Fluid Mechanics

  •  Stress  : Force acting per unit area of a body.
  • Formula: Stress=Force (F)/ Area (A)
  • Unit: Pascal (Pa) or N/m²
  • Types of Stress:
    • Tensile stress – Pulling force
    • Compressive stress – Pushing force
    • Shear stress – Tangential force
  • Strain: Change in dimension per unit original dimension.
  • Formula: Strain=Change in Length (ΔL)/ Original Length (L)
  • Unit: No unit (dimensionless)
  • Types of Strain:
    • Longitudinal strain
    • Volume strain
    • Shear strain
  • Hooke’s Law:  Within the elastic limit, stress is directly proportional to strain.
    • Stress ∝ Strain 
    • Stress/Strain= Constant (K)
    • This constant is the Modulus of Elasticity .
  •  Young’s Modulus of Elasticity (Y) Ratio of longitudinal stress to longitudinal strain.
    • Formula: Y=Longitudinal Stress/ Longitudinal Strain
    • Unit: Pascal (N/m²)
    • Application: Stretching a wire, rods, etc.
  • Bulk Modulus (K) : Ratio of volumetric stress to volumetric strain.
    • Formula:
    • Unit: Pascal (N/m²)
    • Used when: Pressure is applied from all sides (e.g., fluids).
  •  Modulus of Rigidity (Shear Modulus, η or G) : Ratio of shear stress to shear strain.
    • Formula: η=Shear Stress/ Shear Strain
    • Unit: Pascal (N/m²)
    • Used when: Force causes angular deformation (e.g., twisting).
  • Poisson’s Ratio (σ) : Ratio of lateral strain to longitudinal strain.
    • Formula: σ=Lateral Strain/ Longitudinal Strain
    • Unit: No unit (dimensionless)
    • Typical Range: 0 to 0.5
  • Compressibility (β) : It is the reciprocal of bulk modulus. Indicates how easily a substance can be compressed.
    • Formula:
    • Unit: m²/N or Pa⁻¹
    • High β ⇒ Easily compressible (e.g., gases)
  • Thrust: Force exerted perpendicularly on a surface.
    • Thrust=Force=F=m⋅g
    • Unit: Newton (N)
    • Nature: Vector quantity
  • Pressure (दाब) : Thrust per unit area.
    • P=F/A
      • : Pressure
      • : Force
      • : Area
    • SI Unit: Pascal (Pa) = N/m²
    • 1 atm = 1.013×105 Pa
  • Density ρ=m/V
    • : Density
    • : Mass
    • : Volume
    • SI Unit: kg/m³
  • Pascal’s Law : When pressure is applied at any point in a confined fluid, it is transmitted equally in all directions.
    • Used in: Hydraulic press, brake, lift, etc.
  • Buoyancy :  upward force experienced by a body immersed in fluid.
  • Upthrust / Buoyant Force FB=ρ⋅g⋅V
    • : Buoyant force
    • : Density of fluid
    • : Acceleration due to gravity
    • : Volume displaced
    • Unit: Newton (N)
  • Archimedes’ Principle (Law of Floatation) A body immersed in a fluid experiences a buoyant force equal to the weight of the fluid displaced.
    • Buoyant force=Weight of displaced fluid
    • Floating condition: Weight of body=Upthrust
  • Viscosity (श्यानता): Internal friction between fluid layers.
    • Symbol: η (eta)
      SI Unit: Pascal-second)
    • Formula (Stoke’s Law): F=6πηrv 
  • Surface Tension : The force acting per unit length on the surface of a liquid.
    • T= Tension Force/ Length
    • Unit: N/m
  • Cohesion : Attraction between same molecules
  • Adhesion : Attraction between different molecules
  • Capillarity (केशिकता)Rise or fall of liquid in a narrow tube due to surface tension and adhesion.
    •  \[ h = \frac{2T \cos\theta}{\rho g r} \]
      • h: Height of rise
      • T: Surface tension
      • θ: Contact angle
      • r: Radius of tube
  • Bernoulli’s Theorem : For an incompressible, non-viscous fluid in steady flow, the total mechanical energy is constant.
    • \[ P + \frac{1}{2} \rho v^2 + \rho g h = \text{Constant} \]
    • Applications: airplane wings, carburetors, sprayers, chimney draft.
  • Pressure Measuring Instruments
    • Barometer – Atmospheric pressure
    • Manometer – Gas pressure in closed container
    • Sphygmomanometer – Blood pressure
    • Bourdon gauge – High pressure fluids
    • Venturi meter – measure the flow rate of an incompressible fluid in a pipe.
    • Orifice meter- measure fluid flow rate
Fluid Mechanics