Fluid Mechanics: Conservation of Momentum
The conservation of momentum states that any change in momentum within a control volume must be balanced by an external force. This principle is crucial for calculating forces exerted by or on fluids.
Core Principles
- Momentum (M) is defined as mass (m) times velocity (v), M = mv.
- A change in momentum requires a force, as described by Newton's Second Law: F = d(mv)/dt.
- For steady flow, the momentum equation relates forces to the rate of change of momentum flux.
- Forces acting on a control volume can be body forces (like gravity) or forces transmitted through boundaries (pressure, viscous).
- The Steady Flow Momentum Equation is used to analyze forces in fluid systems by considering momentum in and out of a control volume.
- In 2D and 3D, the momentum equation is applied by considering components of force, momentum, and pressure in perpendicular directions (e.g., x and y).
Action Steps
- 1. Draw a control volume and define the positive x-direction.
- 2. Mark pressures on all control volume boundaries (use gauge pressures).
- 3. Indicate momentum fluxes entering and leaving the control volume.
- 4. Write down the steady flow momentum equation for the chosen direction(s).
- 5. Use continuity and/or Bernoulli's equation to find unknown velocities or mass flow rates.
- 6. Substitute values to calculate the external force applied to the fluid (F).
- 7. If needed, determine the force on the surroundings (R = -F).
Formulas
- $M = mv$
- $F = \frac{d(mv)}{dt}$
- $F_{net} = F_x + p_{in}A_{in} - p_{out}A_{out} = \dot{m}_{out}v_{out} - \dot{m}_{in}v_{in}$
- $F_x + \sum (pA)_x = \sum (\dot{m}v)_{x,out} - \sum (\dot{m}v)_{x,in}$
- $F_y + \sum (pA)_y = \sum (\dot{m}v)_{y,out} - \sum (\dot{m}v)_{y,in}$
- $M = \int \rho u^2 dA$
- $M = \beta \rho \pi R^2 \bar{u}^2$
Key Terms
- Momentum: The product of an object's mass and its velocity (M = mv).
- Momentum Flux: The rate at which momentum passes through a given area, calculated by integrating ρu²dA.
- Control Volume: An arbitrary region in space chosen for the analysis of fluid flow.
- Steady Flow: Flow where properties (like velocity, pressure) at any point do not change with time.
- Momentum Correction Factor (β): A factor that relates the momentum flux of a non-uniform flow to that of a uniform flow with the same mass flow rate.
Pro Tips
- Using gauge pressure simplifies calculations by setting atmospheric pressure to zero.
- Choosing the correct coordinate system can significantly simplify 2D and 3D problems.
- Momentum flux (∫ ρu²dA) is not the same as mass flow rate times average velocity (ṁū).
- The momentum correction factor (β) accounts for the difference in momentum flux between uniform and non-uniform flow profiles.
Pitfalls to Avoid
- Forgetting to account for all forces (body forces, pressure forces, viscous forces).
- Incorrectly defining the control volume or its boundaries.
- Using absolute pressure when gauge pressure is more convenient, or vice versa.
- Confusing momentum flux with the product of mass flow rate and average velocity.
- Not considering all surfaces of the control volume when flow is unconstrained.
Myth vs Reality
- Momentum flux is simply mass flow rate multiplied by average velocity.: Momentum flux is calculated by integrating ρu²dA across the flow area, which is generally higher than ṁū due to the velocity squared term.
- Atmospheric pressure always needs to be included in force calculations.: Using gauge pressure effectively sets atmospheric pressure to zero, simplifying calculations. If absolute pressure is used, atmospheric pressure acting on all surfaces must be carefully considered.
Real World Examples
- A jet of water hitting a surface.: Calculating the force exerted by the jet on the surface, or the force required to hold the surface stationary.
- Flow through a pipe bend.: Determining the forces on the pipe bend due to the change in momentum of the fluid.
- A submarine moving through water.: Calculating the drag force on the submarine by analyzing the momentum change in its wake.
People
- Dr Anna Young: Author/Lecturer
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