Steady Flow of a Fluid Stream

A control volume is a defined region, such as a pipe with solid boundaries, through which fluid flows and changes direction. The fluid entering the control volume at time t with velocity v1 equals the fluid leaving at time t + dt with velocity v2. This framework allows engineers to apply impulse and momentum principles to analyze how forces from pipe walls affect fluid motion.

In steady flow, the mass of fluid entering the control volume equals the mass leaving during an infinitesimal time interval dt. The momentum of the fluid within the control volume remains constant over this interval. By applying the principle of linear impulse and momentum for a system of particles and dividing by dt, engineers derive expressions for the resultant external force acting on the fluid.

Mass flow rate, expressed as dm/dt, represents the constant inflow and outflow of fluid per unit time. It is calculated as the product of the fluid's density and its discharge flow, also called volumetric flow. This relationship shows how mass flow depends on both the fluid's density and the volume of fluid passing through the pipe per unit time.

Pipe walls exert impulse on flowing fluid because the fluid changes direction as it moves through the pipe. This directional change requires a resultant force from the pipe walls to alter the fluid's momentum. The impulse from these forces causes the fluid's velocity to change from v1 at the inlet to v2 at the outlet, demonstrating the relationship between force and momentum change.

Linear and angular impulse-momentum equations provide a framework for analyzing fluid dynamics within a control volume. By dividing these equations by dt, engineers derive expressions for the resultant external force and moment about an arbitrary origin point O. These principles enable prediction of fluid behavior and force analysis on pipe systems carrying flowing fluids.

Discharge flow, or volumetric flow, measures the volume of fluid passing through a pipe per unit time, while mass flow measures the mass of fluid passing through per unit time. Mass flow is calculated by multiplying the fluid's density by its discharge flow. For incompressible fluids, knowing volumetric flow allows direct calculation of mass flow using the constant fluid density.

The momentum equation, derived from principle of angular impulse and momentum principles, directly relates the rate of momentum change to external forces acting on the fluid. In steady flow, dividing the impulse-momentum equation by dt yields the resultant external force. This relationship allows engineers to calculate pipe wall forces needed to redirect fluid flow and maintain system equilibrium.