Fluid Properties — Viscosity, Density & Surface Tension

1. Definition & Classification of Fluids

A fluid is defined as a substance that cannot sustain a shear stress in static equilibrium. Unlike a solid (which deforms a fixed amount under shear and then stops), a fluid deforms continuously as long as any shear stress — even infinitesimally small — acts on it.

1.1 Liquids vs Gases

1.2 Types of Fluids (based on viscosity behaviour)

For GATE CE, Newtonian fluids dominate all numerical problems. The Bingham plastic model appears as conceptual/MCQ questions.

2. Density, Specific Weight & Specific Gravity

2.1 Mass Density (ρ)

2.2 Specific Weight (γ)

2.3 Specific Gravity (SG) / Relative Density

2.4 Effect of Temperature on Density

For liquids: density decreases with rising temperature (thermal expansion).
For ideal gases: ρ = p / (RT) — density decreases if pressure is constant and temperature rises.

3. Viscosity — Newton’s Law & Types

Viscosity is the measure of a fluid’s internal resistance to flow (resistance to deformation by shear stress).

3.1 Newton’s Law of Viscosity

3.2 Kinematic Viscosity (ν)

3.3 Variation of Viscosity with Temperature

Empirical formulae:

3.4 No-Slip Condition

At a solid boundary, fluid velocity equals the wall velocity (zero for a stationary wall). This is the no-slip condition — a direct consequence of viscosity and is the basis for the velocity gradient (du/dy) between layers.

4. Surface Tension & Capillarity

4.1 Surface Tension (σ)

Surface tension is the force per unit length acting along the liquid surface due to unbalanced molecular cohesion at the interface.

4.2 Pressure Inside a Droplet / Bubble

4.3 Capillarity

Capillary action occurs in narrow tubes due to the interplay of cohesion (within the liquid) and adhesion (between liquid and tube wall).

5. Compressibility & Bulk Modulus

5.1 Bulk Modulus of Elasticity (K)

5.2 Compressibility (β)

5.3 Speed of Sound in a Fluid

For most civil engineering hydraulics (velocities much less than 1450 m/s), water is treated as incompressible (ρ = constant). Compressibility is important only in water hammer analysis.

6. Vapour Pressure & Cavitation

6.1 Vapour Pressure (p_v)

Vapour pressure is the pressure at which a liquid begins to vaporise at a given temperature. It increases with temperature.

6.2 Cavitation

Cavitation occurs when the local pressure in a flowing liquid drops to or below the vapour pressure. Vapour bubbles form and then implode violently when they move to a higher-pressure region, causing noise, vibration, erosion, and damage to pump impellers and pipe walls.

Prevention: Ensure the minimum pressure in the system stays well above p_v. In pump design, this is quantified by the Net Positive Suction Head (NPSH).

7. Worked Examples (GATE CE Level)

Example 1 — Newton’s Law of Viscosity (GATE CE 2019 type)

Problem: A flat plate moves over a fixed bottom plate with a gap of 2 mm filled with oil (μ = 0.8 Pa·s). If the upper plate moves at 0.5 m/s, find the shear stress on the upper plate.

Answer: τ = 200 Pa

Example 2 — Capillary Rise (GATE CE level)

Problem: Water (σ = 0.073 N/m, ρ = 1000 kg/m³, θ = 0°) rises in a glass tube of diameter 0.5 mm. Find the capillary rise h.

Answer: h ≈ 59.5 mm

Example 3 — Kinematic Viscosity & Specific Gravity

Problem: A liquid has a dynamic viscosity of 5 × 10⁻³ Pa·s and a specific gravity of 0.85. Find its kinematic viscosity in m²/s and in Stokes.

Answer: ν = 5.88 × 10⁻⁶ m²/s = 5.88 cSt

Example 4 — Pressure Inside a Soap Bubble

Problem: Find the pressure difference inside a soap bubble of diameter 4 cm. Surface tension of soap solution = 0.025 N/m.

Answer: Δp = 5 Pa

Example 5 — Bulk Modulus & Volumetric Compression

Problem: A 2 m³ volume of water is subjected to a pressure increase of 5 MPa. Find the decrease in volume. (K_water = 2.1 GPa)

Answer: Volume decreases by ≈ 4.76 × 10⁻³ m³ (4.76 litres)

8. Common Mistakes

Mistake 1 — Confusing Dynamic and Kinematic Viscosity

Error: Using μ and ν interchangeably in the Reynolds number formula or Darcy-Weisbach calculations.
Root Cause: Both have “viscosity” in the name. The Reynolds number Re = VD/ν uses kinematic viscosity, while shear stress τ = μ(du/dy) uses dynamic viscosity.
Fix: Always check units. μ has units Pa·s; ν has units m²/s. Re = VD/ν (no density term needed because ν = μ/ρ already incorporates it).

Mistake 2 — Wrong Surface Tension Formula for Bubbles vs Droplets

Error: Applying Δp = 4σ/d for a soap bubble (should be 8σ/d).
Root Cause: Forgetting that a soap bubble has TWO liquid–air interfaces (inner wall + outer wall), while a liquid droplet has only ONE.
Fix: Droplet = 4σ/d; Soap bubble = 8σ/d; Liquid jet = 2σ/d. Memorise this trio.

Mistake 3 — Capillary Rise Formula: Using Radius Instead of Diameter

Error: Writing h = 4σ cosθ / (ρgr) or getting confused about whether to use r or d.
Root Cause: Some textbooks derive the formula in terms of radius r (giving h = 2σ cosθ / ρgr); others state it in terms of diameter d (h = 4σ cosθ / ρgd). Both are correct but must not be mixed.
Fix: Derive once from force balance: upward force = σ(πd)cosθ; weight of liquid = ρg(πd²/4)h. Equating gives h = 4σ cosθ/(ρgd).

Mistake 4 — Assuming Viscosity of Liquids Increases with Temperature

Error: Believing all fluids become more viscous when heated (correct only for gases).
Root Cause: Everyday intuition — hot things feel different. But liquid viscosity is dominated by intermolecular cohesion which weakens at higher temperatures, making liquids flow more easily.
Fix: Remember the rule: Liquids → μ ↓ with ↑T; Gases → μ ↑ with ↑T.

Mistake 5 — Treating Density of Water as 1000 kg/m³ at All Temperatures

Error: Using ρ = 1000 kg/m³ for water problems regardless of temperature.
Root Cause: Memorising a single standard value without noting its temperature dependence (maximum density occurs at 4 °C).
Fix: For GATE, 1000 kg/m³ is accepted unless a specific temperature is given. If T = 20 °C or 100 °C, use the table value. Always note if the problem specifies temperature.

9. Frequently Asked Questions

Q1. What is the physical significance of kinematic viscosity?

Kinematic viscosity ν = μ/ρ represents the ratio of viscous forces to inertial forces per unit volume — or equivalently, the momentum diffusivity of the fluid. It determines how quickly momentum is diffused through the fluid (how fast a velocity perturbation spreads). Fluids with high ν (like thick oil) diffuse momentum rapidly, dampening flow disturbances; fluids with low ν (like water) resist momentum diffusion, making turbulence easier to sustain. In the Reynolds number Re = VD/ν, a higher ν for the same velocity and geometry means lower Re (tendency toward laminar flow).

Q2. Why does mercury show capillary depression in glass while water shows rise?

The direction of capillary action depends on the contact angle θ between the liquid and the tube material. For water–glass, adhesion (water molecules attracted to glass, which is silica-based and polar) dominates over cohesion, so θ < 90° (≈ 0°), cos θ is positive, and the liquid rises. For mercury–glass, cohesion among mercury atoms (strong metallic bonds) dominates over adhesion to glass, so θ > 90° (≈ 135°), cos θ is negative, and the formula gives a negative h — i.e., capillary depression. This is why mercury thermometers have a convex meniscus while water-filled manometers have a concave one.

Q3. What is cavitation and why is it important in civil engineering hydraulics?

Cavitation is the formation of vapour cavities (bubbles) in a flowing liquid when the local absolute pressure drops to or below the vapour pressure at that temperature. These bubbles collapse violently when they travel into higher-pressure zones, generating intense local shock waves (pressure up to several hundred MPa). In civil engineering, cavitation is critical in:

• Pump design: Cavitation on impeller blades causes pitting, vibration, noise, and efficiency loss. NPSH (Net Positive Suction Head) criteria are used to prevent it.
• Hydraulic turbines: Francis and Kaplan turbines are susceptible near the draft tube exit.
• High-velocity spillways: Chute blocks and flip buckets can experience cavitation at flow velocities above ~12 m/s, requiring aerators or special surface finishes.

Q4. How is Newton’s law of viscosity different for a Bingham plastic?

For a Newtonian fluid: τ = μ (du/dy) — flow begins the moment any shear stress is applied, and the relationship is linear with zero intercept. For a Bingham plastic: τ = τ₀ + μ_pl(du/dy), but only when τ > τ₀. Below the yield stress τ₀, the material behaves like a rigid solid and does not flow. This is why toothpaste or fresh concrete sits still on a nearly flat surface but flows when sufficient force is applied. In geotechnical engineering, this model also applies to soft clays, mud slurries, and grout flow in grouting operations.