# Why do proofs of Bernoulli's equation assume that forces on opposite ends point in different directions?

I've read 4 different books and yet nobody explains why forces $$F_1$$ ($$=p_1A_1$$) and $$F_2$$ ($$=p_2A_2$$) point in different directions. Shouldn't $$F_2$$ point in the same direction as $$v_2$$?

Since we're assuming that parts of fluid between $$a$$ and $$b$$ have the same kinetic and potential energies (same holds for $$c$$ and $$d$$), why do all proofs state that the change in work: $$W_2 - W_1$$ is equal to the change in energy $$E_2 - E_1$$? Work is equal to the change in kinetic energy, so $$W_2 = W_1 = 0$$ (because we assumed that fluid between each pair of points has the same energy).

Then there's the problem of signs, how do we determine which sign to choose and how do potential energies come into the equation?

• What about the pressure from the left side for the surface $A_2$?The entire fluid is the same. – user3711671 Aug 25 '19 at 13:20
@mike stone Does a great job at addressing your first point. To address you second point, it is true that the net work changes the kinetic energy, i.e. $$W_{net}=\Delta K$$. However, we are interested just in the work done by the external forces acting on the fluid. This means that $$W_\text {ext}=\Delta E$$. This is the work done by your forces on either end of the fluid segment.