Physics
Fluid dynamics calculator: Bernoulli, Torricelli, Venturi
Pick the relation and clear the quantity you are after: the calculator inverts the formula for you. The whole chapter comes out of two conservation laws — mass, which gives continuity, and energy, which gives Bernoulli.
Two conservation laws, and everything else
School fluid dynamics comes down to two statements. The first is that mass does not vanish: in a full pipe the same flow rate crosses every section, so where the section narrows the fluid runs faster. The second is that energy does not vanish either: along a streamline the sum of the pressure, the kinetic energy per unit volume and the height term stays the same. From this follows the part that surprises everybody — where the fluid is faster the pressure is lower, not higher. A fast jet of water between two sheets of paper pulls them together instead of pushing them apart, and a wing works for the same reason.
Torricelli and Venturi are not new laws: they are Bernoulli with one term switched off. At a hole in a tank the pressure is atmospheric inside and out, leaving speed and height, and out comes v = √(2gh), the speed of a free fall. In a horizontal pipe the height is the same everywhere, leaving pressure and speed, and out comes the pressure drop at the throat that makes a Venturi meter work. Recognising which term goes to zero is half the work in an exam problem.
The rest of the chapter is there to remind us that the ideal fluid does not exist. Poiseuille puts viscosity back and shows that the flow depends on the fourth power of the radius: a slightly narrower tube resists far more than its width suggests, which matters in plumbing as much as in physiology. The Reynolds number, finally, marks where the tidy formulas run out: past the threshold the motion turns turbulent and none of the earlier relations describes what actually happens.
Common mistakes
- Confusing flow rate with speed. The flow rate is the same all along the pipe, the speed is not: it is precisely because the flow rate is conserved that the speed changes from section to section.
- Thinking there is more pressure where the fluid is faster. It is the other way round: the extra kinetic energy is taken out of the pressure, which therefore falls.
- Applying Bernoulli to a viscous fluid over long distances. In the form used here there is no friction term at all: down a long or narrow tube the pressure falls even at constant speed, and it is Poiseuille that says by how much.
- Forgetting that the radius enters Poiseuille's law to the fourth power. Halving it does not halve the flow: it cuts it to a sixteenth.
- Using the laminar formulas past the transition. Once the Reynolds number runs into the thousands the motion is turbulent and the flow is no longer proportional to the pressure difference.
Frequently asked questions
Why does the fluid speed up where the pipe narrows?
Because nothing accumulates inside: the same flow rate has to cross every section. Halve the area and the speed doubles, because the product A·v stays constant. That is the continuity equation, and it is nothing but conservation of mass.
Why is the pressure lower where the speed is higher?
Because along a streamline the sum p + ½ρv² + ρgh is constant. If the kinetic energy of the fluid goes up, something has to come down, and at constant height that is the pressure. Accelerating the fluid needs a net push, and that can only come from higher pressure behind than in front.
What does Torricelli's law say?
That water leaves a hole at the speed a body dropped from the free surface down to that level would have: v = √(2gh). It does not depend on the density of the liquid or on any mass, exactly as in free fall.
Why does a narrowed artery cut the flow so much?
Because of the fourth power in Poiseuille's law: Q goes as r⁴. A radius reduced by 20% lets through about 41% of the flow, and a radius halved only a sixteenth. That is why a modest stenosis has such disproportionate effects.
What is the Reynolds number for?
For telling whether the motion is laminar or turbulent. It is the ratio of inertial to viscous forces: below about 2000 the layers of fluid slide past one another in order, above 4000 they mix. The laminar-flow formulas, Poiseuille's first among them, only hold below the transition.
How this calculation works
Volumetric flow rate: Q = A·v. Continuity equation for an incompressible fluid: A₁·v₁ = A₂·v₂, that is, Q constant along the pipe. Bernoulli's equation along a streamline, for an ideal fluid in steady flow: p₁ + ½ρv₁² + ρg·h₁ = p₂ + ½ρv₂² + ρg·h₂. Torricelli's law, the special case with atmospheric pressure on both surfaces and a wide tank: v = √(2·g·h). Venturi effect, the special case with h₁ = h₂: Δp = p₁ − p₂ = ½ρ·(v₂² − v₁²). Hagen–Poiseuille law for the laminar flow of a viscous fluid down a cylindrical tube: Q = π·Δp·r⁴/(8·η·L); the hydraulic resistance is R = Δp/Q = 8ηL/(πr⁴) and the mean speed is Q/(πr²). Reynolds number: Re = ρ·v·D/η, dimensionless; the critical speed reported is the one at which Re reaches 2300. Every calculation uses g = 9.81 m/s².