Learning Objectives
- Understand pressure in fluids and Pascal's law
- Learn about buoyancy and Archimedes' principle
- Study Bernoulli's principle and equation of continuity
- Understand viscosity, surface tension, and capillarity
- Apply fluid mechanics concepts to practical problems
Key Concepts
Pressure in Fluids
Pressure: P = F/A (SI unit: Pascal, Pa = N/m²). Other units: 1 atm = 1.013 × 10⁵ Pa = 760 mmHg.
Gauge Pressure: P_gauge = P - P_atm = ρgh (pressure due to fluid column of height h).
Absolute Pressure: P = P_atm + ρgh
Pressure at a point in a fluid is the same in all directions (Pascal's law).
Pascal's Law
A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the container.
Hydraulic lift/press: F₁/A₁ = F₂/A₂. A small force on a small piston produces a large force on a large piston.
Archimedes' Principle and Buoyancy
When a body is immersed in a fluid, it experiences an upward buoyant force equal to the weight of fluid displaced.
Buoyant force: F_b = ρ_fluid × V_displaced × g
If ρ_body < ρ_fluid: body floats; ρ_body = ρ_fluid: body is neutrally buoyant; ρ_body > ρ_fluid: body sinks.
Law of flotation: For a floating body, weight = buoyant force. Fraction submerged = ρ_body/ρ_fluid.
Fluid Dynamics
Streamline flow (laminar): Smooth, orderly flow where fluid layers do not mix. Each particle follows the same path.
Turbulent flow: Irregular, chaotic flow occurring above a critical velocity.
Reynolds Number: Re = ρvd/η. Re < 1000: laminar; Re > 2000: turbulent.
Equation of Continuity
For incompressible fluid: A₁v₁ = A₂v₂ (volume flow rate is constant).
Fluid flows faster where cross-section is smaller.
Bernoulli's Principle
For an ideal fluid in streamline flow: P + ½ρv² + ρgh = constant
Where fluid velocity increases, pressure decreases (and vice versa).
Applications: airplane lift, Venturi meter, atomizer, Magnus effect in sports.
Torricelli's theorem: Speed of efflux from an orifice: v = √(2gh).
Viscosity
Internal friction in fluids that resists flow.
Newton's law of viscosity: F = ηA(dv/dy), where η is the coefficient of viscosity.
SI unit of η: Pa·s or N·s/m² (Poiseuille). CGS unit: poise (1 poise = 0.1 Pa·s).
Stokes' Law: Drag on a sphere: F = 6πηrv
Terminal velocity: v_t = 2r²(ρ_s - ρ_l)g / 9η (when drag + buoyancy = weight).
Surface Tension
Surface tension (S): Force per unit length along the surface. S = F/L (SI unit: N/m).
Alternatively: Surface energy per unit area (J/m²).
Excess pressure: Inside a soap bubble: ΔP = 4S/R. Inside a liquid drop: ΔP = 2S/R.
Contact angle: Angle between the tangent to the liquid surface and the solid surface at the point of contact.
Capillary rise: h = 2S cos θ / (ρgr), where r is the radius of the capillary tube.
For water (θ < 90°): meniscus is concave, liquid rises. For mercury (θ > 90°): meniscus is convex, liquid falls.
Summary
Fluid mechanics covers static and dynamic properties of liquids and gases. Pascal's law governs pressure transmission. Archimedes' principle explains buoyancy. Bernoulli's equation relates pressure, velocity, and height for flowing fluids. Viscosity provides internal resistance to flow, and Stokes' law describes drag on small spheres. Surface tension arises from molecular forces at the surface and causes capillary action.
Important Terms
- Pascal's Law: Pressure change transmitted undiminished through fluid
- Buoyancy: Upward force on submerged object equal to weight of displaced fluid
- Bernoulli's Principle: Increase in fluid speed corresponds to decrease in pressure
- Viscosity: Internal friction in fluid, resistance to flow
- Surface Tension: Force per unit length at the surface of a liquid
- Terminal Velocity: Constant velocity when net force on falling body is zero
Quick Revision
- P = P_atm + ρgh; Pascal's law: F₁/A₁ = F₂/A₂
- Buoyant force = ρ_fluid × V_displaced × g
- Continuity: A₁v₁ = A₂v₂; Bernoulli: P + ½ρv² + ρgh = constant
- Stokes' law: F = 6πηrv; Terminal velocity: v_t = 2r²(ρ_s - ρ_l)g/9η
- Excess pressure: 4S/R (bubble), 2S/R (drop)
- Capillary rise: h = 2S cos θ/(ρgr)