14.12 : Steady Flow of a Fluid Stream

Consider a control volume, such as a pipe with solid boundaries, through which fluid flows and changes direction due to the impulse exerted by the resulting force from the pipe walls. In steady flow, the mass of fluid entering the control volume at a given time, t, with velocity v₁, is equal to the mass leaving after infinitesimal time dt, with velocity v₂.

During this process, the momentum of the fluid within the control volume remains constant over the time interval dt. By applying the principles of linear and angular impulse and momentum and dividing the resulting equations by dt, we derive expressions for the resultant external force and moment about an arbitrary origin point O, respectively.

The rate of change of mass over time represents the mass flow, indicating a constant inflow and outflow of fluid per unit of time. Mass flow is typically expressed as the product of the density of the incompressible fluid and its discharge flow. Discharge flow, also known as volumetric flow, quantifies the volume of fluid passing through the pipe per unit time.

Understanding these principles is crucial in fluid dynamics, helping to analyze and predict the behavior of fluid within a system. The linear and angular impulse and momentum equations provide a comprehensive framework for studying fluid flow dynamics within a control volume.