Unit I: Fluid Properties and Fluid Statics

Unit I – Fluid Properties and Fluid Statics

• Dimensions and units, fluid properties, and fluid statics form the foundation for analyzing fluids in engineering applications.

• Key properties covered: density, specific weight, specific volume, specific gravity, viscosity, compressibility, vapour pressure, capillarity, and surface tension.

• Core concepts in fluid statics: hydrostatic (Pascal’s) law, atmospheric/absolute/gauge/vacuum pressures, measurement of pressure using manometers, buoyancy, and meta-centre concepts.

Fluids

• A fluid is a substance with no definite shape that continuously deforms or flows when subjected to an external force (e.g., water, milk, steam, gases). It cannot preserve a fixed shape unless confined by surroundings.

Types of Fluids

• Ideal Fluid: A hypothetical fluid with no viscosity, no surface tension, and incompressible. Its flow experiences no resistance, determined only by density. (
  Real fluids always have viscosity and often surface tension.)

• Real Fluid: Fluids with viscosity, and often surface tension, in addition to density. All real fluids have these properties to some extent.

• Newtonian Fluids: Fluids that follow Newton’s law of viscosity. Examples: water, kerosene, air. Their shear stress is proportional to the rate of shear strain: \tau = \mu \dfrac{du}{dy}

• Non-Newtonian Fluids: Fluids that do not obey Newton’s law (e.g., slurries, mud, polymer solutions, blood).

• Ideal Plastic Fluid: A fluid where shear stress exceeds the yield value and is proportional to the rate of shear strain.

CLASSIFICATION OF MECHANICS

• Mechanics splits into:

  • Rigid bodies (Solid Mechanics)

  • Deformable fluids (Fluid Mechanics)

• Sub-fields:

  • Statics

  • Dynamics

  • Kinematics

  • Kinetics

Fluid Mechanics Overview

• Fluid Mechanics studies fluids in motion (fluid dynamics/kinematics) or at rest (fluid statics).

• Gases and liquids are fluids (e.g., air, water).

Why Study Fluid Mechanics?

• It is a fundamental engineering discipline bridging physics and core BE courses.

• Applications include dairy plant processes (fluids like milk, water, air, refrigerants, steam), water supply and treatment, pumps handling various fluids, ships, submarines, airplanes, automobiles, storage tanks, piping systems, valves, flow meters, CIP systems, etc.

Units and Dimensions

• A unit of measurement is a definite magnitude of a physical quantity. Systems:

  • SI (International System of Units)

  • CGS (centimetre–gram–second)

  • MKS (meter–kilogram–second)

  • FPS (Foot–Pound–Second)

SI System: Primary and Derived Quantities

• Primary quantities (typical in SI): Length (L), Mass (M), Time (T), Temperature (K), Mole (n).

• Derived quantities include:

  • Force: F\,=\,N\,=\,kg\cdot m\,s^{-2}

  • Pressure: p\,=\,Pa\,=\,N\,m^{-2}

  • Energy/Work: W\,=\,J\,=\,N\cdot m

  • Power: P\,=\,W\,=\,J\,s^{-1}\,=\,W

• SI relations:

  • 1 N = 1 kg·m/s²

  • 1 Pa = 1 N/m²

  • 1 J = 1 N·m

  • 1 W = 1 J/s

• Dimensional analysis foundations help convert between quantities.

SI Unit Prefixes

• Common prefixes:

  • Giga: 10^{9}, Symbol: G

  • Mega: 10^{6}, Symbol: M

  • kilo: 10^{3}, Symbol: k

  • centi: 10^{-2}, Symbol: c

  • milli: 10^{-3}, Symbol: m

  • micro: 10^{-6}, Symbol: \mu

  • nano: 10^{-9}, Symbol: n

Quantities, Dimensions and Units

• Length (L): Dimension L, Unit m

• Time (T): Dimension T, Unit s

• Mass (M): Dimension M, Unit kg

• Area: L^{2}, Unit m^2

• Volume: L^{3}, Unit m^{3}

• Velocity: LT^{-1}, Unit m/s

• Acceleration: LT^{-2}, Unit m/s^2

• Discharge/Flow rate: L^{3}T^{-1}, Unit m^{3}/s

• Force: MLT^{-2}, Unit N

• Pressure: ML^{-1}T^{-2}, Unit Pa

• Shear stress: same as pressure, N/m^{2}

• Density: ML^{-3}, Unit kg/m^{3}

• Specific weight (weight density): ML^{-2}T^{-2}, Unit N/m^{3}

• Energy/Work/Heat: ML^{2}T^{-2}, Unit J

• Power: ML^{2}T^{-3}, Unit W

• Dynamic viscosity: ML^{-1}T^{-1}, Unit Pa\cdot s

• Kinematic viscosity: L^{2}T^{-1}, Unit m^{2}/s

Some Important Units and Conversions

• 1 pound ≈ 0.453 kg

• 1 m ≈ 3.28 ft; 1 m ≈ 100 cm

• 1 ft ≈ 30.5 cm; 1 ft ≈ 12 in; 1 in ≈ 2.54 cm

• 1 km ≈ 0.621 miles; 1 ha ≈ 2.47 acre; 1 acre ≈ 4046.85 m²

• 1 L = 0.264 gal

• Pressure references:

  • 1 atm ≈ 101.325 kPa ≈ 105 Pa? (use 101.325 kPa)

  • 1 bar = 10^5 Pa

  • 1 atm head ≈ 760 mm of Hg ≈ 10.33 m of water

• Dyne to Newton conversion: 1 dyn = 10^{-5} N; 1 dyne·s/cm² is a cgs unit of viscosity.

Density, Fluid Properties, Specific Weight, Specific Volume, Specific Gravity, Viscosity, Compressibility, Vapour Pressure, Capillarity and Surface Tension

Density (ρ)

• Mass density: mass per unit volume. For water, ρ ≈ 1000 kg/m³.

• Example: ρ = m/V.

Weight Density (Specific Weight) ω

• Defined as weight per unit volume: \omega = \rho g

• For water: ρ ≈ 1000 kg/m³, g ≈ 9.81 m/s², so \omega_{water} ≈ 1000 \times 9.81 = 9.81\times 10^{3} \text{ N/m}^3

Specific Volume (v)

• Volume per unit mass: v = \frac{V}{m} = \frac{1}{\rho}

• Unit: m^3/kg

Specific Gravity (SG)

• Definition: SG = density of a liquid / density of reference fluid (water for liquids, air for gases)

• For liquids: SG = \frac{\rho}{\rho_{water}} (dimensionless)

Viscosity

• Viscosity measures resistance to flow (internal friction).

• Dynamic viscosity (μ): units \text{N} \cdot \text{s} / \text{m}^2 = \text{Pa} \cdot \text{s}

• Kinematic viscosity (ν): \nu = \dfrac{\mu}{\rho} with units \text{m}^2/ ext{s}

• Common conversions: 1 Poise (P) = 0.1 Pa·s; 1 Poise = 100 centipoise (cP); 1 cP = 0.001 Pa·s.

• Temperature dependence: For liquids, viscosity decreases with increasing temperature (weaker cohesive forces, easier momentum transfer). For gases, viscosity typically increases with temperature (molecular velocity increases, collisions change).

Compressibility and Bulk Modulus

• Compressible fluids change volume under pressure; incompressible fluids show negligible volume change.

• Bulk Modulus of Elasticity (K): the ratio of change in pressure to the resulting volumetric strain:
  K = -\frac{dP}{dV} \cdot \frac{V}{1} = -V \frac{dP}{dV} = \frac{dP}{-(dV/V)}.

• Practically, for small changes: K \approx -\frac{\Delta P}{\Delta V/V}.

Vapour Pressure

• The tendency of a liquid to vaporize at its surface; the partial pressure exerted by its vapor in equilibrium with the liquid. Higher vapour pressure means a higher rate of molecular escape at the surface.

Capillarity and Surface Tension

• Surface tension (σ) is the cohesive force at a liquid surface that minimizes surface area: units N/m.

• Capillarity (capillary rise/fall) occurs when a liquid rises or falls in a small tube relative to the surrounding liquid level.

• Capillary rise in a tube of diameter d (contact angle θ):
  h = \frac{4 \sigma \cos\theta}{\rho g d}

• For a soap bubble (two interfaces), the pressure difference across the film is:
  \Delta P = \frac{8 \sigma}{d} where d is the bubble diameter. This follows from the single-radius relation with two interfaces.

• Example: In a capillary with water (σ ≈ 0.0725 N/m, θ ≈ 0°, ρ ≈ 998 kg/m³, g ≈ 9.81 m/s², d = 4 mm):
  h = \frac{4 \cdot 0.0725 \cdot \cos 0°}{998 \cdot 9.81 \cdot 0.004} \approx 7.5\text{ mm}.

• For mercury (σ ≈ 0.52 N/m, contact angle θ ≈ 130°), the cosine is cos(130°) < 0, giving capillary depression (negative rise) of a few millimetres depending on tube diameter.

Buoyancy

• Archimedes’ principle: when a body is submerged in a fluid, it experiences an upward buoyant force equal to the weight of the displaced fluid.

• If a body spans multiple fluids with densities ρ1, ρ2, etc., the total upthrust is:
  U = ρ1 g V1 + ρ2 g V2 + \cdots where V_i are the immersed volumes in each fluid.

• The centre of buoyancy may differ from the geometric centre depending on submerged volumes and densities.

Pascal’s Law (Hydrostatics)

• Pressure at a point in a static fluid is the same in all directions:
  Px = Py = P_z.

Atmospheric, Absolute, Gauge, and Vacuum Pressures

• Absolute pressure (P_abs) is measured relative to a perfect vacuum.

• Gauge pressure (Pgauge) is the pressure relative to atmospheric pressure (Patm): P{abs} = P{atm} + P{gauge} \,;\quad P{gauge} = P{abs} - P_{atm}.

• Vacuum pressure (Pvac) is the amount by which atmospheric pressure exceeds the absolute pressure: P{vac} = P{atm} - P{abs} \,;\quad P{abs} \le P{atm} \Rightarrow P_{vac} \ge 0.

Atmospheric Pressure

• Standard atmosphere: Patm ≈ 101.325 kPa, which corresponds to a height of 760 mm of Hg or about 10.33 m of water column.

Measurement of Pressure by Manometers and Gauges

• Pressure measurement devices include manometers and mechanical gauges.

• Manometers balance the pressure against a column of liquid (often with different specific gravities): types include Simple U-tube, inverted U-tube, U-tube with enlarged leg, two-fluid U-tube, and inclined U-tubes.

• Piezometers measure gauge pressures via the height of a liquid column: pressure head h satisfies P = \rho g h .

• Bourdon-tube pressure gauges, dead-weight gauges, diaphragms, and bellows are mechanical gauges handling both gauge and absolute pressures with appropriate reference.

Simple U-tube Manometer (principle)

• Let left limb be connected to pressure P1, right limb to P2, with manometer fluid of density ρ_m and the test fluid density ρ.

• The pressure balance at the same level yields (for the standard case where the left side fluid is the test fluid and the right side is the heavy manometer fluid):
  P1 + \rho g (a+h) = P2 + \rhom g a + \rhom g h.

• Consequently, the pressure difference is approximately
  P1 - P2 = (\rho_m - \rho) g h when the setup is chosen such that the geometry terms cancel appropriately.

Inverted U-tube Differential Manometer

• Used to measure small pressure differences in liquids.

• With a light manometer fluid and the left/right pressures PA, PB, the relation becomes
  PA - PB = (\rho_m - \rho) g h (for appropriate density contrasts and small perturbations).

U-tube with One Leg Enlarged; Inclined U-tube

• These configurations minimize reading displacement (read primarily in the narrow leg or along the incline) while preserving the hydrostatic balance.

Pressure Gauges

• Bourdon-tube gauge is used for steam and compressed gases; it measures gauge pressure, i.e., the difference between the system pressure and atmospheric pressure.

Buoyancy and Immersion

• Upthrust on a submerged body depends on the displaced fluid’s density and the submerged volume; total upthrust is the sum of buoyancies from each immersed portion.

• The principle of buoyancy explains floatation and stability of submerged bodies.

Problem-Solving Examples (Representative)

Capillary Rise in a Tube (Water)

• Given: tube diameter d, surface tension σ, contact angle θ, density ρ, gravitational acceleration g.

• Capillary rise height:
  h = \frac{4 \sigma \cos\theta}{\rho g d}.

• Example result for water in a 4 mm tube with σ = 0.0725 N/m, θ = 0°, ρ = 998 kg/m³, g = 9.81 m/s², d = 0.004 m:
  h \approx 7.5 \text{ mm}.

Soap Bubble Pressure Difference

• For a soap bubble, with two interfaces, the pressure difference across the film is:
  \Delta P = \frac{8 \sigma}{d} where d is the bubble diameter.

• This is because the effective curvature corresponds to a radius R = d/2 and there are two interfaces.

• Example: If σ = 0.0725 N/m and d = 40 mm, then
  \Delta P = \frac{8 \times 0.0725}{0.04} = 14.5 \text{ Pa} (illustrative; actual numeric will depend on the given values).

Bulk Modulus (Problem Example)

• Given a liquid compressed from V = 0.0125 m³ to V = 0.0124 m³ with pressure increasing from 80 N/cm² to 150 N/cm².

• Convert pressures: ΔP = 150 − 80 = 70 N/cm² = 70 × 10^4 Pa = 7.0 × 10^5 Pa.

• ΔV = 0.0124 − 0.0125 = −1.0×10^−4 m³; V ≈ 0.0125 m³.

• Bulk modulus:
  K = - V \frac{\Delta P}{\Delta V} = -0.0125 \cdot \frac{7.0\times 10^{5}}{-1.0\times 10^{-4}} \approx 8.75 \times 10^{7} \text{ Pa}.

• In N/cm², this is ≈ 8.75 × 10^3 N/cm².

Dynamic Viscosity from Plate Friction (Problem Outline)

• For a plate moving relative to a stationary plate with plate separation dy, velocity u, and force per unit area τ, the relation is
  \tau = \mu \frac{du}{dy}.

• Given dy, u, and applied force per area, solve for μ.

Practical Implications

• In dairy plants and processing lines, the control of fluid flow, pressure head, and piping design depends on properly applying fluid statics and viscosity concepts.

• Knowledge of absolute vs gauge pressures is essential when selecting sensors and safety relief devices.

• Capillarity and surface tension influence cleaning (CIP) and separation processes at small scales.

Quick Reference Tables

• ρ (Density) of water ≈ 1000 kg/m³; ω = ρ g ≈ 9.81×10^3 N/m³.

• 1 Pa = 1 N/m²; 1 kPa = 10^3 Pa.

• 1 N = 1 kg·m/s²; 1 J = 1 N·m; 1 W = 1 J/s.

• 1 Poise = 0.1 Pa·s; 1 cP = 0.001 Pa·s.

• 1 stoke = 1 cm²/s = 1×10^−4 m²/s.

This unit provides the essential vocabulary and relationships needed to analyze fluids in static conditions and to prepare for more advanced topics in fluid dynamics and machine design. The key equations in this unit are the hydrostatic/pressure relations, Buoyancy, Pascal’s law, and the definitions and units of fundamental fluid properties.