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Fluid Momentum Equation (Control Volume) Calculator

Relates external forces on a control volume to the rate of change of momentum within it and momentum flux across its surface.

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Net External Force

Formula first

Overview

The Fluid Momentum Equation for a control volume is a fundamental principle in fluid mechanics, representing Newton's second law for a fluid system. It states that the sum of all external forces acting on a defined control volume equals the rate of change of momentum within the control volume plus the net rate of momentum outflow across its control surface. This integral form is crucial for analyzing complex fluid flows, designing hydraulic machinery, and understanding aerodynamic forces without needing to track individual fluid particles.

Symbols

Variables

F_net = Net External Force, = Fluid Density, V_in = Inlet Velocity, A_in = Inlet Area, V_out = Outlet Velocity

Net External Force
Fluid Density
Inlet Velocity
m/s
Inlet Area
Outlet Velocity
m/s
Outlet Area
Turn Angle
deg
Mass Flow Rate
kg/s
Change in Velocity Magnitude
m/s

Apply it well

When To Use

When to use: Apply this equation when analyzing fluid flow problems where forces, momentum changes, or momentum fluxes are involved, especially in situations with complex geometries or unsteady flow. It's ideal for problems involving jets, pipes, turbomachinery, and aerodynamic bodies where a control volume can be effectively defined.

Why it matters: This equation is vital for engineers to predict forces exerted by fluids on solid surfaces (e.g., pipe bends, turbine blades, aircraft wings) and to understand how fluid momentum changes. It underpins the design of propulsion systems, hydraulic structures, and countless other fluid-handling devices, ensuring safety and efficiency.

Avoid these traps

Common Mistakes

  • Incorrectly defining the control volume or control surface boundaries.
  • Missing external forces or momentum fluxes across the control surface.
  • Errors in handling vector quantities, especially dot products for momentum flux.
  • Not accounting for unsteady terms when the flow is time-dependent.

One free problem

Practice Problem

Water flows steadily through a horizontal pipe bend. The inlet velocity is 5 m/s with a cross-sectional area of 0.1 m², and the outlet velocity is also 5 m/s with the same area, but at a 90-degree angle to the inlet. The water density is 1000 kg/m³. Neglecting pressure forces and wall friction, what is the net force exerted by the fluid on the control volume (the pipe bend)?

Fluid Density1000 kg/m^3
Inlet Velocity5 m/s
Inlet Area0.1 m^2
Outlet Velocity5 m/s
Outlet Area0.1 m^2
Turn Angle90 deg

Solve for:

Hint: For steady flow, the unsteady term is zero. Focus on the momentum flux term. The force is .

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Bird, R. Byron, Stewart, Warren E., Lightfoot, Edwin N. "Transport Phenomena.
  2. Incropera, Frank P., DeWitt, David P., Bergman, Theodore L., Lavine, Adrienne S. "Fundamentals of Heat and Mass Transfer.
  3. Wikipedia: Control volume (fluid mechanics)
  4. Britannica: Fluid mechanics
  5. Fox and McDonald's Introduction to Fluid Mechanics
  6. White, Fluid Mechanics
  7. Munson, Young and Okiishi's Fundamentals of Fluid Mechanics
  8. Bird, R. Byron, Stewart, Warren E., Lightfoot, Edwin N. Transport Phenomena. John Wiley & Sons.