Fluid Mechanics: Properties and Pressure

• Fluid is either a liquid or a gas
• Newtonian fluid exhibits a linear relationship between shear stress and velocity gradient, resulting in motion when shear stress is applied

Fluid Mechanics

• Study of fluids in motion (fluid dynamics) or at rest (fluid statics)

Properties of Fluid

• Pressure (p) is compressive forces acting on a unit area, measured as p = F/A
• Atmospheric pressure (patm) is the pressure in the atmosphere, measured by a barometer
• Standard atmospheric pressure values, at 40º latitude at sea level:
  • 101.325 kPa
  • 14.7 psi
  • 29.92 in Hg
  • 760 mm Hg
  • 760 torr
  • 1.013 bar
  • 33.90 ft H2O
  • 10.33 m H2O
  • 1 atm
• Gage pressure (pg) is pressure relative to atmospheric pressure, derived from liquid column, can be written as pg = pgh
• Absolute pressure (pa) is pressure relative to zero in a volume void of molecules(ideal vacuum), can be written as pa = pg + patm
• Density (p) is mass per unit volume, expressed as p = m/V
• Specific volume (υ) is volume per unit mass, expressed as v = V/m ; [υ =1/p]
• Specific weight (ү) is weight per unit volume, expressed as γ = w/V and γ = pg
• Specific gravity is the ratio of a substance's density to water's density, written as SG = Psubstance/pwater
• Viscosity/Dynamic Viscosity/Absolute Viscosity (μ) is a fluid's resistance to flow
  • Determines fluid shear rate (dv/dy) from shear stress (τ), calculated as μ = τ/(dv/dy)
  • 1 centi-Poise(cP) = 10-3 Pa-s
• Kinematic viscosity (μκ) is viscosity divided by density, expressed as μκ = μ/ρ
• Bulk Modulus of Elasticity (B) measures fluid compressibility as the ratio of change in unit pressure to volume change, written as B = ∆p/(ΔV/V)
  • Determines sound speed (c) in liquid: c = (B/p)1/2
  • B in Pascal, p in kg/m³
• Surface tension (σ) is attractive forces between molecules, using formulas:
  • Odroplet=pr/2
  • Obubble=pr/4
• Capillarity is liquid rise/fall in a capillary tube, where height (h) is derived by: h = 4στοςθ/(γα)
• Vapor pressure is equilibrium pressure between escaping and reentering molecules
  • Boiling starts when pressure drops below vapor pressure

Thermodynamic Properties of Fluid

• Ideal gas has no attractive forces between particles, compressed completely
  • Behaves according to ideal gas law, expressed as pV = mRT
• Enthalpy (H, h) is the sum of total internal energy and the product of system pressure and volume
  • Change of enthalpy equals heat transferred at constant pressure, using equations: H = U + pV], dH = 6Q, dh = mCpdT
• Internal Energy (U, u) is energy within the system, excludes kinetic and potential
  • Change in internal energy equals system heat transfer at constant volume, using equations: dU = 6Q, dU = mCvdT
• Specific heat ratio (k) is the ratio of Cp to Cv, expressed as k = Cp/Cv
• Isentropic process is adiabatic and properties remain constant
  • Equation: T2/T1 = (p2/p1)(k-1/k)], applicable for ideal gases with constant specific heats

Viscosity-Torque(T) Relation in Concentric Cylinder Viscometer

• Formula: T = 2πRr²wLµ/h
  • w = angular velocity
  • R = inner cylinder radius
  • r=mean radius
  • L = cylinder length
  • h = gap width

Control Volume

• Fixed region in space for studying mass and energy transfer, defined as an open system with fixed boundaries

Continuity Equations

• Under steady state flow, the formula is m1=m2
• Formula for mass flow rate: P1A1V1=P2A2v2=m
• For liquids with constant density the formula is Q1=Q2
• Formula for volume flow rate A1V1=A2V2=Q
  • (m1, m2): mass flowrate of fluid entering section 1 and leaving section 2
  • (P1-P2): density of fluid at section 1 and 2
  • (A1, A2): cross-sectional area of section 1 and 2
  • (V1,V2): average fluid velocity at section 1 and 2
  • (Q1,Q2): volume flow rate

Forms of Energy in Fluid Flow

• Internal Energy due to temperature, u
• Flow Energy, due to fluid pressure, PV
• Potential Energy due to elevation, gZ
• Kinetic Energy due to fluid velocity, v² / 2
• For most liquid flow problems, change in internal energy is negligible

Bernoulli's Equation (by Daniel Bernoulli, 1738)

• Considered conservation of energy for flowing fluids, measured in unit weight
  • Formula: P1/γ +Z₁+(V1^2)/2g = P2/γ +Z2+(V2^2)/2g ,
• ( Z1, Z2): elevations of points 1 and 2, measured from a common reference
• (P1, P2): fluid pressure at points 1 and 2

Applicability and Assumptions

• Negligible viscous effects
• Fluid density is constant
• Steady flow
• Flow is along a streamline
• Streamline is not accelerating

Modified Bernoulli's Equation (the Energy Equation)

• Formula: P1/γ +Z1+(V1^2)/2g +Hp-Hm-HL = P2/γ +Z2+(V2^2)/2g
  • (Hp)- Pump head, can be obtained using the formula Hydraulics Power pump/ Qy
  • (Hm)- Hydraulic motor head, can be obtained using the formula Hydraulics Power motor/ Qy
  • (HL) Head loss, energy per weight loss due to friction between points 1 and 2

Venturi Effect

• Event of pressure drop will result to increase in velocity and vice versa for sub-sonic flowing fluid with weak compressibility
  • Ideal equation below: P1-P2= ρ/2 (v2^2-v1^2)

Torricelli's Theorem

• States ideally the velocity of a free jet of fluid is equal to the square root of the product of two times the acceleration of gravity times the head producing the jet.
  • Formula: V2=√2 gh

Siphon Effect

• Siphon describes the outflow of fluid from a container in upward direction through a pipe/tube to its free end located lower than the surface of the fluid in the container For fluid to flow out of the free end, the fluid must initially be forced to flow up from the container into the top-center portion U-shape pipe/tube.
• Formula: v2=√2g(h-H₁)