 Okay, so the next speaker is Arun Pandyan. He's going to talk about deterministic and stochastic properties. So really Taylor Miesk sing with a power law space, depending on the acceleration. Hey, so I would like to present this work, which I did with my colleague, Nura Swisher, who unfortunately could not be here, and my advisor, Prof. Nijana, who fortunately could be here. Because now, if you have questions after the talk, she can directly answer it. So the work we did, we wanted to develop a model that describes Rayleigh Taylor mixing when acceleration is non-uniform. And then we also wanted to describe the deterministic and stochastic properties. So just to reiterate, RTI develops when fluids are accelerated against their density gradient and this leads to interfacial RTI mixing. Now, in nature, a lot of times, in RTI flows, the acceleration is variable. But when, at the time we started doing this work, we noticed that a lot of work was done where people are studying acceleration as being uniform. So we wanted to sort of study this non-uniform acceleration where, so we took two cases where the acceleration was a power law function of time or in space. So this work mostly describes the latter. So we used the momentum model of RTI mixing, which was originally developed to describe RTI mixing with uniform acceleration, but we extended it to describe a non-uniform acceleration. So basically for a parcel of fluid per unit mass, its dynamic is determined by two things. One is the momentum that it gains due to bio-NC and the other one is the momentum it loses due to a drag. So per unit mass, you have a resultant acceleration of the rate of momentum gain, which is a power law function in space, as I already mentioned, and the rate of momentum loss, which is given by CV squared over L, where L is the characteristic scale of energy dissipation. In RTI mixing, the dominant scale is the vertical length scale. So this is essentially CV squared over H and we get solutions both in space where time is a parameter and also explicitly in time for both the vertical amplitude as well as velocity. So now if we treat drag as a stochastic quantity, these two equations directly comes from the momentum model and we get this third equation under the assumptions that the drag coefficient C, the stochastic process with stationary log-normal distribution, which it reaches at a characteristic time tau and the fluctuations are generated by a standard Wiener process. Now we also know that there are from this model that there are two sub-regimes depending upon purely the acceleration exponent, M. There is a critical value that's given by this expression where C is the mean value of the drag coefficient. If your acceleration exponent is above this critical value, then your solution exponent and your solution... and then you are in the acceleration-driven regime and your solution exponent and the solution pre-factor in this regime would be both dependent upon that acceleration exponent M. Now if you're in the other regime, which is the dissipation-driven regime, then basically your acceleration exponent M is less than your M critical, then your solution exponent would depend on the drag coefficient and your solution pre-factor will depend upon the initial conditions. These properties actually persist even if you express a solution in space. Once again you have two mixing regimes, the acceleration-driven mixing regime and the dissipation-driven mixing regime, depending upon your acceleration exponent purely. Once again, your solution exponent in the acceleration-driven mixing regime depends upon the acceleration exponent. Your solution pre-factor also depends upon your acceleration exponent and not the drag coefficient. In the dissipation-driven mixing regime, your solution exponent depends on the drag coefficient and the pre-factor depends upon the initial conditions again. As we saw, this is the same case for solutions in time. So we decided to solve these equations numerically. So we wrote a solver, ran it on a supercomputer for about a thousand trajectories, over a million time steps, which took about a week or so. Although I believe now it is faster because essentially these trajectories are independent of each other, so we can sort of run them in parallel. So we tested a broad range of parameters for the acceleration exponent. We let them range from both sides of the M critical so because we wanted to get some cases from the dissipation driven regime as well as the acceleration driven regime. Because for the C mean we chose our M critical was negative eight, so we chose its values from both sides and as I already described our C is a log normal process and it has a log normal distribution and the standard deviation we chose was half of the mean value. So we got solutions that are power loss in, so this is now this describes solutions in time is also the case with solution space. As you can notice the amplitude in both acceleration driven and dissipation driven mixing regimes always increase but in the acceleration driven mixing regime the acceleration dominates as you would affect the dynamics of the system while in the dissipation driven mixing regime the acceleration doesn't play an important role. We also found that for the solution pre-factor which is in green here and in here just scaled by their theoretical values they are more sensitive to the noise than the values of the exponent, the solution exponents. And yes and we were sort of we were able to easily identify both of them. Also we found out that so this is a solution in space. Now here also the solution pre-factor is very more sensitive to noise than the solution exponent but both the solution pre-factor and the exponent are both very more sensitive than the solutions in time. Which is interesting. We also have another result which is that both these regimes have their own characteristic invariant which is a quantity that's dimensionless and in their respective regimes is around one. So now in the dissipation driven mixing regime this value is scaled by V dot it's this expression and in the acceleration driven mixing regime it's scaled by the effect of acceleration G and it's given by this expression and graphically that looks like this. So in the acceleration driven mixing regime the values are very close to one and in the same value of expression and for the same quantity in the dissipation driven regime in this insert here it diverges. And likewise the invariant for dissipation driven regime is very close to one in that regime for MS negative nine. If you remember our critical M critical was negative eight so we are in the dissipation driven regime here and now if you look at the same quantity in the acceleration driven regime it again diverges. So this is a good way of sort of knowing which mixing regime you are currently in. Now, oh yeah so in conclusion like I said we sort of we wanted to develop this model that would describe RT mixing when acceleration is non-uniform specifically when there are power loss solutions in time or in space and we found out that we have self-similar power loss asymptotic solutions both in time and space and whether they are in time or in space the solution pre-factor is more sensitive to noise than the solution exponent and the solution in space is more sensitive to noise than the solutions in time for both the pre-factor and the exponent and then there are two distinct mixing regimes the dissipation driven and the acceleration driven regimes which each of which have their own characteristic invariant value. That's it, thank you.