1. Fundamental Concepts
- Fluid: Substance that flows and takes shape of container
- Density (ρ): Mass per unit volume\[ \rho = \frac{m}{V} \]
- Specific Weight (γ): Weight per unit volume\[ \gamma = \rho g \]
- Specific Gravity: Ratio of density to water density at 4°C
- Viscosity (μ): Resistance to flow, internal friction
- Kinematic Viscosity (ν):\[ \nu = \frac{\mu}{\rho} \]
- Surface Tension: Property of liquid surface to minimize area
- Capillarity: Rise or fall of liquid in narrow tube
2. Pressure and Measurement
- Pressure (P): Force per unit area\[ P = \frac{F}{A} \]
- Pascal’s Law: Pressure transmits equally in all directions
- Hydrostatic Law: Pressure variation with depth\[ \frac{dP}{dh} = \rho g \]
- Absolute Pressure: Measured from absolute zero
- Gauge Pressure: Difference between absolute and atmospheric pressure
- Vacuum Pressure: Pressure below atmospheric
- Manometers: Devices to measure pressure using liquid columns
3. Fluid Statics
- Hydrostatic Force on Plane Surface:\[ F = \rho g A \bar{h} \]where \(\bar{h}\) is depth of centroid
- Center of Pressure: Point where total force acts\[ h_{cp} = \bar{h} + \frac{I_G}{A\bar{h}} \]where \(I_G\) is moment of inertia
- Buoyancy: Upward force on submerged body\[ F_b = \rho g V \]
- Archimedes Principle: Buoyant force equals weight of displaced fluid
- Stability of Floating Bodies: Metacentric height determines stability
4. Fluid Kinematics
- Streamline: Imaginary line tangent to velocity vector
- Pathline: Actual path followed by fluid particle
- Streakline: Locus of particles passed through a point
- Timeline: Instantaneous position of fluid particles
- Types of Flow:
- Steady vs Unsteady
- Uniform vs Non-uniform
- Laminar vs Turbulent
- Compressible vs Incompressible
- Rotational vs Irrotational
- Continuity Equation: Conservation of mass\[ A_1 V_1 = A_2 V_2 \](for incompressible flow)
5. Fluid Dynamics
- Bernoulli’s Equation: Conservation of energy\[ \frac{P}{\rho g} + \frac{V^2}{2g} + z = \text{constant} \]
- Venturimeter: Measures flow rate using pressure difference\[ Q = C_d A_1 A_2 \sqrt{\frac{2g h}{\sqrt{A_1^2 – A_2^2}}} \]
- Orificemeter: Measures flow through orifice
- Pitot Tube: Measures flow velocity\[ V = \sqrt{2g h} \]
- Momentum Equation: Force equals rate of momentum change\[ F = \rho Q (V_2 – V_1) \]
6. Flow Through Pipes
- Reynolds Number (Re):\[ Re = \frac{\rho V D}{\mu} \]
- Re < 2000: Laminar flow
- Re > 4000: Turbulent flow
- Darcy-Weisbach Equation: Head loss in pipes\[ h_f = \frac{f L V^2}{2g D} \]
- Hagen-Poiseuille Equation: Laminar flow in pipes\[ h_f = \frac{32 \mu L V}{\rho g D^2} \]
- Minor Losses: Due to fittings, valves, etc.
7. Dimensional Analysis
- Buckingham π Theorem: Relationship between variables
- Important Dimensionless Numbers:
- Reynolds Number (Re): Inertial/Viscous forces
- Froude Number (Fr): Inertial/Gravity forces\[ Fr = \frac{V}{\sqrt{gL}} \]
- Weber Number (We): Inertial/Surface tension forces
- Mach Number (M): Velocity/Speed of sound
- Euler Number (Eu): Pressure/Inertial forces
8. Important Constants and Values
- Density of water at 4°C: 1000 kg/m³
- Specific weight of water: 9810 N/m³
- Dynamic viscosity of water at 20°C: 1 × 10⁻³ Ns/m²
- Kinematic viscosity of water at 20°C: 1 × 10⁻⁶ m²/s
- Surface tension of water: 0.072 N/m
- Atmospheric pressure: 101.325 kPa or 760 mm Hg
- Acceleration due to gravity (g): 9.81 m/s²
- Specific gravity of mercury: 13.6
9. Formula Sheet
| Concept | Formula |
|---|---|
| Density | \(\rho = \frac{m}{V}\) |
| Specific Weight | \(\gamma = \rho g\) |
| Specific Gravity | \(SG = \frac{\rho}{\rho_{water}}\) |
| Pressure | \(P = \rho g h\) |
| Buoyant Force | \(F_b = \rho g V\) |
| Continuity Equation | \(A_1 V_1 = A_2 V_2\) |
| Bernoulli’s Equation | \(\frac{P}{\rho g} + \frac{V^2}{2g} + z = \text{constant}\) |
| Reynolds Number | \(Re = \frac{\rho V D}{\mu}\) |
| Darcy-Weisbach | \(h_f = \frac{f L V^2}{2g D}\) |
| Venturimeter | \(Q = C_d A_1 A_2 \sqrt{\frac{2g h}{\sqrt{A_1^2 – A_2^2}}}\) |
| Pitot Tube | \(V = \sqrt{2g h}\) |
| Froude Number | \(Fr = \frac{V}{\sqrt{gL}}\) |
- 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
- P=F/A
- 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
- Symbol: η (eta)
- 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
- \[ h = \frac{2T \cos\theta}{\rho g r} \]
- 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