We show that the thickness, h(0), dependence of the effective viscosity, eta(eff)(h(0)), of unentangled random copolymer films of styrene (St) and 4-methoxystyrene (MeOS) supported by silica (SiOx) can be easily tuned by tuning the MeOS concentration, XMeOS, in the copolymer. When XMeOS is increased from 0% to 100%, eta(eff)(h(0)) changes steadily from suppressed to enhanced with decreasing film thickness. At X-MeOS = 10%, eta(eff)(h(0)) displays a nonmonotonic variation. Results obtained by fitting a three-layer model to the eta(eff)(h(0)) data show that eta(eff)(h(0)) of the X-MeOS = 0% film was dominated by the mobility-enhancing effect of the free surface, but as X-MeOS increases, the mobility-slowing effect of the substrate surface increases and dominates. The results also show that the propagation distance, l(t), of the substrate effect increases with increasing XMeOS and is similar to 2.6 to 6 times the radius of gyration of the polymer. We tentatively suggest the long propagation distances to be caused by the long-range van der Waals interactions. Analysis of the van der Waals interfacial potentials of the films supports this interpretation.