 Good morning, welcome back. We are going to look at we are going to extend our discussion of secondary atomization, look at a couple of different modeling approaches and then look at a generalized modeling approach that you can do for not just secondary atomization systems but something beyond that. The simplest of the breakup regimes for secondary atomization is the Weber number base model which is in the first of those is the vibrational breakup regime for Weber numbers about 0 to 10. This is what we found yesterday and for this regime essentially the physics associated with breakup of a drop in a stream of some velocity u is that it starts to oscillate and this oscillation can grow in time and eventually cause the drop to break up. Let us look at and the simplest of models that is used to describe this regime of breakup is called the Taylor analogy breakup model very often called the tab model. The approach in this model is quite simple you take a drop of some radius r and superpose over it a disturbance some x. So, you treat this system as a one dimensional as a one degree of freedom system is an amplitude response to an equivalent spring mass damper system. So, you can think of the drop. So, the moment I have a perturb the drop we know that we have seen videos of this it is going to undergo oscillations and if I impart an impulse force forcing to the drop it is going to oscillate and then come to rest. The reason for the damping is due to the liquid viscosity. So, you essentially I mean it has all the features of a regular spring mass damper system just for the sake of completeness I will write I will draw a schematic for a spring mass damper system one kind of a spring mass damper system and an equation if x is the variable characterizing the degree of freedom then I have mx double dot plus cx dot plus k times x equals some f. So, if I have a force f acting on it. So, I have the drop of a certain mass the surface tension force acts like a spring stiffness in the viscosity acts like a damping force the origination of the viscosity is responsible for damping in the system. So, it has we are going to make this equivalence and write down some scaling variables for each of these. So, I will just for the sake of simplicity I will divide all of this by m and we will write a scaling for each of these f by m goes as the force acting on this drop is due to the relative velocity. So, if half times rho a u squared times pi r squared is the force acting on the drop it is sort of a magnitude of the force divided by the mass of the drop itself which is rho l times 4 pi by 3 r cubed I am not. So, I am going to write this f by m as some cf times rho a u squared by rho l r. Likewise k by m has units of force per unit mass and in this case k has the same units as sigma which is Newton per sigma being surface tension we have used this before. So, sigma divided by rho l r cubed that is the scaling of this k by m or I can write k by m as some ck times this dimensionless group and then I also have c over m c over m has units of or at least c has units of force per unit velocity. So, let us just check the units on c it is kilogram meters per second squared per kilogram. So, per meter per second has units of kilogram per second. So, I am going to reconstruct if I take c to be dependent on the liquid viscosity and so liquid viscosity times r the drop radius has the same units as kilogram per second because the units of liquid viscosity dynamic liquid viscosity is kilogram per meter per second. So, mu l times r has the same units as c divided by our rho l r cubed. So, from here I can write c m is some c mu times mu l over r squared. So, I have now taken a model system which is a spring mass damper system and taken the model constants which are these f by m k by m and c by m and related them to my real system which is an oscillating drop in a wind. I have to have some way of estimating these c l c f c k and c mu that is one of my remaining tasks and second part of the remaining task is what do I do with the results predicted from this model and how do I relate it related back to the drop itself. The way we look at the second part. So, we will now define an eta which is some c b times x over r. I am going to let this eta be my independent dimensionless variable. So, I can effectively take my equation which is and dimensional non dimensionalize this whole thing in terms of eta a cast and replace the f by m k by m with these values which are c prime which are c f k c k and c mu. If I do that what do I find? So, if eta is c b c b is some number c b times x over r. So, it is like saying that when this eta let us say for example, c b is 1. When eta which is my oscillation amplitude reaches 1 the non dimensionalized oscillation amplitude is equal to the radius the drop is likely to break. So, I can have a number for c b which says for example, if the drop reaches 90 percent of the radius it would have it would have it would basically break the drop up or c b is a parameter in the problem, but it is only used to cast our dimensional variable x in terms of a non dimensionalized in a non dimensionalized form. So, let us see what do I find here c b over r. Let me just write rewrite this part eta is c b x over r which automatically means x equals eta r over c b. So, x double dot is r over c b times eta double dot plus c mu nu l over r squared nu l is of course, our kinematic viscosity which is mu l over rho l I am just using our standard nomenclature we followed all along. And then I still have x dot which is r over c b eta dot plus k over m is c k sigma over rho l r cubed times x which is r over c b eta this equals f over m which is my c f rho a rho a u squared rho l over r. I can factor out the r over c b from all of the equation all terms on the right hand side and what do I have now this is a linear second order ordinary differential equation in eta. And if I for the right for the right set of values the system I mean in fact, we do not have to assume say for the right set of values for a typical drop the system is under damped which means if I plot eta as a function of time given my forcing function rho will be able to find the it will for a given forcing function rho a u squared will be able to find eta as a function of time. And you set a threshold value of eta at which the drop is likely to break up. So, this is the simplest of models that you can use to study secondary atomization given the set of fluid properties surface tension density you have a model to related to the break up characteristics. In the higher Weber number regime say in the sheet thinning or multi mode break up in the sheet thinning or the multi mode break up what you essentially have is a drop in a cross flow if the if I do a linearized stability analysis of this drop. So, essentially if I do the full linearized instability analysis of a spherical drop of liquid in a cross flow. You can do exactly what we did with a cylindrical jet or with a planar liquid sheet except the mathematics would be much more cumbersome. In fact, the analytical solution for this flow so consider a drop of radius r of some viscosity nu l in a fluid of some viscosity nu a. The analytical solution is given by what is called the Hadamard Rybczynski solution. So, this is the case of this is an analytical solution for the case of laminar flow past a liquid drop where you set up if you can imagine a vortex inside the drop. So, this solution gives you the complete fluid mechanics of what is happening inside the drop and outside the drop in the laminar regime. So, this is your mean flow condition at this mean flow if you now if you now perturb this mean flow where you you know on the surface you have a certain n in the direction that I have indicated and another m in the. So, it is like a latitude direction and a longitude direction you have a certain m a second azimuthal wave number in the other direction. So, if I do a full three-dimensional linear instability analysis of the Hadamard Rybczynski solution subject to these m n wave numbers you will find one m star and n star which correspond to the most unstable pair of wave numbers ok. What that means is that if I subject this drop to a cross flow and that means I have a certain number of waves that will preferentially grow on this drop and as they grow I might have this wave grow and eventually cause pinch out a drop from the side that corresponds to this wave number. So, this is what we have seen even in the jet breakup problem in the cylindrical jet breakup problem the wave number that corresponds to the most unstable point is responsible for the drop size. Likewise here you can do this entire calculation and show that the wave number that is responsible azimuthal wave number that is responsible for the drop the most unstable point is responsible for the drop size. This is called again linear instability analysis based model for secondary atomization often also called the wave model, but the wave model is a little more than just a linear linearized instability analysis of the Hadamard Rybczynski solution. The wave model makes some assumptions. So, if I take this drop which is initially spherical the red line that I have drawn corresponds to m equal to 2 that is it corresponds to if we go back to our linearized instability corresponds to the case where there are two lobes that are formed and the higher the value of m the smaller is this length scale d this d is proportional to 1 over m or n whichever is the we will use n I think there. So, as n increases or more precisely as n star the most unstable wave number increases the size of the drop that you are pinching off becomes much smaller than the size of the parent drop itself which basically means that the fact that the parent drop has some initial curvature is no longer as important. This is practically like ripples on a lake. So, if I look at a small part here I might as well be looking at ripples on a lake if the wave if that little yellow region contains a large number of ripples already the fact that the drop is curved is not as important. So, this is the assumption underlying this wave model this wave model assumes not a spherical drop, but a cylindrical drop. So, it looks at wave number in just one plane assuming no waves in the other plane and from that gets an m star which is responsible for the most unstable wave number. So, linearized instability analysis is a very powerful tool because it allows you to study perturbations from a mean flow and go all the way to predicting the final performance characteristics which is I mean there is no theoretical reason to believe this we have already talked about this except to say that empirically it seems to hold true in a wide range of these kinds of spray situations. I want to show you a slightly more generalized way of talking about these kinds of modeling. So, we looked at Taylor analogy breakup model which is basically a spring mass damper equivalent. I replace the parameters in my spring mass damper equation with what I know from my drop properties of my drop and from there I am able to predict what is going to happen. So, how can I go about writing a general model for a generalized situation say for example the simplest case would be I will take the example of the sheet thinning regime where I start with a drop in a cross flow this drop seems to produce a sheet like that that is this is further elongated then I may have some drops being shed. I want to now write a model of this process how do I do it the start is of course experiments you have to have some understanding of what is happening. So, the fact that I have drawn these the sequence of cartoons means that I have I know this is what is happening in this Weber number regime which is like let us say about between some 60 and some 300 I know this is something this is been observed to happen in this range of Weber numbers. How do I go about understanding how do I go about writing a model for this the start of writing a model is of course experiments like I said and then we will talk of a tool called time scale analysis ok what do I mean by time scale analysis. There are many different physical processes occurring in this phenomenon each one is happening on a slightly different time scale and we want to get estimates of those time scales and then see how under what time scales can we expect the sheet thinning break up to happen ok. So, if I start with a breeze of velocity u flowing past a circular drop of some radius r and if I am like I said for me to get these observations experimentally I have to be zoomed into a single drop correct for me to be zoomed into a full picture of a single drop I am automatically saying my length scale is the diameter of the drop that is the length scale on which I have to fix my observation to see this phenomenon. If I fix my length scale to be much smaller than this say like I had like the yellow window that I drew in the previous graph if I instead of looking at the whole drop if I only look at a tiny section there I am looking at ripples on a lake I am not looking at the drop breaking up ok. If I look at a much larger length scale I am not looking at the physics of what is happening at the droplet level I am looking at a particle shattering so if I know that at that length scale let us say the radius or diameter does not matter the first cartoon going from this a to b is where I am observing a deformation in this part. So, this part is being dragged forward due to the shear stress from the air ok so that is the start of this sheet thinning at least the way I have written this sequence of images. So, what is the time scale associated with this drag force the time scale associated with this air drag acting on the drop is that I want to have visible deformation on this scale with a velocity u so in other words that red part is moving forward at the velocity u and the on a scale that is approximately the drop size I want to be able to see this. So, I will use r the radius as my length scale so the time scale associated with droplet deformation due to air drag is this r over u ok. Now let us come into the drop and see what this deformation is doing inside the drop so I will take a slightly simpler version of this if I have this drop and this part is moving at a velocity u if I look at what is happening inside the drop in this region initially the velocity everywhere was 0 initially the entire liquid inside the drop was at rest as soon as this starts to become dragged what do you expect will happen I am now writing a zoomed out picture to get what the velocity profile inside here will look like essentially you start to create what looks like a boundary layer now this is still so in other words I am saying that this is my center but it is very rapidly decaying this is a after a short time after I have initiated this process now that means the a short time after I have initiated the process this momentum at the drop free surface is diffused into the drop due to the liquid viscosity. So, this momentum diffusion time scale I will call this T nu is given by r squared over nu where nu is the kinematic viscosity of the liquid that is in a time r squared over nu for whatever r and for whatever nu I would expect diffusion I would expect momentum imparted to the free surface to reach the center of the drop ok. So, if I have a certain velocity imparted to the top of the drop in a time are given by this r squared over nu that momentum would have come into the middle of the drop. So, the middle of the drop would have experienced some effect of the air outside the drop in this time until then if you are in the middle and the free surface on the top of the drop is moving you would not know it that is essentially the meaning of this momentum diffusion time scale at times much less than this r squared over nu if you are in the center of the drop you would not know that there was an air outside that was causing the free surface to deform. There is a third time scale which is due to the oscillation of the drop. So, what is this if I take the drop and if I just give it an impulse that drop is going to oscillate and these oscillations have a certain time scale associated with it ok. How do I get that time scale? I know that this looks now again like our spring mass system. So, if I ignore the damper part of the spring mass damper essentially the oscillation is due to the mass of the liquid inside the drop and surface tension. So, this is given by sigma over rho L r cubed over sigma surface tension. So, let us take our raindrop just estimate these 3 problems. So, we said a raindrop these 3 time scales these were the values that we had going to assume r is about since we had chosen r to be our length scale I will leave it at that. The drag time scale is r over u that is the air outside is trying to drag the free surface and you will be you will see visible deformation in 10 power minus 4 seconds from the start that is the meaning of this. If you take the viscous time scale 1 second the oscillation time scale time scale is this rho L approximately 10 power minus 3 seconds we are doing only an order of magnitude analysis. From these these are the only 3 time scales that matter in the problem in any drop breakup problem ok. So, if you look at what is happening here you will see visible deformation on the drop in 10 power minus 4 seconds surface tension reacts in about 10 power minus 3 seconds viscosity is going to take much much longer to react. So, it is like if you are in the middle of the drop you will hardly know what is happening outside ok. So, if you look at this if you look at just these 3 time scales the relative competitions between these 3 time scales tell you which dimension less parameters are important. If you take the drag time scale and compare that to the oscillation time scale you end up with essentially the Weber number ok when the drag time scales are on the order of the oscillation time scales you end up with the Weber number times density ratio in this case. If you compare the drag time scale to the viscous time scale you end up with the Reynolds number. If you compare the viscous time scale to the oscillation time scale you end up with the Onosorg number. I will leave that to you as a homework to derive these 3 under the situation when the drag time scale is comparable to the viscous time scale or more precisely the ratio of the viscous time scale to the drag time scale is your Reynolds number. So, instead of a 1 micron drop 1 millimeter drop like we had if I take a 0.1 millimeter drop which is a 100 micron drop clearly r is now a factor of 10 lower for the same viscosity nu which means your viscous time scale is 10 power minus 2 seconds. So, that is in 10 power minus 2 seconds the effect of what is happening on the free surface of the liquid drop is felt in the middle ok. If the viscous time scale is the lowest of all these 3 time scales I will just take an extreme case if the viscous time scale is the lowest of all these 3 time scales what you expect is whatever happens at the free surface is immediately felt in the middle whatever happens due to oscillations is transmitted everywhere in the drop immediately ok. That means this is where a situation where the drop is almost behaving like a rigid sphere meaning that there is instantaneous transfer of information due to diffusion, but it is so fast that if I try to move the free surface of the drop the whole drop starts to move ok. So, if the viscous time scale is very small that means you have the case for the viscous forces trying to hold the drop together. If I want to break up a drop I have to create a velocity gradient inside the drop. The moment I create a velocity gradient I have created a stress field inside the drop. If that stress field is not diffused sufficiently fast this drop is likely to break up. If the stress field is diffused sufficiently fast then I cannot break up this drop. The viscosity will spread any kind of a differential motion I create so fast that it is essentially going to bring it back to a rigid body translation motion. So, if I try to move one I have a drop if I try to move one part of the drop preferentially this part also starts to move in the same direction I can never break it up due to aerodynamic forces alone. This is the physical meaning of secondary atomization and these are the contributing time scales that in any problem for any given drop situation look at these three time scales ok. So, if I take let us say a standard PDPA data set I have a data set of some n drops each one moving at some velocity of some diameter in some air velocity. Let us say I know the air velocity from some other source of information say the smallest size class of the drops at that point. Now I can take I can take that information and for each drop in the problem estimate these three time scales and for each drop based on the three time scales I can tell what each of those drops is likely to do is it just going to remain like it is at undergo rigid body translation is it going to break up or is it going is it going to break up if so is it due to vibrational break up which is essentially the oscillation time scale being comparable to the viscous to the drag time scale or is it going to break up by the sheet thinning mode where the viscous time scale is much smaller than the oscillation and the drag time scales that essentially what happens on the surface stays at the surface. If I drag the surface forward that surface is being stretched into a thin sheet with a with the middle of the drop remaining where it is that is what we do as a cartoon for the sheet thinning break up that is where the drag time scale is much faster than the viscous time scale and the oscillation time scale. We will stop here we will move on to a discussion of multi phase flows and understanding sprays as a random process from next time onwards.