Continuity and Bernoulli

What a fluid loses in cross-section it pays back in speed, and speed is bought with pressure.

Continuity

For an incompressible fluid, the volume flow rate Q = Av is conserved: what passes one cross-section each second must pass every other. Squeeze the pipe to half the area and the speed doubles. Track the flow rate in m^3/s, not the speed alone.

A_1 v_1 = A_2 v_2

Bernoulli's equation

Along a streamline in steady, incompressible, nonviscous flow, P + (1/2)rho*v^2 + rho*g*y stays constant: pressure, speed, and height trade off. Where flow is faster, pressure is lower — that is the lift on a wing and the draw of a Venturi nozzle. The equation fails when viscosity or turbulence are significant.

P + \tfrac{1}{2}\rho v^2 + \rho g y = \text{const}

Worked example

A tank drains through a hole 5.0 m below the surface, the hole small enough that the top surface barely moves. Bernoulli between surface and exit gives v = sqrt(2gh) = sqrt(2*9.8*5.0) = sqrt(98), about 9.9 m/s — Torricelli's result, the same speed the water would gain free-falling 5.0 m. The atmospheric pressures on surface and jet cancel.

v = \sqrt{2gh} = \sqrt{(2)(9.8)(5.0)} \approx 9.9\ \text{m/s}

Flashcard

Q = A_1 v_1 = A_2 v_2

Continuity: volume flow rate is conserved

Incompressible fluid cannot pile up or vanish between sections, so area times speed is the same everywhere. Narrow the pipe and the fluid must speed up proportionally.

Check your understanding

Water flows at 1.5 m/s through a pipe of diameter 6.0 cm, then enters a section of diameter 3.0 cm. What is its new speed?

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