Problem 71 - College Physics

Question

The aorta in humans has a diameter of about \(2.0 \mathrm{~cm}\), and at certain times the blood speed through it is about \(55 \mathrm{~cm} / \mathrm{s}\). Is the blood flow turbulent? The density of whole blood is \(1050 \mathrm{~kg} / \mathrm{m}^{3}\), and its coefficient of viscosity is \(2.7 \times 10^{-3} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\).

Step-by-Step Solution

Answer

No, the blood flow is not turbulent as the Reynolds number calculated is less than 2000.

Step 1: Identify given quantities

The given quantities from the problem are: diameter \(d = 2.0 \,cm = 0.02 \,m\), velocity \(v = 55 \,cm/s = 0.55 \,m/s\), density of blood \(\rho = 1050 \,kg/m^3\), and coefficient of viscosity \(\mu = 2.7 \times 10^{-3} \,N.s/m^2\).

Step 2: Calculate Reynolds Number

Substitute the given values into the formula for Reynolds number. \[ Re = \frac{{\rho v d}}{{\mu}} = \frac{{(1050 \,kg/m^3)(0.55 \,m/s)(0.02 \,m)}}{{2.7 \times 10^{-3} \,N.s/m^2}}\]

Step 3: Evaluate the Result

The calculated Reynolds number then helps to determine if the flow is turbulent. If the Reynolds number \( Re > 2000 \), then the flow is considered to be turbulent. If \( Re < 2000 \), then the flow is considered to be laminar, which is a smooth, orderly flow.

Key Concepts

Turbulent Flow

In fluid dynamics, turbulent flow refers to a type of fluid movement characterized by chaotic changes in pressure and flow velocity. Unlike laminar flow, where fluid particles move in organized layers, turbulent flow is unpredictable. This unpredictable nature often leads to swirling eddies and vortices. Turbulent flow is common in various practical situations, from ocean currents to blood flow in large arteries like the aorta.
Some important aspects of turbulent flow include:

• High Reynolds Number: Turbulent flow is typically indicated by a high Reynolds number, generally above 4000. However, in biological systems like blood flow through arteries, turbulence can begin at lower values.
• Chaotic Motion: The movement is highly disorderly and irregular, which can affect how energy is dissipated in the fluid.
• Increased Mixing: The chaotic nature of turbulent flow enhances mixing, which can lead to improved heat and mass transfer.

Understanding turbulent flow is crucial in engineering applications for designing efficient systems that can handle such conditions.
When we calculated the Reynolds number for blood flow in the aorta, we determined whether the flow could be turbulent, providing insights into cardiovascular health and function.

Laminar Flow

Laminar flow is the opposite of turbulent flow. It describes a fluid flow regime where the movement is smooth and orderly. During laminar flow, fluid particles move in parallel layers or "laminae," without disruption between the layers. This type of flow is most commonly found in small or viscous fluid systems where the flow velocity is low.
Key features of laminar flow:

• Low Reynolds Number: Laminar flow occurs when the Reynolds number is less than 2000. This is typical in situations where high viscosity or small diameters limit flow velocity.
• Predictable Flow Patterns: The orderly movement means that flow patterns are quite predictable and easier to analyze mathematically.
• Minimal Energy Loss: Due to the lack of mixing and eddies, laminar flow generally experiences less energy loss compared to turbulent flow.

In the context of blood flow, laminar flow is often favorable as it indicates healthy and non-disruptive circulation through vessels. By comparing the Reynolds number against known thresholds, we can determine whether blood flow in the aorta is in a laminar or turbulent state.

Viscosity in Fluids

Viscosity is a fundamental property of fluids that quantifies their resistance to flow. High viscosity means more friction within the fluid, leading to slower movement and more energy required to move the fluid. Viscosity is crucial in determining the type of flow, either laminar or turbulent.
Here are some important points about viscosity:

• Definition: Viscosity is defined as a measure of a fluid's internal resistance to flow and can be thought of as "thickness" or "stickiness." Higher viscosity implies a thicker fluid.
• Measurement: It is measured in terms of the coefficient of viscosity, often denoted as \( \mu \), with units of \( \text{N} \cdot \text{s/m}^2 \) or \( \text{Pa} \cdot \text{s} \).
• Influence on Flow: High viscosity fluids like honey flow slowly and are more likely to exhibit laminar flow. In contrast, low viscosity fluids like water flow more quickly and may transition to turbulence more easily.

In the context of blood flow, viscosity is crucial because it affects how easily blood can circulate through vessels like the aorta. By knowing the viscosity, one can calculate the Reynolds number to understand the nature of the flow.