Soap films try to minimize their area, thus forming smooth surfaces. In this project, we study what happens when a soap film spans a flexible wire loop. The soap film pulls on the loop, and the loop resists bending, creating a competition between surface tension and elasticity.

For small loops, the system remains a flat circle. As the loop becomes larger, the circular shape becomes unstable, changing into a noncircular but still flat shape. A long-standing question has been: when this flat shape becomes unstable and buckles out of the plane?

Using mathematical analysis, we determined this critical point exactly. We found that the transition to a three-dimensional shape occurs much earlier than previous numerical estimates had suggested. As a result, the noncircular flat shapes exist only over a very narrow range of loop radii. The shape of the buckled loop is a twisted saddle (see adjoining figure).

We see through our results how simple everyday objects such as soap films exhibit rich and surprising behavior that can be understood via mathematics. This work can be read here.

Residual stresses (RS) are 'self-equilibrating' stresses. If a body under RS is dismantled or cut into smaller pieces, they shrink or expand so that the individual parts do not fit together unless deformed. Think for instance a wheel consisting of a rubber tire stretched over a rim. Or onion rings. 

It is important to measure the RS in a body since it affects important properties of the latter such as its strength, durability, etc. My work, done under the guidance of my PhD advisor Prof. Anindya Chatterjee, was motivated by the following question: If we know the value of the RS at N points in a two- or three-dimensional body, N being a finite number, can we use those values to get a reasonable approximation of the RS in the whole body, without knowing how the RS came about or the material properties of the body?

We developed basis functions for RS which allow us to do just that, and much more. These functions are independent of the material properties or the deformation history, and their computation only requires the prescription of the current geometry of the body. This work can be read here and here.

A tabla is an Indian percussion instrument consisting of two drums: a treble drum (dayan) and a bass drum (bayan). The remarkable thing about dayan is that its first nine modes are harmonically arranged, i.e., their frequencies are integral multiples of the lowest frequency. This makes the tabla quite tonic in nature, like the stringed instruments. 

It is well known and studied that the chief reason for this harmonicity is the addition of a black paste (siyahi) made with wheat dough or iron fillings onto the tabla membrane. Alternatively, kettledrums are also harmonic in nature, but due to a different reason: the air trapped inside these instruments. In this work, done after my M. Tech. under the guidance of Prof. Anurag Gupta, we investigated the effect of the combination of siyahi and air loading on tabla. 

We modeled the tabla drums as enclosed cylinders. Air loading was calculated using a Green's functions approach. We showed that in dayan, siyahi is the primary reason for harmonicity, and the air loading merely fine tunes it; in bayan, air loading plays a significant role in improving the harmonicity. Air loading also helps in sustaining the modes for longer. We also showed that the physical parameters that maximize the harmonicity are consistent with those in the actual tabla designs. This work can be read here.

Sweet sorghum (sorghum: jawar) (SS) is a millet with a sweet and juicy stalk. When the juice is heated to a density of about 1.4 kg/l, a red-brown syrup having high nutritional content and a long shelf-life is obtained.

At the Nimbkar Agricultural Research Institute (nariphaltan.org), we developed a semi-mechanized plant for production of SS syrup. All the processes (stripping of SS stalk >> crushing of stalk >> settling of juice >> heating of juice >> cooling of syrup >> storage in cold room) were either mechanized or modified for minimizing time, costs and human interference, while maximizing convenience. In most cases, we first designed prototypes using scrap material and then fabricated/purchased the final product. Some examples of developed products are: a fuel shredder cum feeder, a semi-mechanized syrup cooling device, a scum filter, etc. I also did economic analysis of SS syrup production, which revealed that  farmer growing SS and producing syrup can earn a profit of at least Rs. 6 lakhs per acre-year assuming nominal values of other involved parameters. For comparison, a sugarcane farmer in the Phaltan region earns about Rs. 90,000 per acre-year.

I learnt a great deal about designing and fabrication. I also learnt how to design simple experiments. I owe much of my learnings to Dr. Anil Kumar Rajvanshi, the director of NARI. 

SEIQR models divide the population into the following five subpopulations: susceptible, exposed, infected, quarantined and recovered. In these models, it is generally assumed that all the infected people have the same infectivity. This, of course, is an unrealistic assumption, and data from the first wave of COVID-19 suggested that some people were more likely to spread infection. 

This distribution of infectivity can be incorporated in the model by dividing the infected population into a finite number N of subpopulations, each with different infectivity. My PhD advisor (Prof. Anindya Chatterjee) extended this model by considering the limit of N tending to infinity, physically corresponding to a statistically distributed (continuum) infectivity.

In the highlighted work, we fitted the continuum model on the COVID-19 data from four European countries: Italy, Germany, UK and Spain.  The fits were excellent given that the model had just five free parameters. Our analysis revealed that the actual number of infected people far outnumbered those who were detected, consistent with serological surveys. Another important conclusion was that 'super-spreaders' (people with high infectivity) played a crucial role in COVID-19 transmission. This work can be read here.