info Engineer

Self Assembly – a viable manufacturing technique??

noreply@blogger.com (Balkrishnanitt) — Thu, 26 May 2011 03:57:00 +0000

Self-assembly is scientifically interesting as it occurs inside the most basic form of life – the cell. The cell contains an astonishing range of complex structures such as lipid membranes, folded proteins, structured nucleic acids, protein aggregates, molecular machines, and many others that form by self-assembly. At the same time, self-assembly is technologically important as it provides routes to a range of materials with regular structures: molecular crystals, liquid crystals, and semicrystalline and phase-separated polymers are examples.Self-assembly is a process in which components, either separate or linked, spontaneously form ordered aggregates. Self-assembly can occur with building blocks from molecular to the macroscopic scales, provided that appropriate conditions are met. For customized materials on macroscopic length scales to be made, self-assembly is essential because the number of building blocks that would have to be precisely placed in the absence of a self-directed method are too large and the process is prohibitively time consuming. A Fermi approximation of the time required for building a 1cm cube using 10 nm building blocks with the help of optical tweezers would be in the order of a hundred million days.Self-Assembly is an alternative technique by which materials can be formed. The basic idea behind using self-assembly as a manufacturing technique is to develop building blocks that have sufficient information content inherent in them, such that a desired structure can form spontaneously from a collection of building blocks in a thermal bath.Amongst interesting building blocks that have been used to self-assemble custom materials are colloids. These mixtures of immiscible materials form fine suspensions that have potential photonic applications because the particles that make up the stabilized phase can be tuned to have diameters on the same scale as visible light (500nm) and can form periodic structures. Polystyrene colloidal crystals, consisting of charged monodisperse polystyrene microspheres suspended in water, organize into a face-centered-cubic (fcc) lattices with lattice spacings comparable to the wavelength of light (Clark, N.A. et al, Nature 1979, 281, 57-60). Such colloidal structures may also find applications in nanoscale electronics, miniature diagnostic systems and hierarchically structured catalysts amongst others. However, for the assembly of more complex structures, it has been recognized that anisotropic shape and specific interactions (or “patchiness”) are useful (Glotzer S.C., & Solomon M.J. Nature materials 2007, 6(8), 557-62).

A recent work by Chen et al., Nature (2011), 469 (7330), 381-384, has shown that triblock Janus particles can self-assemble into a Kagome lattice (see Figure 1). A triblock Janus particle has two hydrophobic poles of tunable area and an electrically charged middle band. There are two competing forces that drive self-assembly for suspensions of these particles: the hydrophobic poles agglomerate in aqueous solvents, while the charged middle bands repel. The balance between these forces can be tuned by screening the charged repulsion via the addition of an ionic compound like NaCl, and the particles can stabilize in a kagome lattice. Furthermore, sheets of kagome lattices are reported to stack in parallel layers. An interesting video of this self assembly process can be found at: http://www.nature.com/nature/journal/v469/n7330/extref/nature09713-s3.mov Thus self-assembly seems to offer a viable strategy for generating nanostructures. Recent advances in nanoparticle synthesis have led to the availability of various building blocks including spheres,cubes, rods, and tetrapods at both the nano and micro scales. Specific interactions between these building blocks can also be obtained by modifying chemical composition, changing shape, and by adding functional groups. With such a vast parameter space available to experimentalists, there is much opportunity for simulation to aid in the understanding of how such complicated building blocks assemble.

- N. Khalid Ahmed

PhD Pre-Candidate,

Glotzer Group,

Department of Chemical Engineering,

University of Michigan, Ann Arbor – 48109

nkahm@umich.edu

Solving First-order simultaneous ODEs

noreply@blogger.com (Prem Anand) — Wed, 11 May 2011 00:20:00 +0000

Solving a system of ordinary differential equations (ODEs) has always been a cakewalk. Given a pen and paper, it takes almost no time to solve a simple system such as this:

$ \frac{dx}{dt} + ax + by = 0$

$ \frac{dy}{dt} + cx + dy = 0$

One can introduce the differential operator $D$ and solve as if it were a set of algebraic equations. But when the system of equations gets bigger, things become messy. They can be solved only by an in-built solver routine in a high-level language like MATLAB. However, feeding such a huge system can still be troublesome. The purpose of this post is to expose a cute little trick in mathematics and exploit it to our advantage.

To explain the trick, we consider the above set of differential equations. We rewrite them in matrix form as:

$
\begin{bmatrix}
x'\\
y'
\end{bmatrix}
+
\begin{bmatrix}
a &b\\
c &d
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}
=
\begin{bmatrix}
0\\
0
\end{bmatrix}
$

or simply,

$ X' + AX = 0$

where, $X$ is the unknown vector whose elements are $x$ and $y$.
Now, here is the trick:

Split the matrix $A$ into 3 matrices as $P^{-1} \Lambda P$, where $\Lambda$ is a diagonal matrix, with the eigenvalues of $A$ as the diagonal elements. $P$ is a matrix, which has the eigenvectors of $A$ as its columns.

This is called as Eigenvalue decomposition. This is actually an one-liner in MATLAB. Now, the matrix equation becomes:

$ X' + P^{-1}\Lambda P X = 0$

Pre-multiply by $P$ to get

$ PX' + \Lambda PX = 0$

Now, take $PX$ to be $U$, and owing to the fact that $\Lambda$ is diagonal, we get a system of the form:

$ \frac{du_1}{dt} + \lambda_1 u_1 = 0$

$\frac{du_2}{dt} + \lambda_2 u_2 = 0$

Now that we have separated the dependent variables, solving this should be a no-brainer. Finally, apply the reverse transformation $P^{-1}U$ to get the solution.

This works even when the right-hand side is non-zero. We just have to pre-multiply it by $P^{-1}$. Also, remember to transform the boundary conditions as well.

A variation to this problem may be something like this:

$ AX' + BX = c$

where $A$ and $B$ are square matrices. In such a case, first pre-multiply by $A^{-1}$ and convert it to the 'standard form' discussed above. The rest should be easy.

Always remember: The trick behind the trick is knowing when to use the trick. Else there is no point in knowing the trick.

3D printing

noreply@blogger.com (Balkrishnanitt) — Sun, 10 Oct 2010 07:31:00 +0000

3D printing is a form of additive manufacturing technology where a three dimensional object is created by laying down successive layers of material^[1]. 3D printers are generally faster, more affordable and easier to use than other additive manufacturing technologies. 3D printers offer product developers the ability to print parts and assemblies made of several materials with different mechanical and physical properties in a single build process. Advanced 3D printing technologies yield models that closely emulate the look, feel and functionality of product prototypes.

In recent years 3D printers have become financially accessible to small- and medium-sized business, thereby taking prototyping out of the heavy industry and into the office environment. It is now also possible to simultaneously deposit different types of materials.

3D printers offer tremendous potential for production applications as well.^[2] The technology also finds use in the jewellery, footwear, industrial design, architecture, engineering and construction (AEC), automotive, aerospace, dental and medical industries.

Additive Manufacturing :

Additive manufacturing (AM) is defined by ASTM as the "process of joining materials to make objects from 3D model data, usually layer upon layer, as opposed to subtractive manufacturing methodologies. Synonyms: additive fabrication, additive processes, additive techniques, additive layer manufacturing, layer manufacturing and freeform fabrication"^[1]
The term Additive manufacturing describes technologies which can be used anywhere throughout the product life cycle from pre-production (i.e. rapid prototyping) to full scale production (also known as rapid manufacturing) and even for tooling applications or post production customisation.
Technologies :

One variation of 3D printing consists of an inkjet printing system used by Z Corporation. A 3D CAD file is imported into the software. The software slices the file into thin cross-sectional slices, which are fed into the 3D printer. The printer creates the model one layer at a time by spreading a layer of powder (plaster, or resins) and inkjet printing a binder in the cross-section of the part. The process is repeated until every layer is printed. This technology is the only one that allows for the printing of full colour prototypes. It is also recognized as the fastest method.
Alternately, in DLP, or Digital Light Projection, a liquid polymer is exposed to light from a DLP projector under safelight conditions. The exposed liquid polymer hardens. The build plate then moves down in small increments and the liquid polymer is again exposed to light. The process repeats until the model is built. The liquid polymer is then drained from the vat, leaving the solid model. The ZBuilder Ultra is an example of a DLP rapid prototyping system.
Fused deposition modeling (FDM), a technology developed by Stratasys^[3] that is used in traditional rapid prototyping, uses a nozzle to deposit molten polymer onto a support structure, layer by layer.
Another approach is selective fusing of print media in a granular bed. In this variation, the unfused media serves to support overhangs and thin walls in the part being produced, reducing the need for auxiliary temporary supports for the workpiece. Typically a laser is used to sinter the media and form the solid. Examples of this are SLS (Selective laser sintering) and DMLS (Direct Metal Laser Sintering), using metals.
Finally, ultra-small features may be made by the 3D microfabrication technique of 2-photon photopolymerization. In this approach, the desired 3D object is traced out in a block of gel by a focused laser. The gel is cured to a solid only in the places where the laser was focused, due to the nonlinear nature of photoexcitation, and then the remaining gel is washed away. Feature sizes of under 100 nm are easily produced, as well as complex structures such as moving and interlocked parts.^[4]
Each technology has its advantages and drawbacks, and consequently some companies offer a choice between powder and polymer as the material from which the object emerges.^[5]Generally, the main considerations are speed, cost of the printed prototype, cost of the 3D printer, choice of materials, colour capabilities, etc.^[6]
Unlike stereolithography, inkjet 3D printing is optimized for speed, low cost, and ease-of-use, making it suitable for visualizing during the conceptual stages of engineering design through to early-stage functional testing. No toxic chemicals like those used in stereolithography are required, and minimal post printing finish work is needed; one need only to use the printer itself to blow off surrounding powder after the printing process. Bonded powder prints can be further strengthened by wax or thermoset polymer impregnation. FDM parts can be strengthened by wicking another metal into the part.
The democratization of 3D printing is evolving in two streams, firstly with DIY 3D Printers such as BotMill, MakerBot and RepRap for home 'desktop manufacturing'. The second stream is through online services such as Shapeways or Sculpteo that allow users to upload their designs to have them 3D printed in a wide range of materials (currently 20 material options) and shipped worldwide. The creation of tools that enable 3D printing without the direct use of CAD are also currently being implemented.
Below is the video of MakerBot - A 3D printing robot .
http://www.youtube.com/watch?v=fScRYhq-5M0
The Future :
According to Neil G
ershenfeld ,
who runs MIT's Center for Bits and Atoms,

foresees a time when computers will upgrade from PCs to PFs, or personal fabricators. His book on FAB at home reveals a lot on this topic.

Introduction to ALD and MLD

noreply@blogger.com (Dutchaa aka Sriram) — Tue, 31 Aug 2010 04:47:00 +0000

ALD and MLD refer to "Atomic Layer Deposition" and "molecular layer Deposition". They are used to produce both organic and inorganic polymers. Miniaturization to the nanometer scale has been one of the most important trends in science and technology over the last several years. The chemistry to fabricate nanolayers, the engineering for nanocomposite design and the physics of nanostructure properties have created many exciting opportunities for research. These new interdisciplinary areas in nanoscience and nanotechnology supersede the more traditional disciplines and demand new paradigms for collaboration.

Many of our surface chemistry and thin film growth investigations utilize atomic layer deposition (ALD) techniques. ALD is based on sequential, self-limiting surface reactions as illustrated in the accompanying figure. This unique growth technique can provide atomic layer control and allow conformal films to be deposited on very high aspect ratio structures. ALD methods and applications have developed rapidly over the last few years. In particular, ALD is on the semiconductor road map for high-k gate oxides and diffusion barriers for backend interconnects.

ALD is based on sequential, self-limiting surface chemical reactions. For example, for Al2O3 deposition, the binary reaction: 2Al(CH3)3 + 3H2O -> Al2O3 + 6CH4 can be split into the following two surface half-reactions

A) AlOH* + Al(CH3)3 -> AlOAl(CH3)2* + CH4
B) AlCH3* + H2O -> AlOH* + CH4

where the asterisks denote the surface species. In the (A) reaction, Al(CH3)3 reacts with the hydroxyl (OH*) species and deposits aluminum and methylates the surface. The (A) reactions stops after all the hydroxyl species have reacted with Al(CH3)3. In the (B) reaction, H2O reacts with the AlCH3* species and deposits oxygen and rehydroxlates the surface. The (B) reactions stops after all the methyl species have reacted with H2O. Because each reaction is self-limiting, the Al2O3 deposition occurs with atomic layer control. By applying these surface reactions repetitively in an ABAB... sequence, Al2O3 ALD is achieved with a growth rate of 1.1 Å per AB cycle. We have also extended the ALD method to deposit single-element metal films.

Similar self-limiting surface reactions can be employed for the growth of organic polymer films. This film growth is described as molecular layer deposition (MLD) because a molecular fragment is deposited during each reaction cycle.The precursors for MLD have typically been homobifunctional reactants. A cartoon illustrating the MLD process is shown in the nearby figure. MLD methods have been developed for the growth of organic polymers such as polyamides.The polyamides have been deposited using dicarboxylic acid and diamines as the reactants. New approaches to MLD involve heterobifunctional and ringopening precursors. In addition to organic polymers, the precursors for ALD and MLD can be combined to grow hybrid organic-inorganic polymers.

Source:
One of the research groups that pioneers in this study can be visited from here.

Fluidics

noreply@blogger.com (Balkrishnanitt) — Mon, 30 Aug 2010 19:56:00 +0000

Fluidics (also known as Fluidic logic) is the use of a fluid or compressible medium to perform analog or digital operations similar to those performed with electronics.

The physical basis of fluidics is pneumatics and hydraulics, based on the theoretical foundation of fluid dynamics. The term Fluidics is normally used when the devices have no moving parts, so ordinary hydraulic components such as hydraulic cylinders and spool valves are not referred to as fluidic devices. The 1960s saw the application of fluidics to sophisticated control systems, with the introduction of the fluidic amplifier.

A jet of fluid can be deflected by a weaker jet striking it at the side. This provides non-linear amplification, similar to the transistor used in electronic digital logic. It is used mostly in environments where electronic digital logic would be unreliable (e.g., systems exposed to high levels of electromagnetic interference or ionizing radiation).

Nanotechnology considers fluidics as one of its instruments. In this domain, effects such as fluid-solid and fluid-fluid interface forces are often highly significant. Fluidics have also been used for military applications.

Amplifiers :

The basic concept of the fluidic amplifier is shown here. A fluid supply, which may be air, water, or hydraulic fluid, enters at the bottom. Pressure applied to the control ports C₁ or C₂ deflects the stream, so that it exits via either port O₁ or O₂. The stream entering the control ports may be much weaker than the stream being deflected, so the device has gain.

Given this basic device, flip flops and other fluidic logic elements can be constructed. Simple systems of digital logic can thus be built.

Fluidic amplifiers typically have bandwidths in the low kilohertz range, so systems built from them are quite slow compared to electronic devices.

Systems :

Fluidic components appear in some hydraulic and pneumatic systems, including some automotive automatic transmissions. As digital logic has become more accepted in industrial control, the role of fluidics in industrial control has declined.Fluidic injection is being researched for thrust vectoring in aircraft jet engine nozzles, and for ships. Such systems divert thrust via fluid effects . Tests show that air forced into a jet engine exhaust stream can deflect thrust up to 15 degrees. Such nozzles are desirable for their lower: mass, cost (up to 50% less), inertia (for faster, stronger control response), complexity (mechanically simpler, no moving parts or surfaces), and radar cross section for Stealth. This will likely be used in many unmanned aircraft and 6th generation fighter aircraft.

Professor Nikolai Priezjev of the Department of Mechanical Engineering writes about fluidics and micro - fluidics here .The transport and manipulation of small amounts of fluids are crucial for emerging technologies. A brisk demand for micro-devices, which are used for the transportation of nanovolume liquid samples, gave rise to a new exciting field called microfluidics. This field combines various disciplines including engineering, chemistry, physics, and biology. The long range goal in modern technology is a reduction in size and the further development of microfluidic devices, which could be used for diagnoses of diseases, for the autonomous or remote detecting of biological and chemical agents, and for gene and drug delivery.

A very good introduction to microfluidics can be found at the wiki page. Microfluidics news is another kind of a aggregator for papers published on this subject .

Food for thought :

Is it possible to use constructal theory as a basis to design applications using microfluidics ?? What would be the implications of doing so ?

Sources of information :

Wikipedia and Google .

A not-so-ordinary insight into everything around us ... Constructal theory .

noreply@blogger.com (Balkrishnanitt) — Sun, 08 Aug 2010 15:05:00 +0000

This is one of the most fascinating theories developed in recent times which actually gives a very different insight into nature and its structure . For example in the field in thermodynamics it replaces the black boxes , we as mechanical engineers assume while designing any system by actually delving into the flow configuration.

The core of the theory is the constructal law :

“For a finite-size flow system to persist in time (to survive) its configuration must evolve in such a way that it provides an easier access to the currents that flow through it”

This principle predicts natural configuration across the board: river basins, turbulence, animal design (allometry, vascularization, locomotion), cracks in solids, dendritic solidification, Earth climate, droplet impact configuration, etc. The same principle yields new designs for electronics, fuel cells, and tree networks for transport of people, goods, and information.

An amazing way to describe phenomena around us . To what extent is it adopted by the industry is something we have to wait n watch ....

Cya ... Have a gr8 day ahead ....

Sources :

Constructal theory of generation of configuration in nature and engineering - Adrian Bejan¹ and Sylvie Lorente²

Constructal.org - A site dedicated to it .

The Art of Approximation in Engineering and Science

noreply@blogger.com (Balkrishnanitt) — Wed, 21 Jul 2010 16:10:00 +0000

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

noreply@blogger.com (Krishna Chandran) — Sun, 04 Jul 2010 11:28:00 +0000

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

noreply@blogger.com (Balkrishnanitt) — Sun, 04 Jul 2010 06:56:00 +0000

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