Fluid properties and statics

In short

This topic covers what a fluid *is* (density, viscosity, surface tension, compressibility, vapour pressure) and what a fluid at rest *does* (pressure distribution, forces on dams and gates, flotation and stability). It is the foundation of the whole fluid mechanics paper — every later topic uses ρ, μ and the pressure head defined here. In GATE (ME, CE, XE) it reliably yields 3–6 marks in questions on manometers, centre of pressure and metacentric height, and it is standard viva matter in Mechanical, Civil and Marine labs. On the job it appears in dam and sluice-gate design, pressure-gauge selection and calibration, hydraulic-test pressures for vessels and pipelines, barge and pontoon stability, and in sizing lubrication films — asked by L&T, BHEL, Kirloskar, KSB, Thermax, Cochin Shipyard, IRS, NHPC and state irrigation departments.

Explained

What a fluid is, and the continuum idea

• A fluid cannot resist shear stress at rest — it deforms continuously, however small the shear. A solid takes a finite strain and stops.
• Liquids have a free surface and nearly fixed volume; gases expand to fill the vessel and are highly compressible.
• Continuum assumption: properties are treated as smoothly varying point values, valid when the molecular mean free path is far smaller than the body (fails in high vacuum, rarefied flow).
• No-slip condition: fluid in contact with a solid boundary has the velocity of the boundary — this is why viscous shear exists at a wall.
• Classification by behaviour: ideal fluid (inviscid, incompressible — a model only), real fluid (viscous), Newtonian, non-Newtonian, ideal plastic.

Properties of a fluid

• Mass density ρ = mass/volume, kg/m³. Water ≈ 1000 (998 at 20 °C), mercury ≈ 13,600, air ≈ 1.2 at 1 atm, 20 °C.
• Specific weight (weight density) w = ρg, N/m³. Water ≈ 9810 N/m³ ≈ 9.81 kN/m³.
• Specific gravity (relative density) S = ρ/ρ_water for liquids (ρ/ρ_air for gases) — dimensionless. Mercury 13.6, diesel ≈ 0.83.
• Specific volume v = 1/ρ, m³/kg — used mostly for gases.
• Dynamic (absolute) viscosity μ: resistance to shear, N·s/m² = Pa·s. 1 poise = 0.1 Pa·s; 1 centipoise = 10⁻³ Pa·s (water at 20 °C).
• Kinematic viscosity ν = μ/ρ, m²/s. 1 stoke = 10⁻⁴ m²/s; water ≈ 1.0 × 10⁻⁶ m²/s. ν governs Reynolds number.
• Effect of temperature: liquid viscosity *falls* with temperature (molecular cohesion weakens); gas viscosity *rises* (momentum exchange increases). Pressure effect is small except at very high pressure.
• Measurement: capillary-tube viscometers (Ostwald), Saybolt and Redwood efflux viscometers, rotating-cylinder and falling-sphere viscometers; engine oils are graded by SAE viscosity grade.
• Compressibility and bulk modulus K: K = −dp/(dV/V), units Pa; compressibility = 1/K. Water K ≈ 2.2 GPa, so water needs ~22 MPa to compress 1% — hence "incompressible" for most hydraulics. K also fixes the speed of a pressure wave (water hammer).
• Surface tension σ (N/m): tensile force per unit length in a liquid surface, caused by unbalanced cohesion at the free surface. Water–air ≈ 0.0728 N/m at 20 °C, mercury ≈ 0.48–0.51 N/m; σ decreases with temperature and with surfactants.
• Capillarity: rise or fall in a narrow tube from the balance of surface tension and weight; depends on the contact angle θ (adhesion vs cohesion). Water in glass rises (θ ≈ 0°, wetting); mercury in glass falls (θ ≈ 130°, non-wetting). This is why glass manometers use large-bore tubes.
• Vapour pressure is the pressure at which a liquid flashes to vapour at a given temperature (water ≈ 2.34 kPa absolute at 20 °C). If local pressure in a flowing system falls to vapour pressure, vapour cavities form and collapse — cavitation, which pits pump impellers and turbine blades.

Newtonian and non-Newtonian fluids

• Newton's law of viscosity: τ = μ (du/dy) — shear stress is proportional to the velocity gradient (rate of shear strain). Fluids obeying it are Newtonian: water, air, kerosene, thin oils, mercury.
• Non-Newtonian fluids have τ = A(du/dy)ⁿ + B:
  • Pseudoplastic (shear-thinning, n < 1): paint, blood, polymer melts, milk.
  • Dilatant (shear-thickening, n > 1): sugar/starch slurries, wet sand, quicksand.
  • Bingham plastic: needs a yield stress before it flows — toothpaste, drilling mud, sewage sludge, fresh concrete.
  • Thixotropic: viscosity falls with time of shearing (printer's ink, some paints); rheopectic is the reverse.
• Only Newtonian fluids are used in standard pipe-loss and lubrication formulae; slurry pipelines need rheological data.

Pressure and its measurement

• Pressure p = normal force/area, N/m² = Pa; 1 bar = 10⁵ Pa, 1 kgf/cm² ≈ 98.07 kPa, 1 atm = 101.325 kPa = 760 mm Hg = 10.33 m of water.
• Pascal's law: at a point in a fluid at rest, pressure is the same in all directions (px = py = pz). Basis of the hydraulic press, jack and brake — force multiplication in the ratio of piston areas.
• Hydrostatic law: dp/dz = −ρg, so for constant density p = ρgh (gauge) — pressure depends only on vertical depth, not on vessel shape or volume (the hydrostatic paradox).
• Pressure head h = p/ρg, expressed in metres of the stated fluid; convert between fluids by equating ρh.
• Absolute, gauge, vacuum: p_abs = p_atm + p_gauge; a vacuum (negative gauge) pressure gives p_abs = p_atm − p_vac. Absolute pressure can never be negative.
• Manometers (liquid column, primary standards for low pressure):
  1. Piezometer — open tube tapped into the pipe; only for low, positive liquid pressures.
  2. Simple U-tube — one limb open to atmosphere; mercury for high pressure, lighter liquids for small pressures.
  3. Differential U-tube — measures the difference between two points; inverted U-tube uses a lighter fluid for small differences in liquids.
  4. Single-column (micromanometer) and inclined-tube manometers magnify a small reading by 1/sinθ.
• Reading rule: start at one end with a known pressure, add ρgh going down, subtract ρgh going up, and equate at the common surface in the same continuous fluid.
• Mechanical gauges: Bourdon tube (curved flattened tube uncoils with pressure), diaphragm, bellows and dead-weight tester (calibration standard); electronic strain-gauge and piezoresistive transducers for data logging. Dial gauges in India are covered by IS 3624 (pressure and vacuum gauges) and legal trade use by the Legal Metrology Act, 2009; verify the current revision of any IS number before quoting it.
• Gauges read gauge pressure; always select a gauge whose normal operating point is near mid-scale (roughly 1/3 to 2/3 of range).

Hydrostatic force on surfaces

• Total pressure F on a plane submerged surface = ρgAh̄, where A = area and h̄ = vertical depth of the centroid. Independent of the inclination for a given h̄.
• Centre of pressure (where the resultant acts) is always *below* the centroid: h_cp = h̄ + I_G sin²θ/(A h̄), with I_G the second moment of area about the centroidal axis and θ the angle of the plane with the free surface (θ = 90° for a vertical surface).
• Useful I_G values: rectangle b d³/12 (about the horizontal centroidal axis), circle πd⁴/64, triangle b h³/36.
• As depth increases, h_cp approaches h̄ because I_G/(A h̄) shrinks — a deeply submerged gate is loaded almost uniformly.
• Curved surfaces: resolve rather than integrate.
  • Horizontal component F_H = force on the vertical projection of the surface, acting at that projection's centre of pressure.
  • Vertical component F_V = weight of the fluid above the surface up to the free surface (or the weight of the imaginary volume, acting upward, if the fluid is below).
  • Resultant F = √(F_H² + F_V²), inclined at tan⁻¹(F_V/F_H).
• Design use: moment of the resultant about a hinge gives gate-operating torque and hoist capacity; dam sections are checked for overturning, sliding, tension at the heel and crushing, with uplift and silt pressure added. Indian practice for gravity dams follows the IS 6512 series criteria; radial and vertical-lift gates have their own IS codes.
• Pressure diagrams (triangular for a surface-piercing wall, trapezoidal for a submerged one) give F per metre width as the area of the diagram.

Buoyancy, flotation and stability

• Archimedes' principle: an immersed body experiences an upward force equal to the weight of fluid displaced, F_B = ρ g V_displaced, acting through the centre of buoyancy B (centroid of the displaced volume).
• A body floats when its weight equals the buoyant force; it sinks if its average density exceeds the fluid's. Draft adjusts itself to satisfy W = ρ g V_disp.
• Submerged body: stable if G is below B; unstable if above; neutral if coincident. There is no metacentre for a fully submerged body.
• Floating body: when tilted by a small angle, B shifts; the vertical through the new B cuts the original centre line at the metacentre M.
  • GM > 0 (M above G): stable — restoring couple W·GM·sinθ.
  • GM = 0: neutral; GM < 0 (M below G): unstable — the body capsizes.
• BM = I/V, where I is the second moment of the waterline (water-plane) area about the tilting axis and V the displaced volume; GM = BM − BG. A wide hull has large I, hence large GM.
• Experimental GM: shift a known weight w by distance x across the deck and measure the tilt θ; GM = w·x/(W·tanθ).
• Rolling period T = 2π√(k²/(g·GM)), k = radius of gyration about the roll axis. A very large GM gives a stiff, uncomfortable, fast roll; a small GM gives a tender, slow roll. Passenger vessels therefore use moderate GM, cargo barges larger.
• Free surfaces of liquid in partly filled tanks reduce effective GM (free-surface effect) — a known cause of capsize; hence baffles and sub-divided tanks. Indian ship stability is regulated under the Merchant Shipping Act, 2025 (in force 15 March 2026, repealing the 1958 Act) through the Directorate General of Maritime Administration (DGMA, formerly DG Shipping) and classed by the Indian Register of Shipping.

Fluids in relative equilibrium (rigid-body motion)

• A fluid accelerating uniformly with its container has no internal shear, so hydrostatic methods still apply with gravity replaced by the resultant body force.
• Uniform horizontal acceleration a: the free surface tilts at tanθ = a/g; lines of constant pressure are parallel to it.
• Vertical acceleration: p = ρ(g ± a)h — upward acceleration increases pressure, free fall (a = g downward) gives zero gauge pressure throughout.
• Forced vortex (vessel rotating at ω about its vertical axis): the free surface is a paraboloid, z = ω²r²/(2g) above the vertex; pressure rises radially, p = ρω²r²/2 + ρgh. Used in centrifugal separators, cream separators and to explain the pressure rise in a pump casing.
• Closed, completely filled rotating vessels develop the same parabolic pressure distribution without a visible surface.

Key points

• Fluid — substance that cannot sustain shear at rest; deforms continuously under any shear stress.
• ρ, w, S, v — density (kg/m³), specific weight ρg (N/m³), specific gravity (dimensionless), specific volume 1/ρ.
• Newton's law — τ = μ du/dy; μ in Pa·s, 1 poise = 0.1 Pa·s, 1 stoke = 10⁻⁴ m²/s.
• μ vs ν — dynamic viscosity measures shear resistance; kinematic ν = μ/ρ appears in Reynolds number.
• Temperature effect — liquid viscosity falls, gas viscosity rises, with temperature.
• Bulk modulus K — water ≈ 2.2 GPa; large K means nearly incompressible; 1/K is compressibility.
• Surface tension — Δp = 4σ/d (droplet), 8σ/d (soap bubble), 2σ/d (liquid jet).
• Capillary rise — h = 4σcosθ/(ρgd); negative for mercury in glass (depression).
• Vapour pressure — fall to it and cavitation begins; water ≈ 2.34 kPa abs at 20 °C.
• Pascal's law — pressure at a point is equal in all directions; basis of hydraulic jacks and brakes.
• Hydrostatic law — p = ρgh; depends on depth only, not on vessel shape (hydrostatic paradox).
• 1 atm — 101.325 kPa = 1.01325 bar = 10.33 m of water = 760 mm Hg; p_abs = p_atm + p_gauge.
• Manometer rule — add ρgh downward, subtract ρgh upward, equate across a common level in the same fluid.
• Total pressure — F = ρgAh̄ on a plane surface, using the centroid's vertical depth.
• Centre of pressure — h_cp = h̄ + I_G sin²θ/(A h̄), always deeper than the centroid.
• Curved surface — F_H from the vertical projection, F_V = weight of fluid above the surface.
• Buoyancy — F_B = ρgV_disp acting at the centre of buoyancy B.
• Stability — submerged: G below B; floating: GM = BM − BG = I/V − BG > 0; roll period T = 2π√(k²/(g·GM)).

Formulas and examples

Example 1 — viscous drag on a lubricating film. A flat plate of area 0.5 m² slides at 0.6 m/s over a fixed surface separated by an oil film 1 mm thick; oil viscosity 0.9 poise.

• μ = 0.9 × 0.1 = 0.09 Pa·s; velocity gradient du/dy = 0.6/0.001 = 600 s⁻¹.
• τ = 0.09 × 600 = 54 N/m²; F = 54 × 0.5 = 27 N.
• Power to drag the plate P = 27 × 0.6 = 16.2 W (all of it dissipated as heat in the film).

Example 2 — simple U-tube manometer. A pipe carries water; a mercury U-tube shows a mercury column 300 mm high, with the water column above the mercury in the left limb 200 mm high. Find the pipe pressure (ρ_Hg = 13,600 kg/m³).

• Equating at the lower mercury level: p_A + ρ_w g (0.2) = ρ_Hg g (0.3).
• ρ_Hg g (0.3) = 13,600 × 9.81 × 0.3 = 40,024.8 Pa; ρ_w g (0.2) = 1000 × 9.81 × 0.2 = 1962 Pa.
• p_A = 40,024.8 − 1962 = 38,062.8 Pa ≈ 38.06 kPa gauge = 0.381 bar.
• As head of water: 38,062.8/9810 = 3.88 m of water. Absolute pressure = 38.06 + 101.33 = 139.4 kPa.

Example 3 — vertical sluice gate: force, centre of pressure, hinge moment. A rectangular gate 2 m wide and 3 m deep closes an opening whose top edge is 1 m below the water surface.

• h̄ = 1 + 3/2 = 2.5 m; A = 2 × 3 = 6 m².
• F = ρgAh̄ = 1000 × 9.81 × 6 × 2.5 = 147,150 N = 147.15 kN.
• I_G = 2 × 3³/12 = 4.5 m⁴; A h̄ = 15 m³ → h_cp = 2.5 + 4.5/15 = 2.8 m below the surface.
• About a hinge at the bottom edge (4 m deep), lever arm = 4 − 2.8 = 1.2 m → moment = 147.15 × 1.2 = 176.6 kN·m, the figure that sizes the hoist.

Example 4 — stability of a rectangular barge. A barge 6 m long and 3 m wide floats in fresh water at a draft of 0.9 m; G is 1.0 m above the base and the radius of gyration about the roll axis is 1.1 m.

• Displaced volume V = 6 × 3 × 0.9 = 16.2 m³ → W = 9810 × 16.2 = 158,922 N ≈ 158.9 kN (about 16.2 t).
• Water-plane I = L B³/12 = 6 × 3³/12 = 13.5 m⁴ → BM = 13.5/16.2 = 0.833 m.
• B is at 0.9/2 = 0.45 m above the base, so BG = 1.0 − 0.45 = 0.55 m.
• GM = 0.833 − 0.55 = +0.283 m → stable.
• Roll period T = 2π√(1.1²/(9.81 × 0.283)) = 2π√(1.21/2.779) = 2π × 0.660 = 4.15 s.
• If cargo raised G to 1.30 m above the base, BG = 0.85 m and GM = −0.017 m — the barge would capsize.

Example 5 — surface tension, capillarity and compressibility. σ_water = 0.0728 N/m (θ = 0°), σ_Hg = 0.51 N/m (θ = 130°), K_water = 2.2 GPa.

• Pressure inside a 0.5 mm water droplet: Δp = 4 × 0.0728/0.0005 = 582.4 N/m² above ambient.
• Capillary rise in a 1 mm glass tube (water): h = 4 × 0.0728/(1000 × 9.81 × 0.001) = 0.2912/9.81 = 0.0297 m = 29.7 mm rise.
• Mercury in the same tube: h = 4 × 0.51 × cos130°/(13,600 × 9.81 × 0.001) = −1.3113/133.416 = −9.8 mm (depression).
• Pressure to reduce a water volume by 0.5%: Δp = K × 0.005 = 2200 × 0.005 = 11 MPa (110 bar) — confirming water is effectively incompressible in hydraulics.
• Open cylinder of radius 0.3 m rotated at 150 rpm: ω = 2π × 150/60 = 15.71 rad/s → surface rise at the wall = ω²r²/(2g) = (246.7 × 0.09)/19.62 = 1.13 m above the vertex.

Where it is used

• Water resources and dams: gravity-dam section design, spillway radial gates, sluice and penstock gates, canal regulators — force and centre-of-pressure calculations by CWC, NHPC, state irrigation departments and L&T/Megha-type contractors.
• Pressure instrumentation and calibration: Bourdon and digital gauges, diaphragm seals and transmitters for boilers and process plants; dead-weight testers and NABL-accredited calibration labs; hydrostatic test pressures for vessels and pipelines under pressure-vessel codes.
• Pumps, turbines and lubrication: NPSH and cavitation margins at Kirloskar, KSB and Jyoti; viscosity grades for journal bearings and gearboxes; hydraulic jacks, presses and brake systems using Pascal's force multiplication.
• Shipbuilding and offshore: draft, displacement and metacentric height calculations at Cochin Shipyard, Mazagon Dock and GRSE; inclining experiments and IRS/DGMA (formerly DG Shipping) stability approvals; ballast and free-surface management.
• Process and petroleum industries: tank gauging and level measurement by hydrostatic head, specific-gravity (hydrometer) checks on fuels and battery electrolyte, slurry rheology in mining and cement pipelines.
• Roles: design engineer (hydraulic structures), piping/stress engineer, instrumentation engineer, naval architect, QA engineer running hydro-tests, and lab/testing staff in polytechnics and R&D centres.

Interview questions

• Q: What distinguishes a fluid from a solid? A: A fluid cannot resist shear stress at rest and deforms continuously under any shear, whereas a solid takes a finite elastic deformation and holds it.
• Q: Difference between dynamic and kinematic viscosity? A: Dynamic viscosity μ is the shear stress per unit velocity gradient (Pa·s); kinematic viscosity ν = μ/ρ (m²/s) is the momentum diffusivity used in Reynolds number.
• Q: Why does viscosity of a liquid fall but that of a gas rise with temperature? A: Liquid viscosity comes from intermolecular cohesion, which weakens on heating; gas viscosity comes from molecular momentum transfer, which increases as molecules move faster.
• Q: State Pascal's law and one application. A: At a point in a static fluid, pressure acts equally in all directions; it underlies the hydraulic jack, where a small force on a small piston raises a large load on a large piston.
• Q: Two tanks of very different shape hold water to the same depth — is the bottom pressure the same? A: Yes, p = ρgh depends only on depth, not on shape or volume; this is the hydrostatic paradox (the total force differs because the areas differ).
• Q: Why is the centre of pressure below the centroid? A: Pressure increases with depth, so the lower part of the surface carries more load and the resultant shifts downward by I_G sin²θ/(A h̄).
• Q: How do you find the force on a curved gate without integrating? A: Take the horizontal component as the force on the vertical projection of the surface and the vertical component as the weight of the fluid column above it, then combine vectorially.
• Q: What is metacentric height and why does it matter? A: GM = BM − BG is the distance between the metacentre and the centre of gravity; a positive GM gives a restoring couple W·GM·sinθ, so the floating body returns upright.
• Q: Is a large metacentric height always desirable? A: No — it makes the vessel stiff with a short, violent roll period (T = 2π√(k²/(g·GM))), uncomfortable and hard on cargo; a moderate GM is chosen for passenger ships.
• Q: Why does a submerged body have no metacentre? A: Its displaced volume does not change on tilting, so the centre of buoyancy does not shift sideways; stability depends only on G being below B.
• Q: What is cavitation and which property governs it? A: Formation and violent collapse of vapour bubbles when local pressure falls to the liquid's vapour pressure, causing noise, vibration and pitting of impellers.
• Q: Why is mercury used in manometers rather than water for high pressures? A: Its high density (13.6 times water) keeps the column short and readable, and it does not wet glass, giving a clean meniscus.

Common mistakes

• Mixing gauge and absolute pressure — adding atmospheric pressure twice, or reporting a negative absolute pressure.
• Unit slips: poise taken as Pa·s, stoke as m²/s, 1 kgf/cm² as exactly 1 bar, or mm of mercury left unconverted.
• Using the depth of the *top edge* (or the slant distance along an inclined plane) instead of the vertical depth of the centroid in F = ρgAh̄.
• Forgetting sin²θ in the centre-of-pressure formula for inclined surfaces, or using I about the wrong axis.
• In GM = BM − BG, measuring BG from the water surface instead of from the keel/base, or using the wrong water-plane I (tilting about the long axis uses L B³/12).
• Taking the buoyant force as the weight of the *whole* body volume for a floating body instead of the submerged (displaced) volume.
• Treating the hydrostatic paradox as wrong because "the bigger tank holds more water" — confusing pressure with total force.

Related topics

Fluid flow: Bernoulli and flow measurement · Viscous flow and pipe losses · Hydraulics and pneumatics · Dimensional analysis and similitude · Centrifugal and reciprocating pumps · Strength of materials: thin and thick pressure vessels