Fluid mechanics

The Bernoulli equation

Energy conservation for steady flow: wherever a fluid moves faster, its pressure is lower.

The Bernoulli equation is energy conservation applied to the steady, inviscid, incompressible flow of a fluid along a streamline. It relates pressure, velocity, and height of the fluid at different points on the same streamline.

p + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant along a streamline}

$p$ is static pressure, $\tfrac{1}{2}\rho v^2$ can be thought of as 'dynamic pressure' (related to the moving fluid's kinetic energy), and $\rho g h$ is 'gravitational pressure' (related to potential energy from height $h$ ). The sum of these three terms is constant along a streamline, if viscous friction is neglected.

Narrow the Venturi tube's throat and watch pressure there drop the instant velocity increases — exactly as the Bernoulli equation predicts.

The link with the continuity equation

The Bernoulli equation usually goes hand in hand with the continuity equation $A_1 v_1 = A_2 v_2$ (flow-rate conservation, a consequence of mass conservation for an incompressible fluid): when the pipe's cross-section $A$ shrinks, velocity $v$ must increase to keep the flow rate constant. Combining both for a horizontal pipe ( $h$ constant):

p_1 + \tfrac{1}{2}\rho v_1^2 = p_2 + \tfrac{1}{2}\rho v_2^2 \quad\Rightarrow\quad v_2 > v_1 \implies p_2 < p_1

This is why a smaller cross-section and higher velocity mean lower pressure — the effect used to measure flow rate (the Venturi tube), to generate lift on an airplane wing, and to explain why a car's curtain gets sucked outward at high speed.

Definition: Streamline

A streamline is a curve whose tangent at every point matches the fluid velocity direction there, at a given instant. In steady flow (unchanging over time), streamlines coincide with the actual paths traced by fluid particles.

Example: Pressure at a Venturi throat

Water flows through a horizontal pipe whose inlet area is twice the throat area ( $A_1 = 2A_2$ ). If the inlet pressure is $p_1 = 200$ kPa and inlet velocity $v_1 = 2$ m/s, find the pressure at the throat (ρ = 1000 kg/m³).

Solution

From continuity: $v_2 = v_1 A_1/A_2 = 2 \times 2 = 4$ m/s. From Bernoulli: $p_2 = p_1 + \tfrac12\rho(v_1^2 - v_2^2) = 200000 + 0.5 \times 1000 \times (4 - 16) = 200000 - 6000 = 194{,}000$ Pa.

Quick check

Example: Water speed leaving a tank

A small outlet lies $h=1.25,mathrm m$ below the free surface. For a large tank open to the atmosphere, the surface speed is negligible, and pressure at both the surface and outlet is atmospheric. Bernoulli then gives $v=sqrt{2gh}=sqrt{2(9.8)(1.25)}approx4.95,mathrm{m/s}$ . This is an ideal estimate: viscosity, contraction at the opening, and finite tank size reduce the actual speed.

In a horizontal pipe, if the cross-section shrinks, the fluid pressure there will:

The Bernoulli equation is fundamentally an application of which conservation law?

References

Daniel Bernoulli (1738). Hydrodynamica
Kundu, Cohen, Dowling (2015). Fluid Mechanics