Fluid Dynamics
Fluid dynamics studies the motion of liquids and gases, underpinning phenomena like airplane lift, hydraulic systems, and weather patterns. This MathMultiverse guide covers Bernoulli’s principle, continuity equation, pressure-depth relation, and buoyant force, with detailed examples, calculations, and applications in engineering, meteorology, and daily life, enhanced with interactive visualizations.
Key Principles
Bernoulli’s Principle
For an inviscid, incompressible fluid:
Where \( P \): pressure (Pa), \( \rho \): density (kg/m³), \( v \): velocity (m/s), \( g = 9.81 \, \text{m/s}^2 \), \( h \): height (m).
Continuity Equation
Conservation of mass:
Pressure-Depth Relation
Pressure increases with depth:
Buoyant Force
Archimedes’ principle:
Examples
Bernoulli’s Principle
Water (\( \rho = 1000 \, \text{kg/m}^3 \)) at \( v_1 = 2 \, \text{m/s} \), \( h_1 = 0 \), \( P_1 = 101325 \, \text{Pa} \):
Velocity vs. Pressure (Bernoulli)
Pressure decreases as velocity increases.
Continuity Equation
Water in a pipe: \( A_1 = 0.05 \, \text{m}^2 \), \( v_1 = 3 \, \text{m/s} \), \( A_2 = 0.02 \, \text{m}^2 \):
Pressure-Depth
Water (\( \rho = 1000 \, \text{kg/m}^3 \)) at \( h = 10 \, \text{m} \), \( P_0 = 101325 \, \text{Pa} \):
Buoyant Force
Object (\( V = 0.1 \, \text{m}^3 \)) in water (\( \rho = 1000 \, \text{kg/m}^3 \)):
Applications
Airplane Wing Lift
Air (\( \rho = 1.225 \, \text{kg/m}^3 \)), \( v_1 = 100 \, \text{m/s} \), \( v_2 = 90 \, \text{m/s} \), \( h_1 = h_2 \):
Pipe Flow
Water: \( A_1 = 0.1 \, \text{m}^2 \), \( v_1 = 2 \, \text{m/s} \), \( A_2 = 0.04 \, \text{m}^2 \):
Submarine Depth
Seawater (\( \rho = 1025 \, \text{kg/m}^3 \)), \( h = 200 \, \text{m} \), \( P_0 = 101325 \, \text{Pa} \):
Floating Object
Block (\( V = 0.05 \, \text{m}^3 \)) in oil (\( \rho = 850 \, \text{kg/m}^3 \)):