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Physics 1

Fluid mechanics: hydrostatic pressure, Pascal, Archimedes and Bernoulli

Density and pressure, hydrostatic pressure, Pascal's principle, buoyancy and floating, the continuity equation, Bernoulli's equation and its applications, viscosity.

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Liquids and gases have no shape of their own: they flow. To describe them we do not follow individual pieces, we use quantities defined point by point such as density, pressure and velocity. In this lesson we look at fluid statics (Stevin’s law, Pascal’s and Archimedes’ principles) and at the dynamics of ideal fluids (continuity and Bernoulli), with proofs that use only Newton’s laws and energy conservation.

Density and pressure

Density is mass per unit volume, \rho = m/V: water is about 1000 kg/m³, air at sea level 1.2 kg/m³, mercury 13,600 kg/m³.

Pressure is the perpendicular force per unit area, p = F/A, measured in pascals (Pa = N/m²). Atmospheric pressure at sea level is about 101,325 Pa, that is 1 atm or 1.013 bar. In a fluid at rest the pressure at a point is the same in every direction.

Stevin’s law (hydrostatic pressure)

Proof

Inside a liquid at rest of density \rho, consider a small vertical cylinder of cross-section A and height h. The vertical forces on it are the pressure on the top face, p_1 A downwards, the pressure on the bottom face, p_2 A upwards, and the weight of the liquid, \rho A h\,g. Sideways forces cancel by symmetry. In equilibrium:

    \[ p_2 A = p_1 A + \rho\,A\,h\,g \quad\Rightarrow\quad p_2 = p_1 + \rho\,g\,h \]

Pressure grows linearly with depth and does not depend on the shape of the container. In water it rises by about 1 atm every 10 m: at a depth of 10 m a diver feels twice the pressure at the surface.

Torricelli’s barometer. A tube full of mercury turned upside down in a dish drops to a height h such that \rho_{Hg}\,g\,h = p_{atm}: about 760 mm. Hence the unit mmHg, still used for blood pressure.

Pascal’s principle

A change in pressure applied to an enclosed fluid is transmitted unchanged to every point of the fluid. The hydraulic press uses it: a small piston of area A_1 pushed with force F_1 creates a pressure F_1/A_1 that also acts on the large piston of area A_2, with force

    \[ F_2 = F_1\,\frac{A_2}{A_1} \]

With an area ratio of 100 you lift a car with the force needed to lift a suitcase. No energy is gained: the small piston must travel a hundred times further. It is the principle of hydraulic brakes and jacks.

Archimedes’ principle

A body immersed in a fluid receives an upward push equal to the weight of the fluid it displaces:

    \[ F_A = \rho_f\,V\,g \]

Body immersed in a liquid with its weight pointing down and the buoyant force pointing up
The body of volume V feels its weight mg and the buoyant force F_A. Buoyancy arises because the liquid’s pressure (light blue arrows) grows with depth.

Proof

Take a box-shaped body with horizontal faces of area A and height h, immersed in a fluid of density \rho_f. The forces on the side faces cancel in pairs. On the top face, at depth y, pressure pushes down with p(y)\,A; on the bottom face, at depth y + h, it pushes up with p(y + h)\,A. By Stevin’s law the difference is

    \[ F_A = \big(p(y + h) - p(y)\big)\,A = \rho_f\,g\,h\,A = \rho_f\,g\,V \]

For a body of any shape, argue like this: if fluid were in its place, it would be in equilibrium, so the push of the surrounding pressures is exactly the weight of that fluid. The pressures do not change if the body takes the fluid’s place.

Floating. A body floats if its average density is less than the fluid’s. At equilibrium the submerged part V_{sub} satisfies \rho_f V_{sub} g = \rho_b V g, that is V_{sub}/V = \rho_b/\rho_f. Ice (917 kg/m³) in sea water (1025 kg/m³) has about 89% of its volume under water: the famous tip of the iceberg. A steel ship floats because its average density, air in the hull included, is less than water’s.

Moving fluids: the continuity equation

Consider an incompressible fluid flowing steadily through a pipe of varying cross-section. In a time dt a volume A_1 v_1\,dt passes through section A_1; the same volume must leave through section A_2, because the fluid neither piles up nor compresses:

    \[ A_1\,v_1 = A_2\,v_2 \]

The product Q = A\,v is the flow rate (m³/s). Where the pipe narrows, the fluid speeds up: it is what you do when you squeeze the end of a garden hose with your thumb.

Bernoulli’s equation

For an ideal fluid (incompressible and without viscosity) in steady flow, along a streamline

    \[ p + \tfrac12\,\rho\,v^2 + \rho\,g\,y = \text{constant} \]

Proof

Follow a portion of fluid that, in a time dt, moves from one part of the pipe (section A_1, pressure p_1, speed v_1, height y_1) to another (A_2, p_2, v_2, y_2). The net effect is that a small volume dV = A_1 v_1 dt = A_2 v_2 dt of mass dm = \rho\,dV moves from the first position to the second.

The fluid behind pushes with p_1 A_1 over a distance v_1 dt, the fluid ahead resists with p_2 A_2 over v_2 dt. The work of the pressure forces is W = (p_1 - p_2)\,dV. By the work-energy theorem, writing the work of gravity as a change in potential energy:

    \[ (p_1 - p_2)\,dV = \tfrac12\,\rho\,dV\,(v_2^2 - v_1^2) + \rho\,dV\,g\,(y_2 - y_1) \]

Dividing by dV and collecting the terms with index 1 and 2 gives Bernoulli’s equation. It is energy conservation per unit volume.

Consequences

  • Torricelli’s theorem. From a hole at depth h below the free surface of an open tank, the liquid comes out at v = \sqrt{2gh}, the same speed as a stone dropped from that height: the pressure is atmospheric both at the top and at the outlet.
  • Venturi effect. In a horizontal constriction the speed rises, so the pressure falls. Measuring the pressure difference between two sections gives the flow rate: it is the principle of Venturi meters and of the carburettor.
  • Lift. A wing is shaped so that air flows faster over it than under it: the pressure difference produces an upward force. The full explanation also needs how the wing deflects air downwards, but Bernoulli gives the right order of magnitude.

Limits of the ideal model: viscosity

Real fluids have internal friction, viscosity \eta. In a pipe of radius r and length L in laminar flow, the flow rate follows Poiseuille’s law, Q = \pi r^4\,\Delta p/(8\eta L): halving the radius cuts the flow by sixteen times. That is why a narrowed artery is so dangerous, and why liquid cooling in computers uses tubes that are not too thin. At higher speeds the flow turns turbulent, and the Reynolds number Re = \rho v d/\eta says when: in a pipe, above roughly 2000 to 4000.

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