Fluid Properties, Pressure, Buoyancy, and Fluid Dynamics

1. Density

Density is a fundamental property of matter, describing how much mass is contained in a given volume. It is crucial for understanding fluid behavior and buoyancy.

• Definition: Density (ρ) is the mass per unit volume of a substance.

• Formula:

• Units: kg/m3

• Example: Water has a density of approximately 1000 kg/m3.

2. Pressure

Pressure is the force exerted per unit area. In fluids, pressure plays a key role in determining how forces are transmitted and how fluids move.

• Definition: Pressure (P) is the normal force (F) applied perpendicularly to a surface area (A).

• Formula:

• Units: Pascal (Pa) = N/m2

• Pressure in a fluid at depth:

• = surface/atmospheric pressure

• = depth below surface

• Example: The deeper you go underwater, the greater the pressure due to the weight of the water above.

3. Pascal’s Principle

Pascal’s Principle states that a change in pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid.

• Formula (Hydraulic lift):

• Application: Hydraulic brakes and lifts use this principle to multiply force.

4. Archimedes’ Principle (Buoyant Force)

Archimedes’ Principle explains why objects float or sink in fluids. The buoyant force is equal to the weight of the fluid displaced by the object.

• Formula:

• = displaced fluid volume

• Floating Object Condition: An object floats if its average density is less than that of the fluid.

• Condition for floating:

or

5. Continuity Equation (Ideal Fluid Flow)

The continuity equation expresses the conservation of mass in fluid flow. For an incompressible fluid, the product of cross-sectional area and velocity is constant along a streamline.

• Formula:

• Application: Explains why water speeds up when flowing from a wide pipe into a narrow one.

6. Bernoulli’s Equation

Bernoulli’s Equation relates pressure, velocity, and height in a moving fluid, expressing the conservation of energy for flowing fluids.

• General form:

• Special horizontal-flow case:

• Applications: Airplane wings, venturi tubes, and fluid speed measurement.

7. Torricelli’s Law (Efflux Speed)

Torricelli’s Law gives the speed of fluid flowing out of an opening under gravity, derived from Bernoulli’s equation.

• Formula:

• Application: Used to calculate the speed of water exiting a hole in a tank.

8. Viscosity (for reference)

Viscosity is a measure of a fluid’s resistance to flow. Newtonian fluids have a constant viscosity regardless of the applied stress.

• Formula (for force due to viscosity):

• = viscosity coefficient

• Example: Honey has a higher viscosity than water.

9. Volume Flow Rate

The volume flow rate measures how much fluid passes through a given area per unit time.

• Formula:

• Units: m3/s

10. Absolute, Gauge, and Atmospheric Pressure

Pressure measurements can be referenced to a vacuum (absolute), atmospheric pressure, or the difference between them (gauge).

• Formula:

11. Manometer Relation

Manometers are devices used to measure pressure differences using columns of fluid.

• Formula:

12. Surface Tension

Surface tension is the force per unit length acting along the surface of a liquid, due to molecular interactions.

• Force due to surface tension:

• Pressure for a bubble:

Complete Fluid Formula Table (Condensed Version)

This table summarizes the main formulas and their applications in fluid mechanics.