Fluid Statics and Properties

Fluid at Rest

• When a fluid is at rest, shear force equals zero
• Normal force is not equal to zero
• Compressive force is a negative normal force
• Mohr's circle is a point

Fluid Compressibility

• Liquids are generally incompressible but become compressible during water hammer
• Gases are generally compressible, but behave as incompressible when Ma ≤ 0.3

Fluid Properties

• Density formula: ρ = m/V, Unit: kg/m³, Dimension: ML-3
• Specific weight formula: w = W/V = mg/V = ρg. Unit: N/m³ or kg/m²s², Dimension: ML-2T-2; water = 9.81 × 1000
• Specific gravity formula: SLiquid = (Density of liquid) / (Density of standard substance). It is unitless
• Specific volume formula: v = V/m = 1/ρ , Unit: m³/kg, Dimension: M-1L3
• Bulk modulus formula: K = -dP / (dV/V) = dP / (dρ/ρ). Unit: N/m², Dimension: ML-1T-2
• Compressibility formula: β = 1/K, Unit: m²/N, Dimension: M-1LT2
• Surface tension formula: σ = F/l . At critical point = 0 , Unit : SI - N/m or CGS - dyne/cm, Dimension: MT-2; water/air = 0.0736 N/m
• Vapor pressure formula: pv = Force/Area , Unit : SI - N/m² or CGS - dyne/cm², Dimension: ML-1T-2

Dynamic Viscosity

• Dynamic viscosity formula: τ = μ (du/dy), du/dy: velocity gradient, dθ/dt : shear strain or deformation rate
• Dynamic viscosity relationship: µHg > µH2O and µ H2O > µPetrol > µair
• SI unit: N-s/m² or Pa-s
• CGS unit: dyne-s/cm²
• MKS unit: kgf-s/m²
• 1 poise = 1/10 N-s/m²
• 1 Centipoise = 10-2 Poise
• For liquids, as temperature increases, cohesion decreases, and dynamic viscosity decreases
• For gases, as temperature increases, molecular momentum exchange increases, and dynamic viscosity increases

Kinematic Viscosity

• Kinematic viscosity formula: υ = (Dynamic Viscosity)/ (Mass Density) = μ/ρ
• SI unit: m²/sec
• CGS unit: cm²/sec or stoke
• 1 stoke = 10-4 m²/sec or 1 cm²/sec
• 1 m²/sec = 104 stoke
• Kinematic viscosity relationship: υair > υwater
• For liquids, surface tension decreases with an increase in temperature

Excess Pressure

• Pressure inside a drop (solid-like sphere) formula: p = 4σ/d
• Pressure inside a bubble (soap bubble) formula: p = 8σ/d
• Pressure inside a liquid jet formula: p = 2σ/d
• σ = Surface tension, d = diameter of bubble

Capillarity

• Capillary rise occurs when adhesion is greater than cohesion (e.g., H2O and glass)
• Capillary fall occurs when adhesion is less than cohesion (e.g., Hg & glass)
• Capillarity is due to both cohesion and adhesion
• Rise or depression (h) in a capillary tube: h = (4σ cos θ) / (ρgd)
• θ = 0° for pure water and glass tube
• θ = 128° - 138° for mercury and glass tube
• h is proportional to 1/d
• As diameter (d) increases, h decreases
• For diameter ≤ 6 mm = capillary tube
• 6 mm < diameter < 15 mm a piezometric tube is used
• For diameter ≥ 15mm → Pipe

Fluid Types based on Wetting

• Wetting fluid: Adhesion > Cohesion, Contact angle (θ) < 90° [Acute] (e.g., Water and glass)
• Non-wetting fluid: Cohesion > Adhesion, Contact angle (θ) > 90° [obtuse] (e.g., Mercury and glass)

Types of Fluid

• Ideal Fluid: Incompressible, Non-viscous, Perfectly rigid (β=0, K= ∞), Surface tension σ ≈ 0
• Real Fluid: Possesses viscosity and Compressibility
• Ideal plastic Fluid: Shear stress is more than yield value, shear stress (τ) ∝ du/dy or dθ/dt
• Newtonian Fluid: Shear stress is directly proportional to the rate of shear strain or Newtonian fluid does not change with viscosity or with the rate of deformation or shear strain (e.g., Water, Kerosene, Petrol, Benzene, Ethanol, Alcohol, Mercury)
• Non-Newtonian Fluid: Shear stress is not proportional to the rate of shear strain; does not obey Newton's law of viscosity

Non-Newtonian Fluids - Time Independent

• Dilatant (Shear thickening fluid): τ = μ(du/dy)^n, where n > 1; μ↑ (du/dy)↑ (dθ/dt)↑ (e.g., Slurry, Printing ink, dye, starch, molasses, Aqueous suspension, Quick sand, Sugar Solution, butter)
• Bingham Plastic: Behaves like a Newtonian fluid but after yield stress, τ = τº + μ(du/dy)^n, n = 1; τ º = Yield shear stress or threshold shear stress (e.g., Water suspension of clay, fly ash, Creams, Toothpaste, Drilling Muds)
• Pseudo Plastic (Shear thinning fluid): τ = μ(du/dy)^n, n < 1; μ↓ (du/dy)↑ (dθ/dt)↑ (e.g., Paper pulp, Clay, Polymer solutions, milk, blood, syrup)

Non-Newtonian Fluids - Time Dependent

• Thixotropic: τ = τº + μ(du/dy)^n+ f(time) [decreasing function] (e.g., Lipstick, Printer inks, Enamels Paint, Jelly)
• Rheopectic: τ = τº + μ(du/dy)^n+ f(time) [ Increasing function] (e.g., Gypsum pastes and Bentonite slurry)

Unit of Pressure

• 1 Pascal = 1 N/m²
• 1 bar = 105 Pa = 100 KPa = 0.1 MPa
• 1 atm = 101325 Pa
• 1 kgf / cm² = 9.81×104 N/m²
• 1 Psi = 6894.76 Pa
• 1 Torr = 133.3 Pa =1 mm Hg

Pressure Measurement

• Absolute pressure formula: Pabs = Patm + Pgauge
• Vacuum pressure formula: Pvaccum = Patm – P'abs

Pascal's Law

• Pressure at a point in a fluid system is equally distributed in all directions. Applied to fluid at rest.
• px = py = pz
• Valid when shear stress is zero and only normal force is present

Hydrostatic Law

• The rate of pressure increase in the vertical direction equals the fluid's weight density at that point.
• Formula when downward direction is positive: dP/dz = γ
• Formula when upward direction is negative: dP/dz = -γ
• Conversion of one liquid column to another liquid column: ρ1 h1 = ρ2 h2 also S1 h1 = S2 h2
• Pressure Head formula: h = p/ρg or p/w

Pressure in Accelerated Vessels

• Vertically Accelerated Vessel: p = ρgh (1+ (a/g)) or p = ρgh (1- (a/g))
• Cases for Pascal's law- conditions for no shear: - Moving fluid (Ideal fluid) with μ = 0 : τ = 0 ⇒ pascal's law - Static fluid (Real fluid) : τ = 0 ⇒ Pascal's law - Moving fluid (Real) with constant acceleration ⇒ τ = 0 - Rotating fluid with a constant velocity

Instrument for Pressure Measurement

• Piezometer: Measures only +ve gauge pressure; moderate pressure of liquid only; not for high pressures or gases
• Inverted U tube manometer: Used for very low pressure difference measurement; Hg cannot be used as manometric fluid; Sm{liquid} < S{fluid it is measuring}
• Micromanometer: Used for very high pressure difference
• Total and center of pressure for submerged plane: - Horizontal position, F = pgAX, = wAX, hcp = x - Vertical position, F = pgAX, = wAX, hcp = X + (IG/AX) - Inclined position, F = pgAX, = wAX, hcp = X + (IG Sin² θ /AX)

Hydrostatic Force

• Curved surfaces: FH = ρg ∫ h.dA sin θ, Fν = ρg ∫ h.dA cos θ, FR = √(FH² + Fv²) = wAx
• Note: Location of center of pressure does not depend on the density of fluid but the value of hydrostatic force depends on density of fluid
• Hydrostatic force of curved surfaces in vertical direction: [ FH = ρgAx ] A - Projected Area also x - Vertical distance of center of gravity of body from free surface, Resultant Force 'F' = √(FH² + (Fv)²), Fν– Weight of liquid block above curved surface

Geometry Properties

Rectangle

• Center of Gravity (C.G.) x = d/2
• Depth of center of pressure (C.P.) h = 2d/3

Metacentric Height

• G.M. = B.M. – B.G
• B.M = Imin/Vimmersed -B.G. Where:
• Imin = M.O.I.

Metacentric Height for Ships

• Merchant ship: < 1 m
• Sailing ship: < 1.50 m
• Battle ship: < 2 m
• River boat: < 3.50 m
• Passenger ship: 0.3 to 1.5 m
• Stability relationship: G.M Cargo ship > G.M Passenger ship Therefore, cargo ships are more comfortable

Time Period of Oscillation

• T = 2π * √( k2 / G.M × g)
• k = Least Radius of gyration
• Metacentric height for rolling condition will be less than metacentric height for pitching condition

Rolling and Floating

• Rolling is the most dangerous

Archimedes's Principle

• When a body is immersed wholly or partially in a liquid, it is lifted up by a force equal to the weight of liquid displaced by the body
• FB = Weight of liquid displaced by the body
• FB = ρf × Vfd × g
• Equilibrium condition for Submerged and floating body
• Stable, B is above G, M is above G
• Unstable, B is below G, M is below G
• Neutral, B and G coincide, M and G coincide

Fluid Flow Types

• Steady Fluid
  • Fluid property like density, pressure, velocity does not change with time
• ∂v/∂t = 0; ∂p/∂t = 0; ∂ρ/∂t = 0
• Unsteady Fluid
  • Fluid property changes with time
• ∂v/∂t ≠ 0; ∂p/∂t ≠ 0; ∂ρ/∂t≠ 0
• Uniform Fluid
  • At a given time, fluid property does not change with respect to space
  • (∂v/∂s) t = Constant =0
  • Pipe should be uniform cross section

Fluid Flow Types Continued

• Non-Uniform Flow
  • At a given time, velocity changes with respect to space
  • Diverging and converging
  • (∂v/∂s) t = Constant ≠ 0
• Irrotational Flow
  • Viscous flows are rotational
  • Fluid particle does not rotate about its own axis in both circular as well as straight line motion
• Local Acceleration is the rate of increase of velocity with respect to the time at a given point in a flow field ∂u; ∂v; ∂w / ∂t
• Convective Acceleration is the rate of change of velocity due to the change of position of fluid in a fluid flow

Flows (Laminar, Turbulent, ... )

• Laminar Flow
  • Fluid particles move along well-defined path or stream line and all the stream lines are straight and parallel
  • Adjacent layer does not cross each other
  • Also known as stream line flow or viscous flow
  • Generally occurs at low velocity
• Turbulent Flow Flow
  • Fluid particle moves in a zig-zag or in random order
  • Generally occurs at high velocity
• Compressible Flow Density of fluid changes from point to point or density is not constant in fluid flow i.e. ρ ≠ constant
• Incompressible Flow Density remains constant i.e. ρ = constant

Mach Number

• Velocity of fluid / Velocity of Sound
• Incompressible flow ⇒ MN < 0.3 {Water}
• Compressible flow ⇒
• 0.3 < MN < 1 ⇒ Subsonic flow
• MN = 1 ⇒ Sonic flow
• 1 < MN < 6 ⇒ Supersonic flow
• MN > 6 ⇒ Hypersonic flow

Reynold Number

• ρVd/µ
• Where, ρ = density, V = Average velocity, µ = Dynamic viscosity, d = Characteristics length

Flow Type Reynolds Numbers

• Laminar Flow, Transition Flow, Turbulent Flow
• Pipe Flow, Open channel, b/w Parallel Plates

Flow Equations

• Continuity
  • Mass Conservation In A Flow ∂ρ / ∂t +(∂(ρu)+∂(ρv)+∂(ρw))=0
• Steady & Incompressible Flow Equation
  • ∂u+∂v+∂w=0
• 1D Flow Equation
  • A1V1 = A2V2
• Momentum/Navier stokes
  • Euler

Flow Potential Equation

• u=-∂φ / ∂x, v=-∂φ / ∂y, w= -∂φ / ∂z

Type of Lines - Streamlines

• Streamline
  • It is an imaginary or curved line in space such that a tangent drawn to it at any point gives the direction of velocity
  • Two streamlines can never intersect each other
  • For, steady flow = Shape of streamline does not change
  • As there is no flow possible across the streamline the discharge will remain constant between any two streamline
• Path line
  • Path traced by a single fluid particle at different instant of time
• Streak line
  • The locus of different fluid particles passing through a fixed point

Shear and Rotation

• Shear Strain Rate =1/2 * (∂V/∂x + ∂U/∂y)
• Rotation Rate = Ω, where curl is Ω = (∂U/∂y - ∂V/∂x)

Pressure Conversion

  γ = 0

• Pascal's Law applicable if and only if fluid is Newtonian and Pressure is Constant
• Pressure Head = P/(ρg)

Bernoulli Equation

• Viscosity & Compressibility are zero P+KE+PE+∫ PdV

Steady Flow

• Viscosity plays no role

Elevation Head

• datum should be specified

Forces Acting on Fluid (Reynold's Equation)

• External+Buoyant+Pressure

Mechanical Engineering Capsule 126 YCT

Key Formulas

Inertia Force ~ PAV^2 Viscous Force ~ μVL Gravity Force ~ pALg Pressure Force ~ PA