 Thank you very much for the little introduction. Well, I'm Manigarh. I'm working with Stefano Berti, Enrico Calzverni, and Gilmar Montpia. And today, I would like to discuss about the transport properties of particles in elastic turbulent flows. So in my first few slides, I'll give the motivation behind our work. And in further slides, the introduction to the phenomenology of elastic turbulence, and then finally results in the conclusion part. As you know, that mixing in small size container is very interesting from research point of view and can be relevant to many industrial applications. For example, by engineering devices like lab on a chip, or cooling of computer chips using fluid flow, or manure micro heat exchangers. But it raises a fundamental question. That is how to introduce mixing in these small size containers when it is not possible to increase the Reynolds number. One of the possibility is to resort to the Reynolds, to viscoelastic flows. It has been observed that these viscoelastic flows can introduce nonlinearities in the system, even in the limit of vanishing Reynolds number. By involving another control parameter, that is known as Weisenberg number, which accounts for the polymers elasticity. And these type of nonlinearities can provide us many promising microfluidic applications. A model system, which has these type of elastic property, is known as elastic turbulence. It was experimentally discovered in 2000 by Victor Steinbar and Alexander Grosma. They choose a very simple geometrical setup in their experiments. It was a rotating disc. The lower disc was kept fixed, and the upper was rotating. What they are observing in their experiment, with the addition of the polymer, there was onset of these elastic instabilities. And the transition from the laminar to the chaotic random flow. You can see the snapshot of the flow here. It's not laminar, even when the Reynolds is 0.7. We are in a laminar regime. But the flow has some turbulent structures. And in this turbulent like regime, the velocity spectra is characterized with the power law decay with an exponent, which is much greater than 3. This is similar to what we observe in higher in all turbulent flows. However, the spectra is much steeper. The flow is specially smooth flow. And in their experiments involving different geometries, they observe that elastic turbulence can be proposed as an efficient technique for mixing in microfluidic flows. Also, some of the applications studied recently emphasis is the fact that elastic turbulence can not be only used to enhance mixing in microfluidic channel flows, but can also be used as an efficient heat exchanger or to form emulsifications from immiscible liquids and to extract oil and the gases from the porous structures of the rocks. This is something that interested us a lot. An illustration to such kind of work can be seen as the extraction of the shale gas or just sorry, or oil extraction from the porous rocks. What exactly is happening in US? People are extracting the shale gas from the porous rocks, their thin structures. It is not possible by using a laminar flow. So what they do, they use a mixture of water with the polymers. So we are in a regime of where we have the onset of the elastic instabilities. So we have the onset of this elastic turbulence and this way they extract this oil and the gases from this porous rocks. The question of interest here is the absence of the laboratory clean conditions. In the absence of the laboratory clean conditions, there is a presence of solid impurities, sand dust particles, or maybe chemical additives. So the question is what the point of interest. These impurities may sediment along these thin porous structures of the rocks. As you can see, we have very thin porous structures. These impurities may distribute along these structures and reduce the efficiency of extraction of these gases and the oil. So it is really important and necessary to understand how these solid impurities distribute in elastic turbulent flow. And to achieve this purpose, we study the transport of inertial particles by means of DNS of elastic turbulence. And to observe the phenomenology of elastic turbulence numerically, a model was proposed in 2008 by Stefano Berti and collaborators. The model is based on Aldroid B model for viscoelastic fluids with a Kolmogorov forcing. And with a periodic boundary conditions, the Kolmogorov forcing here corresponds to a parallel flow. And what do we observe in our numerics that beyond a certain value of Weisenberg number, there is an onset of elastic instability. If you look at this instantaneous snapshot of vorticity field in this elastic turbulence regime, the flow is not laminar even when the Reynolds is 0.62. We have some turbulent-like features. And we also show here a velocity spectra, which is characterized by a power law decay with an explained greater than 3, which is similar to what we observe in our experiments. One thing to be keep in mind as our flow is especially smooth flow, so we have typically one time scale. Our viscoelastic Kolmogorov flow can be characterized by the mean profiles. Here, I show you the two profiles which are relevant. This represent my mean base flow, whereas this profile represents the transversal component of the velocity fluctuations. The important thing that we would like to highlight here, the origin of these fluctuations is purely elastic. There is no inertial effect, so if I do not have polymers in my flow, these fluctuations will be 0. And wherever you see these symbols, it means the averages along the direction of the homogeneity and the time. And please keep in mind these profiles because I'll use them to anticipate my results later on. So now, let me allow me to introduce you the dynamics of the impurities that is inertial particles I consider in my flow. We consider the heavy inertial particles whose have a density much greater than the density of the fluid. We assume that the particles are spherical in shape and small as compared to the relevant time scale of the flow. So under these assumptions, one can use this Maxwell equations for describing the dynamics of the inertial particles. And here we consider only the hydrodynamical drags on the particle, which is known as the Stokes drag represented by this parameter tau p. And in this simulation, u is coming from the DNS of elastic turbulence. For a result and analysis part, we can make profit of the studies performed in the context of high Reynolds turbulent flows, where this study that the particles can aggregate in certain regions of the flow and the size of these aggregations or these clusters vary from small scales to the large scales. Hence, we have small scale clustering and the large scale clustering. The mechanism behind these two different processes is completely different. But in any case, the parameter which characterizes this clustering is always defined as a ratio between the Stokes drag and the relevant time scale of the flow. Generally, what happens in high Reynolds turbulent flows? For small scale clustering, the relevant time scale of the flow is chosen to be the dissipative time scale. Whereas for the large scale clustering, it is chosen to be the relevant largest time scale of the flow. But as I told earlier, we are in a scenario. We do not have an error of time scales. We just have one typical time scale of the flow. So for the moment, we decided to fix it as tau gamma dot mean, which is computed from the averages of the strain rate. So from here on, I would like to start discussing the results part. First of all, I would like to discuss the small scale clustering. The small scale clustering is due to the dissipative and the chaotic dynamics of the inertial particles. With the evolution of the time, the particles evolve, and it tends on to the fractal sets. And these fractal sets can be characterized by many fractal dimensions. The one we choose here is an easy and simple approach, which is the correlation dimension d2. And thanks to Grassberg-Precocci algorithm. And what we observe, if you look at this curve, we compute the correlation dimension d2 as a function of stokes. And we found that the maximum of the fractal clustering, which corresponds to the minimum of d2, occurs around stokes is equals to 1. And if you look at this colorful, beautiful snapshot, this represents the instantaneous snapshot of the vorticity field, where the black dots represents the instantaneous special distribution of the particles. We saw that the particles are aligned along these thin filamentary structures presented in my flow. And here, this snapshot corresponds to the case where we have the maximum clustering, that is this point, where we have the dimension around 1. Now the question arises, what happens at large scales, and what do we expect? There have been some studies performed in the context of atmospheric boundary layers and wall-wounded flows, where they observe that at large scales, particles have that tendency to accumulate from high turbulence intensity regions towards the low turbulence intensity regions. And they show that it is a wall presence phenomenon. But recently, this also has been studied in high-enol turbulent Kolmogorov flows, where they show that it can happen also even in the absence of the walls. So we are in a scenario where we do not have walls, and we were interested to see if such kind of large scale inhomogeneities are present in our flow or not. There are several possibilities to answer this question. One of the possibilities is to look at average density profiles of the particles along the direction of the inhomogeneities of the flow. And in our case, the inhomogeneities are present along the transversal, that is the y direction. We looked for these density profiles for approximately 36 values of the stocks, but I showed here just for two values. And these red dots represents my density profiles along the direction of the inhomogeneities of the flow. And we see, yes, there is a presence of large scale inhomogeneities in elastic turbulent flows. And this black curve, sorry, line, this represents a fitting function, which was suggested by recent studies done by Delilo in the context of high-enol turbulent Kolmogorov flow. So we saw, OK, they fit our data pretty well. And then we looked at turbophoresis from another possibility. That is observing the observable key, which accounts for the deviations from the uniform distribution of the particles. This key is a sort of an amplitude. And what we observe, this key first increases as a function of stokes. And it reaches a maximum value. And then it starts decreasing. And again, if you look back to this colorful snapshot of vorticity with the particle distribution over it, what do we observe? That the particles are clustered in certain regions of the flow. Also, they are modulated with the large scale modulation, which is twice of our mean flow. If you correctly remember the profiles that I showed you before for the fluid. So what do we understand till now? In our elastic turbulent flows, there is a presence of large scale inhomogeneities. That is, the particles are accumulating in a regions where we have less turbulence intensity. So if you look at this profile of transversal velocity fluctuations, so you expect that the particles will more particles will be in the regions where my arrows are pointing. That is the regions where we have less velocity fluctuations. However, the relation between the density profiles and the velocity fluctuations is not so linear. But based on some available theories, we can they show that they are related with an exponent alpha. We try to see how it fits with the data. So here, I show again density profile fitting with this prediction for two different value of the stokes. And we found they match pretty well for some fractional value of these exponents. However, this comparison fits pretty well for the small values of the stokes. The reason is, as we are increasing the stokes, we are moving towards stokes around one. We have the presence of the small scale clustering, because it's dominating. So it results the deviation from the uniform sinusoidal profile. The one thing that could be interesting is to know what is the value of this exponent alpha. Is it stokes dependent or is it constant? What exactly it is? We computed the value of the alpha using least care analysis. And we found it is also sort of an amplitude. And its behavior is similar to key. And it increases first increase in stokes. It reaches a maximum. And then it starts decreasing. The question is, how to interpret this behavior? Can we? The answer is obviously yes. There are several possibilities. Either you use a hydrodynamical approach to find the expression for the alpha, or you use a probabilistic approach. So there are some theories available in the context of high renal turbulent flows. I will not go through the derivation part. So we have the expressions for the alpha. And there are some improvements which are also validated recently by Theoretical Wiff by S. Bellan. And they say this is another expression for alpha. And we try to see how it fits with our numerical data. And here are the comparison of these two different predictions with our value of alpha. And we see it doesn't fit our data pretty well. Until unless we reshift our x-axis, reshifting our x-axis means what? We have to reinterpret our stokes. Reinterpretation of stokes means the value of the relevant timescale that we have chosen, that is tau gamma dot mean, is not the appropriate value of the timescale, which relates to these large scale in homogeneities. This is an open question and something interesting to investigate. But somehow we had an idea this timescale might be related to the presence of small filamentary structures in our flow. If you look at this snapshot, so we decided to look at the instantaneous snapshot of the strain rate fields. This is the instantaneous snapshot of the strain rate field. And you see there are the presence of these certain small filamentary structures. These are the regions where the strain rate is maximally concentrated. And we also looked at the y-profiles of these instantaneous snapshots of the strain rate field. If anybody wants, I can show them later on. I do not have in my slides. There we conclude that the local timescale varies over the range 0.0121, which it makes it very evident that our choice for the tau gamma dot mean, which is based on the average of the strain rate, is not a good approximation. So we have to look for a relevant timescale, which is the most valued value of the tau gamma dot. So we looked at the PDF of tau gamma dot, which is computed by averaging over 85 independent snapshots of the strain rate fields. And we choose the most valued value of this tau gamma dot, which corresponds to value 0.02. And we represent it as tau gamma dot peak. So now we reinterpret our stokes based on this new value of this relevant timescale of the flow. Now it's represented by capital S t, which is tau gamma dot tau p divided by tau gamma dot peak. And we looked at our results, they fit pretty well. And we are happy with the results, at least. But there is still more to investigate, because this is something we had an idea that there is a presence of these timescales. But there is a large theory devoted to these presence of small filament structures that is related to the width of the boundary layers and something. But we are still investigating this question. At last I would like to discuss the Weisenberg dependency for both small and the last scale clustering. And here I represent this computation of d2 for as a function of stokes for multiple values of Weisenberg number. And we see there is a very weak dependency of this d2 as a function of Weisenberg number. And this maximum of d2 occurs for stokes around 1. What about the large scale clustering? We observed that there is a rather weak dependency when the Weisenberg is large enough. There might be some interesting features at the smaller Weisenberg number near the range where we have this threshold instabilities. But for larger Weisenberg number, it has a very weak dependency. At last I would like to conclude my work. In our elastic turbulent flows, we have a presence of small scale clustering, which is due to the dissipative and chaotic dynamics of the particles. There is also presence of the phenomenon turbophoresis, which accounts for the large scale in homogeneities. And this is induced due to the local variations of the turbulent intensity. And the function alpha as a function of the stokes, this exponent alpha, which accounts for turbophoresis, can be well described by the theories available in the context of high-reign old turbulent flows. One thing that I forgot to tell you that these models are validated only for small particles, not for the large size particles. And there is the maximum of this large and the small scale clustering occurs for similar value of tau p. And there is a very weak Weisenberg dependency for both small and the large scale clustering. In the future, we are willing to understand or to discuss the quantitatively that this turbophoresis for large size particles. And it's possible also understanding of this polymer statistics along the particle trajectories. Thank you. Yes? Yes. And they showed that the material particles, the tails in the sensitive. Yes. So that means that the dissipation of the small scale filaments in front of particles is most important. Which gives us these results. Yes. And also, we found it experimentally. Yes? Yes. I'll try to think about it. It's because of the elastic forcing. As you can see, we have this elastic forcing, which accounts for these turbulent fluctuations in our flow. It has something similar feature to Kolmogorov spectra, but not exactly. The spectra is much steeper than the Kolmogorov one. That's why we have a specially smooth flow. It's something similar to bachelor's regime. Turbulence, actually, you have a laminar flow. When you add polymers, due to the backstretching of the polymers, you have these turbulent-like states. It's not inertial. When you add polymers, they backstretch. And due to their back effects on the flow, that's why you have these turbulent-like features. It's not a turbulent flow, but it's something similar to turbulent flow.