We study experimentally the dynamics of the spontaneous rise of a meniscus made of polydimethylsiloxane of various viscosities on the outside of a micrometric fiber made of monofilament in polyethyleneterephthalate. Optical methods are used to measure simultaneously the height of the liquid interface and the associated contact angle versus time. Versus the liquid viscosity, we observe asymptotically, for the dynamic contact angle, either a t(-1/2) behavior in agreement with the hydrodynamic description or a t(-1) behavior predicted by the molecular-kinetic theory, in its linear form, thus confirming the existence of at least two major time scales in the rise phenomenon.