Fluid Mechanics

Fluid Mechanics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Fluid mechanics studies liquids and gases at rest and in motion. The two workhorses of CUET UG Physics are the continuity equation and Bernoulli’s equation, both rooted in conservation of mass and energy for an ideal, incompressible, non-viscous fluid.

• Pressure P = F/A (SI: pascal = N/m²); hydrostatic form P = ρgh, growing linearly with depth.
• Buoyancy F_b = ρ_fluid V_submerged g (Archimedes’ Principle).
• Continuity: A₁v₁ = A₂v₂ — narrower pipe ⇒ faster flow.
• Bernoulli: P + ½ρv² + ρgh = constant — speed up ⇒ pressure drops (Venturi effect).

CUET UG tip: roughly 1 MCQ per shift tests numericals on Bernoulli or apparent weight.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Pressure and Pascal’s Law

A fluid is any substance that flows and deforms continuously under an applied shear. Pressure is the normal force per unit area, P = F/A, transmitted undiminished in all directions at a point (Pascal’s Principle). This principle powers hydraulic lifts, where a small force on a small piston produces a large force on a larger piston because the pressure change is identical.

Hydrostatics and Buoyancy

Pressure inside a static fluid rises with depth: P = ρgh. Atmospheric pressure at sea level ≈ 1.013 × 10⁵ Pa acts on every exposed surface. Archimedes’ Principle states the buoyant force equals the weight of displaced fluid: F_b = ρ_fluid V_submerged g. Apparent weight = true weight − buoyant force; a body floats when its average density is less than that of the fluid.

Flow Regimes and Continuity

Streamline (laminar) flow has parallel, non-crossing paths; turbulent flow is chaotic with eddies. The Reynolds number Re = ρvD/η distinguishes them: Re < 2000 ⇒ laminar, Re > 4000 ⇒ turbulent. The continuity equation, A₁v₁ = A₂v₂, follows from mass conservation for incompressible flow.

Bernoulli’s Equation

Along a steady, non-viscous streamline:

P + ½ρv² + ρgh = constant

Higher velocity and higher elevation both lower pressure — the principle behind Venturi meters, atomisers, and aerofoil lift.

Viscosity and Surface Tension

Viscosity η (Pa·s) is internal friction; liquid viscosity drops with temperature, while gas viscosity rises. Surface tension γ (N/m) arises from cohesive forces at a surface, and capillarity makes liquids climb (water in glass) or depress (mercury in glass) in narrow tubes depending on the contact angle.

Quantity	Formula	Variables
Pressure	P = F/A	F (N), A (m²)
Hydrostatic pressure	P = ρgh	ρ (kg/m³), g (m/s²), h (m)
Buoyant force	F_b = ρ_fluid V_submerged g	ρ_fluid, V (m³), g
Continuity	A₁v₁ = A₂v₂	A (m²), v (m/s)
Bernoulli	P + ½ρv² + ρgh = const.	P (Pa), ρ, v, g, h

• Always use absolute pressure (gauge + atmospheric) in closed-vessel Bernoulli problems.
• Cross-sectional area uses radius squared, never diameter, in A = πr².
• Buoyancy direction is always vertically upward, opposite to weight.
• Bernoulli fails for compressible gases at high Mach number or strongly viscous flows.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Limits of the Ideal Model

Bernoulli’s equation assumes steady, incompressible, non-viscous flow along a single streamline. Real fluids deviate: viscous losses produce a pressure drop ΔP = 32 η L v / d² for laminar pipe flow (Poiseuille’s law), and at high Mach numbers compressibility must be included. The Reynolds number Re = ρvL/η (L = characteristic length) sets the regime boundary near 2300–4000 for pipe flow.

Torricelli’s Law and Worked Example

Efflux speed from a tank: v = √(2gh), a direct consequence of Bernoulli applied between the free surface and the orifice. Worked: water (ρ = 1000 kg/m³) fills a tank to h = 5 m above a small hole. Efflux speed v = √(2 × 9.8 × 5) = √98 ≈ 9.9 m/s. Volume flow rate Q = A_orifice × v; multiply by ρg to get the mass-flow rate. Sanity check: doubling h multiplies v by only √2, not 2 — a classic trap.

Common Mistakes

1. Treating gauge pressure as absolute inside sealed containers — add P_atm ≈ 1.01 × 10⁵ Pa when the gas above the liquid is open to the atmosphere only at the surface.
2. Applying Bernoulli across different streamlines or across a pump/turbine — the constant changes whenever energy is added or removed.
3. Swapping ρ_fluid and ρ_object in floating problems; a steel ship floats because its average density (including air cavities) is below water’s density, not because steel is lighter than water.
4. Forgetting that liquid viscosity decreases with temperature (oil thins when hot) while gas viscosity increases (air thickens slightly when hot).
5. Using diameter instead of radius in A = πr² for circular pipes — a factor-of-four error.

Connection to Adjacent Topics

Capillarity links to surface tension; both tie into the Young-Laplace equation ΔP = 2γ/r for a spherical interface. Bernoulli connects to work-energy theorem; continuity to mass conservation; viscosity to momentum diffusion. Buoyancy bridges statics and dynamics via Archimedes’ Principle.

Regimen	Reynolds number (Re)	Character
Laminar	< 2000	Smooth, parallel layers
Transitional	2000–4000	Unstable, intermittent eddies
Turbulent	> 4000	Chaotic mixing, high drag

Practice prompts

1. A horizontal pipe narrows from 8 cm to 4 cm diameter. If the speed in the wide section is 2 m/s and gauge pressure there is 60 kPa, find the speed and gauge pressure in the narrow section (water, ρ = 1000 kg/m³).
2. A wooden block of mass 1.2 kg and density 600 kg/m³ floats in water. What fraction of its volume lies below the surface? If a 0.5 kg mass is placed on top, will the block fully submerge? Show by calculating the new submerged volume.

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