Bernoulli’s Effect and Blood Physics: MDCAT Physics notes

Bernoulli's equation

For an ideal fluid in steady flow, the sum of pressure, kinetic energy per unit volume and potential energy per unit volume is constant along a streamline:

$$P + \tfrac12\rho v^2 + \rho g h = \text{constant}$$

It follows from the law of conservation of energy. For horizontal flow ($h$ constant), $P + \tfrac12\rho v^2$ is constant, so where the speed is high, the pressure is low. This is Bernoulli's effect.

Applications

• Aeroplane lift: the wing (aerofoil) is curved so that air moves faster over the top surface than underneath. The pressure above the wing is therefore lower than below, and the pressure difference gives an upward lift.
• Venturi meter: a narrow throat in a pipe increases speed and lowers pressure; the pressure difference measures the flow speed. Venturi relation: $P_1 - P_2 = \tfrac12 \rho v_2^2$ (when $v_1$ is negligible).
• Atomiser / spray gun and filter pump: a fast jet of air or water creates low pressure that draws liquid or air in.
• Chimney draught: wind blowing across the top lowers the pressure there, helping smoke rise.
• Swing of a spinning ball: the side where air moves faster has lower pressure, so the ball curves towards it.

Blood physics

Blood pressure

Blood pressure is measured with a sphygmomanometer. The cuff is wrapped round the upper arm, at about the level of the heart, and inflated until it stops the flow in the artery. As the pressure is slowly released, blood begins to squirt through the partly closed artery. A normal adult reading is about 120 torr (systolic) over 80 torr (diastolic).

Laminar and turbulent flow

Normally blood flows smoothly in laminar (streamline) layers, which is silent. If the speed rises above a critical value, or the vessel has obstructions and rough walls, the flow becomes turbulent, with eddies that produce sound.

• When the cuff partly squeezes the artery, blood rushes through the narrow gap fast and turbulently; this noise, heard through a stethoscope, marks the systolic reading. When the sounds disappear, flow is laminar again (diastolic).
• In a clogged artery (plaque of fatty deposits, arteriosclerosis) the passage is narrowed and roughened. By continuity the blood speeds up, flow becomes turbulent, and by Bernoulli's principle the pressure in the narrowed region drops. The abnormal sound helps doctors detect the blockage.

Laminar flow favoured by	Turbulent flow favoured by
Low speed	High speed (above critical)
Smooth, unobstructed vessel	Narrowed, obstructed or rough vessel
Higher viscosity	Lower viscosity

Key formulas

• $P + \tfrac12\rho v^2 + \rho g h = \text{constant}$
• Horizontal flow: $P_1 + \tfrac12\rho v_1^2 = P_2 + \tfrac12\rho v_2^2$
• Venturi: $P_1 - P_2 = \tfrac12\rho v_2^2$

Common MDCAT traps

• Faster air over the wing means lower pressure on top, not higher and not zero.
• Turbulence in blood comes from a clogged vessel and high speed, not from low velocity or higher viscosity.
• Bernoulli's equation expresses conservation of energy; continuity expresses conservation of mass.
• Blood pressure is measured at the arm because it is level with the heart, so the $\rho g h$ term does not distort the reading.

Quick revision

• High speed, low pressure.
• Lift arises from the pressure difference across the wing.
• Normal blood pressure is about 120/80 torr.
• Laminar flow is silent; turbulent flow is noisy.
• A narrowed artery raises blood speed and causes turbulence.