Mechanical Properties of Fluids
'Water is the driver of nature.' — Leonardo da Vinci
1. Chapter Overview
FLUIDS (liquids and gases) behave differently from solids — they FLOW and take the SHAPE of their container. This chapter covers both FLUID STATICS (fluids at rest — pressure, buoyancy, Pascal's law) and FLUID DYNAMICS (fluids in motion — Bernoulli's theorem, viscosity). Surface tension — the property that makes water drops spherical — is also covered.
2. Pressure in Fluids
- Pressure P = F/A (force PERPENDICULAR per unit area)
- SI unit: pascal (Pa) = N/m²
- Other units: 1 atm = 1.013×10⁵ Pa = 76 cm Hg = 760 torr
- Pascal's Law: Pressure applied to an enclosed fluid is transmitted UNDIMINISHED to every point in the fluid and to the walls of the container
Hydrostatic Pressure
- P = P₀ + ρgh (pressure at depth h in a fluid of density ρ)
- P₀ = atmospheric pressure at the surface
- Pressure depends ONLY on depth, NOT on the shape of the container
Hydraulic Lift (Pascal's Law Application)
- F₁/A₁ = F₂/A₂ → F₂ = (A₂/A₁)F₁
- A small force on a small piston produces a LARGE force on a large piston
Worked Problem
Q: In a hydraulic lift, the small piston area is 10 cm² and large is 500 cm². Find force needed on small piston to lift 2000 kg. A: F₂ = mg = 20000 N. A₂/A₁ = 500/10 = 50. F₁ = F₂×A₁/A₂ = 20000/50 = 400 N.
3. Archimedes' Principle and Buoyancy
- Archimedes' Principle: When a body is IMMERSED in a fluid, it experiences an UPWARD buoyant force equal to the WEIGHT of the displaced fluid
- F_b = ρ_fluid × V_submerged × g
- Floatation: A body floats if its AVERAGE density ≤ fluid density
- Apparent Weight: Weight in fluid = Actual weight — Buoyant force
Laws of Floatation
- A body floats if ρ_body < ρ_fluid
- A body sinks if ρ_body > ρ_fluid
- At equilibrium for a floating body: Weight of body = Buoyant force = Weight of displaced fluid
4. Bernoulli's Theorem
- Statement: For an IDEAL fluid (incompressible, non-viscous, streamlined flow), the SUM of pressure energy, kinetic energy, and potential energy per unit volume is CONSTANT
- Equation: P + ½ρv² + ρgh = CONSTANT
- OR: P₁/ρg + v₁²/2g + h₁ = P₂/ρg + v₂²/2g + h₂ (Bernoulli's equation in head form)
Applications
- Venturimeter: Measures flow rate. Pressure difference at constriction gives velocity
- Atomiser: Fast-moving air creates low pressure, draws liquid up
- Aeroplane Wing: Air moves faster over curved top → lower pressure → LIFT
- Magnus Effect: Spinning ball curves due to pressure difference
- Torricelli's Law: v = √(2gh) (efflux velocity from a hole at depth h)
Worked Problem
Q: Water flows in a pipe with area 0.02 m² at 2 m/s and pressure 2×10⁵ Pa. At a constriction of area 0.01 m², find velocity and pressure. A: A₁v₁ = A₂v₂ → 0.02×2 = 0.01×v₂ → v₂ = 4 m/s. Bernoulli: P₁ + ½ρv₁² = P₂ + ½ρv₂² (horizontal, h same) 2×10⁵ + ½×1000×4 = P₂ + ½×1000×16 2×10⁵ + 2000 = P₂ + 8000 → P₂ = 1.94×10⁵ Pa.
5. Viscosity
- Definition: The property of a fluid that OPPOSES relative motion between its layers (internal friction)
- Coefficient of Viscosity (η):
- F = ηA(dv/dy) (Newton's law of viscous flow)
- Si unit: Pa·s or poiseuille (1 poise = 0.1 Pa·s)
- Streamline vs Turbulent Flow: Laminar flow at LOW velocity; becomes TURBULENT beyond a critical velocity
Stokes' Law
- F = 6πηrv (viscous drag force on a sphere of radius r moving through a fluid at speed v)
- Terminal Velocity: When viscous force + buoyancy = weight
- v_t = (2/9)(ρ_s — ρ_f)gr²/η
Reynolds Number (R_e)
- R_e = ρvD/η (where D = characteristic dimension, e.g., diameter)
- R_e < 1000: Streamline (laminar) flow
- R_e > 2000: Turbulent flow
- 1000 < R_e < 2000: Transition region
6. Surface Tension
- Definition: The property of a liquid surface to behave like a STRETCHED ELASTIC MEMBRANE
- Surface tension S = F/L (force per unit length)
- SI unit: N/m
- Origin: Molecules at the surface experience a NET INWARD force (cohesive forces unbalanced)
Surface Energy
- Surface energy = S × ΔA (work done to increase surface area by ΔA)
- Liquid drops try to MINIMISE surface area → SPHERICAL shape
Capillary Rise
- h = 2Scosθ/(ρgr) (rise/fall in a capillary tube)
- θ < 90°: Liquid RISES (water in glass)
- θ > 90°: Liquid FALLS (mercury in glass)
- θ = 0°: Maximum rise (perfect wetting)
Applications
- Needle floating on water (surface tension supports it)
- Water drops on lotus leaf (high contact angle, self-cleaning)
- Capillary action in plants (water rises through xylem)
- Detergents REDUCE surface tension for better cleaning
7. Common Mistakes
- Pressure is NOT a vector: It acts EQUALLY in all directions (scalar)
- Buoyant force depends on displaced volume, NOT the object's volume: If an object is partially submerged, only the submerged volume counts
- Bernoulli's equation applies ONLY to ideal fluids: Real fluids have viscosity and energy losses
- Surface tension decreases with temperature: Hot water has lower surface tension, cleans better
- Terminal velocity increases with r²: Larger drops fall faster (important in rain)
8. CBSE Exam Focus
- Pascal's law — hydraulic lift numerical (3-mark)
- Bernoulli's theorem derivation (5-mark) and applications
- Viscosity — Stokes' law and terminal velocity (3/5-mark)
- Surface tension — capillary rise (5-mark)
- Venturimeter and aerofoil lift — explanation
9. Key Formulas
- P = P₀ + ρgh
- F₁/A₁ = F₂/A₂ (Pascal's law)
- F_b = ρVg (Buoyancy)
- P + ½ρv² + ρgh = constant
- A₁v₁ = A₂v₂ (Continuity equation)
- F = ηAdv/dy (Viscosity)
- v_t = (2/9)(ρ_s — ρ_f)gr²/η
- h = 2Scosθ/(ρgr)
- R_e = ρvD/η
10. Self-Test (5+ Q&A)
Q1: Calculate the pressure at the bottom of a 10 m water tank (ρ = 1000 kg/m³, P₀ = 1×10⁵ Pa, g = 10 m/s²). A: P = P₀ + ρgh = 10⁵ + 1000×10×10 = 2×10⁵ Pa = 2 atm.
Q2: A sphere of radius 1 mm falls through water (η = 0.001 Pa·s). Find terminal velocity. (ρ_s = 5000 kg/m³, ρ_f = 1000 kg/m³, g = 10) A: v_t = (2/9)(5000-1000)×10×(0.001)²/0.001 = (2/9)×4000×10×10⁻⁶/0.001 = (2/9)×40 = 8.89 m/s.
Q3: Water rises to 5 cm in a capillary tube. If the tube radius is halved, what will be the rise? A: h ∝ 1/r. If radius is halved, height DOUBLES = 10 cm.
Q4: An object weighs 50 N in air and 30 N in water. Find its volume and density. A: Buoyant force = 20 N = ρVg. V = 20/(1000×10) = 0.002 m³. Density = 50/(10×0.002) = 2500 kg/m³.
Q5: Why does a spinning ball curve in flight? A: The Magnus effect: spinning drags air faster on one side, creating a pressure difference that deflects the ball.
11. Conclusion
Fluid mechanics governs everything from the flow of blood in your veins to the flight of aeroplanes. Pascal's law explains hydraulic machinery, Bernoulli's theorem explains lift and atomisers, viscosity determines how fluids flow, and surface tension explains capillary action and droplet shape. These concepts have ENORMOUS practical importance in engineering, meteorology, and biology.
