Fluid Dynamics and Pressure

Introduction to Fluids

Fluids, which include both liquids and gases, are substances that can flow and take the shape of their container. Their behavior is governed by properties such as pressure, density, and flow rate.

• Pressure is defined as the force exerted per unit area. In fluids, pressure acts equally in all directions at a given point.

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

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

Pressure in Fluids

Pressure at a certain depth in a fluid is determined by the weight of the fluid above that point.

• Hydrostatic Pressure Equation: Where is the pressure at the surface, is the fluid density, is acceleration due to gravity, and is the depth below the surface.

• Atmospheric pressure at sea level is about N/m2 (1 atmosphere).

• Pressure units include Pascal (Pa), bar, atm, dyne/cm2, mm-Hg, and torr.

Gauge Pressure and Atmospheric Pressure

Gauge pressure is the pressure relative to atmospheric pressure. Absolute pressure is the total pressure, including atmospheric pressure.

• Standard atmospheric pressure does not crush us because our bodies exert an equal internal pressure.

• Barometers and manometers are used to measure atmospheric and gauge pressure.

Buoyancy and Archimedes' Principle

Buoyant Force

When an object is submerged in a fluid, it experiences an upward force called the buoyant force, due to the pressure difference between the top and bottom of the object.

• Archimedes' Principle: The buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by the object.

• Buoyant force formula:

• If the object's density is less than the fluid's, it will float; otherwise, it will sink.

Floating Objects and Density

The fraction of a floating object submerged is determined by the ratio of its density to that of the fluid.

• For a floating object:

• Example: If an object floats with half its volume submerged, its density is half that of the fluid.

Applications of Buoyancy

• Ships float because their average density (including air inside) is less than water.

• Helium balloons rise because helium is less dense than air.

• Water level changes when objects are added or removed from a floating vessel, depending on whether they float or sink.

Fluid Flow and Continuity

Laminar and Turbulent Flow

Fluid flow can be laminar (smooth, orderly) or turbulent (chaotic, with eddies).

• Laminar flow occurs at low velocities and in smooth pipes.

• Turbulent flow occurs at high velocities or with obstacles.

Equation of Continuity

For incompressible fluids, the mass flow rate must be constant throughout the flow.

• Equation: Where is cross-sectional area and is fluid velocity.

• If the pipe narrows, velocity increases to conserve mass flow rate.

Bernoulli’s Principle

Bernoulli’s Equation

Bernoulli’s principle relates the pressure, velocity, and height in a moving fluid, based on conservation of energy.

• Bernoulli’s Equation:

• Where fluid velocity is high, pressure is low, and vice versa.

Applications of Bernoulli’s Principle

• Airplane wings generate lift due to pressure differences above and below the wing.

• Sailboats move against the wind using pressure differences across the sail.

• Spinning balls curve due to pressure differences created by varying air speeds on each side.

• Chimneys draw smoke upward more rapidly when wind increases air speed above the chimney, lowering pressure.

• Venturi meters measure fluid flow by comparing pressure differences in constricted sections.

Viscosity

Viscosity and Internal Friction

Viscosity is a measure of a fluid’s resistance to flow, caused by internal friction between layers.

• Viscous force formula: Where is the viscosity coefficient, is area, is velocity, and is the distance between layers.

• High viscosity fluids (like honey) flow slowly; low viscosity fluids (like water) flow easily.

Summary Table: Key Fluid Concepts

Additional info: Some context and equations have been expanded for clarity and completeness, based on standard college physics curriculum.