Fluids

Notes

Density and pressure

• Density $\displaystyle \rho = \frac{m}{V}$. Water is 1,000 kg/m³.
• Pressure $\displaystyle P = \frac{F}{A}$, in pascals (1 Pa = 1 N/m²). Atmospheric pressure is about $1.0 \times 10^5$ Pa.
• Pressure in a fluid increases with depth: $P = P_0 + \rho gh$. Gauge pressure is $\rho gh$ (pressure above atmospheric).
• Pressure at a given depth is the same in all directions and doesn't depend on the container's shape.

Buoyancy

Archimedes' principle: the buoyant force equals the weight of the fluid displaced:

• An object floats if its average density is less than the fluid's. A floating object displaces its own weight of fluid.
• Fraction submerged $\displaystyle = \frac{\rho_{object}}{\rho_{fluid}}$.

Fluids in motion

• Continuity equation (incompressible flow): $A_1v_1 = A_2v_2$. A narrower pipe means faster flow.
• Bernoulli's equation (energy conservation for fluids): $\displaystyle P_1 + \rho gy_1 + \frac{1}{2}\rho v_1^2 = P_2 + \rho gy_2 + \frac{1}{2}\rho v_2^2$. Where fluid moves faster at the same height, the pressure is lower.
• Torricelli's theorem: water leaving a hole a depth $h$ below the surface comes out at $\displaystyle v = \sqrt{2gh}$.