A common parameter that can be used to predict turbulence in...

A common parameter that can be used to predict turbulence in fluid flow is called the Reynolds number. The Reynolds number for fluid flow in a pipe is a dimensionless quantity defined as $$\operatorname{Re}=\frac{\rho v d}{\eta}$$ where $\rho$ is the density of the fluid, $v$ is its speed, $d$ is the inner diameter of the pipe, and $\eta$ is the viscosity of the fluid. The criteria for the type of flow are as follows: $\cdot $ If Re $<2300$, the flow is laminar. $\cdot $ If $2300<\mathrm{Re}<4000$, the flow is in a transition region between laminar and turbulent. $\cdot $ If $\operatorname{Re}>4000,$ the flow is turbulent. (a) Let's model blood of density $1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and viscosity $3.00 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}$ as a pure liquid, that is, ignore the fact that it contains red blood cells. Suppose it is flowing in a large artery of radius $1.50 \mathrm{~cm}$ with a speed of $0.0670 \mathrm{~m} / \mathrm{s}$ Show that the flow is laminar. (b) Imagine that the artery ends in a single capillary so that the radius of the artery reduces to a much smaller value. What is the radius of the capillary that would cause the flow to become turbulent? (c) Actual capillaries have radii of about $5-10$ micrometers, much smaller than the value in part (b). Why doesn't the flow in actual capillaries become turbulent?

A common parameter that can be used to predict turbulence in fluid flow is called the Reynolds number. The Reynolds number for fluid flow in a pipe is a dimensionless quantity defined as $$\operatorname{Re}=\frac{\rho v d}{\eta}$$ where $\rho$ is the density of the fluid, $v$ is its speed, $d$ is the inner diameter of the pipe, and $\eta$ is the viscosity of the fluid. The criteria for the type of flow are as follows:
$\cdot $ If Re $<2300$, the flow is laminar.
$\cdot $ If $2300<\mathrm{Re}<4000$, the flow is in a transition region between laminar and turbulent.
$\cdot $ If $\operatorname{Re}>4000,$ the flow is turbulent.
(a) Let's model blood of density $1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and viscosity $3.00 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}$ as a pure liquid, that is, ignore the fact that it contains red blood cells. Suppose it is flowing in a large artery of radius $1.50 \mathrm{~cm}$ with a speed of $0.0670 \mathrm{~m} / \mathrm{s}$ Show that the flow is laminar.
(b) Imagine that the artery ends in a single capillary so that the radius of the artery reduces to a much smaller value. What is the radius of the capillary that would cause the flow to become turbulent?
(c) Actual capillaries have radii of about $5-10$ micrometers, much smaller than the value in part (b). Why doesn't the flow in actual capillaries become turbulent?

Key Concepts

Scaling effects in fluid flow

Scaling effects refer to how changes in the dimensions of a flow system, such as the transition from a large artery to a much smaller capillary, impact the fluid dynamics. A reduction in the characteristic length scale, such as the vessel's radius, can lower the Reynolds number, which is crucial in determining whether the flow remains laminar or transitions to turbulent behavior.

Viscosity

Viscosity is a measure of a fluid's resistance to deformation and flow. It plays a key role in damping fluid motion and is responsible for the viscous forces in a flow. Higher viscosity tends to suppress turbulent fluctuations, promoting laminar flow, while lower viscosity can allow inertial forces to dominate, potentially leading to turbulence.

Characteristic length scale

The characteristic length scale, such as the inner diameter of a pipe or blood vessel, is a critical parameter in fluid dynamics that influences the Reynolds number. It provides a measure of the size of the system or geometry and directly affects the balance between inertial and viscous effects in the flow.

Flow regimes (laminar, transitional, turbulent)

Flow regimes describe the state of fluid motion. In laminar flow, the fluid flows in parallel layers with minimal mixing, whereas turbulent flow is characterized by chaotic and irregular fluctuations. The transitional regime marks the intermediate phase where the flow shows features of both laminar and turbulent behavior, typically determined by specific Reynolds number thresholds.

Reynolds number

The Reynolds number is a dimensionless parameter defined as the ratio of inertial forces to viscous forces in a fluid flow. It is calculated using quantities like fluid density, velocity, a characteristic length (such as the diameter of a pipe), and viscosity. This number is central in predicting whether a fluid flow will be laminar, transitional, or turbulent.

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