Learn first: Hydrostatics and aerostatics
Hydrodynamics studies the motion of liquids and gases. A moving fluid “behaves” differently from a fluid at rest: in a narrow part of a pipe it flows faster and its pressure drops. This simple-looking effect explains the lift on an aircraft wing, sprays, carburettors, pumps, water mains and even why ships sailing side by side are pulled together. The topic continues hydrostatics and applies the conservation of energy to fluids.
Real fluid flow is complicated, so a simplified model is used. An ideal fluid:
If the flow is steady, the velocity at each point does not change with time (though it may differ from point to point).
Streamlines are lines whose tangent at every point shows the direction of the velocity there. Where streamlines crowd together, the speed is high. A “tube” bounded by streamlines is a stream tube.
In slow flow the layers do not mix: the flow is laminar. At high speed eddies appear and the flow becomes turbulent (a fast river, smoke rising from a chimney). The ideal-fluid model works well mainly for laminar flow.
An incompressible fluid cannot “pile up” or “vanish” in a pipe, so the same volume passes every cross-section in the same time:
S₁v₁ = S₂v₂
Volume flow rate: Q = Sv (m³/s); mass flow rate: Q_m = ρSv (kg/s)
Where a pipe narrows, the speed rises; where it widens, the speed falls. The area is proportional to the square of the diameter (S = πd²/4): halving the diameter makes the area 4 times smaller and the speed 4 times larger. That is why water squirts further when you cover the end of a hose with your thumb.
Water flows at 1 m/s in a pipe of diameter 4 cm, which then narrows to 2 cm. What is the speed in the narrow part?
The Swiss scientist Daniel Bernoulli (1738) wrote the conservation of energy for a flowing fluid. Consider a section of a stream tube: the work done by pressure forces changes the kinetic and potential energy of the fluid. Per unit volume:
p + ρv²/2 + ρgh = const
For a horizontal pipe (h = const):
p₁ + ρv₁²/2 = p₂ + ρv₂²/2
So where the speed is higher, the pressure is lower. This is a “surprising” result: in the narrow part the fluid pushes less on the walls. The reason is that the fluid speeds up as it goes from wide to narrow, and for that the pressure behind must be greater than the pressure ahead.
For a fluid at rest (v = 0) Bernoulli’s equation becomes the hydrostatic formula p = p₀ + ρgh.
A horizontal pipe narrows from 20 cm² to 5 cm²; the water speed in the wide part is 1 m/s. What is the pressure difference between the wide and narrow parts?
A suction pump is 5 m above the water surface and connected to it by a pipe; p₀ = 100 kPa. What is the greatest possible speed of water in the pipe?
The speed of liquid leaving a hole in the side of a wide open tank, at depth h below the surface, follows from Bernoulli’s equation (the pressure is atmospheric at the surface and at the hole, and the surface speed is almost zero):
Torricelli’s formula: v = √(2gh)
This is the same speed as a body falling freely through h. The jet then moves along a parabola, like a body thrown horizontally. The deeper the hole, the faster the jet but the nearer the floor; for a tank standing on the floor, the jet travels furthest from a hole halfway down the column of liquid.
A hole is 0.8 m below the water level in a tank and 0.45 m above the floor. Find the speed of the jet and where it hits the floor.
Real liquids and gases have internal friction between layers, called viscosity. Honey or oil is far more viscous than water. In a pipe the layer touching the wall is almost at rest and the middle flows fastest, so pumping water over long distances needs a pump doing work all the time.
A body moving through a fluid feels a drag force. At low speeds it is roughly proportional to speed, at high speeds to the square of speed. That is why a falling raindrop or a skydiver stops accelerating after a while: the drag becomes equal to the weight. Streamlined shapes of cars and aircraft reduce drag.