This paper presents an advection-diffusion model that describes normal grai
n growth in 'size-sides' space. Ordinary differential equations governing t
he self-similar distributions at the steady state of normal grain growth ar
e derived. By solving numerically the continuity equations (time-dependent)
and the corresponding ordinary differential equations (time-independent),
we get the self-similar grain size distributions in time-dependent and time
-independent form. The two sets of distributions have nearly the same shape
, confirming the self-consistency of the model. Some comparisons with other
models and simulations are given.