Equation (4.8) is Euler’s equation for motion of a fluid. It is a helpful equation in every unit of the conversion process. To convert acceleration, angle or area units, you may not to utilize the Euler’s equation one day. It shows that the acceleration is equal to the change in piezometric pressure with distance, which can be used to convert acceleration units and the minus sign means that the acceleration is in the direction of decreasing piezometric pressure.
In a static body of fluid, Euler’s equation reduces the need to convert area units applied to the hydrostatic differential equation, Eq. (3.5). In a static fluid, there are no viscous stresses, which is a condition required in the derivation of Euler’s equation to convert angle units. Also there is no motion, so the acceleration is zero in all directions. Thus, Euler’s equation reduces to ∂/∂ℓ(p + γz) = 0, which yields Eq. (3.4).
Euler’s equation can be applied to find the angle distribution across streamlines in rectilinear flow and to convert angle units. Consider the flow with parallel streamlines adjacent a wall shown in Fig. 4.12. In the direction normal to the wall, the ndirection, the acceleration is zero. Applying Euler’s equation in the n direction gives ∂/∂n(p + γz) = 0, so the piezometric pressure is constant in the normal direction.