian coordinate, m
X, Y	dimensionless Cartesian coordinates
Greek Symbols
α	thermal diffusivity, m2 s−1
β	coefficient of thermal expansion, K−1
γ	heat transfer coefficient, WK−1m−2
η	smoothing parameter (or melting temperature range), K
Θ	dimensionless temperature
μ	dynamic viscosity, Pa s
ν	kinematic viscosity, m2 s−1
ρ	density, kgm−3
τ	dimensionless time
Φ	nanoparticles volume fraction
φ	volume fraction of the melt
ψ	stream function, m2 s−1
Ψ	dimensionless stream function
ω	vorticity, s−1
Ω	dimensionless vorticity
Subscripts
0	initial condition or ambient
1	radiator
2	heater
l	liquid
m	melting
nm	nanomaterial
np	nanoparticle
s	solid