Bipolar Equation

2014-11-01 00:00:00

Hello. My name is Bernoulli. Bernoulli Equation, to be more precise. One of the best established and most beautiful equations in the history of science. I am the spirit of the elegant curves on the aircraft. I am at work in every breath living creatures take. I am at the heart of the mighty winds of cyclones.

...

I am sorry, give me a minute.

...

You see, I easily lose my head. So, I need to use medications to balance my mood. Oh my God! That was supposed to be a secret, and it just slipped from my mouth to an admirer. Look, I suspect that once you learn my most-hidden aspects, my reputation with you will be ruined, and you won't love me anymore. And it really distresses me to have failed your high expectations of an equation like me.

Nevertheless, it is such a heavy burden to suppress one side of mine while constantly showing the other. Yes, being an equation, I am well balanced and very clear about my views. I am one of those lucky theories that have a sound mathematical foundation, and this is why even the psychologists could not suspect a grain of problem due to my childhood. But I am telling you: I am bipolar....

Well, wait a minute! Who are you, and why would I share my secret with you? Go away with your own business, and leave me alone. I mean, I don't want to disrespect you, but this is my private life, you know! ...

No, no, wait, wait... Give me a second... Let me take my medication!

I know it sounds ridiculous if an equation complains of a disorder. How can you be an equation if you have a disorder, and how can you be in disorder if you are an equation? Well believe it or not, that's exactly what I have been suffering from for such a long time. I am both orderly and disorderly; just like light is both wave and particle at the same time....

Hey, why are you looking at me like that? I told you to leave me alone. It was you who insisted on waiting and learning my secret. I don't care if your designs are ruined or whatever. Actually, I'm the one who should be blamed. Why did I trust a person whom I met just a few minutes ago? I am like a fish that is eager to eat the worm on the hook. Such an idiot, I am...

Hello? Are you still there? I think I am losing my head again. Where are my pills?

Look, this is not easy for me to do, because in my routine life, people are obsessed with my equation aspect. And, they are passing this obsession from generation to generation. Every time I meet a young brain, my dream to finally meet an unconditioned mind fails. I am always too late to reach the fresh and eager minds. People always manage to find them before me, and instruct them to treat me as a well-established equation. No matter how much I want to show otherwise, they refuse to see my imperfect side, even if the experiments hit them in the face.

Why do they keep doing it? Much like a missionary, they are converting people into the belief of the Bernoulli Equation. I am telling you guys: I am bipolar, and I have imperfections, too!

What am I doing?! You are not going to believe me anyway. I am telling you my deepest secret, and making myself vulnerable as can be, but I still can't wake you from a dream. You don't want to wake up anyway; why would you? Do you have any other theory to believe in? Do you have another equation with which to orient yourself in an ocean of unknowns? Go on with your sleep, have nice dreams...

Unfortunately, I don't have the same luxury as you. My continual tumbling between the two aspects of my reality never allows me to dream... Sometimes, I feel so energetic, so elated. I feel part of everything in the universe, and fall into an ecstasy by realizing that I contain everything in me. And the flow of time stops; we all become one and at peace with each other. No need to rush anywhere, no need to do anything... Just be...

But then my other side kicks in. I feel depressed under the burden of my duties and deadlines. I find myself in an ever faster pace of life, where there is no time to "just be". There is always an action being commanded; there is always a motivation behind exchanges with others. The universe appears to be made of distinct individuals, like broken pieces of glass. Each is headed to a target of its own, unaware of any union. When life is filled with such a merciless momentum, what is more meaningful than getting rid of my life altogether?

Then, in that gloom, as my dizziness fades, my energetic side starts shining. By virtue of having been created in the same story, I focus on myself to read the universe. I realize that I carry the traits of anything and everything else around me. Once again, I start breathing the life that is gushing forth from me.

As I lose my conscience by getting high in life, I get stuck with the fact that losing my conscience defeats the purpose of being whole. You cannot hold onto the entities whose existence you fail to recognize. I start criticizing myself for disrespecting those around me. I blame myself for being such an addict and a loser; the shame of creation...

You see, there is no end to my ups and downs. As a remedy to these continuous head losses, I am using medications, but they have side effects. I am constantly growing fat, and losing my beauty. I hope one day, someone is going to be inspired with another equation that can take over my duties. Then I can retire from this bipolar life, and be mentioned in the scientific stories as a respectable grandparent...

The Bernoulli Equation given in its simplest form contains a pressure term (P) and a velocity term (1/2 (pV)^2), the sum of which is constant along a streamline:

P+1/2 pV^2=P_0

Thus, pressure and velocity work inversely with each other. As one increases, the other decreases.

The Bernoulli Equation can be derived in two independent ways. First, one can start with the conservation of energy (1st law of thermodynamics), and then follow some assumptions to end up with the Bernoulli Equation. Energy has a scale but no direction, and can be transformed from one form to another. Therefore, everything in the universe can essentially be converted into each other, just like the pressure and velocity terms in the Bernoulli Equation.

The second way to reach the Bernoulli Equation is based on the momentum equation (Newton's second law). Again using few assumptions, one can achieve the nice and simple Bernoulli Equation. Unlike energy, momentum has both scale and direction. Any changes to this directionality require a forceful interaction between two objects.

Having its base on two fundamental laws of physics, hence reflecting both energy and momentum, makes the Bernoulli Equation one of the strongest and most beautiful equations of science, but not perfect!

The underlying theory of the Bernoulli Equation requires that the density (p) of the fluid be constant - in other words, incompressible. Therefore, the Bernoulli Equation fails to accurately explain the flow of gases at high speeds, which involves density changes. Similarly, the effects of viscosity are neglected in the foundations of the Bernoulli Equation. This means that the variation of pressure and velocity near a solid object cannot be explained by the Bernoulli Equation.

As a result of these imperfections, the Bernoulli Equation describes a fluid as a pile of metal sheets, and explains a flow as the sliding and bending of these metal sheets around each other. Clearly, this is not what a fluid is and not what happens in a real flow. Therefore, predictions of the Bernoulli Equation are inherently wrong, especially when there are sharp turns or obstructions in the way of the fluid.

In order to compensate for some of these deficiencies, a concept known as head loss (hloss) is introduced. Head loss is experimentally measured and then artificially added to the Bernoulli Equation, which then becomes:

P+1/2 pV^2+pgh_(loss,turns)+pgh_(loss,obstructions)+pgh_(loss,wall shear)=P_0

The final result is an equation that is useful but repulsively oversized, and it is certainly not well-established in theory.

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