The Art of Approximation in Engineering and Science

Approximation. Aren't we supposed to be exact ? This counter intuitive statement raises a lot of doubts and questions . Like , how can approximate models be at all useful? or what makes some models more useful than others?
For the first one , an approximate model is only something our systems , intellect can decipher . So when we represent or model any process in the world , we need to throw away things that dont matter . Say for example , if you want to analyse the trajectory of a piece of stone thrown at an angle of 45 degrees , including air viscosity , wind speed , coriolis force , etc. all are fine but simply way to difficult to comprehend and also a waste of time distracting the observer from the main problem .
In this post lets explore this art and apply it to a certain physical phenomenon .
One of my favourites : Drag.
a sphere of radius $R$ falling through a fluid of viscosity $\nu$ .
Lets start out by describing the physics exactly. The terminal velocity of the sphere can be calculated by solving the partial differential equations for fluid flow - namely the navier stokes equations . For incompressible flow ,
\[ \frac{\partial v}{\partial t} + (\nabla . v) = - \frac{1}{\rho} \nabla p + \nu \nabla^{2} v\]
\[\nabla . v = 0\]
Here $v$ is the fluid velocity , $p$ is the pressure , $\rho$ is the density and $\nu$ is the kinematic viscosity . All the equations are coupled partial differential equations and three of them are non - linear . Closure to these equations can be described if the boundary conditions are accurately known , which can also be written in terms of primitive variables . Further more someone did require a decent program to discretize the equations and solve it , here in this case the famous SIMPLE(Semi-Implicit Method for Pressure Linked Equations) algorithm .All this for a simple analysis of a sphere through a fluid !!!!!!
Relax there is a naive but practical way out .

Dimensional Analysis : To use dimensional analysis follow the usual steps of the buckingham-pi theorem which can be found in any standard fluid dynamics textbook . Make the appropriate groups and finally you will get a dimensionally balanced equation of the form :
speed = some function of groups formed during the Buckingham - Pi analysis . Performing a couple of experiments one can determine the conversion factors at a reasonably accurate value .

A Simpler Approach : (Dont confuse this with the previous SIMPLE algo ..)
We have to calculate the terminal velocity or speed . The 'terminal' word suggest at a final instant i.e. after long long time . It indicates that the speed has become constant , i.e. no net force acts on it . That is weight balances the drag and buoyancy forces at this time . So ,
\[ F_{gr} = \frac{4\pi}{3}r^{3}g \rho_{sphere} \]
In accordance with Archimedes principle ,the buoyant force is ,
\[ F_{bu} = \frac{4\pi}{3}r^{3}g \rho_{fluid} \]
The drag calculation can be easily done with another dimensional analysis with the Buckingham - Pi theorem the force can be estimated to be a function of viscosity , speed , density of fluid and radius of sphere . Fortunately , the British Mathematician Stokes , the first to derive its value , found that ,
\[ F_{dr}=6\pi\eta r v\]

Now lets do the approximation part .. ;) .
To balance out the forces its clear that ,
\[ F_{gr}=F_{dr}+F_{bu}\]
Lets ignore their constant terms , namely $6 \pi$ etc. To specify how accurate we are , we can assume to be first order accurate .
Hence ,
\[ F_{gr} \approx F_{dr}+F_{bu} \]
Rearranging the terminal speed is then ,
\[ v \approx \frac{gr^{2}}{\nu} ( \frac{\rho_{sp}}{\rho_{fl}} - 1 )\]

This is a very decent approximation to our system . A factor of 2/9 is achieved if the constants from Stokes equation , etc. are not neglected . However this gives a fairly reasonable value to the terminal speed .

The above is just an example of how approximation , dimensional analysis can solve certain problems in engineering and science . However this method is used just for a preliminary design of a system to get a feel of the behaviour of the system . Empirical data and equations should not be neglected at any cost .

Source of Information : MIT OCW .
Image Sources : Here , here and here .

Wireless and Beyond

My first post is about something very simple, in the news today, and extremely compelling. Wireless Electricity. It had a typical story-book conception, an MIT professor who was irritated by his wife's cellphone beeping on low battery, and a group of brilliant MIT students helping him realize his dream. Although the MIT experimental prototype was relatively bulky, its tremendous potential was immediately realized and, in the great American fashion, a big company called WiTricity Corp. was set up.


The concept is pretty simple- Resonant Energy Transfer. The key was the working of a transformer, in which energy is transferred from the primary coil to the secondary coil without actual contact, ie. magnetic coupling. But the primary and secondary coils are wound extremely close to each other, their insulation touching, so as to avoid losses. In resonant energy transfer, the primary coil, which we will call the transmitter, and the secondary coil, the receiver, are tuned to a mutual resonant frequency. Thus when the transmitter transmits magnetic waves at this frequency, the receiver picks it up. But even the resonance effect cannot help transfer energy over larger distances, so you'd be mistaken if you're imagining charging your cellphone with the transmitter in the next room, or across the same room. It can transmit energy to a distance a few times the size of the transmitter. So all it does, basically, is eliminate the need for cables. They also talk about the environment, and how the need for batteries is eliminated.


What we have in the end, is an extremely commercial product that will, probably, hit households at large within the next couple of decades. The following video shows a demonstration:





But what really caught my eye when I was reading about wireless electricity was, initially, the work of Nikola Tesla and later on, in fact currently, the work of Prof. Konstantin Meyl. Tesla had the vision of a global power grid based on the facts that:
1. The Earth is a conductor.
2. Higher atmosphere is a conductor.
Hence, there is a small insulating patch of atmosphere between the earth and the conducting atmosphere. Tesla proposed long distance transmission of Electricity through a spherical transmitter that transmitted Electric potential waves from transmitter to receiver, which are longitudinal, or scalar, as opposed to conventional electromagnetic waves, which are transverse. Thus, in Tesla's vision, energy was transferred from transmitter to receiver much like the vibration of a guitar string. The following figure shows Tesla's transmitter.
The existence of such transverse electromagnetic waves, or more popularly known as scalar waves, was disputed by classical physicists as not adherent to Maxwell's equations. But Prof. Meyl successfully constructed a working prototype of a scalar wave transmitter and receiver, and showed that electricity can, thus, be transmitted over longer distances, irrespective of the size of the transmitter. Meyl also propounded a theory based on Faraday's laws, as opposed to Maxwell's equations, to prove the existence of a potential vortex, which is a longitudinal potential wave, while still adhering to the laws of classical physics (for info about Prof. Meyl's work, click here). He demostrates his prototype in the following video:



(for more videos, click here.)


The idea of a worldwide wireless power grid is breathtaking, to say the least. Thinking this, I continued reading about scalar waves, and I chanced upon a most curious page about Keylontic Science, which was defined as "the Point of Union between Scientific and Spiritual perspective, through which we can begin to understand the reality of our connection to the Divine and to comprehend the purposes for and the processes of our Personal Evolution." Scalar waves are, apparently, very important in Keylontic Science, as they are essentially standing energy waves, ie. each point in a scalar wave only stands and oscillates. Keylontic science calls these points 'static points of light', and hence 'flashing points of consciousness'. It was a fascinating read, although thankfully, I'm cynical enough to dismiss it as merely interesting and hence did not go through a spiritual-scientific awakening. If you want in on the mysteries of the universe, click here.


May the force be with you.

Chaotic Advection

Over the years the importance of mixing in industrial applications has driven researchers to study and implement several new methods and enhancements to conventional mixing methods. It has been seen that mixing is quantitatively better in the turbulent flow regime. But the control prolems assosiated with the turbulent flow has rendered it improbable for practical applications. As the microfluid analysis has gained momentum, the new challenge was to improve the mixing in laminar regime. The idea of lagrangian chaos and chaotic advection has made this a reality.Intense research on this matter was carried out by Ottinow and Wiggins. Despite the very low Reynold's number and the laminar regime, the fluid particle ends up being chaotic. Following the premise set we can now dwell into the details of the Chaotic Advection/Mixing and then see how an engineer can exploit this to the benefit of the society.

The first and foremost step for understanding this phenomenon is tho know what chaos is ? So answering that that question as an engineer, it is nothing but the sensitivity to initial conditions. The best example in real life is the day today weather. Say a meteorologist is following the temperature of a particular place. He can follow the temperature for 10 years but still cannot come up with a prediction for the next day temperature. Reason the system we are considering is totally chaotic. There are several books available on this that you can go through. Coming to a specific example, The Gas Chamber.

Construct a simple system: take a box, a simple solid rectangular solid. Within this box, place a gaseous substance. Heat the box, sit back, and watch. What happens to the gas? Everyone knows that warm gases rise while cooler gases sink; and initially, the portions of the gas closest to the walls of the box will become heated and rise. At certain temperatures, the gas will begin to form cylindrical rolls spaced like jelly rolls lying lengthwise in the box. On one side of the box, the gas rises, and on the other, it sinks; the rising gases move to one side and carry warmer gases up with them; as the gas cools, it falls on the other side of the box. With a regularly applied temperature, a smooth box interior, and a system completely closed with regards to the gas itself, it might be expected that the circular motion of the moving gas should be regular and predictable. Nature, however, is neither regular nor predictable. It turns out that the motion of the gas is chaotic. The rolls do not simply roll around and around in one direction like a steamroller; they roll for a while

in one direction, and then stop and reverse directions. Then, seemingly at random, the gas reverses direction again; these changes continue at unpredictable times, at unpredictable speeds.

How can Fluid Mechanics enthusiast exploit this? We must be familiar with the term Stream function. Its a general thumb of law that the stream function is time dependant, then system invariably turns out to be chaotic. I will explain this with a specific example.

We all know about the RayleighBenard Convection. There are 2 parallel plates with the hot one at the bottom and the cold one at the top. The convection currents formed in between the plates is known as the Rayleigh Bernard convection. To see how this is realised, we can See the contour plots of the 2 stream functions with and without chaos. Illustration 1 give the contour of a stream function that is time independent convective currents given by,
$ \phi (x,y) = K \sin x \sin y $

To differentiate it from a system with inherent chaos let us see the contours formed by a time

dependent model of the stream function for RB convection current.
\[ \phi (x,y,t) = K_{1} \s in \left (x + K_{2} \cos t \right ) \sin y \]

have a look at illustrations 2 and 3.

Illustration 1: Stream function of system without chaos

Analyse the contour in illustration 2 and 3. The seperatrix (neutral point) oscillates about a position x=3.14 in the system analysed. The position of the seperatrix is dependent on time. So what is the use of this system being chaotic?

Its the ease of mixing. The basic reason why we turn to chaotic advection is that it brings about good mixing in laminar. Microfluidics is a fast growing research field for applied sciences. With nanotechnology getting a huge boost for R&D purpose, such efficient mixing at very small channels. (as diameter reduces drastically, so does the reynolds number). What I have given here is just a introduction to a vast topic of chaotic advection. One might wonder how having a time dependent stream function will affect the mixing this much? Its for you to explore. People who are interested to do further research in this Field can contact me for papers. One of the pioneers in this field is one Dr. John Otttino. He is from the Northwestern university. More info here .

Illustration 2: Contours of a time dependent stream function at t=0 .








Illustration 3: At a later time t=40