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Chapter 10 · Part I · Mechanics
Revised on ____________________
Chapter 10 · Fluid Mechanics

Pressure, buoyancy and the price of going faster

2 sheets
2 diagrams
29 results

Formulas and conditions only — no derivations, no solved numbers. The diagrams are the reader's own.

I

Hydrostatics

DEPTH ONLY — SHAPE NEVER MATTERS
I.Pressure  P = F⊥A scalar; acts equally in all directions
II.With depth  P = P0 + ρ g h
III.Gauge pressure  Pg = P − P0 = ρ g h
IV.Pascal's law  F1A1 = F2A2 the hydraulic press
V.Force on a wall  F = 12 ρ g h² L acting at 2h/3 below the surface
VI.Accelerating tank  tan θ = ag the free surface tilts
VII.Rotating tank  the surface is a paraboloid, y = ω² r²2g
VIII.Manometer  equal heights ⇒ equal pressure, in the same connected fluid
II

Buoyancy

THE FLUID YOU PUSHED OUT OF THE WAY
I.Archimedes  B = ρfl Vdisp g acts at the centre of buoyancy
II.Apparent weight  Wapp = W − B
III.Floating  ρbody V = ρfl Vsub
IV.Fraction submerged  VsubV = ρbodyρfl ice in water: 0.9
V.Relative density  RD = WairWair − Wwater
VI.In a lift with a  replace g by (g + a) in both B and W the fraction floating is unchanged
BWVsubV
III

Flow: continuity and Bernoulli

IDEAL, STEADY, INCOMPRESSIBLE, NON-VISCOUS
I.Continuity  A1 v1 = A2 v2 narrow means fast
II.Bernoulli  P + 12 ρ v² + ρ g h = constant along one streamline
III.Torricelli  v = √(2 g h) the same as a free fall from h
IV.Venturi meter  v1 = A2 √( 2 (P1 − P2)ρ (A1² − A2²) )
V.Dynamic pressure  12 ρ v² the pitot tube reads this
VI.Reynolds number  Re = ρ v Dη turbulent above ≈ 2000
v = √(2gh)h
IV

Viscosity

REAL FLUIDS DRAG
I.Newton's law  F = − η A dvdy η in Pa s
II.Stokes' law  F = 6 π η r v small sphere, laminar flow
III.Terminal velocity  vt = 2 r² (ρ − σ) g9 η ∝ r²
IV.Poiseuille  Q = π r4 ΔP8 η L halve the radius, lose 15/16 of the flow
V.Temperature  η falls for liquids, rises for gases
V

Surface tension

ENERGY PER UNIT AREA
I.Definition  S = FL = energyarea N m−1 = J m−2
II.Excess pressure, drop  ΔP = 2 SR one surface
III.Bubble in air  ΔP = 4 SR two surfaces — the classic trap
IV.Capillary rise  h = 2 S cos θρ g r falls if θ > 90°
Where marks are lost in this chapter
1Treating ρgh as the absolute pressure. It is gauge — absolute is p0 + ρgh.
2Applying Bernoulli between points not on one streamline. Trace the streamline before you equate.
3Halving a pipe's radius and halving the flow. Q ∝ R4: it falls by a factor of 16.
4Using the total volume in Archimedes for a floating body — only Vsub counts.