let us, for once, streamline this question and only keep it specific t othe case in which surrounding temp. is lower than 37 C. applying elementary fluid mechanics AV is a constant ( equation of continuity) as the aperture (mouth) has different cross-sections for both cases the velocity will change accordingly. hence, in a wide opening, the amount of air per unit volume will be lesser hence, causing little scattering of the original coat of air present around the hair, hence, a small increase in temp. but, a uniform blow from a small opening will intially interact wid only a small section og the hand and then cause movement in the rest of the cover of air. due to the stream of fast CO2 moving outside da rate of evaporation is greater for the moisture hence, cooling is caused.

note: in both cases the initial relaxation of the lungs is taken as the same for application of eqn of  continuity.

another point, the application of da adiabatic process cud b used for the sheath of air around the hand. as the air from da narrow opening comes out it has more k.e. hence, vit displaces the air around da hand over a larger area hence, lowering da temp, which isnt done in da bigger opening. this case can b seen as one of perfect overlap in da case of the wide opening and of irregularity in da case of da narrow 1.