 Good morning, we are going to start off from where we left at the last class, we were looking at a cylindrical jet, we were looking at applying linear instability analysis to a cylindrical jet and the jet itself is moving with the velocity u2 and the fluid outside in general could be moving with the velocity u1 and we wanted to see how this extends to, how the stability analysis that we discussed extends to this case. By the way this, so once we start off, we ended by writing the by actually not just writing but deriving this normal mode assumption. So, the fact that the Eigen modes of this problem are orthogonal in that space comes out of the realization that we are dealing with a strum-lieuval problem which is basically it is a Laplace, the pressure field in each phase is governed by a strum-lieuval problem and because of that we went through this process and showed that the modes have to be of the form e power im theta and e power ikz which automatically mean that they are going to be orthogonal with respect to the weighting function that comes from that strum-lieuval problem which in this case is I think 1. So, given that, so these are the, this is the form of the Eigen function that we came up with for the pressure field P1 and P2 and from this we can go back to the differential equation that we have and solve for the velocity field Ui. We can also from this actually find that any sort of a surface perturbation, the cylindrical interface perturbation also has to be of the same Eigen mode form. Now, so we now need to formulate a set of boundary conditions to solve, to complete this Eigen value problem. The first set of boundary conditions that we wrote down are what we called the kinematic boundary conditions. Now, I want to make sure we, I am going to repeat some of the material from the earlier linear instability analysis just to make sure we have a complete grasp of the physical meaning of these boundary conditions. The kinematic boundary condition, say if I now take I am going to write the cylindrical polar coordinates in this fashion. So, this is our jet that is moving with the linear velocity U2 and the outside fluid is moving with the velocity U1. The, if I take this is eta as a function of z and time, eta meaning the, this is coming from the normal mode. In fact, z, theta and time. So, if I take a material point on this free surface and I am going to use a different color for that. If I take a material point that is on the liquid jet surface, this material point inside the liquid surface has a radial velocity U r which is given by the perturbation field U i dotted with E r. So, the magnitude of this radial velocity U r is given by U i dotted with this radial unit vector in the radial direction E r. This U r is the Eulerian velocity inside the jet that is essentially what we find that also since the material point happens to be on the interface. The rate of change of eta should equal that material velocity or the Eulerian velocity at the interface. So, what do we mean by that? This interface point can exhibit a radial velocity due to two reasons. One, eta itself undergoes an increase with respect to time at a given location. So, we have to go back to our fundamental understanding of a partial derivative. So, at an r and theta or at a z and theta location on the jet interface, d eta dt or partial derivative of eta with respect to time gives me the rate of change of eta with respect to time. So, that in some sense is like a radial velocity at that point. The second part corresponding to U i partial derivative of eta with respect to z accounts for the fact that even if there is no growth of eta with respect to time, but eta has a sinusoidal motion and that sinusoidal form is very being advected downstream. So, if I take a sinusoidal form and if this sinusoidal form sometime later is at this location, the same material point which was there is now here. So, the material point essentially has moved down that much distance in that period of time that this waves are shown the two snapshots that represent these two waves. So, even if the amplitude of this sinusoidal wave did not change, the fact that the material point moves with the interface is basically what this kinematic boundary condition implies in physics terms. Now, the rate of movement. So, this wave itself has moved a distance U i times delta t. So, if I took the material point as I have shown here just inside the interface inside the jet, then that wave is being advected with a velocity U 1 which is inside the U 2 which is the jet fluid velocity. I can draw this exact same pictures with the orange dot and the pink dot just above the meniscus in the other fluid. When I do that, the material velocity in the axial direction of that point is U 1. So, as viewed from fluid 1, the waves are being advected with a velocity U 1. As viewed from fluid 2, the wave is being advected with a velocity fluid 2. Consequently, the radial velocity in fluid 1 would be different from the radial velocity in fluid 2. So, we are used to understanding this from a flat meniscus. Saying if I have a flat meniscus and if let us say v 2 is the radial velocity in this part of the fluid, that has to exactly equal v 1 which is the radial velocity in the other part of the fluid. But if I am dealing with a curved meniscus, especially if the curvature is moving. So, if the curved meniscus is still stationary, then the radial velocity has to exactly be continuous. It is sort of analogous to our heat flux boundary condition at an interface between two materials. It does not even if the curved meniscus or the interface between the two fluids is curved, it does not matter. But if this curved meniscus is being advected, then the advection velocity is different in the fluid 1 because of the in versus the fluid 2. Now, this is essentially coming from our idea that I am going to throw out all the terms that are not order 1 in this equation. In writing this equation, I am going to throw out all the terms that are not order 1 and I am only going to retain the terms that are order 1. Because of that, the interface is assumed flat or unperturbed for all practical purposes. So, if I have this interface and it is in the unperturbed condition, the fluid the material particle inside the jet is moving with the velocity u 1. The material particle outside the jet is moving with the velocity u 2. So, this u 1 and u 2 being different at the interface is accounted for in this kinematic boundary condition. So, we are going to have to write something like this time and again every time we do a linear instability analysis of any kind of a problem. So, we need to understand where this comes about, what is the physical significance of this and just to make sure that we are able to make sense of it. The second set of boundary conditions are what we call the dynamic boundary conditions. So, if I now again take go back to the cylindrical jet, this is the unperturbed form, the perturbed form has this sinusoidal wave. Now, the pressure inside here is p 2 and the pressure here is p 1. We are only talking of the perturbations from the unperturbed condition. So, these lower case p 1 and p 2 are the pressure additional pressures due to additional pressure fields due to the perturbation. Capital p 1 and capital p 2 were our mean pressures before the perturbation. So, if I now look at this unperturbed or the perturbation pressure fields, the perturbation pressure fields give us require that the forces at this interface remain in balance. So, if I take a short patch of this meniscus, so right here, there is surface tension forces acting in all four directions. Now, because that interface is curved in both those directions, it has a curvature in both those directions. So, because of that, we have to define what is called a principal radii of curvature, which may not necessarily be in the r and theta directions per se. If this, if for a very general free surface, but if I am dealing with a free surface that was originally cylindrical and I am only going to retain terms that are order epsilon or order eta in the perturbation quantities, then those quantities are order eta. So, in that condition, the radii of curvature are in the coordinate directions. So, you have a radius of curvature in the r direction and you have another curvature in the theta direction and that essentially comes to, as you will recognize m is the circumferential wave number and k is the axial wave number, r is the unperturbed jet radius as shown in this figure. So, if I write the complete equation p 2 minus p 1, if I write that the total pressure in fluid 2 minus the total pressure in fluid 1 has to equal sigma times this 1 over r 1 plus 1 over r 2. But, we also know from this, from matching the mean pressures, the unperturbed pressures, this is coming from the mean pressures being matched. So, if I use that, what I end up getting is this. Now, I should be clear that this is p 2 minus p 1 evaluated at the unperturbed jet radius. So, if you go back we did find p 2 and p 1 as a function of r, t, z and theta. All we are saying is that at this meniscus, the pressure field just inside and the pressure field just outside on an infinitesimal strip of this, on an infinitesimal piece of the interface, balance out while including surface tension force. That is the physical meaning of this dynamic boundary condition. So, since the mass of that liquid or mass of both the fluids put together right at that meniscus is so small that when I say f equal to m a or sum of all forces on that infinitesimal piece of material is equal to m a. That m is so small that the sum of forces has to be equal to 0. We are essentially taking a thin slice right at the interface where fluid 2 is pushing from the inside, fluid 1 is pushing from the outside. So, I can take the p 2 and p 1 forms that we saw in this here. Evaluate them for the condition that this lower case r equal to capital R. So, that gives us the pressure inside fluid 2 and fluid 1 at the coordinate r equal to capital R. Now, I want to also make sure we understand one more concept here. When I take this and draw a free body diagram of this, I will erase this part. I have just drawn this is the pressure field p 1. I will use another color for the pressure field p 2 and the unperturbed interface is here. Now, if I really want to understand, if I really have to write this force balance, if I have p 2 as a function of r theta z n time which I do and if I have p 1 as a function of r theta z n time which I also do, I have to evaluate this at the interface location at the perturbed interface location not at r equal to r. So, this coordinate this unperturbed location is my r equal to r, but so this is let us say I will write this at the interface for now and I will show you how this comes about. If I take p 2 as an example which is inside the jet p 2 at the interface which is r the interface location is given by r plus eta which is a function of z n time the radial location of the interface is capital R plus this perturbation eta which is a function of z n time eta given by or in fact z theta n time. We are all we are doing a full three dimensional analysis here. I will write this more compactly as r plus eta, you know that eta is the interface location. I can write p 2 as a p 2 at r plus eta to be equal to p 2 at r plus eta times d p 2 d r evaluated at r equal to r plus order of r. Order eta squared terms simple Taylor series. Now, p 2 itself is a perturbation quantity p 2 itself is order eta or order epsilon quantity. So, this itself so this term is order eta squared already. So, technically if I was to write this next term it would be order eta cubed because p 2 is a small quantity eta is another small quantity d p 2 d r the derivative of p 2 with respect to r is also a small quantity. So, this already is a second order effect second order correction to p 2. So, to the order in to which we are maintaining the fidelity of our equations which is order epsilon. The pressure field at r plus eta is equal to the pressure field at r especially since we are only dealing with the perturbation pressure field. So, this perturbation pressure field I can now take even though I do rigorously if I have to do this to higher orders I do need to retain these terms up to a higher order. But to the order in to which we are retaining these equation to we are writing these equations we can write this we can evaluate these perturbation pressures at the unperturbed interface and still get away with it. Now, this process of doing this Taylor series expansion and then checking the terms that are required to be included in the dynamical pressure matched condition is very important. Because they are situations and one simple situation is where you have a swirling jet if the cylindrical jet had a mean swirl velocity which we are considering two cases where the fluids are simply co-flowing. If the inner fluid had a swirl velocity to it this order epsilon correction order epsilon terms itself would require the second part where the perturbation pressure would now have you would have terms coming from here which are order epsilon. So, without I do not want to go into the details of it we will probably take that up as a homework problem later on. But we now have this pressure matched condition we will call this equation 6 this is 7 or rather I want to make this 7 8. So, if I take all of these equations and go through the process of casting the Eigen value problem that is we ask the question what values of omega k and m yield non-trivial solutions. The answer to that is the dispersion relation that you get which in this case would be of the form the detail derivation of this is shown in this paper by Yang from the journal physics of fluids 1992. So, let us look at this for a moment and try to extract some physics out of it the equation as you will see is quadratic. So, we have it is we have one term that is order omega square that is of the form omega squared omega plus a constant term here. So, in general if I want to complete this write down the terminology that has been adopted here rho 1 m is equal to k times i m k r divided by i m prime k r times rho 1 rho 2 m is k times minus k times k m k r divided by k m prime k r times rho 2 rho 1 and rho 2 are the material density quantities k m is the Bessel function of the second kind modified Bessel function of the second kind prime k m prime and i m prime denote differentiation with respect to the argument of the function. So, if I go back look at this there is this as the for a given k and m this is a quadratic equation in omega which means in general it has two complex routes. The one complex route that gives us the higher real part the one complex route that has the higher real part is the one that is going to dictate the growth of that particular imposed wave. So, the wave itself in this case is characterized by two wave numbers an axial wave number k and a circumferential wave number m. So, these circumferential and axial wave numbers have a characteristic growth rate which is given by the real part of the omega of the real part of the omega and the one of the two that has the higher growth rate is the one that is going to be significant. So, in general if I take any particular is instance and plot this omega as a function of k and in I will plot this in some dimensionless sense I will plot it with respect to k r which is saying my length which is like non dimensionalizing my length unit the radius of the jet to 1. In general you will find this to be the that you will have the characteristic value omega as a function of k r take on a graph that looks like this. So, this end here is a k cut off beyond which if I extend this graph beyond the k cut off both the routes of that quadratic equation for m equal to 0 and k value greater than this k sub c both the routes of that quadratic equation have real parts negative meaning that any perturbation that is that has a circumferential wave number m equal to 0 and an axial wave number greater than k sub c if you could generate that kind of a perturbation on the meniscus is going to completely die down exponentially die down in time and any perturbation with k less than k sub c and m equal to 0 is going to grow in time with a growth rate given by that particular value. So, if for a given k r this is the growth rate that corresponds to that imposed disturbance. Now as you can see each of these disturbances is going to grow exponentially. So, if I had a source of perturbations that spanned all k r going from 0 up to infinity all k r that are that have that are where k is greater than this k sub c are going to be stable. So, this is my neutral stability boundary now all values between 0 and this k sub c are actually unstable meaning the real part is greater than 0, but there is one particular value which achieve which takes on an even greater significance owing to the fact that its growth rate is the highest among all of the wave numbers that are possibly unstable. And because each of these is growing exponentially in time very quickly you see that this particular value of k sub m is going to dominate in amplitude over all other wave numbers that are imposed. So, this is a characteristic of most linear systems and weakly non-linear systems that we see like for example, we looked at the Kelvin Helmholtz instability on a lake and this we said explains why all the ripples on a lake are nearly of the same wavelength. Now, I want to take one more instance we will look at this equation and simplified by setting both the u 1 and u 2 velocities to be 0. So, I am just now looking at a cylindrical column of liquid that is somehow stationary in air with no gravity. Now, what simply going through the process of scratching out the terms that do not apply now this is actually a beautiful result due to Lord Rayleigh which describes the instability of a cylindrical liquid column in vacuum because we said rho 2 equal to 0 due to surface tension alone because u 1 and u 2 are both equal to 0. What he found was that if I plot this omega as a function of k r like I said you will get something like that and this k r at which you get the maximum growth rate k is very nearly equal to 1 which if I translate into physics into real into the physical system if k equal to 1 that means this wave length lambda is equal to 2 pi. So, what he showed was that if I took a jet of diameter r this cylindrical column would break up into drops at equal intervals in length of the cylindrical jet equal to 2 pi r and so this if I took this cylindrical column of liquid and sort of sheared off pieces that are 2 pi r long r diameter and let each blob each of these cylindrical blobs go back to its lowest energy shape which is a sphere that would give us the size of the drop that would form and he showed experimentally that all of this mathematical analysis and the experimental observation of the actual size of a drop formed from a dripping faucet from just past a dripping faucet where you have a cylindrical column of liquid coming out a slow moving cylindrical column of liquid coming out of a bathroom faucet the drops formed from that are very close to this prediction which is quite remarkable because if you go back the whole analysis is based on the fact that I have a unperturbed cylindrical column that I introduce at small infinitesimal perturbation and look at the growth rate of that infinitesimal perturbation in the neighborhood in time of when the perturbation was introduced to start with. So, we are only looking at the case where I have introduced a perturbation and we are just watching it grow, but to then take the results that come out of that calculation and compare to an experiment where clearly the meniscus is well past the infinitesimal perturbation stage where at a case where I am able to visibly see the sinusoidal perturbations on the meniscus to show that the linear instability analysis also holds all the way to the stage where the perturbations can grow and grow to a much larger amplitude than infinitesimal than infinitesimal than being infinitesimal in nature all the way to where the amplitudes are comparable to the jet radius itself and in that instance this kind of an analysis the results from this kind of an analysis still hold good. There is no theoretical reason to believe that they should hold good, but in the fact that they do compare well with experiment is in is an indirect testament of the fact that the nonlinearity present in these equations is weak in comparison to the linear effects ok. So, we are we have been saved by the saved by the grace of Navier here in the sense that for the case of jet break up problems where linear instability analysis has performed remarkably well in compare in being able to predict drop sizes being able to predict the atomization characteristics and at the in fact quantitatively, but at the very least qualitatively where for example if I look at this exact same situation and see what effect surface tension may have on this you will find that qualitatively the predictions of this model hold good you know when you compare the results to experiment. Likewise if you go back to the dispersion relation now in the most general quadratic form that we had if I choose to retain u 1 and u 2 right here I can investigate the individual effects of u 1 and u 2 like for example if I increase u 1 what would that do to the drop size if I increase u 2 what would that do to the final drop size obtained those kinds of qualitative analysis are the trends that you that this analysis would predict are very close to what experiments have shown ok. So, this case where we took a linear stability result and we are able to predict a drop size from knowing that the volume of the blob formed is pi r squared times lambda which in this case happens to be pi r squared times 2 pi r now if I take this. So, this gives me 4 pi r squared r cubed. So, if I take an equivalent spherical drop radius. So, if I take this column of liquid and allow it to coalesce into a spherical blob. So, the drop formed from a cylindrical jet of radius r is going to be cube root of 3 pi times the radius of the cylindrical jet itself. So, you can take at the very least you can say that the size of the drops formed are going to linearly scale with the cylindrical jet for the case of a Rayleigh type or surface tension driven break up process we will stop here we will continue this discussion in the next class.