 I'm Vasily Jakovsky from the Dukhov Research Institute of Automatics and today I'd like to show you our recent results related to formation, instability and fragmentation of shock-producing jets. So you know that typically jets in our daily life is just a string of water which looks quite unstable, cylindrical jets, but today I'll show you a more microscopic level of jet which are formed usually after arriving of shock compression, shock wave to the replete groove with surface of material, hard material, metals usually. So it's important for some technical application and also has some fundamental problem here, still, because it looks like a rich-marmish co-instability in the limit. So shock-produced ejector from the tin we simulated with two different methods, molecular atomistic, molecular dynamic method and smooth particle hydrodynamic code on very different special scale, atomistically 100 nanometers and 100 micrometer and hydrodynamic scale. But the flow, hydrodynamic flow, obtained by both method are quite similar. So first I will discuss about simulation of cylindrical jets and their disintegration to the serious droplets. They are very well-known capillary instability, plateau relay instability. Then I will talk about simulation of fragmentation of planar jets. So that more complicated thing and that still there is some basic equation remains unclear. So the main issue of this and this topic that the planar liquid jet and planar liquid sheet were stable in normal conditions. So however, some experiment shows very likely that the fast interior internal fragmentation of planar jet may happen and happen at much bigger planar jet. Then we may usually obtain in molecular dynamic simulation where the jet becomes sheet, planar sheet becomes unstable when it reaches a thickness of less than one nanometer. Of course, it's not realistic in experimental condition and these issues still wait. So about setup, what's the typical scheme of the experiment and simulation and theory where the such jets may be formed. So shock wave comes to the real surface of material target and if material surface is not plain, some grooved surface or maybe a replete surface depending on geometry, we may call it different way. Then the jets are formed from the minimum of this surface and the characteristic depth of this groove is A and length between the neighbors of lambda and then the dimensionless amplitude of this groove can be represented as a product A and key. So for larger A key, they have a deeper groove. So you see that this picture quite similar to what happened if we have a replete interface between two materials with different density and Riff-Miramish coven stability will develop after passing of shock wave through this interface. But here we have usually here in experimental condition a vacuum or very rare gas. Okay, I want to say also that due to geometry, so if it's a groove, I mean just linear groove, we should have planar jet here, but if it's just look like cone on the surface, it will be a liquid cylindrical jet and they decay in very different ways. So we know very well how decay, how the cylindrical jet decays, but planar jet may need some additional study. So it shows a molecular dynamic simulation and the smooth particle hydrodynamic simulation in dimensionless coordinate and time and you see that the pressure density maps are represented very well and similar way in atomistic and hydrodynamic simulation. It indicated hydrodynamic or works very well until say 10 nanometers special scales dimension and then we may do some scaling of this motion to get more detailed information from the atomistic scale and compare with the SPH simulation. I would like to also underline that these two simulations are consistent in sense of the equation of state was calculated firstly by atomistic simulation and then this equation of state was used in smooth particle hydrodynamic. So this is the reason why they provide us very similar results. Only difference you may see that the surface tension was not included in smooth particle hydrodynamic, which is why here we have not so well pronounced droplet here and droplet. Okay, so for different AK, so roughness of the surface, you see here very nice experimental results in shadow of these jets obtained recently by William Butler and David Orwell. So they show that the for larger AK, we have a very nice very long jets, but we're for smaller one, smallest one here. So jets are mostly controlled by the plasticity if the material was not mailed by shockwave, then plasticity control development, controls development of these jets and even indeed we cannot obtain it because material strength is too high and the prevent formation of this jet. So this is why we mostly are interested in such very nice jet for not so small roughness of the surface. Okay, here the comparison of two simulation approach, MD and SPH and experimental results. So you see across here and it shows experimental point, then SPH, different SPH code shows that just it's almost the same, these circles, closed circles here and different MD. So different MD simulation with the size of the sample, which initially was small one, 50 nanometers up to 150 nanometers, I mean size of the cross section of the target. So that with increasing of size we almost approach, actually just approach to the exact, not exact, this approach MD, SPH, hydrodynamic results with magenta diamond here. So the biggest simulation. So experiment includes a million atoms and 10 million atoms was used in a hydrodynamic simulation. The hydrodynamic code was worked as a sample of the dimension as an experimental condition. So I told about the generation of shock or generation of jets and now let's talk about instability of planar jet. So in order to show some basic ideas of instability, first I'll show experimental, well, I mean on daily life experiment actually. I don't know, just first it's experiment from the fragmentation of liquid jet from the team. So experimentally told that the jet was liquid and expand very quickly from the target from here and but for very short time it will be, it was destroyed on very small filaments black line here, black line, not white black line. So small filaments and droplets. So it's quite unusual for us because in contrast to MD simulation where the jet is very, planar jet is very stable. Here we have a very quick disintegration of this planar jet and the main the problem to find is to find the mechanism of such instability. We don't know still what the mechanism drive such fragmentation of the planar jet. So it worked down just recently three years ago and that's obtained by the holography in the ultraviolet line. So very nice experiment and now about daily life. So instability of liquid jets was studied very long time actually already 200 years starting from the Savarin stability. He was first 200 years ago French scientists who found that the planar jet fragments into the droplets. So if a kind of beam or stream of water heated the some stopper here, a plane stopper then you see jet as a white cell. So the age of this jet is quite unstable and many droplets just move out from here like this. So there is a scheme show that the water comes to this plateau and then moves almost horizontally because the varieties are actually negligible here if the speed is enough. So the unstable thing here not the jet itself, cylindrical planar jet, but only edge of this jet. So the rim edge or rim is unstable through the formation of such kind of casp and casp itself is fragmented to the droplets. So here then another type of instability like flagging stability on the wind which we will not discuss here. And it happens when the flow of the this flow happens in the gas quite dense gas. So we mostly interesting in expansion of the jet micro jet into the vacuum. Okay so another very nice picture we show the instability of the age is a bubble, soft bubble. And then it's stable itself all this whole time except the case when that some small hole is formed on the this sheet of the bubble. So and then you see the edge which propagate to the left side and the many many filaments are formed on this jet. So then this filament itself unstable due to platorally instability, cylindrical jet instability and many droplets finally forward. Okay simulation of the simulation of plan sorry cylindrical jet or liquid cylinder. So it's well known theory of platorally which was almost 120 years ago was developed by Rayleigh. So this theory is based on the Laplace pressure expression for the curved surface. So which in synergy depends on the surface here along the filament along the jet and itself radial curvature of the cylinder itself. Because they are may fluctuate or perturbate then the pressure will fluctuate inside such cylinder. So if the cross section becomes smaller then the pressure here may be may reach a higher value than here and then material will move to the side and then we will finally obtain kind of droplet and which size actually does not depend on surface tension shown here. So this is just I wrote the dispersion relation here and the solution of this equation obtained from the equation for inviscid flow. Earlier equation actually so the positive one will represent will give us a region of the where the jet is unstable. It's this represented by here this curve. The maximum unstable mode shown here it gives us about two droplets of the radius of two of initial cylinder and the characteristic time of development of such instability shown here which depends on density initial radius of cylinder and surface tension. Okay we will check this we checked such kind of prediction but not in the steady cylinder but in extending cylinder in jet expanding in the length. So it means the theory may not work if the expansion rate of characteristic time of expansion is shorter than the characteristic time of development of proline stability. So and this show here and here two different initially different filament with different radius and then the droplet inside droplet radius format inside and the end droplet format on the end of the filaments. So they are quite well agree with the theory but remember that the theory as I said developed from the deformation rate is zero. Okay so no no it's just simulation atomistic molecular dynamic simulation. It depends on the radius was initial radius and the surface tension. So here the moment is we may found that they just when the such kind of small radius filament appear and then we do not control condition for the wraps. We just absorb it in simulation. It's not theory it's simulation. So it's ruptured and at some time and we may say that the rupture time is in a degree with theory theoretical estimation as I said on this one characteristic time. So it's ruptured just exponentially by say three times of this characteristic time. Yeah three times this characteristic time in exponent. So then we may expect it will yeah yes there is a number of course sure sure sure. So and you may see I show this number here so actually size. So third I mean it's simulation simulated size here but really here predict a little bit higher radius of droplets here and and droplets almost agree very well. So and applicability for sorry early instability or platorally more exactly instability is under question because this time break time of the droplet and expansion characteristic time of expansion are almost same I mean same same order. So we cannot really apply to predict but we may predict only order of magnitude of this instability. By the way instability theoretical fear of such instability is not developed yet. I mean expanding jets expanding filaments. Okay here another example we show that the droplet size does not depend on the almost does not depend on surface tension. Here expansion atomistic simulation of expansion of is a larger surface tension and normal actually surface tension of liquid tin here four times less. So you see that the platorally instability will delay a very long time because surface tension is quite small. Actually it will be if it will be zero if the platorally instability will not develop at all. Okay so let's go ahead. So it's the most interesting fee for us I suggest instability of planar jets. In order to give you idea how it's possible to destroy liquid sheet how to make it unstable I just showed the movie downloaded from the internet with from this slow down show channel YouTube. So it shows initially that the bubble very soft bubble very stable itself if there is no hole no anything so it may leave very long time may leave yeah and even stable if you have a soap soapy hand move in and move out it it survives very easily. So remains as a wall however if you just touch with bubble with dried finger it's immediately destroyed and many many droplets what kind of instability leads to this such fragmentation. So you see small bullet moving with a speed about 10 meters per second. So it creates a very small hole here and then the edge of this bubble sheet propagates to the left and there's only and the remnant of this edge is a liquid droplet. So sintering also so these two bubbles satellite bubbles survive in this experiments. So the idea of this instability is fragmentation start from the hole some small pin from on this sheet. Okay I show here more details here so it's touch surface and then this edge moves to the left and from this edge we may see there are a lot of filaments which itself are unstable due to just standard platural instability this fragment to the this filament fragments to a lot of small actually not small quite large droplets large means a larger thicker than the thickness of this initial bubble sheet. Okay many times about 10-20 times. So let's talk about some theory of this instability. So the casp which I told you before is format on the edge of this liquid sheet or jet. So and the material moves to this rim and collected or accumulated within of this casp and casp generate a new secondary cylindrical jet with cylindrical jet unstable and fragments to the small droplets. Okay the scheme is shown here we have let's consider the system just at the rest of the rim at the rest of the rim then sheet inflow material inflow into this rim happens at some angle say and then because tangential velocity should be conserved this velocity just stop it to zero so this is conserved then we obtain such condition where the material inflow forms say inflow inside the rim so to the to toward the caps from here and here so sheet comes to the rim just rotate or how say make angle and move along the rim like river which collect all all small springs all sorts of water and then here we have a accumulation of material which form secondary jet here cylindrical jet which will destroy it on the droplets these are just simple representation of what happened on the sheet so this is MD simulation so the first question how the such casp on the edge of the sheet can be formed it's easy just any fluctuation of thickness of the sheet will result in the formation of this casp we have a sample inflow there's a planter of planter jet which has here thin thickness of this sheet here thick and thin again so we have here a small velocity of the rim here large velocity in the system of the uh the rest of the uh this rim the material comes to uh rim move along the rim to the casp and accumulate here and forms uh second here is just droplet big droplet but later let me uh i'll show you uh long secondary jet will be formed so more details is a side view so thin thin very thin planter jet then here we have a rim and this rim form this kind of uh secondary uh i i may say cylindrical but of course it's not exactly cylindrical than just maybe elliptoid elliptoidal jets and the end of jets we have a formation of the big droplets finally then they expand in time and will detach from this sheet so more interesting to say here we have a velocity 220 meters here because this sheet is quite thin about 10 millimeters so the for 10 millimeters we have a rim speed speed of rim is about 20 200 meters but for one micrometer size of the bubble sheet the speed of the rim is just about 10 meters per second which is why i before show like bullet with the speed of 10 nanometers 10 meters per second has almost the same speed as a uh destroy destroying front of this of this bubble so material inflow to the rim and may uh obtains because with uh oblique inflow then the this flow can serve tangential velocity here and the red one is just moving to the central part green one is moving opposite to the left side to the center and then they accumulate there okay this is a molecular dynamic simulation of very big system it shows that the planter jet form the this secondary plan of jet through the rim forms cylindrical jet initially of course it it became becomes unstable when it's a richer enough length but initially it was quite stable here but later time when the distance lens of this jets becomes enough to develop for development of lateral instability some droplets are just detached from from the end of this jet this is a basic mechanism which we just found it's quite not a really new result because in hydrodynamic simulation it's quite known but here the material was thin and the sheet full transition from the sheet jet in form of sheet to the jet in form of cylinder was absorbed and simulated here okay so you saw that the jet actually jetting formation of droplets in secondary jets it's quite a quasi stationary process so escaping just initial time for formation of this jet so here I show one typical period on later time how the droplets form it and new droplets appear I mean end of the cylindrical jet appear and then it repeat again I mean detach and detach so statistic shown here that the initially small droplets detach we just neglected so and then the very big droplet in the size was format and the end of cylindrical jet and the speed of this was about 70 meters 70 meters per second but later we have a just repeating k repeating situation where the jets are format and detached format and detach with the average velocity about 100 meters so there is a correlation smaller size of droplets a larger velocity so that we may say there is some correlation quasi steady region so okay so in summary what we did just first we made consistent sph and atomistic simulation of shock produced shock induced liquid jet from the team solid team shock wave was melting show and it was our kind of work from first time I think say I think so so what we show that the cylindrical jet disintegrates to the droplets or series of droplets we are well known platorial instability it's not simulated yet I guess in the atomistic scale but of course simulated by the hydrodynamic codes but how hydrodynamic and molecular dynamic simulation quite well agree and we believe that expand our smooth particle hydrodynamic simulation to a real system with surface tension which we'll hope to include soon and then planeral liquid jets are internally stable as you saw from the movie and our simulation we also did some kind of perturbation of the planer jets in simulation but nothing happened it's quite stable and only way to fragment it it's just a form the age a hole any form of age on this on this sheet and due to perturbation of the thickness of the sheet that the oblique flow may results in formation of the rim and rim itself will lead to the formation of secondary jets formation of the caps and secondary jet cylindrical jet and which the last one is unstable due to just normal platorial instability and in in in tin such edge fragmentation moves on the tin liquid tin sheet with a speed about 10 meters per second for one micrometer for thickness of one micrometers it's okay I mean this is a kind of way to destroy such planer jet however in experimental condition the time of destruction is one word one order order shorter so it means when I saw initially I mean of second third slide experimental observation of fragmentation of liquid jet planer jet that cannot I mean this mechanism cannot be utilized to explain this so it means so the mechanics of inner I mean inside planer fragmentation of planer liquid jet if it's existing actually it's maybe some mistake of experimentalists so how are we here if they are right though these mechanics remain unclear and this is of course a subject of future research thank you very much