Fluid dynamics

NUST NET (Engineering) · Physics · Mechanics. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

Viscosity is friction inside a fluid. A small sphere moving slowly through a fluid feels a drag given by Stokes' law, which grows with speed. A falling sphere therefore speeds up until drag balances its weight, then falls at a steady terminal velocity.

For an ideal fluid (incompressible, non-viscous, in steady streamline flow) two conservation laws do all the work. Conservation of mass gives the equation of continuity: the fluid moves faster where the pipe is narrower. Conservation of energy gives Bernoulli's equation: where the speed is higher, the pressure is lower. This explains the lift on a wing, the Venturi meter and the speed of water leaving a hole in a tank.

Rules to remember

Common mistake

Halving the diameter of a pipe and doubling the speed. Area depends on diameter squared, so half the diameter means a quarter of the area and four times the speed.

Worked example

Water (density 1000 kg m−3) flows at 2.0 m s−1 through a horizontal pipe of diameter 4.0 cm, which narrows to a diameter of 2.0 cm. By how much does the pressure fall in the narrow part?

  1. Half the diameter gives a quarter of the area, so v2 = 4 × 2.0 = 8.0 m s−1.
  2. P1 − P2 = ½ρ(v22 − v12) = ½ × 1000 × (64 − 4).
  3. = 500 × 60 = 30 000 Pa.

Answer: 3.0 × 104 Pa

Practice questions

Try each one, then open the answer.

1. The SI unit of the coefficient of viscosity is

  1. A
    N m−2 s−1
  2. B
    N m s−1
  3. C
    kg m s−1
  4. D
    N s m−2
Show answer

Answer: D. From F = 6πηrv, η = F/(6πrv) has the unit N ÷ (m × m s−1) = N s m−2, also written Pa s.

2. Bernoulli's equation for a fluid in steady flow is a statement of the conservation of

  1. A
    mass
  2. B
    energy
  3. C
    momentum
  4. D
    volume
Show answer

Answer: B. P + ½ρv2 + ρgh = constant says that pressure energy, kinetic energy and potential energy per unit volume add up to a constant. The equation of continuity comes from conservation of mass.

3. A pipe of cross-sectional area 20 cm2 carries water flowing at 3.0 m s−1. The volume of water it delivers per second is

  1. A
    6.0 × 10−3 m3
  2. B
    6.0 × 10−1 m3
  3. C
    60 m3
  4. D
    6.0 × 10−5 m3
Show answer

Answer: A. Volume flow rate = Av = (20 × 10−4 m2) × 3.0 m s−1 = 6.0 × 10−3 m3 s−1, that is 6 litres per second. Using 10−2 for cm2 gives 6.0 × 10−1.

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