Mechanical Properties of Fluids

• 1. Hydrostatics (Fluids at Rest)

  • Pressure & Density
    • Fluid Pressure: $P = \frac{F}{A}$ (S.I. Unit: Pascal / $N/m^2$)
    • Density: $\rho = \frac{m}{V}$
    • Gauge Pressure: $P_{gauge} = P - P_0 = \rho gh$
    • Absolute Pressure: $P = P_0 + \rho gh$
    • Measurement: Barometer (Atmospheric), Manometer (Gauge)
  • Pascal's Law
    • Pressure applied to enclosed fluid is transmitted equally in all directions
    • Applications: Hydraulic Lift, Hydraulic Brakes
  • Archimedes' Principle
    • Buoyant Force = Weight of fluid displaced by submerged body
    • Law of Floatation: Body floats when buoyant force equals its weight

• 2. Hydrodynamics (Fluids in Motion)

  • Types of Flow
    • Streamline/Laminar Flow: Smooth, non-crossing flow lines
    • Turbulent Flow: Chaotic, irregular flow above critical speed
    • Reynolds Number ($R_e$): Predicts flow nature
      • $R_e < 1000$: Streamline
      • $R_e > 2000$: Turbulent
  • Equation of Continuity
    • Based on Conservation of Mass
    • Formula: $A_1 v_1 = A_2 v_2 = \text{constant}$
  • Bernoulli's Principle
    • Based on Conservation of Energy
    • Formula: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$
    • Applications:
      • Venturimeter: Measures fluid speed in pipes
      • Torricelli's Law: Velocity of efflux ($v = \sqrt{2gh}$)
      • Dynamic Lift: Airplane wings, Magnus Effect in spinning balls

• 3. Viscosity

  • Viscous Force
    • Internal friction between fluid layers resisting motion
  • Newton's Law of Viscosity
    • Viscous Force: $F = -\eta A \frac{dv}{dx}$
    • Coefficient of Viscosity ($\eta$): S.I. Unit is $Pa\cdot s$ or $N\cdot s/m^2$
  • Stokes' Law
    • Viscous drag on a moving sphere: $F = 6\pi \eta r v$
  • Terminal Velocity ($v_t$)
    • Maximum constant velocity reached by an object falling through a fluid
    • Formula: $v_t = \frac{2}{9} \frac{r^2 (\rho - \sigma) g}{\eta}$

• 4. Surface Tension

  • Concept & Cause
    • Intermolecular Cohesive Forces pulling surface molecules inward
    • Surface behaves like a stretched elastic membrane
    • Formula: $S = \frac{F}{L}$ (S.I. Unit: $N/m$)
  • Surface Energy
    • Work done to increase surface area: $W = S \times \Delta A$
  • Excess Pressure inside Curved Surfaces
    • Liquid Drop: $P_{excess} = \frac{2S}{R}$
    • Soap Bubble: $P_{excess} = \frac{4S}{R}$ (Two free surfaces)
  • Angle of Contact ($\theta$)
    • Angle between tangent to liquid surface and solid surface
    • Acute Angle ($\theta < 90^\circ$): Liquid wets solid (e.g., Water-Glass)
    • Obtuse Angle ($\theta > 90^\circ$): Liquid does not wet solid (e.g., Mercury-Glass)
  • Capillarity
    • Phenomenon of rise or fall of liquid in a narrow tube
    • Ascent Formula: $h = \frac{2S \cos\theta}{r \rho g}$