(II) Poiseuille's equation does not hold if the flow...

(II) Poiseuille's equation does not hold if the flow velocity is high enough that turbulence sets in. The onset of turbulence occurs when the Reynolds number, Re, exceeds approximately 2000 . Re is defined as $$R e=\frac{2 \overline{v} r \rho}{\eta}.$$ where $\overline{v}$ is the average speed of the fluid, $\rho$ is its density, $\eta$ is its viscosity, and $r$ is the radius of the tube in which the fluid is flowing. $(a)$ Determine if blood flow through the aorta is laminar or turbulent when the average speed of blood in the aorta $(r=1.2 \mathrm{cm})$ during the resting part of the heart's cycle is about 40 $\mathrm{cm} / \mathrm{s}$ (b) During exercise, the blood-flow speed approximately doubles. Calculate the Reynolds number in this case, and determine if the flow is laminar or turbulent.

(II) Poiseuille's equation does not hold if the flow velocity is high enough that turbulence sets in. The onset of turbulence occurs when the Reynolds number, Re, exceeds approximately 2000 . Re is defined as
$$R e=\frac{2 \overline{v} r \rho}{\eta}.$$
where $\overline{v}$ is the average speed of the fluid, $\rho$ is its density, $\eta$ is its viscosity, and $r$ is the radius of the tube in which the fluid is flowing. $(a)$ Determine if blood flow through the aorta is laminar or turbulent when the average speed of blood in the aorta $(r=1.2 \mathrm{cm})$ during the resting part of the heart's cycle is about 40 $\mathrm{cm} / \mathrm{s}$ (b) During exercise, the blood-flow speed approximately doubles. Calculate the Reynolds number in this case, and determine if the flow is laminar or turbulent.

Key Concepts

Turbulent Flow

Turbulent flow is characterized by chaotic, irregular fluid motion with eddies and vortices. This regime occurs when inertial forces outweigh viscous forces, typically at high Reynolds numbers, leading to enhanced mixing, fluctuating velocities, and a departure from the simple behavior described by Poiseuille's law.

Laminar Flow

Laminar flow is a regime of fluid motion where the fluid moves in smooth, parallel layers with minimal mixing between them. It is characterized by streamlined motion and stable flow patterns, usually occurring at low Reynolds numbers, where viscous forces dominate and suppress disturbances.

Poiseuille's Equation

Poiseuille's equation describes the pressure-driven flow of viscous fluids through a cylindrical pipe under laminar flow conditions. It relates the volumetric flow rate to the pressure difference, the fluid’s viscosity, the length of the pipe, and the radius of the tube, and assumes a parabolic velocity profile typical of laminar flow.

Reynolds Number

The Reynolds number is a key dimensionless parameter in fluid dynamics that quantifies the relative importance of inertial forces compared to viscous forces in a flowing fluid. It is used to predict the transition between laminar and turbulent flow regimes, with lower values indicating laminar flow and higher values indicating potential turbulence.

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