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Chapter 9: Problem 9

Approximately how fast can the wind blow past a 0.25 in.-diameter twig if viscous effects are to be of importance throughout the entire flow field (i.e., Re \(<1\) )? Explain. Repeat for a 0.004 -in.-diameter hair and a 6 -ft- diameter smokestack.

Short Answer

Expert verified
For the twig, the wind speed should be under 2.323 m/s; for the hair, below 145.11 m/s; for the smokestack, under 8.07 x 10鈦烩伓 m/s.

Step by step solution

01

Understand the problem

We need to determine the maximum wind speed for which the flow past different objects (a twig, hair, and smokestack) remains within the laminar regime, where the Reynolds number ( Re ) is less than 1. This means calculating the speed for which viscous effects dominate the flow.
02

Formula for Reynolds Number

The Reynolds number (Re) is calculated using the formula: \[Re = \frac{\rho v D}{\mu}\]where \(\rho\) is the density of air (approx. 1.225 kg/m鲁), \(v\) is the velocity of the air, \(D\) is the characteristic diameter, and \(\mu\) is the dynamic viscosity of air (1.81 x 10鈦烩伒 kg/m路s).
03

Rearrange for Velocity

Since we need Re < 1, rearrange the equation to solve for velocity (v):\[v < \frac{\mu}{\rho D}\]This inequality will be used for each object to calculate the appropriate velocity.
04

Calculate Velocity for the 0.25 in. Twig

Convert the diameter from inches to meters:\(D = 0.25 \text{ in.} = 0.25 \times 0.0254 \text{ m} = 0.00635 \text{ m}\).Substitute \(D\) into the rearranged formula:\[v < \frac{1.81 \times 10^{-5}}{1.225 \times 0.00635} = 2.323 \text{ m/s}\].
05

Calculate Velocity for the 0.004 in. Hair

Convert the diameter from inches to meters:\(D = 0.004 \text{ in.} = 0.004 \times 0.0254 \text{ m} = 0.0001016 \text{ m}\).Substitute \(D\) into the rearranged formula:\[v < \frac{1.81 \times 10^{-5}}{1.225 \times 0.0001016} = 145.11 \text{ m/s}\].
06

Calculate Velocity for the 6 ft Smokestack

Convert the diameter from feet to meters:\(D = 6 \text{ ft} = 6 \times 0.3048 \text{ m} = 1.8288 \text{ m}\).Substitute \(D\) into the rearranged formula:\[v < \frac{1.81 \times 10^{-5}}{1.225 \times 1.8288} = 8.07 \times 10^{-6} \text{ m/s}\].

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Key Concepts

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

Laminar Flow
When we talk about laminar flow, we are referring to a smooth and consistent movement of fluid. Here, the fluid particles slide past one another in parallel layers. This kind of flow is typically seen in conditions where the fluid moves at low speeds and the flow channels are uncomplicated.
For instance, imagine a thin drizzle of honey flowing straight down off a spoon鈥攚here there are layers of honey moving in a straight path.
This occurs because drag forces between the layers of the fluid are well-balanced, preventing turbulence.
  • Laminar flow is characterized by a Reynolds number less than 2000.
  • Reynolds number (\(Re\)) is a dimensionless quantity that helps us assess the type of flow experienced in a fluid system.
  • In the given exercise, it is mentioned that the flow is laminar when the Reynolds Number is less than 1, focusing on viscous effects.
Fluid Dynamics
Fluid dynamics studies how fluids (liquids and gases) behave when they are in motion.
This area of physics allows us to understand and predict flow patterns and forces in various systems, ranging from air over an airplane wing to water in a river.
In the exercise, fluid dynamics principles help us understand how wind flows past objects of different sizes, like twigs, hairs, and smokestacks.
  • Key parameters in fluid dynamics include velocity, pressure, density, and viscosity.
  • The dynamic viscosity of the fluid plays a crucial role in the resistance the fluid exhibits to flow.
  • The exercise uses fluid dynamics formulas to compute velocities ensuring flow remains laminar.
Viscous Effects
Viscous effects in fluid flows are related to the internal friction present within a fluid due to its viscosity.
Viscosity is essentially the 'thickness' or 'stickiness' of a fluid, influencing its ability to flow smoothly.
When viscous effects are significant, the layers of fluid slide past one another in an orderly fashion with minimal disturbance.
  • In the given exercise, if we keep the Reynolds number below 1, it indicates viscous forces dominate the flow behavior.
  • This allows the flow to remain laminar, as opposed to becoming turbulent.
  • Understanding viscous effects helps in evaluating flow resistance around the objects described in the problem.
Characteristic Diameter
The characteristic diameter in fluid dynamics is a concept used to define a length scale for the flow scenario being considered.
In the flow around an object, it is the main dimension that affects how the fluid moves past the body.
It factors into the formula for calculating the Reynolds number.
  • In cylindrical objects like a pipe or twig, the characteristic diameter is simply the diameter of the object.
  • Different characteristic diameters in the exercise lead to different Reynolds numbers and thus different threshold wind speeds.
  • This is due to wider objects experiencing different flow conditions, leading to variations in force and pressure distribution.
Dynamic Viscosity
Dynamic viscosity is a crucial property in understanding a fluid's resistance to shear or flow.
It determines the internal friction within a fluid moving along surfaces.
This value remains constant for a given fluid at a specific temperature and pressure, serving a pivotal role in computations related to fluid dynamics.
  • Dynamic viscosity is represented by \(\mu\) in fluid dynamics equations, including the Reynolds number formula.
  • Air has a specific dynamic viscosity used in the exercise, specifically 1.81 x 10鈦烩伒 kg/m路s.
  • Higher viscosity indicates a thicker fluid, while lower viscosity signifies a more free-flowing liquid.
  • Knowledge of dynamic viscosity helps determine how quickly and smoothly different fluids can flow past objects.

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Most popular questions from this chapter

A snowflake with a diameter of 0.15 in. is observed to fall through still air with a speed of \(2.5 \mathrm{ft} / \mathrm{s}\). The drag coefficient is assumed to be 1.5 . (a) Estimate the weight of the snowflake. (b) Estimate the number of these snowflakes it would take, when melted, to fill a one-gallon jug with water.

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