-
-
Notifications
You must be signed in to change notification settings - Fork 1
/
introduction-to-dimensional-analysis.html
305 lines (291 loc) · 18.3 KB
/
introduction-to-dimensional-analysis.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
<!DOCTYPE html>
<html lang="en">
<head>
<title>Blueschisting</title>
<link rel="icon" type="image/png" href="/images/favicon/favicon-32x32.png" sizes="32x32" />
<link rel="icon" type="image/png" href="/images/favicon/favicon-16x16.png" sizes="16x16" />
<link href='//fonts.googleapis.com/css?family=Open+Sans:400italic,600italic,700italic,400,600,700' rel='stylesheet' type='text/css' />
<link href='//fonts.googleapis.com/css?family=Merriweather:300' rel='stylesheet' type='text/css'/>
<link href='//fonts.googleapis.com/css?family=Source+Code+Pro:200,400,700' rel='stylesheet' type='text/css'/>
<link rel="stylesheet" type="text/css" href="/theme/css/icons.css"/>
<link rel="stylesheet" type="text/css" href="/theme/css/styles.css"/>
<meta charset="utf-8" />
</head>
<body id="index">
<!-- header -->
<header class="siteheader">
<!-- site image -->
<div class= "siteimage">
<a class="nodec" href=/images/escape_of_the_core.png>
<img width="200" height="200" src=/images/escape_of_the_core.png>
</a>
</div>
<div class = "sitebanner">
<h1><a class="sitetitle nodec" href="/index.html">Blueschisting</a></h1>
<h3 class ="sitesubtitle">Thoughts on planetary science, fluid dynamics, transit, and scientific computing</h3>
<!-- nav -->
<nav class="menu">
<ul>
<!-- menu items-->
<li><a class="nodec" href="/pages/about.html">Ian Rose</a></li>
<li><a class="nodec" href="/blog_index.html">blog</a></li>
<!--pages-->
<!-- services icons -->
<li><a class="nodec icon-github" href="http://github.com/ian-r-rose"></a></li>
<li><a class="nodec icon-twitter" href="http://twitter.com/IanRRose"></a></li>
</ul>
</nav>
</div> <!-- sitebanner -->
</header>
<!-- content -->
<div id="main">
<div id="content">
<section class="content">
<h3 class="posttitle">
<a class="nodec" href="/introduction-to-dimensional-analysis.html" rel="bookmark" title="Permalink to Introduction to dimensional analysis">
Introduction to dimensional analysis
</a>
</h3>
<div class="postinfo">
<p class="published" title="2015-09-08T00:00:00-07:00">
Tue 08 September 2015
</p>
</div><!-- .postinfo -->
<div class="article">
<p>This is the beginning of what I intend to be a multi-part series on dimensional analysis.
The goal for this post is to set the stage for why dimensional analysis is a powerful tool,
and to ask the questions which will lead us to the Buckingham Pi theorem.</p>
<p>The Buckingham Pi theorem is probably the most important result in the field of dimensional analysis.
That being said, many descriptions of the theorem leave something to be desired, and when
I used to read them I would come away as confused as when I started.<br>
The <a href="https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem">Wikipedia article</a> on the topic describes it thusly:</p>
<blockquote>
<p>...if there is a physically meaningful equation involving a certain number <span class="math">\(n\)</span> of physical variables,
then the original equation can be rewritten in terms of a set of <span class="math">\(p = n − k\)</span> dimensionless parameters
<span class="math">\(\pi_1\)</span>, <span class="math">\(\pi_2\)</span>, ..., <span class="math">\(\pi_p\)</span> constructed from the original variables. (Here <span class="math">\(k\)</span> is the number of
physical dimensions involved; it is obtained as the rank of a particular matrix.)</p>
</blockquote>
<p>This is a true statement, but it is essentially the punchline of the theorem. Without the
context for what <span class="math">\(p\)</span>, <span class="math">\(n\)</span>, and <span class="math">\(k\)</span> are, how they relate to the <span class="math">\(\pi_p\)</span>, and why we care
about any of it, it is very difficult to understand.</p>
<p>I would like to develop some of that context, and to begin we will ask what is meant by
dimensional analysis in the first place. A lot of what follows will be similar to first-rate
treatements of dimensional analysis by <a href="https://archive.org/details/dimensionalanaly00bridrich">Percy Bridgman</a>
and by <a href="http://www.cambridge.org/us/academic/subjects/mathematics/mathematical-modelling-and-methods/scaling?format=PB">G.I. Barenblatt</a>.
Both of these (short) books are excellent, and I have shamelessly borrowed from them.
As an aside, Bridgman was a physicist at Harvard who did foundational work on high pressure
physics and materials science. Recently the most abundant mineral on Earth (in Earth?),
<a href="https://en.wikipedia.org/wiki/Silicate_perovskite">bridgmanite</a>, was named in his honor.</p>
<h2>Physical equations</h2>
<p>Central to dimensional analysis is the concept of a physical equation.
A physical equation is something which is has the following properties:</p>
<ul>
<li>It involves an equals sign (<span class="math">\(=\)</span>).</li>
<li>All the terms in the equation have physical dimensions.</li>
<li>The physical dimensions of these terms are the same.</li>
</ul>
<p>The first point simply tells us we are dealing with equations, and the second part
tells us that we are dealing with equations regarding measurable things in the reals world.
This may sound trivial, but we will see that many of the statements in dimensional analysis
which sound trivial are, in fact, quite slippery. When a quantity has physical dimensions it
means that there is some standard that we have agreed upon to measure that quantity. We then can
determine any quantity of that type in multiples of the standard. This is quite abstract,
but it means that we can, for instance, define a <a href="https://en.wikipedia.org/wiki/Kilogram">standard kilogram</a>,
and then measure the mass of any item by determining how many of that standard have
an equal mass to the item.</p>
<p><img alt="kilogram" src="articles/dimensional_analysis/images/standard_kilogram.jpg" title="The standard kilogram"></p>
<p>As another example, the initial definition of the meter had it as one ten-millionth
of the distance from the equator to the north pole. With that definition, anybody could
(at least in principle), construct a meter-stick and measure the lengths of things to
their hearts content.</p>
<p>Okay, so this standardization of units is all well and good, but at the end of the day, the natural world
should not care at all what system of units we use for measuring things. There is nothing
special about one ten-millionth of one-fourth of the circumference of Earth
(except, perhaps, when performing navigation or geography).
Since the choice of unit is arbitrary, we should be able to convert from one
system of units to another. For instance, if we are measuring length in inches, we can
convert to centimeters by multiplying our values in inches by 2.54.</p>
<p>This gets us to the third point in the above list.
In a physical equation, both sides of the equation must have the same units.
People who have some physics background may find this statement to be obvious,
but when pressed, they may also have a hard time expressing why it has to be so.
To others it may not even make sense. Unfortunately, some fairly
<a href="https://en.wikipedia.org/wiki/Free_abelian_group">deep mathematics</a> is involved,
most of which I am not really qualified to address, so I will proceed with an example.</p>
<p>A common example given for why the units in a physical equation must match is something
along the lines of "you can't add apples and oranges!" </p>
<p><img alt="apple_plus_orange" src="articles/dimensional_analysis/images/apple_plus_orange.jpeg" title="Was this worth my time? Probably not..."></p>
<p>This is kind of an unfortunate argument, since there are conceivably situations
where it makes perfect sense to add apples and oranges. What if I am trying to
organize a school lunch program, and there are bushels (how's that for a unit?)
of apples and oranges in the cafeteria? How many pieces of fruit do I have to
distribute? I challenge you to figure it out without adding apples and oranges.</p>
<p>So we are going to try a more extreme example. What would it mean to to have an equation
that added minutes and meters which combined to make a new unit, say minumeters?
Something along the lines of</p>
<div class="math">\begin{equation}
x \; \mathrm{minutes} + y \; \mathrm{meters} = z \; \mathrm{minumeters}
\label{minumeters}
\end{equation}</div>
<p>However, we know that physics should not care about the units that we choose,
so what happens if we want to use seconds, feet, and a new (fictitious) unit, feeconds?
Well, we know that we can convert meters to feet by multiplying by ~3.3
(the number of feet in a meter), and we can convert minutes to seconds by multiplying
by 60 (the number of seconds in a minute).
Furthermore, since we presume minutmeters and feeconds to be valid units, there should
be a scale to convert them between each other (both being the result of adding length and time together),
which we denote by <span class="math">\(s\)</span>.</p>
<p>So we should be able to convert Equation \eqref{minumeters} to feeconds by
scaling all the terms in it appropriately:
</p>
<div class="math">\begin{equation}
60 x \; \mathrm{seconds} + 3.3 y \; \mathrm{feet} = s\;z \;\mathrm{feeconds}
\label{feeconds}
\end{equation}</div>
<p>Here is where we run into trouble. Inspecting Equations \eqref{minumeters} and
\eqref{feeconds}, we begin to see that the unit conversion on the left hand side
is not a simple scaling operation. Indeed, there is no single value for <span class="math">\(s\)</span> which
can satisfy Equation \eqref{feeconds} for all values of <span class="math">\(x\)</span> and <span class="math">\(y\)</span> (give it a shot!).</p>
<p>We have arrived at a contradictory result, so there must have been a problem somewhere
in the analysis. This problem, of course, is that we should not have added quantites
of different dimensions. Do that, and you quickly run afoul the "physics does not care
about our choice of measures" rule.</p>
<h2>Example: a simple pendulum</h2>
<p>The requirement that all the terms of the equation have the same units is actually stronger
than it may seem at first, and it lies at the heart of dimensional analysis.
If we have a physical system that we are trying to understand, it likely has some parameters which characterize it.
Any physical equation that we cook up must satisfy the rules of dimensional analysis, and therefore
the parameters have to be combined in such a way that they have the right units.
Any other combination of these parameters has to be nonsense.</p>
<p>For instance, let us say that we are trying to understand how the pendulum in a grandfather clock works.
It does a very good job of marking time, and it occurs to us that somebody has
designed the pendulum so that its period is exactly one second.
We decide to figure out what sets the period (<span class="math">\(T\)</span>) of the pendulum.
<img alt="clock" src="images/grandfather_clock.jpg" title="It felt like there was a lot of text, here is a clock!">
After studying the clock for a few minutes, we decide that the most important parameters
must be the mass (<span class="math">\(M\)</span>) of the bob, which makes it swing back and forth, and the length (<span class="math">\(L\)</span>) of the pendulum.
Therefore, we begin looking for an equation that relates the period to the mass and length, or
</p>
<div class="math">\begin{equation}
T = f( M, L )
\label{incorrect-period}
\end{equation}</div>
<p>
Seems simple enough. But as soon as we look at the units, we run into a problem.
The unit of <span class="math">\(T\)</span> is seconds, the unit of <span class="math">\(M\)</span> is kilograms, and the unit of <span class="math">\(L\)</span> is meters.
There is simply no way to combine the parameters <span class="math">\(M\)</span> and <span class="math">\(L\)</span> to make something in seconds,
so you cannot actually find a relation like \eqref{incorrect-period}.</p>
<p>So what was our mistake? Well, it turns out that our accounting of the important parameters missed one.
A pendulum also needs gravity, otherwise there is no reason for it to swing back towards the ground.
Therefore we add the acceleration due to gravity <span class="math">\(g\)</span> to our list of parameters, which has units of <span class="math">\(m/s^2\)</span>.
</p>
<div class="math">\begin{equation}
T = f( M, L, g )
\label{correct-period}
\end{equation}</div>
<p>
Now we have a parameter that has seconds as a component, so it may be possible to combine the units
to match the units of the period <span class="math">\(T\)</span>.
Indeed, after some playing around, we figure out that there is really only one way to combine
the units to make seconds, which is to say
</p>
<div class="math">\begin{equation}
T \sim \sqrt{ \frac{L}{g} }
\label{period}
\end{equation}</div>
<p>
Not only is this the only way to make something with units of seconds, but there is no place for the mass of the bob on the pendulum.
This turns out to be correct: for simple pendula, the mass is irrelevant: only gravity and the length matter.
Now, the <a href="https://en.wikipedia.org/wiki/Pendulum_%28mathematics%29#Simple_gravity_pendulum">full formula</a>
for the period of a simple pendulum has a factor of <span class="math">\(2\pi\)</span>, so we were
not 100% correct, but we got pretty close, and we captured the essential physics of the pendulum without
actually solving any equations. Instead, all we considered were the dimensions of the thing we were looking for,
and the dimensions of the parameters.</p>
<h2>Asking the right questions</h2>
<p>So all this is well and good, but the initial guess about the relevant parameters forgot about gravity,
and it included a superfluous parameter. We had to play around with the parameters a bit in order to figure
out how to combine them in the correct way. And even then, it is not necessarily obvious that there is only
one correct way to combine them.
It would be nice to have a way to go about this more systematic and rigorous way.</p>
<p>We now have the context to ask more specific questions.
Given a system to study and some relevant paramters, we want to answer:</p>
<ul>
<li>Can these parameters represent the physics we are interested in?</li>
<li>How many ways can we combine them to get the correct units?</li>
<li>What are allowable functional forms of the solution according to the rules of dimensional analysis? </li>
</ul>
<p>These are the fundamental questions of dimensional analysis:
questions which are more or less answered by the Buckingham Pi theorem.
Along the way we will encounter nondimensional numbers, which turn out to be, in a sense,
the most natural system of units to represent the physics of the system.
And hopefully at the end we will be able to go back to the above Wikipedia quote and understand what it means.</p>
<p>In my <a href="nondimensional-numbers-and-the-buckingham-pi-theorem.html">next post</a>,
I will be taking a qualitative look at the content of the Buckingham Pi theorem.
<img alt="new-cuayma" src="articles/dimensional_analysis/images/new-cuyama.jpg" title="My kind of place"></p>
<script type="text/javascript">if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) {
var align = "center",
indent = "0em",
linebreak = "false";
if (false) {
align = (screen.width < 768) ? "left" : align;
indent = (screen.width < 768) ? "0em" : indent;
linebreak = (screen.width < 768) ? 'true' : linebreak;
}
var mathjaxscript = document.createElement('script');
mathjaxscript.id = 'mathjaxscript_pelican_#%@#$@#';
mathjaxscript.type = 'text/javascript';
mathjaxscript.src = 'https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML';
mathjaxscript[(window.opera ? "innerHTML" : "text")] =
"MathJax.Hub.Config({" +
" config: ['MMLorHTML.js']," +
" TeX: { extensions: ['AMSmath.js','AMSsymbols.js','noErrors.js','noUndefined.js'], equationNumbers: { autoNumber: 'AMS' } }," +
" jax: ['input/TeX','input/MathML','output/HTML-CSS']," +
" extensions: ['tex2jax.js','mml2jax.js','MathMenu.js','MathZoom.js']," +
" displayAlign: '"+ align +"'," +
" displayIndent: '"+ indent +"'," +
" showMathMenu: true," +
" messageStyle: 'normal'," +
" tex2jax: { " +
" inlineMath: [ ['\\\\(','\\\\)'] ], " +
" displayMath: [ ['$$','$$'] ]," +
" processEscapes: true," +
" preview: 'TeX'," +
" }, " +
" 'HTML-CSS': { " +
" styles: { '.MathJax_Display, .MathJax .mo, .MathJax .mi, .MathJax .mn': {color: 'inherit ! important'} }," +
" linebreaks: { automatic: "+ linebreak +", width: '90% container' }," +
" }, " +
"}); " +
"if ('default' !== 'default') {" +
"MathJax.Hub.Register.StartupHook('HTML-CSS Jax Ready',function () {" +
"var VARIANT = MathJax.OutputJax['HTML-CSS'].FONTDATA.VARIANT;" +
"VARIANT['normal'].fonts.unshift('MathJax_default');" +
"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
"});" +
"MathJax.Hub.Register.StartupHook('SVG Jax Ready',function () {" +
"var VARIANT = MathJax.OutputJax.SVG.FONTDATA.VARIANT;" +
"VARIANT['normal'].fonts.unshift('MathJax_default');" +
"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
"});" +
"}";
(document.body || document.getElementsByTagName('head')[0]).appendChild(mathjaxscript);
}
</script>
</div><!-- .content -->
</section>
</div>
</div>
<!-- footer -->
<footer>
<p>
Copyright © Ian Rose.
Generated by <a href= "http://docs.getpelican.com/">Pelican</a> with
<a href="http://github.com/ian-r-rose/pelican-blueschisting">blueschisting</a> theme.
</p>
</footer>
</body>
</html>