Suppose that in a container of volume V we pour V1 of methanol and V2 of water (V2>V1), supposing that V2 is big enough that all the methanol will be dissolved, do we have to mix them mechanically or the diffusion will be enough if we want them to mix in seconds?
As already mentioned by @Zhe, the rate of the mixing of methanol in water (or the other way around) depends (among other parameters) on the rate of addition, the geometry of the vessel, if / how the solutions are mixed by a stirrer. In addition, this process diffusion is dependent on temperature and gradients of concentration. One approach to quantify diffusion is based on the differential Fick's laws by determination of the diffusion coefficient.
As difficult it may appear, the diffusion of methanol in water was determined experimentally, as here and here, for example. If you do not have access to a university library, NIH / HHS made a more recent reference available here, and a more general value is given here