  We study the cosmic velocity-density relation using the spherical collapse
model (SCM) as a proxy to non-linear dynamics. Although the dependence of this
relation on cosmological parameters is known to be weak, we retain the density
parameter Omega_m in SCM equations, in order to study the limit Omega_m -> 0.
We show that in this regime the considered relation is strictly linear, for
arbitrary values of the density contrast, on the contrary to some claims in the
literature. On the other hand, we confirm that for realistic values of Omega_m
the exact relation in the SCM is well approximated by the classic formula of
Bernardeau (1992), both for voids (delta<0) and for overdensities up to delta ~
3. Inspired by this fact, we find further analytic approximations to the
relation for the whole range delta from -1 to infinity. Our formula for voids
accounts for the weak Omega_m-dependence of their maximal rate of expansion,
which for Omega_m < 1 is slightly smaller that 3/2. For positive density
contrasts, we find a simple relation div v = 3 H_0 (Omega_m)^(0.6) [
(1+delta)^(1/6) - (1+delta)^(1/2) ], that works very well up to the turn-around
(i.e. up to delta ~ 13.5 for Omega_m = 0.25 and neglected Omega_Lambda). Having
the same second-order expansion as the formula of Bernardeau, it can be
regarded as an extension of the latter for higher density contrasts. Moreover,
it gives a better fit to results of cosmological numerical simulations.
