Fluids: Properties, Pressure, and Dynamics

9.1 Properties of Fluids

Definition and Types of Fluids

Fluids are substances that can flow and take the shape of their container. Both liquids and gases are classified as fluids, but they have distinct molecular arrangements and properties.

• Gases: Molecules are far apart, making gases compressible. Gas molecules move freely and occasionally collide with each other or the walls of the container.

• Liquids: Molecules are close together, making liquids essentially incompressible. Molecules have weak bonds but can slide past each other, allowing flow.

Volume and Density

Volume is the amount of space a system occupies, measured in cubic meters (m3). Density (\( \rho \)) is the ratio of mass to volume:

$\rho = \frac{m}{V}$

SI units for density are kg/m3, but g/cm3 is also common.

9.2 Pressure in Fluids

Definition of Pressure

Pressure is the ratio of force to the area over which the force is exerted:

$p = \frac{F}{A}$

Pressure is a scalar quantity and is measured in pascals (Pa), where 1 Pa = 1 N/m2.

Pressure in Liquids and Gases

• In liquids, pressure increases with depth due to gravity.

• In gases, pressure is nearly uniform in small containers but decreases with height in the atmosphere.

Causes of Pressure

• In a weightless environment, liquids do not exert pressure on container walls, but gases do due to molecular collisions.

• On Earth, gravity causes liquids to exert pressure on the bottom and sides of containers, and causes a slight density gradient in gases.

Atmospheric Pressure

Atmospheric pressure decreases with altitude. At sea level, the standard atmospheric pressure is:

$1\ \text{atm} = 101,000\ \text{Pa} = 101\ \text{kPa}$

Pressure Forces in Fluids

Fluids exert pressure in all directions, resulting in balanced forces on submerged objects. Devices like suction cups work by creating a pressure difference.

Pressure in Liquids: Hydrostatic Pressure

The pressure at depth d in a liquid is given by:

$p = p_0 + \rho g d$

where $p_0$ is the pressure at the surface, $\rho$ is the liquid's density, and $g$ is the acceleration due to gravity.

Hydrostatic Equilibrium

• In a connected liquid at rest, the pressure is the same at all points on a horizontal line.

• Liquids rise to the same height in all open regions of a container.

Gauge Pressure and Barometers

Gauge pressure is the pressure in excess of atmospheric pressure. Many devices, such as tire gauges, measure gauge pressure. To find absolute pressure, add atmospheric pressure to the gauge reading:

$p_g = p - 1\ \text{atm}$

Barometers measure atmospheric pressure using the height of a liquid column:

$p_{\text{atm}} = \rho g h$

9.3 Buoyancy and Archimedes' Principle

Buoyant Force

Fluids exert an upward force on submerged objects called the buoyant force. This force arises because pressure increases with depth, so the bottom of an object experiences more pressure than the top.

Archimedes' Principle

Archimedes' principle states that the buoyant force on an object is equal to the weight of the fluid displaced by the object:

$F_B = \rho_f V_f g$

where $\rho_f$ is the density of the fluid and $V_f$ is the volume of fluid displaced.

Applications: Floating and Sinking

• An object sinks if its average density is greater than the fluid's density.

• An object floats if its average density is less than the fluid's density.

• Neutral buoyancy occurs when the object's average density equals the fluid's density.

9.5 Fluids in Motion

Laminar and Turbulent Flow

Fluid flow can be laminar (smooth and steady) or turbulent (chaotic and irregular). Most introductory analysis assumes laminar flow and incompressible fluids.

Equation of Continuity

For an incompressible fluid, the volume flow rate is constant throughout a tube:

$v_1 A_1 = v_2 A_2$

where $v$ is the fluid speed and $A$ is the cross-sectional area. The volume flow rate $Q$ is:

$Q = vA$

9.6 Ideal Fluid Dynamics and Bernoulli's Equation

Bernoulli's Equation

Bernoulli's equation relates pressure and velocity in an ideal (nonviscous, incompressible) fluid:

$p_1 + \frac{1}{2} \rho v_1^2 = p_2 + \frac{1}{2} \rho v_2^2$

Pressure is higher where the fluid moves slower, and lower where it moves faster.

9.7 Viscous Fluid Dynamics

Viscosity and Poiseuille's Equation

Viscosity (η) is a fluid's resistance to flow. For viscous fluids, a pressure difference is required to maintain flow. The volume flow rate for a viscous fluid in a tube is given by Poiseuille's equation:

$Q = \frac{\pi R^4}{8 \eta} \frac{\Delta p}{L}$

where $R$ is the tube radius, $\Delta p$ is the pressure difference, $L$ is the tube length, and $\eta$ is the viscosity.

Turbulence and Reynolds Number

Flow becomes turbulent when the Reynolds number (ratio of inertial to viscous forces) exceeds a critical value. Turbulent flow is characterized by chaotic changes in pressure and velocity.

Summary Table: Pressure Units

Unit	Abbreviation	Conversion to Pa	Uses
pascal	Pa	1 Pa = 1 N/m2	SI unit, most calculations
atmosphere	atm	1 atm = 101 kPa	General
millimeters of mercury	mm Hg	1 mm Hg = 133 Pa	Gases, blood pressure, barometric pressure
pounds per square inch	psi	1 psi = 6.89 kPa	U.S. engineering and industry