If pressure is increased while vessel radius remains constant, how does flow rate respond?

If pressure is increased while vessel radius remains constant, how does flow rate respond?

• A. Flow rate remains constant regardless of pressure.
• B. Flow rate decreases as pressure increases.
• C. Flow rate increases with the square of pressure.
• D. Flow rate increases linearly with pressure.

Answer: (D)

Driving flow in a long, straight vessel is governed by Poiseuille’s law: the flow rate is proportional to the pressure difference across the vessel and to the radius raised to the fourth power, while it’s inversely proportional to viscosity and vessel length. If you hold the radius constant, the ratio simplifies to flow rate ∝ pressure difference. That means increasing the pressure difference will increase the flow rate in direct, linear proportion. Doubling the pressure difference doubles the flow rate, assuming viscosity and length stay the same. So the best statement is that flow rate increases linearly with pressure. The other ideas—flow constant regardless of pressure, flow increasing with the square of pressure, or flow decreasing as pressure rises—don’t fit the direct, proportional relationship described by the governing equation when the radius is fixed.

Driving flow in a long, straight vessel is governed by Poiseuille’s law: the flow rate is proportional to the pressure difference across the vessel and to the radius raised to the fourth power, while it’s inversely proportional to viscosity and vessel length. If you hold the radius constant, the ratio simplifies to flow rate ∝ pressure difference. That means increasing the pressure difference will increase the flow rate in direct, linear proportion. Doubling the pressure difference doubles the flow rate, assuming viscosity and length stay the same.

So the best statement is that flow rate increases linearly with pressure. The other ideas—flow constant regardless of pressure, flow increasing with the square of pressure, or flow decreasing as pressure rises—don’t fit the direct, proportional relationship described by the governing equation when the radius is fixed.