Fluid Mechanics Formula Sheet

Fluid Mechanics deals with the behaviour of liquids and gases at rest and in motion. Fluids cannot withstand shearing stress and begin to flow when such stress is applied.

1. Pressure

Pressure is the force exerted per unit area by a fluid on any surface in contact with it.

P = F / A

• SI unit: Pascal (Pa)
• 1 bar = 10⁵ Pa
• 1 atm = 1.013 × 10⁵ Pa = 760 mm Hg
• Pressure is a scalar quantity
• At a point in a fluid, pressure acts equally in all directions

2. Density

Density is mass per unit volume.

ρ = m / V

• Relative density = Density of substance / Density of water at 4°C
• Density decreases with increase in temperature
• Variation with pressure: ρ = ρ₀ (1 + ΔP / B)

3. Pressure Due to Liquid Column (Hydrostatic Pressure)

p = p₀ + ρgh

• p₀ = atmospheric pressure
• ρ = density of liquid
• h = depth below free surface
• Gauge pressure: p_g = ρgh

Atmospheric pressure decreases exponentially with height:

P = P₀ e^−h/8.55

4. Liquid in an Accelerated Container

• If container accelerates horizontally with acceleration a, free surface tilts
• tan θ = a / g

5. Force Exerted by a Liquid

• On base due to gauge pressure:

F = ρgV (equal to weight of liquid)

• On vertical wall of height H:

F = ½ ρg bH²

Resultant force acts at a depth 2H / 3 from the free surface.

6. Pascal’s Law

Any change in pressure applied to a confined fluid is transmitted undiminished throughout the fluid.

Hydraulic lift:

F / A = f / a

• Used as a force-multiplying device
• Multiplication factor = A / a

7. Buoyancy and Archimedes’ Principle

A body immersed in a fluid experiences an upward force equal to the weight of fluid displaced.

B = Weight of displaced fluid = ρVg

Apparent weight:

W′ = W − B

• If ρ_body > ρ_liquid → body sinks
• If ρ_body < ρ_liquid → body floats partially
• If ρ_body = ρ_liquid → body floats fully submerged

8. Fluid Flow

• Steady flow: velocity constant with time
• Streamline: tangent gives direction of velocity
• Irrotational flow: no net angular velocity
• Turbulent flow: velocity changes randomly

9. Equation of Continuity

For steady flow of incompressible fluid:

ρAv = constant

For incompressible fluid:

Av = constant

10. Bernoulli’s Theorem

For streamline flow of an ideal fluid:

P/ρ + v²/2 + gh = constant

Applications

• Dynamic lift of aeroplane wings
• Velocity of efflux: v = √(2gh)
• Liquid jet trajectory

11. Viscosity

Internal resistance to flow between fluid layers.

F = −ηA (dv/dx)

• η = coefficient of viscosity
• SI unit: N·s m⁻²

12. Flow Through a Narrow Tube (Poiseuille’s Formula)

V/t = (πPr⁴) / (8ηl)

13. Stokes’ Law

Viscous force on a sphere:

F = 6πrηv

14. Terminal Velocity

When viscous force balances apparent weight.

v_t = (2r²g(ρ − σ)) / (9η)

15. Reynolds Number

Re = ρvD / η

• Re < 2000 → Streamline flow
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