I am having trouble understanding the relationship between fluid velocity and pressure in a defined space such as a hose, pipe, etc. I understand that, by Bernoulli's Principle, the pressure of a given fluid decreases proportionally with an increase in velocity, and vice verse. I also understand that, by the equation of continuity $A_1\cdot V_1=A_2\cdot V_2$, fluid velocity in a defined space must increase if there is a proportional decrease in area of such space. My question is, if one were to, for example, put a nozzle on a water hose, thus forcing the water into a smaller space, would both velocity and pressure change, and if so, would they increase or decrease? Would one not change at all? I feel that it is natural that the pressure exerted on the nozzle would increase if you forced the same amount of water out over the same period of time as you were without a nozzle.
Your first equation shows you that $v$ is greater if the area decreases: $A_1 v_1 = A_2 v_2$.
Now, if you introduce this result into Bernoulli's equation, yo get that higher velocity requires less pressure... if heights are the same. Of course this could be false if the entrance and the exit are at very different heights from the ground, so that pressure changes due to change of potential energy too.
For a horizontal hose, however, pressure decreases, since "the rest of pressure is the one exerted by fast particples", which you're loosing though the nozzle, so you feel the slower ones.
This can be easily shown in everyday's life: just put a finger on the tap/faucet.