Fluid Mechanics: Density, Pressure, and Fluid Flow

Fluid Statics and Properties

Fluid Statics: Equilibrium and Compressibility

Fluid statics deals with fluids at rest and the forces and pressures associated with them. Key concepts include equilibrium, compressibility, and incompressibility.

• Equilibrium: A fluid is in equilibrium when it is at rest and the net force on any part of the fluid is zero.

• Compressible vs. Incompressible: Liquids are generally considered incompressible, while gases are compressible.

Additional info: Incompressibility simplifies many fluid calculations, as density remains constant.

Density and Mass Properties

Density (ρ, or d, or m/V)

Density is a measure of mass per unit volume and is a fundamental property of fluids.

• Definition: $\rho = \frac{m}{V}$, where $m$ is mass and $V$ is volume.

• SI Unit: $\mathrm{kg/m^3}$

• Common Densities:

  • Water: $\rho_{\text{water}} = 1000\ \mathrm{kg/m^3}$

  • Air: $\rho_{\text{air}} = 1.29\ \mathrm{kg/m^3}$

  • Mercury: $\rho_{\text{Hg}} = 13,600\ \mathrm{kg/m^3}$

  • Iron: $\rho_{\text{Fe}} = 7,900\ \mathrm{kg/m^3}$

• Example: Calculating the mass of air in a classroom:

  • Given $V = 12 \times 10 \times 3 = 360\ \mathrm{m^3}$, $\rho_{\text{air}} = 1.29\ \mathrm{kg/m^3}$

  • $m = \rho V = 1.29 \times 360 = 464\ \mathrm{kg}$

Pressure in Fluids

Definition and Units

Pressure is the amount of force exerted per unit area by fluid particles.

• Formula: $P = \frac{F}{A}$

• SI Unit: Pascal ($\mathrm{Pa}$), where $1\ \mathrm{Pa} = 1\ \mathrm{N/m^2}$

• Other Units:

  • 1 bar = $10^5$ Pa

  • 1 atm = $1.013 \times 10^5$ Pa

  • 1 mmHg = 133 Pa

• Example: Atmospheric pressure at sea level is approximately $1.013 \times 10^5$ Pa.

Pressure Variation with Depth

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

• Formula: $P = P_0 + \rho g h$

• Where:

  • $P_0$ = pressure at the surface

  • $\rho$ = fluid density

  • $g$ = acceleration due to gravity

  • $h$ = depth below the surface

• Example: Pressure at 10 m depth in water:

  • $P = 1.013 \times 10^5 + (1000)(9.8)(10) = 1.013 \times 10^5 + 98,000 = 199,300$ Pa

Laws and Principles in Fluid Mechanics

Pascal's Law

A pressure change in a confined, incompressible fluid is transmitted undiminished in all directions throughout the fluid.

• Formula: $\Delta P = \rho g \Delta h$

• Hydraulic Press: $\frac{F_1}{A_1} = \frac{F_2}{A_2}$

• Application: Used in hydraulic lifts and brakes.

Archimedes' Principle

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

• Buoyant Force: $F_b = \rho_{\text{fluid}} V_{\text{disp}} g$

• Application: Explains why objects float or sink.

Fluid Flow

Continuity Equation

For an incompressible, steady flow, the mass flow rate is constant throughout the pipe.

• Formula: $A_1 v_1 = A_2 v_2$

• Where:

  • $A$ = cross-sectional area

  • $v$ = fluid velocity

• Application: Used to determine changes in velocity or area in pipe systems.

Bernoulli's Equation

For an incompressible, frictionless fluid in steady flow, the sum of pressure energy, kinetic energy, and potential energy per unit volume is constant along a streamline.

• Formula: $P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$

• Application: Explains phenomena such as lift in airplane wings and the Venturi effect.

• Example: Calculating pressure at different points in a pipe with varying cross-sectional area and height.

Flow Rate

The volumetric flow rate is the volume of fluid passing through a cross-section per unit time.

• Formula: $Q = \frac{\Delta V}{\Delta t} = A v$

• Where:

  • $Q$ = flow rate ($\mathrm{m^3/s}$)

  • $A$ = area ($\mathrm{m^2}$)

  • $v$ = velocity ($\mathrm{m/s}$)

• Example: For a pipe of radius $r = 0.05$ m and $v = 6$ m/s:

  • $A = \pi r^2 = \pi (0.05)^2 = 0.00785\ \mathrm{m^2}$

  • $Q = 0.00785 \times 6 = 0.0471\ \mathrm{m^3/s}$

Summary Table: Key Fluid Properties and Equations

Property/Principle	Equation	SI Unit	Notes
Density	$\rho = \frac{m}{V}$	$\mathrm{kg/m^3}$	Mass per unit volume
Pressure	$P = \frac{F}{A}$	$\mathrm{Pa}$	Force per unit area
Hydrostatic Pressure	$P = P_0 + \rho g h$	$\mathrm{Pa}$	Pressure increases with depth
Pascal's Law	$\frac{F_1}{A_1} = \frac{F_2}{A_2}$	-	Hydraulic systems
Archimedes' Principle	$F_b = \rho_{\text{fluid}} V_{\text{disp}} g$	$\mathrm{N}$	Buoyant force
Continuity Equation	$A_1 v_1 = A_2 v_2$	$\mathrm{m^3/s}$	Conservation of mass
Bernoulli's Equation	$P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$	$\mathrm{Pa}$	Energy conservation in fluids