Understanding Force Formula: ρ AV1 (V1 - V2 cos θ) vs ρ A(V1 - V2 cos θ)

[hotjohn]

1. why the Fx is given by the formula = ρ AV1 ( V1 - V2 cos θ ) , why not = ρ A( V1 - V2 cos θ ) ?

 

[Aniket1]

The force is given by ρQ(v1 - v2cosθ) where Q is the rate of flow. You can understand this by Newton's second law by observing the rate of change of momentum with time. Note that this rate of flow will remain constant throughout the cross-section (which implies that the cross-section after the bend is different than that before the bend because the fluid has different velocities in the two regions). Thus it suffices to calculate the flow rate (Av) in the upper part where the cross-section and velocity is known, Q = A1v1.
Thus you have your desired result.