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 .