(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 $(r=0.80 \mathrm{cm})$ during the resting part of the heart's cycle is about 35 $\mathrm{cm} / \mathrm{s}$ . (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 $(r=0.80 \mathrm{cm})$ during the resting part of the heart's
cycle is about 35 $\mathrm{cm} / \mathrm{s}$ . (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

Poiseuille's Equation

Poiseuille's equation describes the volumetric flow rate of a viscous, incompressible fluid through a long, cylindrical pipe under laminar flow conditions. This equation assumes that the flow is steady and laminar; it breaks down when turbulence is present, as turbulent flow introduces complex behaviors not accounted for by the formula.

Critical Reynolds Number

The critical Reynolds number is the threshold value at which the fluid flow transitions from laminar to turbulent. In many practical situations, including flow through cylindrical tubes, a Reynolds number above this critical value—often around 2000—indicates the onset of turbulence.

Turbulent Flow

Turbulent flow refers to a state where the fluid experiences chaotic fluctuations and irregular mixing. This flow regime typically occurs at high Reynolds numbers when inertial forces overcome viscous forces, leading to eddies and vortices that cause unpredictable flow behavior.

Laminar Flow

Laminar flow is characterized by smooth, orderly layers of fluid where parallel streamlines are maintained. It typically occurs at low Reynolds numbers, where viscous forces dominate and the fluid moves predictably without significant mixing between layers.

Reynolds Number

The Reynolds number is a dimensionless quantity in fluid mechanics that helps predict flow regimes by comparing inertial forces to viscous forces. It is used to determine whether the flow will be laminar or turbulent and plays a crucial role in analyzing the behavior of fluids in different conditions.

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