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Flow in a streamtube

The kinetic energy of an infinitesimal small volume ΔV which moves with speed v is given by Ekin(ΔV) = (mass of ΔV) ×v2/2 = ρfluid ΔV ×v2/2 Consequently Ekin(t+Δt) - Ekin(t) = ρfluid (ΔV2×v22 -ΔV1×v12)/2 where v1 and v2 represent the speeds of respectively the volumes of fluid ΔV1 and ΔV2. Furthermore, the amount of fluid which enters the streamtube on the lefthand side must be the same as the amount of fluid which leaves the streamtube at the righthand side, since the fluid is incompressible. Hence, ΔV2 = ΔV1 When we define ΔV=ΔV2=ΔV1, then we arrive at Ekin(t+Δt) - Ekin(t) = ρfluid ΔV (v22 -v12)/2

streamtube