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Poiseuillé's law describes the velocities of fluids flowing in a tube-for example, the flow of blood in a vein. (See Figure 5.104.) This law applies when the velocities are not too large-more specifically, when the flow has no turbulence. In this case the flow is laminar, which means that the paths of the flow are all parallel to the tube walls. The law states that $$ v=k\left(R^{2}-r^{2}\right), $$ where \(v\) is the velocity, \(k\) is a constant (which depends on the fluid, the tube, and the units used for measurement), \(R\) is the radius of the tube, and \(r\) is the distance from the centerline of the tube. Since \(k\) and \(R\) are fixed for any application, \(v\) is a function of \(r\) alone, and the formula gives the velocity at a point of distance \(r\) from the centerline of the tube. a. What is \(r\) for a point along the walls of the tube? What is the velocity of the fluid along the walls of the tube? b. Where in the tube does the fluid flow most rapidly? c. Choose numbers for \(k\) and \(R\) and make a graph of \(v\) as a function of \(r\). Be sure that the horizontal span for \(r\) goes from 0 to \(R\). d. Describe your graph from part c. e. Explain why you needed to use a horizontal span from 0 to \(R\) in order to describe the flow throughout the tube.

Step by step solution

01

Understanding the Problem

Poiseuillé's law describes the velocity of fluid flow in a tube. The formula is given by \( v = k(R^2 - r^2) \), where \( v \) is the velocity, \( k \) is a constant, \( R \) is the radius of the tube, and \( r \) is the distance from the centerline. We need to analyze this equation to solve various parts of the question.

02

Part a - Velocity at the Tube Walls

To find \( r \) at the walls of the tube, note that the walls are at the radius: \( r = R \). Substitute in the formula: \( v = k(R^2 - R^2) = k(0) = 0 \). Thus, the velocity at the wall is zero.

03

Part b - Maximum Flow Velocity

The fluid flows most rapidly where \( v \) is maximum, which occurs when the term \( R^2 - r^2 \) is maximized. This occurs at \( r = 0 \). Substitute \( r = 0 \) into the equation to find \( v = kR^2 \).

04

Part c - Graphing v(r)

Choose values for \( k \) and \( R \), for example, \( k = 1 \) and \( R = 5 \). The equation becomes \( v = (25 - r^2) \). Plot \( v \) as a function of \( r \) from \( r = 0 \) to \( r = 5 \). The graph is a downward-facing parabola.

05

Part d - Describing the Graph

The graph is a parabola with its peak at \( r = 0 \), where the velocity is maximum, and it decreases to zero at \( r = 5 \), along the tube walls.

06

Part e - Horizontal Span Explanation

The horizontal span needs to be from \( 0 \) to \( R \) because \( r \) is the radial distance from the centerline of the tube and must be less than or equal to the radius, \( R \). It represents the entire range where the fluid flows within the tube.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fluid Dynamics

Fluid dynamics is the branch of physics that concerns the behavior of liquids and gases in motion. It explores how fluids move and how forces affect them. This field helps us understand complex phenomena like weather patterns, ocean currents, and blood flow in our veins.

In fluid dynamics, various principles and equations are used to predict and describe the flow characteristics of fluids. These include the continuity equation and the Navier-Stokes equations, which are fundamental to understanding different types of fluid flow, including laminar and turbulent flows. In the context of tubes or pipes, Poiseuillé's Law becomes particularly important as it defines the velocity profile of a fluid moving through a cylindrical pipe under laminar conditions.

Velocity of Fluids

The velocity of a fluid describes how fast the particles of the fluid are moving. This can vary depending on different factors, such as the fluid's density, the pressure gradient, and the nature of the flow - whether it is laminar or turbulent.

According to Poiseuillé's Law, when a fluid moves through a tube, its velocity ( v ) is described by the equation v = k(R^2 - r^2) . Here, R is the tube's radius, k is a constant related to fluid and tube properties, and r is the distance from the tube's center. This equation indicates that velocity is highest at the center ( r = 0 ) and decreases towards the walls ( r = R ), where it becomes zero.

In real-world applications, understanding fluid velocity is crucial for systems like pipelines and cardiovascular networks, as it affects both efficiency and resistance encountered by the moving fluid.

Laminar Flow

Laminar flow is a flow regime characterized by smooth and orderly fluid motion, with layers of fluid sliding past each other in parallel paths. This is opposed to turbulent flow, where the fluid undergoes chaotic mixing and irregularities.

Poiseuillé's Law assumes laminar flow, where there is no turbulence. The conditions for laminar flow typically involve lower velocities and certain geometrical conditions of the flow path, such as uniform pipe diameter. Under these circumstances, the fluid motion is predictable and described neatly by mathematical equations.

Laminar flow is important in both engineering and natural systems. In human-made systems, achieving laminar flow can reduce energy consumption and wear on system components. In nature, examples include the blood flow in small vessels and the slow movement of lava.

Mathematical Modeling

Mathematical modeling involves using mathematical equations and representations to simulate and analyze real-world systems.

In fluid dynamics, mathematical modeling allows scientists and engineers to predict fluid behavior in various scenarios using differential equations that govern flow properties. Poiseuillé's Law is an example of mathematical modeling in action, providing a specific solution for the laminar flow of fluids in tubes.

Mathematical models help in designing systems that manage fluid flow efficiently. They are especially useful in fields like aerodynamics, meteorology, and engineering, where predicting the behavior of fluids under different conditions is essential for design, safety, and analysis.