Fluid Mechanics

Introduction

Fluid mechanics is the branch of physics concerned with the behavior of fluids (liquids and gases) at rest and in motion. It explains phenomena such as floating, sinking, and flow patterns in nature and technology.

• Fluids at rest are studied under hydrostatics.

• Fluids in motion are analyzed in fluid dynamics.

Density

Density is a fundamental property of matter, defined as mass per unit volume.

• Definition: , where is mass and is volume.

• SI unit: kilogram per cubic meter (kg/m3).

• Homogeneous materials have uniform density throughout.

• Example: Steel wrench and steel nail have different masses and volumes but the same density if made of the same material.

Densities of Common Substances

Pressure in a Fluid

Pressure is the force exerted perpendicular to a surface per unit area within a fluid.

• Definition: , where is the normal force and is the area.

• SI unit: pascal (Pa), where .

• Pressure is a scalar quantity and is independent of surface orientation.

Pressure at Depth

Pressure increases with depth in a fluid due to the weight of the fluid above.

• Equation:

• is the pressure at the surface, is fluid density, is acceleration due to gravity, is depth.

• Pressure at the bottom of connected columns is the same if the height is equal, regardless of shape.

Pascal's Law

Pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid and to the walls of the container.

• Equation:

• Used in hydraulic systems to multiply force.

Absolute Pressure and Gauge Pressure

• Absolute pressure: Total pressure including atmospheric pressure.

• Gauge pressure: Pressure above atmospheric pressure.

• If pressure is below atmospheric, gauge pressure is negative.

Pressure Gauges

• Bourdon-type gauges measure gauge pressure in systems such as gas lines.

• 1 bar = Pa.

Blood Pressure

• Measured as maximum (systolic) and minimum (diastolic) gauge pressures in arteries.

• Units: mm Hg or torr.

• Varies with vertical position; standard reference is upper arm at heart level.

Archimedes's Principle

Describes buoyancy: the upward force exerted by a fluid on an immersed object.

• Principle: An object immersed in a fluid experiences an upward force equal to the weight of the fluid displaced.

• Objects less dense than the fluid float; more dense objects sink.

• Equation:

Surface Tension

• Surface of a liquid acts like a stretched membrane due to molecular attraction.

• Surface tension allows insects like water striders to walk on water.

• Molecules at the surface are pulled into the bulk, minimizing surface area.

Fluid Flow

• Flow line: Path of a fluid particle.

• Steady flow: Flow pattern does not change with time; all particles follow the same path.

• No fluid crosses the side walls of a flow tube in steady flow.

Laminar vs. Turbulent Flow

• Laminar flow: Smooth, orderly flow; layers slide past each other.

• Turbulent flow: Chaotic, irregular flow; pattern changes continuously.

The Continuity Equation

Expresses conservation of mass for incompressible fluids in flow.

• Equation:

• Volume flow rate:

• Explains why a stream of honey narrows as it falls.

Bernoulli's Equation

Relates pressure, velocity, and height in steady, incompressible, non-viscous fluid flow.

• Equation:

• Used to analyze fluid flow in pipes, airplane wings, and blood circulation.

Applications

• Venturi meter: Measures flow speed using pressure differences.

• Lift on airplane wing: Faster flow above wing creates lower pressure, resulting in lift.

• Giraffe blood pressure: High pressure needed to pump blood to the brain due to height.

Viscosity

• Viscosity: Internal friction in a fluid; resistance to flow.

• Velocity is zero at pipe walls and maximum at center (parabolic profile).

• Viscosity decreases with increasing temperature (e.g., lava flows more easily when hot).

Turbulence

• At low speeds, flow is laminar; above a critical speed, flow becomes turbulent.

• Turbulent flow is noisy and irregular.

• In medicine, turbulent blood flow can indicate pathology and is detected with a stethoscope.

Additional info: These notes cover the essential concepts of Chapter 12: Fluid Mechanics, including definitions, equations, and real-world applications relevant for college-level physics students.