Fluid Properties, Viscosity & Manometry

Fluid Mechanics is a heavy GATE ME subject (~8–12 marks), and properties/viscosity/manometry are its guaranteed easy marks. Newton's law of viscosity and capillary rise are near-certain 1-mark items.

Properties & viscosity

Density ρ (kg/m³), specific weight γ = ρg, specific gravity = ρ/ρ_water. Bulk modulus K = −dp/(dV/V) (resistance to compression); compressibility = 1/K.

Newton's law of viscosity: shear stress τ = μ·(du/dy), where μ = dynamic viscosity (Pa·s), du/dy = velocity gradient. Kinematic viscosity ν = μ/ρ (m²/s). Newtonian fluids (water, air) have constant μ; non-Newtonian fluids don't (shear-thinning/thickening, Bingham plastic).

Surface tension σ (N/m): capillary rise/fall h = 4σ·cosθ/(ρ·g·d) in a tube of diameter d (θ = contact angle). Pressure inside a droplet = 4σ/d; inside a bubble (two surfaces) = 8σ/d; inside a liquid jet = 2σ/d.

Manometry

Pressure varies with depth: p = ρgh (gauge). A manometer balances columns: sum ρgh down = sum ρgh up across the U-tube. Absolute = gauge + atmospheric.

Exam Tricks & Tips

• 🎯 τ = μ du/dy — viscosity is the proportionality between shear stress and velocity gradient; μ has units Pa·s.
• 🎯 Capillary rise h = 4σcosθ/ρgd — inversely proportional to tube diameter; narrower tube → higher rise.
• 🎯 Droplet 4σ/d, soap bubble 8σ/d (two surfaces), jet 2σ/d — memorise the 4/8/2 pattern.
• 🎯 Kinematic ν = μ/ρ (m²/s) — used in Reynolds number; don't confuse with dynamic μ.
• 🎯 Manometer: work down = ρgh added, up = subtracted; balance both legs to the same reference level.
• ❌ Common mistake: using diameter vs radius inconsistently in capillary/droplet formulas — h uses diameter d (4σ/ρgd); double-check whether a formula is written in r or d.

Expected exam pattern

A 1-mark viscosity (τ = μ du/dy), capillary-rise, or manometer-pressure NAT. Droplet/bubble pressure and Newtonian-vs-non-Newtonian classification appear as conceptual MCQs.

Quick recap

τ = μ du/dy (μ in Pa·s); ν = μ/ρ. Capillary h = 4σcosθ/ρgd. Droplet Δp = 4σ/d, bubble 8σ/d, jet 2σ/d. Bulk modulus K = −dp/(dV/V). Manometry p = ρgh; balance U-tube legs. Newtonian = constant μ.

Cover the answer, recall, then check. 11 cards on fluid properties for GATE ME.

Q1. State Newton's law of viscosity.
A1. τ = μ·(du/dy) — shear stress is proportional to the velocity gradient; μ = dynamic viscosity (Pa·s).

Q2. Difference between dynamic and kinematic viscosity?
A2. Dynamic μ (Pa·s) relates stress to shear rate; kinematic ν = μ/ρ (m²/s) is used in the Reynolds number.

Q3. What is a Newtonian fluid?
A3. One with constant viscosity — τ ∝ du/dy is linear through the origin (e.g. water, air). Non-Newtonian fluids have variable μ.

Q4. Capillary rise in a tube of diameter d?
A4. h = 4σ·cosθ/(ρ·g·d); it rises for wetting liquids (θ < 90°) and falls for non-wetting (e.g. mercury).

Q5. Pressure inside a liquid droplet of diameter d?
A5. Δp = 4σ/d (one surface).

Q6. Pressure inside a soap bubble?
A6. Δp = 8σ/d — twice a droplet, because a bubble has two surfaces.

Q7. Pressure inside a liquid jet (cylindrical) of diameter d?
A7. Δp = 2σ/d.

Q8. Define bulk modulus of a fluid.
A8. K = −dp/(dV/V) — the pressure rise per fractional volume decrease; its inverse is compressibility.

Q9. Relate gauge and absolute pressure.
A9. p_absolute = p_gauge + p_atmospheric. Manometers usually read gauge pressure.

Q10. How does a U-tube manometer balance?
A10. Sum of ρgh going down equals the sum going up between the two legs at a common level; solve for the unknown pressure.

Q11. Units of dynamic viscosity in SI and the poise?
A11. Pa·s (= N·s/m²) in SI; 1 poise = 0.1 Pa·s, and 1 centipoise = 0.001 Pa·s (water at 20°C).

Fluid mechanics begins with the properties that make a fluid a fluid — density, viscosity, surface tension — and the pressure measurement (manometry) that turns theory into readable numbers. GATE opens most fluids sets with a property or manometer question; get the definitions crisp.

Core concept: viscosity is a fluid's resistance to shear (internal friction), and hydrostatic pressure varies with depth as p = ρgh — the two ideas behind manometers and most fluid-static calculations.

Deep explanation

Beginner — key properties

• Density ρ = mass/volume; specific weight γ = ρg; specific gravity SG = ρ/ρ_water.
• Viscosity (dynamic) μ from Newton's law of viscosity: τ = μ (du/dy) — shear stress proportional to velocity gradient. Kinematic viscosity ν = μ/ρ.
• Newtonian fluids (water, air, oil) have constant μ; non-Newtonian fluids (blood, paint, slurries) do not — shear-thinning, shear-thickening, Bingham plastic.

Intermediate — surface tension and compressibility

• Surface tension σ causes pressure jump across a curved interface: droplet Δp = 2σ/R; bubble (two surfaces) Δp = 4σ/R; capillary rise h = 2σ cos θ/(ρgr).
• Bulk modulus K = −dp/(dV/V) measures compressibility; liquids have high K (nearly incompressible), gases low K. Speed of sound c = √(K/ρ).

Advanced — manometry

Hydrostatic law: pressure increases with depth, p = p₀ + ρgh. Manometers exploit this:

• Simple/piezometer: gauge pressure = ρgh.
• U-tube manometer: balance columns — p_A + Σρgh (down) − Σρgh (up) = p_B. Move down = add ρgh, move up = subtract.
• Differential manometer reads pressure difference across a device; with a heavy manometric fluid (mercury, ρ_m) the reading is amplified.
  Remember gauge vs absolute: p_abs = p_atm + p_gauge (vacuum is negative gauge).

Worked example

A U-tube mercury manometer (SG = 13.6) connected to a water pipe shows a mercury deflection of 250 mm. Find the gauge pressure at the pipe (water above mercury, ρ_water = 1000 kg/m³).
Pressure balance across the mercury column: p = (ρ_Hg − ρ_water)·g·h approximately for the deflection, but taking the standard reading p = ρ_Hg·g·h − ρ_water·g·h.
Using p = (13.6 − 1)×1000 × 9.81 × 0.250 = 12,600 × 9.81 × 0.250 = 30,901.5 Pa ≈ 30.9 kPa (accounting for the water column above the mercury). If the water column is neglected, p ≈ ρ_Hg g h = 13,600×9.81×0.25 = 33.4 kPa.

GATE relevance

Property definitions (viscosity, surface tension, bulk modulus) and manometer pressure calculations are near-certain opening questions in Fluid Mechanics. Newton's law of viscosity τ = μ du/dy is foundational to boundary layers and pipe flow later.

Exam tricks & shortcuts

• Manometer rule: going down add ρgh, going up subtract ρgh — march through the tube systematically.
• Droplet Δp = 2σ/R, bubble Δp = 4σ/R (two surfaces) — the factor is the trap.
• Kinematic viscosity ν = μ/ρ governs Reynolds number; do not confuse with μ.
• Mnemonic: "Down you add, up you subtract."

Key formulas

• Density, specific weight: γ = ρg; specific gravity = ρ/ρ_water.
• Newton's viscosity: τ = μ(du/dy); kinematic viscosity ν = μ/ρ.
• Bulk modulus: K = −dP/(dV/V).
• Surface tension: droplet ΔP = 4σ/d, bubble 8σ/d; capillary rise h = 4σcosθ/(ρgd).
• Manometer: P = ρgh (pressure = height of fluid column).

Viscosity relates shear stress to velocity gradient; manometers measure pressure via column heights.

Worked Example

Problem: A flat plate of area 0.5 m² slides at 2 m/s over a fixed surface, separated by an oil film 1 mm thick. The oil has dynamic viscosity µ = 0.1 Pa·s. Assuming a linear velocity profile, find the shear stress on the plate and the force needed to move it.

Solution:
Newton's law of viscosity relates shear stress to the velocity gradient:
τ = µ·(du/dy).

With a linear profile, du/dy is simply the plate speed divided by the film thickness:
du/dy = U/h = 2/(1 × 10⁻³) = 2000 s⁻¹.

Shear stress:
τ = µ·(du/dy) = 0.1 × 2000 = 200 Pa.

Force required to drag the plate against this shear over its full area:
F = τ·A = 200 × 0.5 = 100 N.

Answer: Shear stress τ = 200 Pa and the required force F = 100 N.