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ENGR5002 / Week 1
L2 slide 11: Modelling pressure with depthFrom lectureCurrent spec2:110 marks12 min

L2 - Pressure and its measurement, slides 6-14 (The life of a diver; how can we model this pressure mathematically?)

A diver at the surface of a lake feels only atmospheric pressure. As the diver descends vertically to 25 m, the weight of the water column above increases and the pressure rises. The diver then swims 10 m horizontally at the same depth; the lecturer says "the depth remains constant, I don't expect a change in the pressure".

The lecturer then asks: we have seen that the diver feels more pressure going deeper, but how can we model this pressure mathematically? (Answer on the slide: by a balance of forces.)

Formulas you may need
  • Differential hydrostatic equation: dpdz=−ρg\dfrac{dp}{dz} = -\rho g for zz measured upwards, equivalently dpdz=+ρg\dfrac{dp}{dz} = +\rho g for zz measured downwards (depth); ∂p∂x=∂p∂y=0\dfrac{\partial p}{\partial x} = \dfrac{\partial p}{\partial y} = 0 (on the formula sheet, derived in slides 12-13: be able to derive it)
  • Pressure at depth: p=patm+ρghp = p_{atm} + \rho g h (on the formula sheet)
  • Gauge pressure: pgauge=pabs−patmp_{gauge} = p_{abs} - p_{atm} (learn this)
  1. (a)
    By a force balance on a small vertical cylinder of fluid at rest (cross-sectional area AA, height dzdz), derive the differential equation for the variation of pressure in the vertical direction. State your sign convention for zz.
    [4]
  2. (b)
    By a force balance on a small horizontal cylinder of fluid at rest, show that pressure does not vary in the horizontal direction.
    [2]
  3. (c)
    Integrate your result from (a), assuming constant density, to find the pressure at a depth hh below a free surface at atmospheric pressure patmp_{atm}. State the hypothesis this needs.
    [2]
  4. (d)
    [Added, to put numbers on the diver] Take ρ=1000 kg m−3\rho = 1000\ \mathrm{kg\,m^{-3}}, patm=101.3 kPap_{atm} = 101.3\ \mathrm{kPa} and g=9.81 m s−2g = 9.81\ \mathrm{m\,s^{-2}}. Calculate the absolute and gauge pressure on the diver at 25 m, and the absolute pressure after swimming 10 m horizontally at that depth.
    [2]