Covariant cosmological evolutions and constraints equations of the conventional cosmology are considered in brane-world cosmology. Following the approach of Senovilla, Sopuerta and Szekeres they are used to investigate in the framework of the brane-world cosmology the validity of the conjecture of the conventional cosmology stating that every shear-free and geodesic perfect fluid must be either expansion-free or rotation-free. Due to the presence of the electrical part of the 5-dimensional Weyl tensor representing the free gravitational effect of the bulk making the effective Einstein Field Equations on the brane not closed, the analysis is complicated and inconclusive. Instead, when the analysis is restricted to a purely Coulomb-like case, it is shown explicitly that, for a perfect fluid with vanishing acceleration the theorem holds equivalently in this new cosmology too. When the pressure and the effective total pressure are constants the result is independent of the effective energy density on the brane arising from the free gravitational field in the bulk. The theorem is valid in the case of variable pressures unless the aforesaid effective energy density is positive-definite.