Fluid properties: viscosity, surface tension, compressibility

Why it matters

Viscosity sets the friction loss in every pipe, the drag on a vehicle and the oil-film torque in an engine bearing. Surface tension controls fuel atomisation, capillary rise in thin passages and bubble formation in pumps, while compressibility decides whether you may treat a fluid as incompressible (most liquid flows, air below about Mach 0.3) or must use gas dynamics. Every later topic in fluid mechanics assumes you know which of these properties matters and in what units.

Key ideas

• Fluid and continuum. A fluid deforms continuously under any shear stress, however small. We treat it as a continuum, so properties such as density ρ, pressure p and velocity V are smooth functions of position. This fails only in very rarefied gases.
• Density, specific weight, specific gravity. ρ in kg/m³; specific weight w = ρg in N/m³; relative density (specific gravity) S = ρ/ρ_water, with ρ_water = 1000 kg/m³ at about 4 °C.
• Viscosity. Resistance to the rate of shear deformation. Newton's law of viscosity says shear stress is proportional to the velocity gradient normal to the flow: τ = μ·du/dy. Fluids obeying this with constant μ (water, air, most oils) are Newtonian.
  • Dynamic viscosity μ (Pa·s = N·s/m²; 1 poise = 0.1 Pa·s). Kinematic viscosity ν = μ/ρ (m²/s; 1 stokes = 10⁻⁴ m²/s).
  • Temperature effect: liquid viscosity falls with temperature (weaker intermolecular cohesion); gas viscosity rises (more molecular momentum exchange). That is why engine oil is graded for cold start and hot running.
  • Non-Newtonian fluids: shear-thinning (pseudoplastic, e.g. paints, polymer solutions), shear-thickening (dilatant), Bingham plastic (needs a yield stress before flowing, e.g. toothpaste). Power-law model τ = K·(du/dy)ⁿ with n < 1 shear-thinning, n > 1 shear-thickening.
  • An ideal fluid has zero viscosity and is incompressible — a model, not a real fluid.
  • For a thin film with linear velocity profile (gap h much smaller than other dimensions), du/dy ≈ U/h.
• Surface tension σ (N/m). Unbalanced cohesive forces at a liquid–gas interface make the surface behave like a stretched membrane. σ is force per unit length of a line on the surface, or energy per unit area (J/m²). It decreases with temperature and with surfactants. Water–air at 20 °C: σ ≈ 0.073 N/m.
  • Pressure jump across a curved interface (inside higher): droplet or liquid jet, bubble in liquid, and soap bubble (two surfaces) differ — see Formulas.
  • Capillarity: liquid rises in a thin tube if it wets the wall (contact angle θ < 90°, water–glass θ ≈ 0) and is depressed if it does not (mercury–glass, θ ≈ 130–140°).
• Compressibility. Bulk modulus K measures resistance to volume change; compressibility β = 1/K. Water K ≈ 2.2 GPa, so a 10 MPa rise compresses it under 0.5 % — hence "incompressible". For a gas K depends on the process: K = p (isothermal), K = γp (isentropic). Speed of sound c = √(K/ρ) links compressibility to Mach number; flow may be treated as incompressible when Ma < about 0.3 (density change below about 5 %).
• Vapour pressure. The pressure at which a liquid boils at the given temperature. If local pressure drops below it, vapour cavities form — the root of cavitation in pumps and turbines (later topics).

Formulas

τ = μ·du/dy τ shear stress (Pa), μ dynamic viscosity (Pa·s), du/dy velocity gradient (s⁻¹). Newtonian fluid, laminar shear.

τ = μ·U/h U relative velocity of plates (m/s), h film thickness (m). Thin film, linear profile.

ν = μ/ρ ν kinematic viscosity (m²/s), ρ density (kg/m³).

T = μ·π²·D³·N·L / (120·h) Viscous torque on a journal of diameter D (m), length L (m), speed N (rpm), radial clearance h (m); torque T in N·m. Thin concentric film; power P = T·ω with ω = 2πN/60 (rad/s).

Δp = 2σ/R (liquid droplet or bubble in liquid) Δp = 4σ/R (soap bubble, two surfaces) Δp = σ/R (cylindrical liquid jet) Δp inside minus outside pressure (Pa), σ surface tension (N/m), R radius (m).

h = 4·σ·cosθ / (ρ·g·d) Capillary rise (negative = depression) h (m) in a tube of bore d (m); θ contact angle; g = 9.81 m/s².

K = −dp / (dV/V) = dp / (dρ/ρ) K bulk modulus (Pa), dp pressure change (Pa), dV/V volumetric strain (dimensionless). The minus sign makes K positive because volume falls as pressure rises.

c = √(K/ρ) and for an ideal gas c = √(γ·R·T) c speed of sound (m/s), γ ratio of specific heats, R gas constant (J/kg·K), T absolute temperature (K).

Worked examples

Example 1 (standard) — sliding plate on an oil film. Given: plate area A = 0.2 m², speed U = 1.2 m/s, oil film h = 1.5 mm, μ = 0.12 Pa·s, ρ = 880 kg/m³. Find shear stress, force, power and ν.

1. Shear stress: τ = μ·U/h = 0.12 × 1.2 / 0.0015 = 96 Pa.
2. Force: F = τ·A = 96 × 0.2 = 19.2 N.
3. Power: P = F·U = 19.2 × 1.2 = 23.04 W.
4. Kinematic viscosity: ν = μ/ρ = 0.12/880 = 1.364 × 10⁻⁴ m²/s. Answer: F = 19.2 N, P ≈ 23.0 W, ν ≈ 1.36 × 10⁻⁴ m²/s (1.36 stokes).

Example 2 (GATE level) — journal bearing power loss. Given: shaft D = 80 mm, bearing length L = 120 mm, N = 1500 rpm, radial clearance h = 0.1 mm, μ = 0.05 Pa·s. Find the power lost in viscous friction.

1. Surface speed: U = πDN/60 = π × 0.08 × 1500/60 = 6.283 m/s.
2. Shear stress: τ = μU/h = 0.05 × 6.283 / 0.0001 = 3141.6 Pa.
3. Wetted area: A = πDL = π × 0.08 × 0.12 = 0.030159 m².
4. Force: F = τA = 3141.6 × 0.030159 = 94.75 N.
5. Torque: T = F·D/2 = 94.75 × 0.04 = 3.790 N·m.
6. Angular speed: ω = 2π × 1500/60 = 157.08 rad/s.
7. Power: P = Tω = 3.790 × 157.08 = 595.3 W. Answer: P ≈ 595 W.

Example 3 (quick) — capillary rise and compression. Water in a clean glass tube of bore 2 mm (σ = 0.0728 N/m, θ = 0): h = 4 × 0.0728 / (1000 × 9.81 × 0.002) = 0.01484 m = 14.8 mm. Water (K = 2.2 GPa) pressurised by 20 MPa: ΔV/V = −Δp/K = −20 × 10⁶ / 2.2 × 10⁹ = −0.91 %.

Common mistakes

• Using kinematic viscosity where dynamic is needed (τ = μ du/dy, not ν du/dy), or mixing poise and Pa·s (1 P = 0.1 Pa·s).
• Using diameter instead of radius in Δp = 2σ/R, or forgetting that a soap bubble has two surfaces (4σ/R).
• Taking the bearing film thickness as the diametral clearance when the radial clearance is required.
• Saying viscosity of gases falls with temperature — it rises.
• Dropping the minus sign in K = −dp/(dV/V) and reporting a negative bulk modulus.
• Forgetting cos θ in capillary rise for non-wetting liquids (mercury gives a depression).

For GATE ME

Expect short numericals on shear stress in a thin film, torque and power in a journal bearing or rotating disc/cone viscometer, capillary rise or depression, excess pressure in drops and bubbles, and bulk-modulus volume change. Conceptual MCQs test Newtonian versus non-Newtonian behaviour, the temperature trend of viscosity for liquids and gases, and when flow can be treated as incompressible. Practise unit conversion (mm, rpm, poise, stokes) — most lost marks come from there.

Quick check

1. What is the SI unit of kinematic viscosity, and how many m²/s is 1 stokes?
2. How does the viscosity of air change when it is heated?
3. What is the excess pressure inside a soap bubble of radius 10 mm if σ = 0.03 N/m?
4. Why does mercury show a capillary depression in glass?
5. For water with K = 2.2 GPa, what pressure rise reduces its volume by 0.5 %? Answers: 1. m²/s; 10⁻⁴ m²/s. 2. It increases. 3. 4σ/R = 4 × 0.03/0.01 = 12 Pa. 4. It does not wet glass (θ > 90°, cos θ < 0). 5. 0.005 × 2.2 GPa = 11 MPa.