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User talk:Social Alterations/Social Alterations

The Social Alterations education lab stays where it is: at www.SocialAlterations.com. So why are we here on Wikiversity? The Social Alterations education lab was designed to work as an open-source project. Here, on Wikiversity, we can link you to our lessons,and to the lessons of individuals and organizations we believe to be valuable, but also invite, encourage and support you to upload your own resources to share within the community of responsible design. (Social Alterations 23:59, 13 October 2010 (UTC))

Together we will transform this industry.

Disclaimer // The views presented here, by Social Alterations, may in no way reflect the views of the individuals or organizations listed below. Likewise, the views any individual or organization listed below may not reflect the views of Social Alterations. (Social Alterations 23:38, 13 January 2011 (UTC))

Fibre Analysis // Understanding the social and environmental impacts of natural and synthetic fibres
•	Acrylic •	Bamboo •	Cotton •	Fairtrade Cotton •	Low-chemical Cotton •	Low-water Cotton •	Organic Cotton •	Hemp •	Linen •	Lyocell •	Naturally Coloured Fibre •	Nylon •	PLA •	Polyester •	Recycled Fibre •	Silk •	Wild Silk •	Soya •	Viscose •	Wool •	Organic Wool (Social Alterations 23:03, 13 October 2010 (UTC))
 * Fibre Analysis: Possible Social and Environmental Impacts, Social Alterations: An Education Lab for Socially Responsible Fashion Design Click here to access. Fibre analysis includes:

 * Fabric and Fashion Glossary Index, Organic Clothing: All about organic, natural & sustainable clothing and eco-fashion Click here to access. This glossary was designed to act as "a reference that explains many terms that you might encounter when purchasing clothing to help buyers better understand what they are purchasing." (Organic Clothing, Fabric and Fashion Glossary Index) (Social Alterations 22:34, 13 January 2011 (UTC))

 * Organics and Low Toxicity, Textile Environmental Design (TED). Click here to access.

•	Facts •	Key Organizations •	Reading Materials •	A selection of Companies and Designers working within the field (Social Alterations 23:21, 13 January 2011 (UTC))

What is 'ecofashion'?
[Lesson 1] Shifting Through the 'Ecofashion Lexicon', Social Alterations: An Education Lab for Socially Responsible Fashion Design Click here to access the lesson.

This lesson introduces the following concepts:
 * consumer choice
 * designer choice
 * the ‘Ecofashion Lexicon’
 * greenwashing
 * unintelligent design
 * cradle to cradle design theory

Introduction // Both consumers and designers alike have been left to fend for themselves when it comes to understanding the social issues and environmental concerns increasingly associated with the fashion industry. Signals of deception, such as greenwashing, as well as unintelligent designs that have created products with hidden ingredients, known as products plus, have seemingly hijacked the potential for any real choice to exist at all. (Social Alterations 23:17, 13 October 2010 (UTC))

Key Players
[Lesson 2] Connect // Key Players Social Alterations: An Education Lab for Socially Responsible Design. Click here to access the lesson.

Identifying the social, cultural, environmental and economic impacts of the fashion, textile and apparel industries requires interdisciplinary research. There are countless touch points along these supply chains, as well as within each lifecycle stage of the products they produce. While the impacts of these industries may be positive and/or negative, it is important to recognize their complicated nature. Internationally, key players are working tirelessly to facilitate interdisciplinary research and learning for change. This lesson hopes to introduce you to these players. Reach out to them and connect. (Social Alterations 06:56, 19 October 2010 (UTC))

Fair Trade and Ethical Production, Textile Environmental Design (TED). Click here to access.

 * Facts
 * Key Organizations
 * Reading Materials
 * A selection of Companies and Designers working within the field (Social Alterations 23:12, 13 January 2011 (UTC))

Global Governance and the Corporation
[Lesson 3] Global Governance and the Corporation Social Alterations: An Education Lab for Socially Responsible Fashion Design. Click here to access the lesson.

This lesson introduces the following concepts:

 * Corporate-Led Globalization
 * The Condition of Globalization
 * Human Rights
 * The United Nations Global Compact
 * Democracy
 * Sovereignty
 * Security
 * Ethics

Globalization has created a new world order; one where traditional notions of state-led governance have been replaced with corporate mandates. There is a grave need for international corporate governance.

In an attempt to curb the corporations’ seemingly unyielding power, the United Nations created the Global Compact (UNGP). The UNGP promotes international corporate governance, but is it working? Is it enough? (Social Alterations 07:07, 19 October 2010 (UTC))

Corporate Social Responsibility (CSR)
[Lesson 4] Corporate Social Responsibility (CSR), Social Alterations: An Education Lab for Socially Responsible Fashion Design. Click here to access this lesson.

This lesson covers the following topics: •	How long has CSR been around? •	Corporate Citizenship •	Corporate Responsibility •	Sustainability •	The 3 P’s [People, Planet & Profit] •	The Triple Bottom Line
 * What is CSR?

•	Internal CSR Programs •	Multi-stakeholder Initiatives •	Business Initiatives
 * Why ‘do’ CSR?
 * Approaches to CSR

 * Difficulties with CSR (Social Alterations 08:02, 19 October 2010 (UTC))

Reading

 * Social Alterations: An Education Lab for Socially Responsible Fashion Design “Social Alterations [Reading List]” Click here to access. (Social Alterations 23:01, 13 January 2011 (UTC))

 * Fashioning an Ethical Industry, A Labour Behind the Label Project “Books” Click here to access. (Social Alterations 23:01, 13 January 2011 (UTC))

 * Textile Environment Design (TED) “Eco Textile Book List” Click here to access. (Social Alterations 23:01, 13 January 2011 (UTC))


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Talk:Performance analysis in sport

= Feedback = If you have feedback, suggestions or other comments on this Unit, please leave them in this section:

= Development notes =

14 October 2010
Leigh Blackall and I discussed copying this unit from its official location in the University of Canberra Moodle site, to this Wikiversity site, to further develop the unit for online, distance and more flexible study potential. Keanewheeler 23:15, 13 October 2010 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Performance analysis in sport


text:
User:Keanewheeler

Keane is a sports scientist working at the University of Canberra, National Institute of Sport Studies (UCNISS). Keane specialises in biomechanics, performance analysis and strength and conditioning. Keane's research interests are in the biomechanics of rugby union; playing structures in team sports; variations of running technique in sport; and appropriate training methods in sport. Keane is a level 2 accredited rugby union, cricket as well as strength and conditioning coach.

keanewheeler@gmail.com

= Projects =

Sport Coaching and Exercise Science

 * Introduction to Sport Biomechanics
 * Applied Sport Biomechanics
 * Clinical Exercise Biomechanics
 * Sport Performance Analysis
 * Introduction to Sport Coaching
 * Assessment of Sport Coaching
 * Sport Strength and Conditioning
 * Introduction to Sport Psychology
 * UCNISS Performance analysis in sport

Rugby Research
 * Oceania Rugby Sevens - Darwin 2010
 * Agility Skill Execution in Rugby Union (PhD)

title: User:Keanewheeler


title: File:Keane-wheeler2.JPG


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User:Ramiamro

=Project Report for User:Ramiamro=

For Introduction to Numerical Analysis, Fall 2010.

Introduction
the project intended to clarify the concept of the stability of ordinary differential equations' Methods, and especially Runge-Kutta methods of order four RK4,it is also discussing the shape of the absulute stability regions for each method, many examples, quizes, and exercises have been established on wikiversity and wikipediastiff equation, this establishment focused on users who have little experience with stability, such that thay can grasb the concept of absolute stability as well as A/L- stablility, more over the project studies particular examples of finding the characteristic polynomial for the multistep methods.

To operate numerical methods on stiff equations and get good approximation for the exact solution; methods with special stability are required, like A-stable or L-stable methods, if the method is applied with h>0 to the test equation then for A-stable method, it will produce approximation function $$R(\lambda h)$$ to the exact solution which will be proportional to $$ exp(\lambda h)$$, the approximation function will be bounded by 1 in the left half of the complex plane, which gave us the stability region, and so we can chose the step size which suitable for a certain kind of equation using that method.

Initial Experience

 * During the course and in courses before I introduced to many types of stabilities, but each time I have those we talked about similar diffenetions of stability in different topics, like stability for ODE's and stability of nonlinear systems, as well the stability for physical systems.
 * To study stability it is important to define the area you study in, the stability diffinitions are similar but at the same time time they are significantly different in details; in numerical analysis we talk about the stability of the method when we apply it to a certain type of equations. i.e. the test equation, our cocentration during this course was on the stiff equation, which is a differential equation that shows wide variance in the results as a result of small changes in the argument (the independent variable), some of the numerical methods shows very good stability when treating such kind of equations, specially the implicit methods, we will see during the project how the explicit methods Rk4, and others how they behave when they employed to simulate the exact solutions of those equations.these modifications manifisted in posting some diffinitions and an example on the wikipedia page stiff equation, and some in quizes and excercices on my wikiversity user  page, these changes took place because I found they are not clearly discussed on the wikipedia page, although my changes my be in the hand or the minds of others, but I beleive I made a better way of  understandig the concept of stability for the Numerical methods more clearer, and easily comprehended especially by using the conceptual quiz I posted on wikiversity down.

Motivation
During the couse, we covered so many topics, and I barely remeber that any class passed without asking or discussing the error propagations in the numerical method which is implicitly implies whether the numerical method is stable or not, on the other hand during the second half of the course these cries increased and I felt that the stability topic is not clear for some of the students, due to the time limitations of the class and the poorley expressed ideas on wikipedia I made my point to clarify this topic more and make it easier to comprehend, so I established my project with a hard base to study the stability of the numerical method such that I introduced the reader to basic diffinitions in this topic, so he can cope up with the flow of the material later on, and made sure the reader will not losely use the concept later on, then moved to show some examples "mine and others on wikipedia" so that he/she get experience with the difinitions

Proposed Changes
my proposed changes were considering,changes to the following subtopics:
 * Stability regions for RK methods.
 * Stability regions for multistep method.
 * The characteristic functions and how to find it.
 * Solving a system of differential equations using Rk4 method.
 * plotting the stability regions.
 * Quizes related to the topic: stability of ODEs' methods.
 * Excercises related to the same topic.

Actual Changes
I almost covered my plan, but I did too much in RK methods than the implicit multistep analysis, one thing that I didn't cover is the application of RK4 to a system of differential equations, the idea is fancy to be discussed on wikipedia, since very little details written about it, and it is worthy to show some hidden details, most books they just give the general formula to apply that method to a system of differential equations.

I starrted my project with a number of definitions such as consistency, convergense, and absolute stability,then followed by a theorem which uses those concepts, the theorem shows some results concerning stability and convergence of the numerical methods both single and multistep methods, this is an excution for what I proposed about the multistep method, in addition to excercises that covers finding the characteristic polynomial for RK2 method, and a nother method.

The rest of my proposed changes has been covered and an example to show how to find the stability polynomial for five types of RK4, including the classical case, A matlab code was shown and tested to generate the stability region for the five cases,In fact I generated the stability regions for the five cases and compared the with other resources, An image of the generated stability region was published on stiff equation, all the changes have been published to the wikipedia. concerning the quizes part I managed to write 13 conceptual quizes, since finding numerical quiz in this topic is really long way, so I just posted the 13 questions on wikiversity below, any user can also access them from user:Mjmohio, under Numerical ordinary differential equations

Excercises have been developed, I published basically tow questions, divided them to five guiding excercises, and arranged them in chronological order in terms of solving for the stability region, and testing wether the the numerical method is stable or not.During the exercises I clarified the L-stability concept by solving and discussing an L-stable method,

Definitions
let $$\tau_i$$ is the local truncation error, and k is the number of time steps, then:   The numerical method is consistent with a differential equation if

 * $$\lim_{h \to 0} \operatorname{max}|\tau_i|=0$$ over $$1\leqslant i \leqslant k$$.

according to this definition Euler's method is consistent.

 A numerical method is said be convergent with respect to differential equation if

 * $$\lim_{h \to 0}|x(t_i)-y_i|=0$$ over $$1\leqslant i \leqslant k$$;

where $$y_i$$ is the approximation for $$x(t_i)$$.

 A numerical method is stable if small change in the initial conditions or data, produce a correspondingly small change in the subsequent approximations.

Theorem: For an initial value problem
 * $$x'=f(t,x)$$ with $$t\in[t_0,t_0+\alpha]$$

and certain initial conditions, $$(t_0,x_0)$$, let us consider a numerical method of the form
 * $$(y_0=x_0)$$ and $$y_{i+1}=y_i+h\phi(t_i,y_i,h)$$.

If there exists a value $$h>0$$ such that it is continuous on the iterative domain, $$\Omega$$ and if there exists an $$L>0$$ such that
 * $$|\phi(t,y,h)-\phi(t,y^*,h)|\leqslant L |y-y^*|$$ for all $$(t,y,h),\,(t,y^*,h)\in \Omega$$,

the method fulfills the Lipschitz condition, and it is stable and convergent if and only if it is consistent. That is,
 * $$\phi(t,x,0)=f(t,x)$$ for all $$t \in \Omega$$.

For a similar argument, one can deduce the following for multi- step methods:   The method is stable if and only if all roots, $$\lambda$$, of the characteristic polynomial satisfy
 * $$|\lambda|\leqslant 1$$,

and any root of
 * $$|\lambda|=1$$

is simple root.  one more result is that if the method is consistent with the differential equation, the method is stable if and only if it it is convergent.  see

stability polynomials of Runge-Kutta methods
The Runge–Kutta methods are very usefull in solving systems of differential equations, it has wide applications for the scientists and the engineers, as well as for the economical models, the recognized with their practical accuracy where we can use and get very good results and approximations when solving an ODE problem, RK has the general form :$$ y_{n+1} = y_n + h\sum_{i=1}^s b_i k_i$$,

Example:finding the stability polynomial for RK4's methods
for RK4" case$$2_a$$", which characterizes by $$c_2= \frac {-1}{4}$$ which has the form: the stability region is found by applying the method to the linear test equation $$y'=\lambda y$$
 * $$ y_{n+1} = y_n + \frac{h}{6}k_1+0k_2+\frac {2k_3}{3}+\frac {k_4}{6} $$
 * the stability region is found by applying the method to the linear test equation $$y'=\lambda y$$

 * $$ \displaystyle \ k_1=hf(t_n,y_n)$$
 * $$k_2=hf(t_n+\frac{h}{2},y_n-\frac{k_1}{2})$$
 * $$k_3=hf(t_n+\frac{h}{2},y_n+\frac{3 k_1}{4}-\frac{ k_2}{4})$$
 * $$k_4=hf(t_n+h,y_n- 2k_1+k_2+2k_3)$$

using the linearized equation $$f(t,y)=\lambda y$$, we get
 * $$k_1=h\lambda y_n$$
 * $$k_2=h\lambda(1-\frac{h\lambda}{2})y_n$$
 * $$k_3=h\lambda(1+\frac{3 h\lambda}{4}-\frac { h \lambda}{4}(1-\frac{h\lambda}{2}))y_n$$
 * $$k_4=h\lambda(1-2h\lambda+ h\lambda(1-\frac{h\lambda}{2})+2h\lambda(1+\frac{3 h\lambda}{4}-\frac { h \lambda}{4}(1-\frac{h\lambda}{2})))))y_n$$

substitue these back in $$y_{n+1}$$, yields $$y_{n+1} = [1+(h\lambda)+\frac{(h\lambda)^2}{2}+\frac{(h\lambda)^3}{6}+\frac{(h\lambda)^4}{24}]y_n=R(h\lambda)y_n $$
 * and so the characteristic polynomial

$$ P(z)=z-R(z); z=h\lambda $$ for the absolute stability region for this method, set |R(z)|<1, and so we get the region in figue1. case$$2_a$$

the table below shows the final forms for the stability function for different forms of RK4, these RK4's are different in the values of $$ b_j$$, and they are fullfilling the consistencty requirement for the method i.e :$$\sum_{j=1}^{i-1} a_{ij} = c_i\ \mathrm{for}\ i=2, \ldots, s.$$

Plotting the stability region
In order to plot the stability region, we can set the stability function to be bounded by 1 and solve for the values of z, then draw z in the complex plane. Since R(z) is the unit circle in the complex plane, each point on the boundary can be represented as $$e^{i \theta}$$ and so by changing $$\theta$$ over the interval$$[0,2\pi]$$, we can draw the boundaries of that region. The following OCTAVE/Matlab code does this by plotting contour curves until reaching the boudaries: The figure at right shows the absolute stability regions for RK4 cases which is tabulated above

Quizes
{ the A-stability is characterized by: - all the points in the left half of the complex plane. - it a special case of the absolute stability. - inludes all $$\{ z \in \mathbb{C} | \mathrm{Re}(z) < 0 \}$$, + all of the above
 * type=""}

{Euler method is: - A-stable + not A-stable
 * type=""}

{the absolute stability region for RK4 is greater than the A-stability region for the same method:
 * type=""}

+ false - True - True for certain values of $$ b_j$$.

{A numerical method is stable if : + small change in the initial condition will produce small change in subsequent steps. - small change in the initial condition will produce huge change in subsequent steps. - big change in the initial conditions produce oscillatory solution at the end.
 * type=""}

{$$ \lambda =1$$ is always a root of the characteristic polynomial of the multistep method:
 * type=""}

+ True - False - usually not the case

{the root of the C.P. can be real or complex,and the method still be stable: + True - False
 * type=""}

{A multi- step method is strongly stable if : - $$ \lambda =1$$ is the only root of magnitude 1. - all other roots has magnitude <1 + the first and the second sentence - just one root inside the unit circle if it has more than one.
 * type=""}

{if more than one root has magnitude equal to 1, and the others are less than one, the method is - strongly stable. - A-stable + weakly stable. - Unstable.
 * type=""}

{The numerical method is unstable if - $$ \lambda > 1$$ for at least one root - $$ Re(\lambda) > 1$$ for at least one root - $$ Re(\lambda) \leqslant 1$$ for at least one root + the firs and the second sentences.
 * type=""}

{the absolute stabity region for the explicit Euler's method is + unit cicle in the complex plane, its center shifted to the left, by one unit. - unit cicle in the complex plane, its center shifted to the rigt, by one unit. - the whole left side of the complex plane. - the method is unstable.
 * type=""}

{ None of the RK methods is A-Stable: + True - False
 * type=""}

{All explicit methods are : - A-Stable + Not A-Stable. - Unstable. - None of the above
 * type=""}

{implicit multistep methods are A-stable if the have order at most -1 +2 -3 -they are always A-stable.
 * type=""}

Ex:1
find the stability function for RK2 which is given by: $$ y_{n+1}=y_{n}+hf(t+\frac {h}{2},y_{n}+\frac {hf(t_n,y_n)}{2})$$

Solution: applying this method to the test equation $$ y'=\lambda y$$
 * we get $$ \begin{align}

y_{n+1} &=y_n+h \lambda y_n+\frac {(h \lambda)^2}{2}y_n \\ \Rightarrow \end{align}$$ the stability polynomial $$ \phi(z)=1+h \lambda +\frac {(h \lambda)^2}{2}$$

Ex:2
find the absolute stability region for RK2. Solution: by setting $$ z = h \lambda $$
 * $$ \Rightarrow $$ the abs.stability region is given by $$\{z\in \mathbb{C}||1+ h \lambda + \frac {(h \lambda)^2}{2}| < 1 \}$$

Ex:3
find the characteristic polynomial for RK2. Solution: it is $$ r^{i+1}=(1+z+\frac {z^2}{2})r^i$$ divide both sides of the equation by $$ r^i;r \ne 0$$ you get $$ r^{i+1}=(1+z+\frac {z^2}{2})r^i$$
 * $$ \Rightarrow \phi(r,z)=r-1+z+\frac {z^2}{2}$$

Ex:4
is RK2 stable, if it is what type of stability. Solution: you get $$ r^{i+1}=(1+z+\frac {z^2}{2})r^i$$ by setting z=0,
 * $$ \Rightarrow r=1$$
 * so the method is strongly stable since r=1, is the only root, and has a value of 1.

Ex:5
Determine the stability of Back ward Euler method. Solution: $$ y_{n+1}=y_n+hf(t_{n+1},y_{n+1})$$
 * by applying this method to the test equation
 * then $$ y_{n+1}=y_n+h\lambda y_{n+1}$$
 * and so $$ y_{n+1}=\frac{1}{1+h\lambda }y_{n}$$
 * call $$ z=h\lambda$$ and so:
 * $$ G(z)\longrightarrow 0$$ as $$ Re(z)\longrightarrow \infty$$
 * and so this is L-Stable method when applied to stiff equation.

Conclusions
During this project, I searched about five resources and other internet websites to weave my ideas and make them as clear as possible, I used examples to clarify the definitions and the theory, I found that Rk methods of order one and the higher orders are cositent with stiff problems, and so they are convergent, see the proof on , and so when we use RK methods we will not be worried if the method is consistent or not. Using any method to simulate the exact solution for any kind of ODE's is dependent on the absolute stability region, and so the choice of the time step will depend on the roots of the characteristic polynomial, if the complex number $$ z=h\lambda$$ should satisfy the condition that the absolut value of the stability polynomial should be strictly less than one, otherwise the method will not be stable, and the results will be far from the accuracy.

On the other hand, theory should be clarified by examples, it wasn't easy to find the stability polynomial for Rk methods, one should be very carefull when use the parantheses, see the example above, missing of one parantheses will leed to a different stability polynomial and so leads to a different stability region, which in role means different method.

Conceptual quiz was very neat to clarify the concepts, and so I included so many of them, they expressing the definition of the absolute stability, A-stability, strong and weak stability, and L-stability; see exercises above, they inclded more about stability properties of RK methods, and recruiting the attention of the reader to some general rules and corollaries that is not of first look to catch and comprehend. I finished my main modifications on wikipedia by showing a sample matlab code how to plot the stability regions,the code is working efficiently, and gives accurate results, see.

Note: it is worthy to mention, that during my writings I noticed that some resources define the stability region to be $$\{z \in \mathbb{C} | |\phi(z)|\leqslant 1 \}$$. But on [w:stiff equation] the definition doesn't include 1

title: User:Ramiamro


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User:Social Alterations/Responsible Fashion

What is 'responsible fashion'? How does it differ from ‘ecofashion’ or ‘ethical fashion’? Are these terms all part of the same meaning?

Responsible fashion must respect both people and planet, through each stage of the supply chain, with each phase of the lifecycles of a fashion product and/or system.

Visit Social Alterations: An Education Lab for Socially Responsible Fashion Design to learn more. (Social Alterations 23:51, 13 October 2010 (UTC))


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User:Ottava Rima/To Autumn FAR

Large amounts of cited text were removed, uncited text was added, claims about Byron that are neither in the body of the text or in any major scholarly work were added to the lead, and other misleading things were added. Here are more specific problems:

misleading attribution to add a fringe item into the lead "...while barred clouds bloom the soft dying day". Despite the emphasis on long vowels, there is almost an absence of hiatus where two adjacent vowels occur without a separating consonant.[28]
 * 1.
 * 2. The statement about Byron is original research and not in the body of the text
 * 3. Removing a description of an image for the reason that the drawing isn't what his real hair color is, but the description is about the image and not reality.
 * 4. He removed the third party summary of the poems and turned it into a copy and paste of the poem, which is inappropriate
 * 5. ""To Autumn" is a poem of three stanzas, each of eleven lines. Like others of Keats's odes written in 1819, the structure is that of an odal hymn, having three clearly defined sections corresponding to the Classical divisions of strophe, antistrophe, and epode. [23" that is not found in the source. The source was only used to cite "Like many of Keats's 1819 odes, the structure of the poem is that of an odal hymn." which doesn't say anything about the amount of lines or how odal hymns are broken down and odal hymns do not have three sections, that is an -ode-.
 * 6. Flesch, William. Companion to British Poetry, 19th Century. Facts on File, 2009. ISBN 978-0816058969 is an unreliable source. Sparksnote is also an unreliable source.
 * 7. Keats characteristically uses monosyllabic words such as "...how to load and bless with fruit the vines that round the thatch-eaves run." The words are weighted by the emphasis of bilabial consonants (b, m, p), with lines like "...for Summer has o'er-brimm'd their clammy cells." There is also an emphasis on long vowels which control the flow of the poem, giving it a slow measured pace:
 * Those aren't examples given in the source and there are no examples of monosyllabic, making it OR. The example of the bilabials is not provided in the text, making it OR and the statement about the long vowels is wrong too. Those aren't long vowels as Keats's vowels are different from modern vowels and the expression is unique to Keats.

 * 8. There is not "almost an absence" but a 100% absence of hiatus, he changed it from saying that, thus making it not reflective of the source. Bate p. 183 - "hiatus is non-existent" image from page 183 verifying this quote.
 * 9. "Like the other odes, "To Autumn" is written in iambic pentameter with five stressed syllables to a line, each usually preceded by an unstressed syllable.[27" That is wrong, 13% of the poem is spondee, making it not qualified to be iambic pentameter. The percentage of spondee cited by Bate was removed and that above was added in its place, and it is cited to sparksnotes. Bate doesn't once say it was iambic and he literally wrote the book on the meter of Keats's.

The original version did no have these problems or paragraphs of uncited text. Ottava Rima (talk) 01:38, 14 October 2010 (UTC)

New statements
In response to the list here, I will cite the statement and respond:

 * 1. "The fact that it is in iambic pentameter"
 * This is incorrect. 14% of the poem is spondee, meaning that there are 1.5 spondees per line. Iambic pentameter is impossible with that high rate. There are also two initial inversions of beat in lines, which makes the poem even less able to be "iambic pentameter". As Jack Stillinger (major critic) on page 15 points out (regarding "To Autumn"): "of course the contrasts and interchanges originate spontaneously, according to ear rather than principle. The same is true of the rhythmial qualities of the lines. Departures from the metrical norm occur almost everywhere..." There is no "normal" when it comes to the poem, and removing of the percentages means that there is no honest discussion of the meter.

 * 2. "The fact that the major poetic tool used here is Personification. (It's the most famous and most cited example in English)"
 * It is not the most famous or the most cited example in English. It is actually obscure and most critics ignore it because the term adds nothing to discussion. If anything, "The Cloud" is a far more famous example just in the same time period, and that is still obscure.

 * 3. "The important reference to the similarity to the painted works of Keats' contemporary John Constable."
 * This is outright false. There is no relationship. The page now claims: "It has parallels in the rural landscapes of the English painter [[John Constable]". The source does not say that: "The poem was a result of having lived in the world, and a farewell to it; at the same time it was a world unto itself, inhabiting fully its autumnal canvas, richer than any Constable painting." The only one linking the two is an obscure critic in a Singapore review, and only saying that Keats is far better than Constable. This is also unique information put in the lead, violating MoS standards.

 * 4. "The comparison between the England-based description by Keats and the Grand Tour views of his contemporaries Byron and Shelley."
 * There is no relationship between them. No major critic tries to argue such. As you can see, I used every major critic on the poem. There is no comparison by critics of any of these poems because there is no match. "Ode to the West Wind" would be compared to "Ode to a Nightingale" if anything. Regardless, it is new information added to the lead that lacks a source.

 * 5. "The sorting out of the fact that the sense of "taste" referred to in the old intro does not exist anywhere in the poem and that was is referrenced is in fact the "tactile" sense."
 * Taste does exist in many of the images. The word "ripe" appeals to taste. "Sweet kernel" also appeals to taste. The word "referrenced" is inappropriate and no a real word. You meaned "referred to".

 * 6. "The pertinent reordering of statement about the form of the poem, giving examples of spondee and so on."
 * Form always comes before summary as is appropriate for critical works and literature article standards. The poem section also needs to have third party summaries or it is inappropriate. We do not just copy and paste from Wikisource. Furthermore, you cannot just -add- examples of your own of the use as that is original research. Many of your examples were -wrong-, like when it came to "long vowels" which has a specific definition under Keats and used a different pronunciation system that you probably don't know about because you need Bate's book to see the list.

 * 7. "The filling out of the description of the poem as a Classical Ode by showing the way in which it does resemble that poetic form"
 * It isn't a classical ode!!! It is a odal hymn. That means it is a hymn with aspects of ode, but not the three that you wish. The three parts mark beginning, middle, end, not call and response. A classical ode would by cyclical whereas this poem is purposefully linear.

- Ottava Rima (talk) 14:23, 15 October 2010 (UTC)

More responses
The individual responds here in "Cut and paste of Ottava Riva's comments". Here are responses:

 * 1. "Large amounts of cited text were not removed. They were rearranged more logically."
 * By removing summaries of the poem from Harold Bloom, the top literary critic in the world, he introduced summary into a themes section which does not deal with summaries. He also removed percentages and other figures which were essential to understanding the structure of the poem.

 * 2. "If this refers to the similarity of the poem to John Constable's rural landscapes, then it is not a "fringe item. It is significant context."
 * As pointed out above, the claim about Constable was a complete misreading nor is it pertinent to the poem. There is no real comparison between the two.

 * 3. ":The contrast with Byron is pertinent, and needs to be in the body of the text as well. Yes, it requires appropriate referencing. But noone who knows anything about Romantic poetry would consider a comparison between Byron and Keats as OR."
 * That is the very definition of original research. The third party sources do not link the two because they are -not- linked. Keats did not care about Byron at the time nor does it have anything to do with the European works.

 * 4. "How childish! The hair in the image is drawn with black lines. that doesn't make it "black hair"."
 * The description is given for blind people who need to know what the image -looks like-. Drawn with black lines by definition makes it an image of black hair.

 * 5. "It was a most inadequate summary."
 * It was a completely accurate summary of the statements by Harold Bloom, the top literary critic in the world. The arrogance of the statement is startling.

 * 6. "The link to odal hymn contains the information. Yes, the three parts need referencing. One will almost certainly be located."
 * It doesn't exist. I have on my shelf over 70 books devoted to Keats's poems by the top critics in the field. You are confusing things. Furthermore, the wiki link is not to "odal hymn" so it is not correct.

 * 7. "The Sparknotes ref has been replaced with a much better one."
 * Actually, the source says: "using iambic pentameter as the starting point", thus reaffirming what was quoted above about it having irregular meter that changes throughout the poem.

 * 8. "I am sure that the references must give some examples, but I don't have those references to hand."
 * Yet you constantly rewrote things to make them opposite of what the source says. You claimed above about

 * 9. "Is it really OR to state that this is an example of monosyllabic writing"
 * It doesn't need an example, it is rather obvious. And yes, by definition that is original research.

 * 10. "The bilabial consonants need to be stated, otherwise your reader has to page-hop to find out what a bi-labial consonant actually is."
 * Doesn't matter if they "page hop or not". Wikilinks are there so you don't clutter up everything. You complained that the page is "essayish" but using all of these examples turns it into an essayish page.

 * 11. "This statement by Ottava Rima is ridiculous and shows a lack of understanding as to what constitutes a "long vowel". Again, why was no example given from the referenced source?"
 * Keats has his own definition of Long Vowels. If you knew Keats you would have known this. You do not know Keats nor do you know early 19th century vowel pronounciation.
 * To quote from Bate 1962 p. 51: "(Benjamine Bailey to Lord Houghton) "One of Keats's favorite topics of discourse was the principle of melody in verse, upon which he had his own motives, particularly in the management of open & close vowels ... Keats's theory was that the vowels should be so managed as not to clash with one another..." then from Bate: "The precise meaning of 'open' and 'close' vowels, in this suggestive passage, is difficult to ascertain. Bailey's marking of the opening lines of Hyperion, moreoever, which he appends to his statements, is of little help. The only inference of reasonable certainty is that Bailey was not using the terms acording to strit phonetic usage of the present day. This inference is (p. 52) further borne out, it would seem, by the lack of any particular patterning of vowels in even the maturest verse of Keats when analyzed according to modern phonetic terms of 'open' and 'close.' Professor Cabell Greet, whose expert knowledge of phonetics has greatly helped me, informs me that the term 'open,' as loosely used by Bailey, was possibly equivalent to the diphthongs of 'day' [ei], 'go' [ou], 'fly' [ai], 'how' [au]... and to the historically 'long' vowels of 'see' [i:], 'father' [a:], 'saw' [e:], 'too' [u:], and 'bird' [e:].... If this conjecture is assumed, it is possible to detect a rather unusual and even frequent interplay of vowels int he verse which Keats wrote between the autumn of 1818 and the following May. Thus, allowing a to represent here and subsequent the diphthongs and historically 'long' vowels which Bailey possibly meant by 'open,' ...."
 * What that is explaining is that the vowels are not today's vowels but 200 years ago vowels, and if you knew anything about linguistics you would know that the Great Vowel Shift is constantly happening.

 * 12. "Check the poem. The poem uses "O'er", "whoever" and "winnowing". You could eliminate "O'er" and "...owing" as diphthongs, but that still leaves you with "who/ever", two clearly defined vowels with no consonant between them."
 * The source says NO HIATUS. That means that you cannot deviate from that and make your own original claim that shows no understanding of 19th-century British vowels. p. 183: "Rigorous structural care is once again apparent at every hand: hiatus is non-existent". Walter Jackson Bate was the top Keats scholar and won a Pulitzer for his biography on Keats. This work is -the- work on analyzing the language of Keats's poetry.

 * 13. "However, regardless of whether you have found a reference to iambic pentameter in Bate or not, the poem is in iambic pentameter"
 * Even your own better source says it is not - it merely started in it but quickly deteriorates.

Ottava Rima (talk) 16:13, 15 October 2010 (UTC)

Personification
By the way, the user mentioned that the poem uses personification. No. Personification is to give inanimate objects human attributes. Instead, this uses anthropomorphization, which is to make non-human things -human-. Autumn is described as a female. That is an anthromorphized character. Ottava Rima (talk) 17:47, 15 October 2010 (UTC)

SandyGeorgia
SandyGeorgia wrote: "Which person opining to keep this article, co-nominated by an ItsLassieTime sock, has checked all of the sources for copyvio and plagirism? Arguing over the prose and which version to keep without checking sources is somewhat pointless. Doing it via proxy even more so."

ItsLassieTime did not provide any sources. Kathyrine/ItsLassieTime came to the artile during its Good Article Nomination as a reviewer. She spent quite a bit of time copyediting the writing and sourcing that I alone provided. She only reworked my content. There is no possible chance for her to have plagiarised.

Amandajm
Responses to this section.

1. "Lacked continuity ie. jumped from form to theme to context withing a single paragraph." Absurd claim - the context is all in background, the form is all in structure, and the themes were all in themes. This is based on standard structure of poetry articles. You cannot randomly remove that pattern, especially to an FA and without discussion.

2. "Clearly challengeable such as analysing percentages of times a form occurred." Walter Jackson Bate, Pulitzer Prize winner for his biography on Keats and the most famous Keats scholar, devoted a whole book to those percentages. They are 100% necessary to have full coverage of scholarship on the matter.

3. "NOTE: "tastes" is incorrect." This assumes that "ripeness" is no longer taste. The dictionary corrects Amandajm. The emphasis on food is taste.

4. "The second sentences here refers to "diction symbolism and literary devises" as if they occurred in the third stanza only. " Clear misreading. It does not imply that they are only in the third stanza, but that the third stanza is only about them.

5. ""Negative connotations" is not a good description. " It is talking about death. It isn't positive.

6. Additions "NOTE: 1. "rural landscape" should read "the works of rural landscape painters"." This is pure original research that shows that Amandajm does not read sources carefully and uses inadequate sources. Even Alan pointed out problems with this. They -wish- it were true so insist it is added to the lead, a place that forms only a summary.

7. "Personification. This is the major poetic tool used in this poem. "To Autumn" is the iconic example. It had been omitted from the FA entirely!!!"

Allegory does not require you to say either "metaphor" or "personification". An allegory is far more complex than such simplistic statements desired by Amandajm. Regardless, if you want to dumb down an audience by adding one word that has no real impact on a page, that does not justify the destruction of the page.

8. "Moved further down the page, after the Theme." Before you can describe a poem you must break down its form for the audience. Structure must always go before the plot summary or the plot summary doesn't make sense.

9. "The thematic divisions that were between the stanzas have been put into the Theme section and expanded." A major violation of standards. A. You cannot have just raw text, B. the section is for summary of the poem and the language is added in its entirety because of its brevity, and C. it was moved down in a very sloppy and inappropriate way.

10. "These progressions are joined with a shift from the tactile sense to that of sight and then of sound, creating a three part symmetry which is missing in Keats's other odes.[11]" She added a source to statements without having the source. The original version is in the second paragraph - it is more structured, unified, and intelligent. It is contrasting To Autumn with other odes and shows how the poem as a whole works.

11. "3. *Most importantly*, the Personification of Autumn is introduced into the discussion of theme and this continues into the subsequent descriptions of the stanzas." Adding something the sources do not constitutes as original research.

12. "NOTE: 1. the line in the FA version about "unlike the following which deal more with sensual observations" is out of place in describing the first Stanza. It would be more useful in describing the second stanze"

Harold Bloom thinks otherwise, as it was patterned off his paragraph discussion stanza one (not stanza two): "The first stanza is natural process; the remaining two stanzas are sensuous observations of the consequences of that process: first, sights of the harvest in its final stages; then, post-harvest sounds, heralding the coming-on of winter. The sequence of the three stanzas then is pre-harvest ripeness, late-harvest repletion, and post-harvest natural music." Bloom 1971 p. 432

13. "The second stanza presents the personification of Autumn as the harvester" - Personification is used inappropriately. Personification is to add human emotions and attributes, not occupations. Anthropomorphization or allegorization are the only appropriate terms. But listing any terms is intellectually inappropriate.

14. "1. I have not the faintest clue what the FA edtior meant by "The second stanza reverses the images of the first stanza". It isn't explained! "

Maybe if you read the complete sentence you would see: "The second stanza reverses the images of the first stanza and describes the process of harvesting." Harvest is the reversal of growth.

15. "The editted version picks up the themes of "Personification", the "advance of the season", the "advance of the day into afternoon", as well as including and further expounding the "stasis and motion" mentioned in the FA version."

All without having the sources, thus adding original research, contradicting what the sources say, and putting in things that are simplistic and make this more of an essay than a quality article.

16. "NOTE: 1. "Within the final moments of the poem, there is an introduction of the harvest and Autumn is manifested in the role of a harvester". This is plainly erroneous. The harvest and manifestation of Autumn as Harvester are not "within the final moments of the poem". They are the whole theme of the second stanza! "

Wrong. The 2nd stanza clearly said: "is not actually harvesting but exists in a stasis". Autumn doesn't harvest in stanza two. She is "sitting careless on a granary floor". The third stanza is the stanza of death - i.e. the reaping or the harvest aka DEATH: "the coming death of the year, and of course the familiar archetypal relevance of the association to our feelings of sequence in our lives" - Bate 1963 p. 583

17. Additions - "he last stanza contrasts Autumn's sounds with those of Spring. The sounds that are presented are not only those of Autumn but essentially the gentle sounds of the evening." and "The full-grown lambs, like the grapes, gourds and hazel nuts will be harvested for the winter. "

Original research. Bate never discusses a contrast with spring or the sounds being part of the evening. What he does suggest is that the images aren't necessarily autumnal except for two - "stubble plains" and the "full-grown lambs". He also doesn't say anything about them being harvested for winter!

18. "The references to Spring, the growing lambs and the migrating swallows remind the reader that the seasons are a cycle."

Original research and not cited. The implication would be that Keats was expecting a rebirth in his own life, which no critic since 1950 was willing to consider. If anything, this is a poem about his own artistic career and him trying to come to terms with his own life, and it was discussed in the theme section when saying a poet is parallel to the farmer.

19. "This section has been enlarged, and reoganised and has had examples drawn from the text added to it, illustrative of the forms and figures of speech described."

The previous version was to the point, informative, and didn't include original research.

20. " having three clearly defined sections corresponding to the Classical divisions of strophe, antistrophe, and epode."

Original research and wrong. Odal hymns do not have clearly defined strophes, antisrophes, and epodes. That is why they are odal hymns and not odes.

21. "2. Discussion of "Ubi sunt" device." A low quality source for something that is unnecessary.

22. "The editted paragraph orders the discussion of structure, defining first the Rhythm. The Rhythm is iambic pentameter, and this poem is cited in various sources as an example. It is the first thing that should be to be mentioned, after the number and length of stanzas. "

Except that this poem is the least iambic of his later works.

23. "if the FA editor, let alone the reader, fails to comprehend that the cited work means that there are 13.9% spondees within a poem that is otherwise composed of iambics, "

It isn't just "iambs" verse "spondees". There are many, many other variations of rhythm. The sheer amount of spondees is completely unique in English poetry.

24. "some reciters will emphasise them and others will treat all but the most obvious as iambics."

Probably why she thinks it is iambic when critics say no.

25. Original research. "The poem doesn't lack hiatus. There are at least three examples of it." Not according to Walter Jackson Bate, who won the Pulitzer on the matter.

26. "NOTE: 1. This is an expansion. " Translation Original research.

title: User:Ottava Rima/To Autumn FAR


text:
Ideas in Geometry/Instructive examples/Proving Pythagorean Theorem Using Similar Angels

On the picture of the triangle with the segment of ab cut in the middle of the triangle, we see that the small triangle with sides bc, b^2 and ab were multiplied by b. If we divide this triangle by b again, the sides would become side c, side b and side a. The other small triangle that is attached was multiplied by a, forming sides ab, ac, and a^2. Again if we divide these triangles up by a, the sides would be side b, side c and side a. The angle across from Side length of b should both have the same angle. Same goes for the angle across from side length a, and c. For example, the angles across from sides b in both the small triangles should have the same angle. The angle across from sides a in both the triangles should also have the same angles. Same goes for angles across from side c.

Since the angles across from sides a, and the angles across from sides b are the same, the angles across from side C should be 90 degrees. This means that there are angles of 30, 60 and 90 degrees in each of the small triangles, which proves that this that the segment ab splitting the whole triangle is actually cutting a 90 degrees angle. This proves that the triangle is 90 degrees. Then we can assume that the sides of the whole triangle are equal to the triangle of the big triangle. There for the hypotenuse of the triangle with the segment ab, should be a^2 + b^2, which should be equal to the hypotenuse of the other large triangle which is c^2. So a^2+b^2=c^2

Completed by: Joe Delinski joe.delinski, Cynthia Gonzalez cgnozal4, Suzie Moon smoo1244, Charmaine Ramos cramos2, Susan Zurawski sue652


text:
User talk:Geodad6

}

title: User talk:Geodad6


text:
User talk:Denyse~enwikiversity

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title: User talk:Denyse~enwikiversity


text:
Motivation and emotion/Book/2010/Student motivation theories/Definitions of key terms

Key Terms Defined:

Amotivation: A psychological state of diminished energy and focus. Amovational state is the polar opposite to a motivated state. Amotivational syndrome

Autonomy: A sense of independence and self-regulation where choices are free of external control

Attribution Theory: How individuals internalised perception of external events effect behaviour Attribution theory

Behaviouristic Perspective: A psychological school of thought that states that only objective behaviours can be studied empirically rather than inferences gathered from philosophising about unconscious states. Behaviorism

Competence: A persons perceptions of their repertoire of skills in reflection of a task. Similar to Maslow’s (1970) notion of self-efficacy.

Drive: A psychological process that energises and directs behaviour towards or away from an event. Similar to the notion of motivation.

External Incentive: An externally influencing event or reward offered to direct an individual’s behaviour in a specific manner i.e. shaping targeted behaviour through offering external rewards

Extrinsic Motivation: Behaviour that is directed and energised through an external incentives. For example, the student who remains on-task in order to receive a gold star from the teacher is demonstrating extrinsic motivational behaviour.

Flow: A state or condition an individual experiences when there is optimal match between the demands of a task and the individual’s skills need to fulfil those requirements (while at the same time the task offers a challenge but is not over demanding or boring)

An energised and focused state or condition an individual experiences when there is optimal match between the demands of a task and the individual’s skills need to fulfil those requirements (while at the same time the task offers a challenge but is not over demanding or boring)

Humanistic Perspective: A theoretical perspective, introduced in the early 1960’s, that emphasises the human nature of individuals such as the importance of self-actualization, choice, individual’s core value system (for example). Humanistic psychology

Intrinsic Motivation: Behaviour that is motivated by inherently intrinsic reinforcements such as interest, curiosity, pleasure and joy that a student may feel when working on an educational activity

Learning: The process of acquiring new skills, information, knowledge and is primarily facilitated by a motivational response. Learning

Learned helplessness: In an educational setting, an amotivational state where an individual (following a numerous experiences of uncontrollable events), gives up on a task or experiences because they have learnt that there efforts are often punished or go unrewarded.

Self-Efficacy: Bandura’s (1980) notion of an internalized process where an individual assess their skills in reflection of the task they are experiencing. Self-efficacy is part of the process of regulting and monitoring oneself.

Student Motivation: A process, where behaviours are directed and energise towards engagement in the educational experience. For example, a motivated student will demonstrate on-task behaviours, be responsive to others in the educational setting and strive to meet educational goals.

Student Engagement: Where an individual is seen to be committed to the educational process they are experiencing such that behaviours seem ‘on-task’. Some examples of student engagement is seen when a student appears attentive to class instruction, contributes to class discussions, adheres to school policy, concentrates on assigned projects, strives to meet educational goals, monitors own progress, shows initiative in asking for further information

Task Engagement: A student’s behaviour towards a specific task that demonstrates they are focused on meeting the demands of that task

Theory: In psychology, a theory is body of principles or idea presented in a proposed model that demonstrates relationships among proposed constructs, comprising of a premises, sometimes philosophically based, but nevertheless poses testable hypothesis(ses) that is based on observations of human behaviour.


text:
Western Philosophy: A History

Introduction
This course introduces the student to the subject of western philosophy, how it started and how it has progressed through to the current day. The aim is to give a sound and well-rounded understanding of philosophy methods and thinkers, in order to provide a base for future learning.

Ancient Philosophy

 * /Presocratic Philosophy/; /Socrates/; /Plato/; /Aristotle/
 * /Epicurus/; /Stoicism/; /Ancient skepticism/; /Judaism and Christianity/; /Neoplatonism/

 * /Ancient logic/; /Ancient epistemology/; /Ancient physics/; /Ancient metaphysics/; /Ancient philosophy of mind/; /Ancient ethics/; /Ancient theology/

Medieval Philosophy

 * /Augustine/; /Boethius/; /Late antiquity/; /Medieval Muslim and Jewish philosophy/; /Avicenna/; /Anselm/; /Abelard/; /Averroes/; Maimonides
 * /Robert Grosseteste and Albert the Great/; /Bonaventure//, /Thomas Aquinas/; /Siger of Brabant/; /Roger Bacon/; /Duns Scotus/; /William Ockham/; /Oxford Calculators/; /John Wyclif/
 * /Renaissance Platonism/; /Renaissance Aristotelianism/

 * /Medieval logic and language/; /Medieval epistemology/; /Medieval physics/; /Medieval metaphysics/; /Medieval philosophy of mind/; /Medieval ethics/; /Medieval theology/

Modern Philosophy

 * /16th-century philosophy/; /Giordano Bruno/; /Galileo/; /Francis Bacon/
 * /René Descartes/; /Thomas Hobbes/; /Cambridge Platonists/; /John Locke/
 * /Blaise Pascal/; /Nicolas Malebranche/; /Baruch Spinoza/; /Gottfried Leibniz/; /George Berkeley/
 * /David Hume/; /Adam Smith/; /Thomas Reid/; /Enlightenment/; /Jean-Jacques Rousseau/; /Christian Wolff/; /Gotthold Ephraim Lessing/
 * /Immanuel Kant/; /Johann Gottlieb Fichte/; /Friedrich Wilhelm Joseph Schelling/; /Georg Wilhelm Friedrich Hegel/

 * /Early modern epistemology/; /Early modern physics/; /Early modern metaphysics/; /Early modern philosophy of mind/; /Early modern ethics/; /Early modern political philosophy/; /Early modern theology/

 * /Jeremy Bentham/; /John Stuart Mill/; /Arthur Schopenhauer/; /Søren Kierkegaard/; /Dialectical Materialism/; /Charles Darwin/; /John Henry Newman/; /Friedrich Nietzsche/
 * /Charles Sanders Peirce/; /Gottlob Frege/; /William James/; /British Idealism/; /Bertrand Russell/; /Ludwig Wittgenstein/; /Logical positivism/
 * /Sigmund Freud/; /Edmund Husserl/; /Martin Heidegger/; /Jean-Paul Sartre/

 * /Modern logic/; /Modern philosophy of language/; /Modern epistemology/; /Modern metaphysics; /Modern philosophy of mind/; /Modern ethics/; /Modern aesthetics/; /Modern political philosophy/; /Modern theology/

Related Materials

 * Aristotle

title: Western Philosophy: A History


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Fluid Mechanics for MAP/Dimensional Analysis

>back to Chapters of Fluid Mechanics for MAP

>back to Chapters of Microfluid Mechanics

Motivation

Consider the steady flow around a submerged sphere. Say we are interested in the drag force on the sphere.

$$F_D=f(U_\infty, D, \rho, \mu)$$

Let's say 10Bold text measurement points are enough for one curve, which shows the effect of one parameter while keeping the others constant. In order to monitor the effect of all variables, $$10^4$$ separate measurements are needed. When each measurement takes 1/2 hr, all test series need 2.5 years.

By using dimensional analysis, we can reduce $$F_D$$ to an equivalent form:

$$\frac{F_D}{\rho U_\infty^2 D^2}=g\left(\frac{\rho U_\infty D}{\mu}\right)\ \ \ \text{or commonly written as} \ \ \ \underbrace{C_D}_{Drag\ coefficient}= \underbrace{g\left(\frac{\rho U_\infty D}{\mu}\right)}_{Reynolds\ number}$$

Both sides are dimensionless. Function $$g$$ is different than $$f$$ but it contains all necessary information.

Full scale models might be impossible or prohibitively expensive. There might be numerous number of parameters to be tested.

Under which conditions these tests are realistic? Are the both flows dynamically similar?

“Dimensional analysis is a method which reduces number and complexity of experimental variables which affect a given physical phenomenon, using a sort of compacting technique.” (White 1979)

Dimensional analysis (DA) helps us to formulate dimensionless forms of governing equations and simplify them by the determination insignificant terms:

$$\rho\left[\frac{\partial U_{j}}{\partial t} + U_{i}\frac{\partial U_{j}}{\partial x_{i}}\right] = -\frac{\partial P}{\partial x_{j}} + \frac{\partial}{\partial x_{i}}\left[\mu\left(\frac{\partial U_{j}}{\partial x_{i}} + \frac{\partial U_{i}}{\partial x_{j}}\right) - \frac{2}{3} \delta_{ij}\mu \frac{\partial U_{k}}{\partial x_{k}}\right]\rho g_{j}$$

$$\rho^{*}\left[St\frac{\partial U_{j}^{*}}{\partial t^{*}} + U^{*}_{i}\frac{\partial U_{j}^{*}}{\partial x_{i}^{*}}\right] = -Eu\frac{\partial P^{*}}{\partial x_{j}^{*}} + \frac{1}{Re}\frac{\partial}{\partial x^{*}_{i}}\left[\mu^{*}\left(\frac{\partial U^{*}_{j}}{\partial x^{*}_{i}} + \frac{\partial U_{i}^{*}}{\partial x^{*}_{j}}\right) - \frac{2}{3}\delta_{ij}\mu^{*}\frac{\partial U_{k}^{*}}{\partial x_{k}^{*}}\right] + \frac{1}{Fr}\rho^{*}g_{j}^{*}$$

$$\rho^{*} \left[St\frac{\partial U^{*}_{j}}{\partial t^{*}} + U_{i}^{*}\frac{\partial U^{*}_{j}}{\partial x^{*}_{i}}\right] = -\frac{\partial P^{*}}{\partial x^{*}_{j}}$$

This is the non-dimensional form of Euler equation.

Benefits of dimensional analysis (DA)
One obvious goal of our efforts should be to obtain the most information from the fewest experiments. Dimensional analysis (DA) is an important tool that often helps us to achieve this goal.

The benefits of this dimensionless representation are:

 * 1) We do not have to vary each variable: It is enough that we vary only Re and measure CD. As a result, we have enormous time and money saving.
 * 2) Dimensional analysis (DA) provides “scaling laws” which can convert data from a cheap small model to provide information about an expensive large prototype.
 * 3) DA helps our thinking and planning of an experiment or a theory.
 * 4) *It suggests dimensionless ways of writing equations.
 * 5) *One can find variables which can be discarded.

As a result, we can avoid to waste money and time while we use experimental and computational resources.

Derivation of dimensionless numbers

Buckingham Π theorem provides a method for computing sets of dimensionless numbers from the given variables, even if the form of the equation is still unknown.

Consider the steady flow around a sphere, which shows no separation. We will find out the dimensionless numbers governing this flow:

Step 1: List all the parameters involved. Let $$n$$ be the number of parameters.

$$F_D, U_\infty, D, \rho, \mu \ \rightarrow \ \ n=5$$

Step 2: Select a set of primary dimensions, e.g. $$M L tT q$$ (mass, length, time, temperature, elec. charge or $$FLtTq$$ (Force, length, ...).

$$M,L\ \text{and}\ t\ \ \ \text{are selected to be the primary dimensions.}$$

Step 3: List the dimensions of all parameters. Let $$r$$ be the number of primary dimensions.

$$F,\ U_\infty,\ D,\ \rho,\ \mu \ \ \rightarrow\ \  \frac{ML}{t^2},\ \frac{L}{t},\ L,\ \frac{M}{L^3},\ \frac{M}{Lt}\ \ \ \ \rightarrow \  \ r=3\ \text{primary dimensions}$$

Step 4: Select $$r$$ number of parameters, including all the primary dimensions. No two parameters should have the same net dimension differing by only a single exponent: for example do not include both $$L$$ and $$L^4$$.

$$U_\infty,\ D,\ \rho \ \ \rightarrow \ \ m = r = 3 \ \ \text{repeating parameters}$$

Step 5: Set up dimensional equations combining the parameters selected in Step 4 with each of the other parameters in turn to form $$n-m$$ dimensionless groups.

$$n-m=2$$ dimensionless groups will result. The dimensional equations are:

$$ \Pi_1\ =\ U_\infty^aD^b\rho^cF= \left(\frac{L}{t}\right)^a \left(L\right)^b \left(\frac{M}{L^3}\right)^c\left(\frac{ML}{t^2}\right)=M^0L^0t^0$$

$$ \Pi_2\ =\ U_\infty^dD^e\rho^f\mu= \left(\frac{L}{t}\right)^d \left(L\right)^e \left(\frac{M}{L^3}\right)^f\left(\frac{M}{Lt}\right)=M^0L^0t^0 $$

solving above equations delivers $$a=-2,b=-2,c=-1,d=-1,e=-1\ \text{and}\ f=-1$$. Hence

$$ \Pi_1=\frac{F}{\rho U_\infty^2 D^2}\ \ \ \text{and}\ \ \ \Pi_2=\frac{\mu}{U_\infty D \rho}=\frac{1}{Re} $$

The functional relationship $$\Pi_1=f(\Pi_2)$$ should be determined experimentally and/or numerically.

Flow around a cylinder and a sphere

In a very large Re range the wake is unsteady, quasi time-periodic.

Important dimensionless numbers are:

$$Re = \frac{\rho U d}{\mu} = \frac{U d}{\nu}$$ where $$\nu = \frac{\mu}{\rho}$$ is the kinematik viscosity.

$$St = \frac{f d}{U} = \frac{w d}{2\pi U}$$ where f is the dominant frequency of the wake structures.

$$C_{D} = \frac{F_{D}}{0.5\rho U^{2}A}$$ where A is the projected area to the flow.

$$\frac{L}{d}$$ is also important for flow around a cylinder.

Lift and drag forces on airfoils
$$C_{L} = \frac{F_{L}}{0.5 \rho U^{2}A} \rightarrow C_{L} = f(\alpha,Re)$$

$$C_{D} = \frac{F_{D}}{0.5 \rho U^{2}A} \rightarrow C_{D} = f(\alpha,Re)$$

Non-dimensionalization of Governing Equations
An isothermal and newtonian fluid flow is governed by the following equations

Conservation of Mass (continuity equation):

$$\frac{\partial \rho}{\partial t} +\frac{\partial \rho U_{i}}{\partial x_{i}} = 0$$

Conservation of Momentum

$$\rho\left[\frac{\partial U_{j}}{\partial t} + U_{i}\frac{\partial U_{j}}{\partial x_{i}}\right] = -\frac{\partial P}{\partial x_{j}} + \frac{\partial}{\partial x_{i}}\left[\mu\left(\frac{\partial U_{j}}{\partial x_{i}} + \frac{\partial U_{i}}{\partial x_{j}}\right) - \frac{2}{3} \delta_{ij}\mu \frac{\partial U_{k}}{\partial x_{k}}\right]\rho g_{j}$$

One can normalize each variable with a characteristic parameter, which has the same dimension

$$\ x_{i} = L_{c}x_{i}^{*}\ ,t = t_{c}t^{*}\ ,U_{j} = U_{c}U^{*}_{j}\ ,\ \rho = \rho_{c}\rho^{*}\ ,\ g_{j} = g_{c}g^{*}_{j}\ ,\ \mu = \mu_{c}\mu^{*}\ ,\ P=\Delta P_{c}P^{*}$$

Inserting these into the continuity equation

$$\frac{\partial \rho_{c}\rho^{*}}{\partial t_{c}t^{*}} + \frac{\partial \rho_{c}\rho^{*}U_{c}U^{*}_{i}}{\partial L_{c}x^{*}_{i}} = 0 \rightarrow \frac{\rho_{c}}{t_{c}}\frac{\partial \rho^{*}}{\partial t^{*}} + \frac{\rho_{c}U_{c}}{L_{c}}\frac{\partial \rho^{*}U_{i}^{*}}{\partial x^{*}_{i}} = 0$$

Dividing by $$\frac{\rho_{c}U_{c}}{L_{c}}$$

$$\underbrace{\frac{L_{c}}{t_{c}U_{c}}}_{St}\frac{\partial \rho^{*}}{\partial t^{*}} + \frac{\partial \rho^{*}U_{i}^{*}}{\partial x_{i}^{*}} = 0$$

where St: Strouhal Number

$$St=\frac{L_{c}}{t_{c}U_{c}} = \frac{\left(\frac{U_{c}}{t_{c}}\right)}{\frac{U_{c}^{2}}{L_{c}}} = \frac{local\ acceleration}{spatial\ acceleration}$$

Similarly, we can apply this technique to make the momentum equation non-dimensional:

$$\rho_{c}\rho^{*}\left[\frac{U_{c}}{t_{c}}\frac{\partial U_{j}^{*}}{\partial t^{*}} + \frac{U_{c}^{2}}{L_{c}}U_{i}^{*}\frac{\partial U_{j}^{*}}{\partial x^{*}_{i}}\right] = -\frac{\Delta P_{c}}{L_{c}}\frac{\partial P^{*}}{\partial x_{j}^{*}} + \frac{\mu_{c}U_{c}}{L_{c}^{2}}\frac{\partial}{\partial x_{i}^{*}}\left[\mu^{*}\left(\frac{\partial U_{j}^{*}}{\partial x_{i}^{*}} + \frac{\partial U_{i}^{*}}{\partial x_{j}^{*}}\right) - \frac{2}{3}\delta_{ij}\mu^{*} \frac{\partial U_{k}^{*}}{\partial x_{k}^{*}}\right] + \rho_{c}g_{c}\rho^{*}g^{*}_{j}$$

Dividing by $$\frac{\rho_{c}U_{c}^{2}}{L_{c}}$$

$$\rho^{*}\left[\underbrace{\frac{L_{c}}{t_{c}U_{c}}}_{St}\frac{\partial U_{j}^{*}}{\partial t^{*}} + U_{i}^{*}\frac{\partial U_{j}^{*}}{\partial x^{*}_{i}}\right] = \underbrace{-\frac{\Delta P_{c}}{\rho_{c}U_{c}^{2}}}_{Eu}\frac{\partial P^{*}}{\partial x_{j}^{*}} + \underbrace{\frac{\mu_{c}}{\rho_{c}U_{c}L_{c}}}_{\frac{1}{Re}}\frac{\partial}{\partial x_{i}^{*}}\left[\mu^{*}\left(\frac{\partial U_{j}^{*}}{\partial x_{i}^{*}} + \frac{\partial U_{i}^{*}}{\partial x_{j}^{*}}\right) - \frac{2}{3}\delta_{ij}\mu^{*} \frac{\partial U_{k}^{*}}{\partial x_{k}^{*}}\right] + \underbrace{\frac{g_{c}L_{c}}{U_{c}^{2}}}_{\frac{1}{Fr}}\rho^{*}g^{*}_{j}$$

Eu: Euler Number $$=\frac{\Delta P_{c}}{\rho_{c}U_{c}^{2}} = \frac{static\ pressure}{dynamic\ pressure}$$

Re: Reynolds Number $$=\frac{\rho_{c}U_{c}L_{c}}{\mu_{c}} = \frac{\frac{\rho_{c}U_{c}^{2}}{L_{c}}}{\frac{\mu_{c}U_{c}}{L_{c}^{2}}} = \frac{inertial\ forces}{viscous\ forces}$$

Fr: Froude Number $$=\frac{U_{c}^{2}}{g_{c}L_{c}} = \frac{\frac{\rho_{c}U_{c}^{2}}{L_{c}}}{\rho_{c}g_{c}} = \frac{inertial\ forces}{gravitational\ forces}$$

The resulting dimensionless conservation of mass and of momentum equations are:

$$St\frac{\partial \rho^{*}}{\partial t^{*}} + \frac{\partial \rho^{*}U_{i}^{*}}{\partial x_{i}^{*}} = 0$$

$$\rho^{*}\left[St\frac{\partial U_{j}^{*}}{\partial t^{*}} + U_{i}^{*}\frac{\partial U_{j}^{*}}{\partial x^{*}_{i}}\right] = -Eu\frac{\partial P^{*}}{\partial x_{j}^{*}} + \frac{1}{Re}\frac{\partial}{\partial x_{i}^{*}}\left[\mu^{*}\left(\frac{\partial U_{j}^{*}}{\partial x_{i}^{*}} + \frac{\partial U_{i}^{*}}{\partial x_{j}^{*}}\right) - \frac{2}{3}\delta_{ij}\mu^{*} \frac{\partial U_{k}^{*}}{\partial x_{k}^{*}}\right] + \frac{1}{Fr}\rho^{*}g^{*}_{j}$$

These equation indicates that solution of similar problems for two problems, say A and B, can only be same when the dimensionless numbers are equal:

$$St_{A} = St_{B},\ Eu_{A} = Eu_{B},\ Re_{A} = Re_{B},\ Fr_{A} = Fr_{B}$$

Furthermore, we should warrant the similarity of the boundary conditions and this requires geometric similarity.

Selection of characteristic variables and Simplifications to governing equations
Consider the momentum transport equation:

$$\rho^{*}\left[St\frac{\partial U_{j}^{*}}{\partial t^{*}} + U_{i}^{*}\frac{\partial U_{j}^{*}}{\partial x^{*}_{i}}\right] = -Eu\frac{\partial P^{*}}{\partial x_{j}^{*}} + \frac{1}{Re}\frac{\partial}{\partial x_{i}^{*}}\left[\mu^{*}\left(\frac{\partial U_{j}^{*}}{\partial x_{i}^{*}} + \frac{\partial U_{i}^{*}}{\partial x_{j}^{*}}\right) - \frac{2}{3}\delta_{ij}\mu^{*} \frac{\partial U_{k}^{*}}{\partial x_{k}^{*}}\right] + \frac{1}{Fr}\rho^{*}g^{*}_{j}$$

For very low St, i.e. when tc < Uc/Lc, in other words when local acceleration much smaller than spatial acceleration, the flow can be accepted to be steady. Thus the first term at LHS can be neglected, i.e. the momentum equation reads

$$\rho^{*}U_{i}^{*}\frac{\partial U_{j}^{*}}{\partial x^{*}_{i}} = -Eu\frac{\partial P^{*}}{\partial x_{j}^{*}} + \frac{1}{Re}\frac{\partial}{\partial x_{i}^{*}}\left[\mu^{*}\left(\frac{\partial U_{j}^{*}}{\partial x_{i}^{*}} + \frac{\partial U_{i}^{*}}{\partial x_{j}^{*}}\right) - \frac{2}{3}\delta_{ij}\mu^{*} \frac{\partial U_{k}^{*}}{\partial x_{k}^{*}}\right] + \frac{1}{Fr}\rho^{*}g^{*}_{j}$$

When we select: $$\Delta P_{c} = \rho_{c}U_{c}^{2} \rightarrow Eu = 1$$

$$\rho^{*}\left[St\frac{\partial U_{j}^{*}}{\partial t^{*}} + U_{i}^{*}\frac{\partial U_{j}^{*}}{\partial x^{*}_{i}}\right] = -\frac{\partial P^{*}}{\partial x_{j}^{*}} + \frac{1}{Re}\frac{\partial}{\partial x_{i}^{*}}\left[\mu^{*}\left(\frac{\partial U_{j}^{*}}{\partial x_{i}^{*}} + \frac{\partial U_{i}^{*}}{\partial x_{j}^{*}}\right) - \frac{2}{3}\delta_{ij}\mu^{*} \frac{\partial U_{k}^{*}}{\partial x_{k}^{*}}\right] + \frac{1}{Fr}\rho^{*}g^{*}_{j}$$

In one phase flows, this kind of selection of the characteristic pressure is common. Thus, one can see that the pressure gradient can not be neglected.

For flows in which cavitation occurs, one can use the vapor pressure of the fluid in the characteristic pressure term:

$$\Delta P_{c} = P - P_{v} \rightarrow Eu = Ca = Cavitation\ Number = \frac{P - P_{v}}{\rho_{c}U_{c}^{2}}$$

$$\rho^{*}\left[St\frac{\partial U_{j}^{*}}{\partial t^{*}} + U_{i}^{*}\frac{\partial U_{j}^{*}}{\partial x^{*}_{i}}\right] = -Ca\frac{\partial P^{*}}{\partial x_{j}^{*}} + \frac{1}{Re}\frac{\partial}{\partial x_{i}^{*}}\left[\mu^{*}\left(\frac{\partial U_{j}^{*}}{\partial x_{i}^{*}} + \frac{\partial U_{i}^{*}}{\partial x_{j}^{*}}\right) - \frac{2}{3}\delta_{ij}\mu^{*} \frac{\partial U_{k}^{*}}{\partial x_{k}^{*}}\right] + \frac{1}{Fr}\rho^{*}g^{*}_{j}$$

The Froude number is important when gravity starts to play an important role, i.e. when buoyancy effects occurs due to density differences and, especially for multi-phase flows. For example, we deal with a two-phase flow, for which two fluids can be accepted to be immiscible, like water and air in the spill way of a dam. In order to introduce the effect of density into the Fr number, the characteristic gravity in the Fr number can be selected to be:

$$g_{c} = g \frac{\rho_{2} - \rho_{1}}{\rho_{2}}$$ where $$\rho_{2}>\rho_{1} \rightarrow Fr = \left(\frac{1}{g(\rho_{2} - \rho_{1})}\right)\left(\frac{\rho_{2}U_{c}^{2}}{L_{c}}\right) = \frac{inertial\ forces}{bouyancy\ forces}$$

Thus, when the density difference becomes negligible, Fr becomes very large and the momentum equation reduces to:

$$\rho^{*}\left[St\frac{\partial U_{j}^{*}}{\partial t^{*}} + U_{i}^{*}\frac{\partial U_{j}^{*}}{\partial x^{*}_{i}}\right] = -Eu\frac{\partial P^{*}}{\partial x_{j}^{*}} + \frac{1}{Re}\frac{\partial}{\partial x_{i}^{*}}\left[\mu^{*}\left(\frac{\partial U_{j}^{*}}{\partial x_{i}^{*}} + \frac{\partial U_{i}^{*}}{\partial x_{j}^{*}}\right) - \frac{2}{3}\delta_{ij}\mu^{*} \frac{\partial U_{k}^{*}}{\partial x_{k}^{*}}\right]$$

The Froude number can be important also for one phase flows with large density gradients. In order to introduce this effect, the characteristic gravity in the Fr number can be written by using the density gradient:

$$g_{c} = g\left(\frac{dp}{dx}\right)_{c} \frac{L_{c}}{\rho_{c}} \rightarrow Fr = \left(\frac{1}{g\left(\frac{dp}{dx}\right)_{c}L_{c}}\right)\left(\frac{\rho_{c}U_{c}^{2}}{L_{c}}\right) = \frac{initial\ forces}{bouyancy\ forces}$$

This form of the Fr number can be used in the field of convective heat transfer at which the characteristic velocities are very low such that buoyancy dominates the flow motion. In the mixing of fluids having different densities, this form can also be one of the important non-dimensional numbers.

Another simplification can be made: when the Re number of the flow is very high, the viscous stresses can be eliminated in regions where the velocity gradients are low. For example, for the cruising aircrafts the Re number is in the order of millions. Therefore, it is common approach, to use the inviscid form of Navier Stokes equations for flow simulations around cruising aircrafts, except in the vicinity of the aircraft body. For such a flow, we can neglect the gravitational effects, since Fr becomes very large. Moreover, Eu can be accepted to be 1 as shown before. Thus, the governing momentum equation reduces to the non-dimensional form of Euler equations:

$$\rho^{*}\left[St\frac{\partial U_{j}^{*}}{\partial t^{*}} + U_{i}^{*}\frac{\partial U_{j}^{*}}{\partial x^{*}_{i}}\right] = -\frac{\partial P^{*}}{\partial x_{j}^{*}}$$

Similarity considerations
Once the variables are selected and the dimensional analysis is performed, the experimenter seeks to achieve similarity between the model tested and the prototype to be designed.

Flow conditions for a model test are completely similar if all relevant dimensionless parameters $$(\Pi_{1},\Pi_{2},...,\Pi_{n})$$have the same corresponding values for the model and the prototype (White):

$$\Pi_{1model} = \Pi_{1prototype}, \Pi_{2model} =\Pi_{2prototype},..., \Pi_{n\ model} = \Pi_{n\ prototype}$$

But this is easier said than done.

Instead of complete similarity, the engineering literature speaks of particular types of similarity:

 * Geometric similarity
 * Kinematic similarity
 * Dynamic similarity
 * Thermal similarity

Geometric similarity
A model and a prototype are geometrically similar if and only if all body dimensions in all three coordinates have the same linear-scale ratio.

All angles are preserved in geometric similarity. All flow directions are preserved. The orientations of model and prototype with respect to the surroundings must be identical.

Kinematic similarity
The motions of two systems are kinematically similar if homologous particles lie at homologous points at homologous times.

In other words, kinematic similarity requires that the model and prototype have dependent length-scale and time-scale ratios. This requires, for example, similarity in Re or Mach number. The result is that the velocity-scale ratio will be the same for both:

One exception, in which the length and the time-scales becomes independent is the frictionless low-speed flow, without any free-surface.

$$\displaystyle D_{model} = \alpha D_{prototype}$$

and

$$\displaystyle V_{model} = \beta V_{prototype}$$

For a frictionless two-phase flow, the Froude numbers should be same, thus

From:

$$Fr_{model} = \frac{V^{2}_{model}}{gL_{model}} = \frac{V^{2}_{prototype}}{gL_{prototype}} = Fr_{prototype}$$

and,

$$L_{model} = \alpha L_{prototype} \rightarrow V_{model} = \sqrt{\alpha}V_{prototype}$$

we obtain:

$$\frac{L_{model}}{L_{protorype}} = \alpha$$

and,

$$\frac{T_{model}}{T_{prototype}} = \frac{L_{model}/V_{model}}{L_{prototype}/V_{prototype}} = \sqrt{\alpha}$$

Dynamic similarity
Dynamic similarity exists when the model and the prototype have the same length scale ratio, time-scale ratio, and force-scale (or mass-scale) ratio.

Mathematically, Newton’s law for any fluid particle requires that the sum of the pressure force, gravity force, and friction force equal the acceleration term, or inertia force:

$$\displaystyle F_{inertia} = F_{body} + F_{pressure} + F_{viscous}$$

The dynamic-similarity ensure that each of these forces will be in the same ratio and have equivalent directions between model and prototype.

Again geometric similarity is a first requirement. Then dynamic similarity exists, simultaneous with kinematic similarity, if the model and prototype force and pressure coefficients are identical. This is ensured if:

 * 1) For compressible flow, the model and prototype Reynolds number and Mach number and specific-heat ratio are correspondingly equal.
 * 2) For incompressible flow
 * 3) *With no free surface: model and prototype Reynolds numbers are equal.
 * 4) *With a free surface: model and prototype Reynolds number, Froude 	number, and (if necessary) Weber number and cavitation number are correspondingly equal.

Incomplete similarity

Not always, it is easy to obtain complete dynamic similarity.

Consider the drag force on the surface of the ship shown. To have complete dynamic similarity between test model and prototype:

$$Fr_{model} = Fr_{prototype},\ Re_{model} = Re_{prototype}$$

For a 100 times downscaled test model, the Fr number similarity requires:

$$L_{model} = \frac{1}{100}L_{prototype} \rightarrow \frac{100V^{2}_{model}}{gL_{prototype}} = \frac{V^{2}_{prototype}}{gL_{prototype}} \rightarrow V_{model} = 10V_{prototype}$$

Thus, the Re number similarity can only be obtained at the same time

$$Re_{model} = \frac{V_{model}L_{model}}{\upsilon_{model}} = \frac{10V_{prototype}L_{prototype}}{\upsilon_{model}100} = \frac{U_{prototype}L_{prototype}}{\upsilon_{prototype}} = Re_{prototype}$$

$$\rightarrow \upsilon_{model} = 0.1\upsilon_{prototype}$$

Only liquid which has less kinematic viscosity than water is mercury, and even with that fluid Re similarity can not be established. Since Froude number becomes more important than Re, in the experiments scaling is done using only Fr number.

Flow Visualization in Inhalers: Inhaler in the Water Tunnel

The flow in the Handihaler is unsteady. Thus, visualization is preferred in order to understand the flow character within the inhaler. The up-scaled models in water are used to slow down the speed of the flow so that we can record streaklines with a high-speed camera.

Flow in the Handihaler® and Respimat®

After preliminary visualization experiments, scenarios were planned. The flow region was divided into sub-regions. Dye injection holes were built in these sub-regions. Around 100 visualizations were made for original and modified geometries of Handihaler and Respimat.

The mechanism of particle discharge from the capsule is discovered.

With the help of the prototypes we could easily study the effect of different flow and geometrical parameters on the unsteady flow within the Handihaler

The visualization is used to validate numerical simulations.

Experiments versus simulations
Complete design of inhalers can be made by the utilization of non-dimensional numbers.

Conclusions

 * Dimensional analysis and similitude are two important tools in experimental research.
 * While analyzing a phenomenon one can use the dimensionless numbers to judge the importance of the governing effects.
 * Similarity considerations should be carefully performed to obtain useful results, especially when similarity is not complete.


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Fluid Mechanics for MAP/Energy Considerations

>back to Chapters

Losses and Energy consideration
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Remember the conservation of energy equation:

$$\dot{Q} + \dot{W}_{schaft} + \dot{W}_{shear} + \dot{W}_{other} = \frac{\partial}{\partial t}\int_{CV}{e\rho dV} + \int_{CS}{\left(e + \frac{P}{\rho}\right)\rho U_{i}n_{i}dA}$$

where

$$e = u + \frac{U^{2}}{2} + gx_{2} = u+\frac{U_{i}U_{i}}{2} + gz$$

The velocity does not have necessarily a plug profile at the inlet and outlet of a control volume and the cross sectional areas might also differ at the inlet and outlet of a CV. The variations in the cross sections of flow conduits and the velocity profiles have to be accounted to calculate the correct pressure drop due to the viscous losses. Therefore, the surface integrals has to be calculated by considering these variations:

$$\dot{Q} = \int_{CS}{\left(e + \frac{P}{\rho}\right)\rho U_{i}n_{i}dA} = \dot{m}(u_{2} - u_{1}) + \dot{m}\left(\frac{P_{2}}{\rho} - \frac{P_{1}}{\rho}\right) + \dot{m}g(z_{2} - z_{1}) +$$

$$\int_{A_{2}}{\frac{U_{2}^{2}}{2}\rho U_{2i}n_{2i}dA} - \int_{A_{2}}{\frac{U_{1}^{2}}{2}\rho U_{1i}n_{1i}dA}$$

For the calculation of the profile effects one can use the kinetic energy coefficient $$\displaystyle \alpha$$ as follows:

$$\int_{A}{\frac{U^{2}}{2}\rho U_{i}n_{i}dA} =\alpha \dot{m}\frac{\bar{U}^{2}}{2}$$

For fully developed laminar velocity profile, where

$$U = 2\bar{U}\left[1 - \frac{r}{R}^{}2\right]$$

$$\displaystyle \alpha = 2$$

For fully developed turbulent flow, due to relatively flat velocity profile, $$ \displaystyle \alpha \approx 1$$

$$\dot{Q} = \dot{m}(u_{2} - u_{1}) + \dot{m}\left(\frac{P_{2}}{\rho} - \frac{P_{1}}{\rho}\right) + \dot{m} \left(\alpha_{2}\frac{\bar{U}^{2}_{2}}{2} - \alpha_{1}\frac{\bar{U}^{2}_{1}}{2}\right)$$

Rearranging the energy equation:

$$\underbrace{\frac{P_{1}}{\rho} + \alpha_{1}\frac{\bar{U}^{2}_{1}}{2} + gz_{1} = \frac{P_{2}}{\rho} + \alpha_{2}\frac{\bar{U}^{2}_{2}}{2} + gz_{2}}_{A} + \underbrace{(u_{2} - u_{1}) - \frac{\dot{Q}}{\dot{m}}}_{B}$$

The conversion of mechanical energy (A) and the irreversible conversion and loss of mechanical energy with viscous dissipation (B) can be better distinguished.

Total pressure drop
Finally the total pressure drop due to viscous effects can be written as

$$\underbrace{P_{1} + \alpha_{1}\rho\frac{\bar{U}^{2}_{1}}{2} + \rho gz_{1} = P_{2} + \alpha_{2}\rho\frac{\bar{U}^{2}_{2}}{2} + \rho gz_{2}}_{Mechanical\ Energy} + \underbrace{\Delta P_{v}}_{Pressure\ drop\ due\ to\ viscous\ losses}$$

$$\Delta P_{v} = (P_{1} - P_{2}) + \rho \left(\alpha_{1}\frac{\bar{U}^{2}_{1}}{2} - \alpha_{2}\frac{\bar{U}^{2}_{2}}{2}\right) + \rho g(z_{1} - z_{2})$$

Major and minor pressure losses
The total pressure drop consist of major losses, which are due to frictional effects in fully developed flow in constant area tubes, and minor losses, which occur at entrances, sudden area changes, bends, elbows, fittings, valves, contraction and diffusers, etc..

Major losses in pipe flows

The pressure drop along a pipe, which has no elevation change, can be written as:

$$\displaystyle \Delta P_{v} = (P_{1} - P_{2})$$

Nondimensionalizing with the kinetic energy per unit mass of the flow:

$$\zeta = \frac{\Delta P_{v}}{0.5 \rho\bar{U}^{2}} = \frac{L}{D}\ f\left(Re, \frac{e}{D}\right)$$

where $$\zeta$$ is the loss coefficient and f is the friction factor.

Non-circular ducts
For non circular ducts, which appears in air conditioning, heating and ventilating applications, an equivalent diameter is calculated so that correlations for circular pipes can be utilized.

This equivalent diameter is called as hydraulic diameter:

$$\displaystyle D_{h} = \frac{4 \times Area}{Perimeter} = \frac{4A}{P}$$

Thus for a rectangular duct with a width of b and height of h:

$$D_{h} = \frac{4bh}{2(b + h)}$$

And for a square duct

$$\displaystyle D_{h} = h$$

Minor losses
For the other components which might cause pressure drop, the loss coefficient (resistance coefficient) can similarly be defined:

$$\zeta = \frac{\Delta P_{v}}{0.5\rho \bar{U}^{2}}$$

However, for the vast variety of geometries, it is not possible to obtain relations as for the pipe flows. Thus, $$\zeta$$ for each component is measured and documented.

Calculation of total loss in a single pipe system
A single pipe system may have many minor losses. Since all are correlated with $$0.5\rho\bar{U}^{2}$$, they can be summed into a single total system loss if the pipe has constant diameter:

$$\Delta P_{total} = \Delta P_{pipe} + \sum \Delta P_{components} =$$

$$0.5\rho\bar{U}^{2}\left[\frac{L}{D}f + \sum_{i}\zeta_{i}\right]$$

However, we must sum the losses separately, if the pipe size changes, i.e. mean velocity changes.

Multiple path system

The solution for pipe network problems is often carried out by use of node and loop equations similar in many ways to that done in electrical circuits.

 * 1) The net flow into any junction must be zero.
 * 2) The net head loss around any closed loop must be zero. At each junction there is one single pressure.
 * 3) All head losses must satisfy the major and minor-loss friction correlations.

In such problems the pipes and components might have different areas, thus pressure loss at each pipe and component should be calculated separately. Moreover, there can be components with unknown properties.

For such cases it is found useful to formulate the pressure loss as a function of the flow rate:

$$\Delta P = \zeta 0.5 \rho \bar{U}^{2} = \underbrace{\zeta \frac{0.5\rho}{{A}^2}}_{K}Q^{2} = KQ^{2}$$

Resultant loss coefficients, Kr, for serial and parallel connections
Serial connection: Flow rate is the same for all components

$$\displaystyle K_{r} = K_{1} + K_{2}$$

$$\displaystyle \Delta P_{r} = K_{r}Q^{2} = \Delta P_{1} + \Delta P_{2} = (K_{1} + K_{2})Q^{2}$$

Parallel connection: Pressure drop is the same inlet and outlet junctions

$$\displaystyle K_{r} = \frac{K_{1}K_{2}}{K_{1} + K_{2} + 2\sqrt{K_{1}K_{2}}}$$

$$\Delta P_{r} = K_{r}Q^{2}_{total} = K_{1}Q_{1}^{2} = K_{2}Q^{2}_{2}$$

$$\displaystyle Q_{total} = Q_{1} + Q_{2}$$

Design of a dry powder inhaler
The utilization of the explained concepts will be demonstrated in the class by developing an inhaler from scratch. The basic steps of the design are as follows:
 * 1) Make a conceptual design
 * 2) Decide pressure drop-flow rate characteristic of the inhaler (Inhaler patient matching)
 * 3) Dimension each component of the inhaler to ensure
 * 4) *Pressure drop-flow rate characteristic
 * 5) *Full emptying of the blister
 * 6) *Sufficient dispersion of the powder composition

An overview of dry powder inhaler design could be found in in the following publication

Reference

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