Problem 84 - Physics

Question

Poiseuille's law remains valid as long as the fluid flow is laminar. For sufficiently high speed, however, the flow becomes turbulent, even if the fluid is moving through a smooth pipe with no restrictions. It is found experimentally that the flow is laminar as long as the Reynolds number Re is less than about 2000: \(\mathrm{Re}=2 \bar{v} \rho R / \eta .\) Here \(\bar{v}, \rho,\) and \(\eta\) are, respectively, the average speed, density, and viscosity of the fluid, and \(R\) is the radius of the pipe. Calculate the highest average speed that blood \(\left(\rho=1060 \mathrm{~kg} / \mathrm{m}^{3}, \eta=4.0 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}\right)\) could have and still remain in laminar flow when it flows through the aorta \(\left(R=8.0 \times 10^{-3} \mathrm{~m}\right)\).

Step-by-Step Solution

Answer

The maximum average speed is approximately 0.47 m/s for the flow to remain laminar.

Step 1: Understand Poiseuille's Law and the Reynolds Number

Poiseuille's equation is not directly needed for this particular calculation, as it deals with flow rate through a pipe based on viscosity, length, and pressure difference. However, its context alerts us that laminar vs. turbulent flow pertains to factors like flow rate and pressure. The turndom from laminar to turbulent is determined by the Reynolds number (Re). The Reynolds number is given by the formula \( \operatorname{Re} = \frac{2 \bar{v} \rho R}{\eta} \), which relates the speed of a fluid to its density, pipe radius, and viscosity.

Step 2: Rearrange the Reynolds Number Formula

To find the maximum average speed \( \bar{v} \) where the flow remains laminar, we need to rearrange the formula for the Reynolds number to solve for \( \bar{v} \). Thus, \( \bar{v} = \frac{\eta \times \operatorname{Re}}{2 \rho R} \).

Step 3: Substitute Values into the Rearranged Equation

Using \( \operatorname{Re} = 2000 \) (the threshold for laminar flow), \( \rho = 1060 \ \mathrm{kg/m^3} \), \( \eta = 4.0 \times 10^{-3} \ \mathrm{Pa \cdot s} \), and \( R = 8.0 \times 10^{-3} \ \mathrm{m} \), substitute into the equation: \[ \bar{v} = \frac{4.0 \times 10^{-3} \times 2000}{2 \times 1060 \times 8.0 \times 10^{-3}} \].

Step 4: Calculate the Maximum Average Speed \( \bar{v} \)

Calculate the expression: \( \bar{v} = \frac{4.0 \times 10^{-3} \times 2000}{2 \times 1060 \times 8.0 \times 10^{-3}} \). This simplifies to: \[ \bar{v} = \frac{8.0}{16.96} \], which results in \( \bar{v} \approx 0.47 \ \mathrm{m/s} \).

Key Concepts

Laminar Flow

In fluid dynamics, laminar flow occurs when a fluid flows in parallel layers with no disruption between them.
This type of flow is smooth, orderly, and described as streamlined.
Each layer in laminar flow moves over the adjacent ones without mixing, creating a predictable pattern. Some key features of laminar flow include:

• Stable and regular flow pattern
• Dominated by viscous forces, meaning viscosity plays a key role in maintaining flow stability
• Low velocity compared to turbulent flows

For example, blood flowing through capillaries behaves in a laminar manner.
Reynolds number helps differentiate between laminar and turbulent flows. For laminar flow, the Reynolds number is typically less than 2000.
This threshold indicates the smooth manner in which fluid particles slide past each other without exerting too much chaotic influence on one another.
Understanding laminar flow is crucial in designing systems like pipelines and air conditioning ducts where stable fluid behavior is desired.

Turbulent Flow

Turbulent flow is characterized by chaotic property changes and occurs at high fluid velocities.
Unlike laminar flow, where fluid motion is smooth, turbulent flow involves irregular movements and vortices. This type of flow experiences increased pressure drops due to:

• High velocity and chaotic eddies
• Irregular mixing of adjacent layers
• Significant energy dissipation

The transition from laminar to turbulent flow involves crossing a critical Reynolds number, usually around 2000 for internal flows, such as those within pipes.
In engineering and scientific applications, managing turbulent flows can be challenging because predicting them requires complex calculations and models.
However, controlling turbulence is important in applications like aircraft design to minimize energy losses and enhance efficiency in transport systems.
Despite its unpredictability, turbulent flow is often more efficient at mixing substances, which can be advantageous in contexts like combustion engines.

Fluid Dynamics

Fluid dynamics is the study of how fluids (liquids and gases) move and interact with various forces.
This branch of physics involves understanding both theoretical and applied aspects to control or predict how fluids behave in different scenarios. Fluid dynamics covers concepts such as:

• Pressure and how it affects fluid flow
• Velocity fields and streamlines
• Effects of viscosity and density
• The impact of forces like gravity and friction

Fluid dynamics principles are applied in numerous fields, including meteorology, oceanography, and engineering.
For example, understanding fluid dynamics is essential for constructing efficient systems in hydrodynamics and aerodynamics.
Key equations, such as the Navier-Stokes equations, help describe how velocity, pressure, temperature, and density of a fluid are related.
Overall, fluid dynamics provides essential tools and insights for solving complex fluid-related problems across various industries.