Fluid Properties, Statics, and Dimensional Analysis
Key Takeaways
- Start every fluid problem by choosing the property model: incompressible liquid, ideal gas, saturated vapor, or property supplied directly in the question.
- Hydrostatic pressure changes with elevation only, so velocity and pipe-loss ideas do not belong in a static-fluid model.
- Gauge pressure, absolute pressure, specific weight, density, and viscosity are separate quantities; mixing them is a common FE trap.
- Hydrostatic resultants on submerged areas use pressure at the centroid for magnitude and center of pressure for line of action.
- Dimensional analysis problems reward recognizing nondimensional groups such as Reynolds number, Froude number, Mach number, and friction factor.
- In USCS fluids, watch the mass-force distinction and conversion constants whenever lbm, lbf, ft, and seconds appear together.
Choose the fluid model first
Fluid mechanics starts with properties, not formulas. For liquid water or oil in ordinary FE problems, the usual model is incompressible: density is treated as constant and pressure changes do not noticeably change volume. For gases, check whether the problem gives density directly, asks for ideal gas behavior, or requires absolute pressure and absolute temperature in PV = mRT. Do not use gauge pressure in an ideal gas equation unless the problem has already converted it to absolute pressure.
Key property pairs are easy to confuse. Density rho is mass per volume. Specific weight gamma is weight per volume and equals rho g in SI. Specific gravity is a ratio to a reference density, usually water for liquids. Dynamic viscosity mu has units such as Pa*s, while kinematic viscosity nu equals mu/rho and has units of m^2/s. Reynolds number uses nu as VD/nu or rhoVD/mu; do not put both viscosity forms into the same version.
Hydrostatics is not pipe flow
A fluid at rest has no shear from flow and no velocity head. Pressure varies with depth according to p = p0 + gamma h when density is constant. That one equation handles open tanks, submerged surfaces, and many manometers if the sign convention is kept consistent. Move downward in a static fluid and pressure increases; move upward and pressure decreases.
| Situation | Model choice | FE trap |
|---|---|---|
| Open water tank | Gauge pressure is zero at free surface | Adding atmospheric pressure when answer asks gauge |
| Closed gas space over liquid | Start with gas pressure at interface | Forgetting absolute pressure if gas law appears |
| U-tube manometer | Step pressure through connected columns | Using height difference with wrong fluid density |
| Submerged plane gate | F = p_centroid A | Placing resultant at centroid instead of center of pressure |
For a submerged plane surface, the magnitude of the hydrostatic force is pressure at the centroid times area. The line of action passes through the center of pressure, which lies below the centroid for a vertical or inclined surface because pressure grows with depth. If the question asks only for total force, centroid pressure is enough. If it asks for torque or support reaction, center of pressure matters.
Buoyancy and stability
Buoyant force equals the weight of displaced fluid. The object density determines whether it floats, sinks, or is neutrally buoyant, but the force itself depends on displaced volume and surrounding-fluid specific weight. Floating problems often require setting object weight equal to displaced-fluid weight. Submerged-object problems may ask for apparent weight: actual weight minus buoyant force.
Dimensional analysis
Dimensional analysis is a model check and sometimes the whole problem. Buckingham Pi problems ask how many nondimensional groups can be formed from variables and fundamental dimensions. More often on FE Mechanical, you identify what a nondimensional number means.
| Group | Typical use |
|---|---|
| Reynolds number | Inertial force versus viscous force; laminar or turbulent tendency |
| Froude number | Free-surface and gravity-wave similarity |
| Mach number | Compressibility and speed relative to sound |
| Euler number | Pressure force compared with inertial effect |
| Friction factor | Pipe-loss correlation input |
Unit discipline
SI fluids are usually direct: Pa is N/m^2, and rho g h gives Pa when rho is kg/m^3, g is m/s^2, and h is m. USCS problems need more care. Pressure may be psi while pipe formulas want ft of head; area may be in in^2 while force is lbf. Convert deliberately. If an answer is off by 144, suspect psi-to-psf. If it is off by about 32.2, suspect lbm-lbf handling.
A sealed tank gauge reads 35 kPa at the gas space above a liquid. If local atmospheric pressure is 101 kPa, what pressure should be used in an ideal-gas calculation for the gas?
Oil flows in a 50 mm diameter pipe at 1.2 m/s with kinematic viscosity 6.0e-5 m^2/s. Which Reynolds number and flow classification are closest?
For a vertical rectangular gate submerged in water, which statement is most accurate for the hydrostatic resultant?