Book cover for Fluid Mechanics

Fluid Mechanics

Frank M. White

ISBN #9789385965494

8th Edition

1,418 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Dimensional analysis is a powerful and efficient tool that simplifies the study of complex physical phenomena by reducing the number of variables involved into a set of dimensionless parameters. This simplification, notably achieved through the Buckingham Pi theorem, enhances the ability to compare, scale, and predict behaviors in both experimental and computational fluid dynamics. Key dimensionless groups such as the Reynolds number exemplify how different flows can be analyzed on a common basis, thereby facilitating universal design and scaling laws.

Learning Objectives

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Key Concepts

CONCEPT

DEFINITION

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Example Problems

Example 1

An idealized velocity field is given by the formula $$\mathbf{V}=4 t x \mathbf{i}-2 t^{2} y \mathbf{j}+4 x z \mathbf{k}$$ Is this flow ficld steady or unsteady? Is it two- or threedimensional? At the point $(x, y, z)=(-1,1,0),$ compute (a) the acceleration vector and $(b)$ any unit vector normal to the acceleration.

Example 2

Flow through the converging nozzle in Fig. P4.2 can be approximated by the one-dimensional velocity distribution $$u \approx V_{0}\left(1+\frac{2 x}{L}\right) \quad v \approx 0 \quad w \approx 0$$ (a) Find a general expression for the fluid acceleration in the nozzle. ( $b$ ) For the specific case $V_{0}=10 \mathrm{ft} / \mathrm{s}$ and $L=6$ in, compute the acceleration, in $g$ 's, at the entrance and at the exit.

Example 3

A two-dimensional velocity field is given by $$\mathbf{V}=\left(x^{2}-y^{2}+x\right) \mathbf{i}-(2 x y+y) \mathbf{j}$$ in arbitrary units. At $(x, y)=(1,2),$ compute ( $a$ ) the accelerations $a_{x}$ and $a_{y},(b)$ the velocity component in the direction $\theta=40^{\circ},(c)$ the direction of maximum velocity, and $(d)$ the direction of maximum acceleration.

Example 4

A simple flow model for a two-dimensional converging nozzle is the distribution $$u=U_{0}\left(1+\frac{x}{L}\right) \quad v=-U_{0} \frac{y}{L} \quad w=0$$ (a) Sketch a few streamlines in the region $0 < x / L < 1$ and $0 < y / L < 1,$ using the method of Sec. $1.11 .(b)$ Find expressions for the horizontal and vertical accelerations. (c) Where is the largest resultant acceleration and its numerical value?

Example 5

The velocity field near a stagnation point may be written in the form $$u=\frac{U_{0} x}{L} \quad v=-\frac{U_{0} y}{L} \quad U_{0} \text { and } L \text { are constants }$$ (a) Show that the acceleration vector is purely radial. (b) For the particular case $L=1.5 \mathrm{m},$ if the acceleration at $(x, y)=(1 \mathrm{m}, 1 \mathrm{m})$ is $25 \mathrm{m} / \mathrm{s}^{2},$ what is the value of $U_{0} ?$
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Step-by-Step Explanations

QUESTION

How can you derive dimensionless parameters from a set of physical variables?\nStep-by-step Answer:\nStep 1: Identify all the relevant physical variables affecting the phenomenon (e.g., velocity, density, viscosity, characteristic length).\nStep 2: Express each variable in terms of the fundamental dimensions (typically mass [M], length [L], time [T], and temperature [\u03b8]).\nStep 3: Count the total number (n) of variables and determine the number (k) of independent fundamental dimensions present.\nStep 4: Apply the Buckingham Pi theorem to conclude that the problem can be described completely by (n - k) independent dimensionless groups.\nStep 5: Form combinations of the variables that yield dimensionless products (\u03c0 groups) by ensuring that the overall dimensions cancel out.\nFinal Answer: The resulting set of (n - k) dimensionless groups can be used to represent the behavior of the system in a compact, scale-independent manner.\n\n"

STEP-BY-STEP ANSWER:

Step 1: Identify all the relevant physical variables affecting the phenomenon (e.g., velocity, density, viscosity, characteristic length).\nStep 2: Express each variable in terms of the fundamental dimensions (typically mass [M], length [L], time [T], and temperature [\u03b8]).\nStep 3: Count the total number (n) of variables and determine the number (k) of independent fundamental dimensions present.\nStep 4: Apply the Buckingham Pi theorem to conclude that the problem can be described completely by (n - k) independent dimensionless groups.\nStep 5: Form combinations of the variables that yield dimensionless products (\u03c0 groups) by ensuring that the overall dimensions cancel out.\nFinal Answer: The resulting set of (n - k) dimensionless groups can be used to represent the behavior of the system in a compact, scale-independent manner.\n\n"
Final Answer: The resulting set of (n - k) dimensionless groups can be used to represent the behavior of the system in a compact, scale-independent manner.\n\n"

"- Topic: Deriving Dimensionless Groups using Buckingham Pi Theorem \nQuestion: How can you derive dimensionless parameters from a set of physical variables?\nStep-by-step Answer:\nStep 1: Identify all the relevant physical variables affecting the phenomenon (e.g., velocity, density, viscosity, characteristic length).\nStep 2: Express each variable in terms of the fundamental dimensions (typically mass [M], length [L], time [T], and temperature [\u03b8]).\nStep 3: Count the total number (n) of variables and determine the number (k) of independent fundamental dimensions present.\nStep 4: Apply the Buckingham Pi theorem to conclude that the problem can be described completely by (n - k) independent dimensionless groups.\nStep 5: Form combinations of the variables that yield dimensionless products (\u03c0 groups) by ensuring that the overall dimensions cancel out.\nFinal Answer: The resulting set of (n - k) dimensionless groups can be used to represent the behavior of the system in a compact, scale-independent manner.\n\n"

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Common Mistakes

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