 So, let me now try to go to the convergent-divergent nozzle and discuss what happens there. Very roughly what you can think about is that we already have a convergent nozzle. What we end up doing is that we add to this a divergent part and then at the end of this divergent part we have the back pressure which will be reduced from the value of P naught. So, let me try to draw the sketch quickly for this convergent-divergent nozzle. So, again I am dealing with a axisymmetric situation. Again, sorry I have, this is my exit plane, here is the back pressure which I am able to lower as I want, here is the throat, so this I will call a C D nozzle, here is the inlet plane as it was before and again the conditions just before the inlet plane are the stagnation values corresponding to which the velocity of the flow is equal to 0 essentially and then I have the flow going from left to right, this is my, sorry, exit plane already mentioned. So, up to the throat we have exactly what we just discussed namely a purely convergent nozzle and then I have a divergent part attached to it. So, again here when I want to start reducing the back pressure, if the back pressure is equal to the upstream stagnation value P naught then again there is no pressure difference to drive the flow and there is absolutely no flow. As you keep on reducing the back pressure initially the entire nozzle will behave in a subsonic fashion because the pressure difference is not large enough for the flow to reach the value of 1 in terms of the Mach number at the throat. So, let me try to draw this again in that pressure distance plot. So, we start with P over P naught equal to 1 and this is the exit plane, back side is where we are maintaining the back pressure and lowering it and here additionally I will add that throat location because now we are dealing with convergent divergent nozzle. So, as far as starting from P b equal to P naught is concerned when we have P b equal to P naught this corresponds to no flow as you start reducing the back pressure the flow will initiate and initially as I said if the difference between P naught and P b is not large enough everything will behave subsonicly and going back to that area velocity relation what I have is if everything is behaving subsonicly the convergent part will be such that the velocity will increase the pressure will decrease and the divergent part will be such that the velocity will decrease and the pressure will increase. Let us say this was the value of the back pressure that I was maintaining. So, it will smoothly come back to that back pressure. So, this is all entire subsonic. So, this will keep on happening until a value of the back pressure is reached corresponding to which the throat pressure becomes exactly equal to P star. So, you are coming from the top by reducing the back pressure gradually you reach a value of the back pressure such that for the first time in the throat the value of the pressure that is achieved is P star. So, then what happens is here is my pressure dropping from P naught to P star. Remember, it is still the limiting case of this entire subsonic case. So, therefore I will slowly go back again this will still be subsonic. However, I will call this as a limiting subsonic case. Why limiting subsonic is because the value of the pressure in the throat for the curve that I just drew corresponds to the critical pressure or the P star value corresponding to your P naught value. So, this particular pressure is a special value in the back pressure because corresponding to this particular value I am reaching P equal to P star at the throat and the operating condition in the nozzle is something that I would like to call a limiting subsonic. So, you remember that A over A star relation that we had discussed for every value of A over A star not equal to 1 we said that there are two isentropic solutions possible one was subsonic and one was supersonic. So, in particular now consider the ratio exit area divided by A star. If you look at this exit area to A star you have certain value for this. If you go to the A over A star relation corresponding to this particular value and look up for example, in the isentropic flow table you will come up with two values of the Mach number as I said in the isentropic flow table one of which is what we have just drawn. So, the limiting subsonic case corresponds to a completely isentropic flow, but fully subsonic through the entire nozzle and will reach some subsonic Mach number at the exit plane here where your back pressure is getting maintained. Now, while we are on the topic of that A over A star and the two limiting cases let me also draw the other supersonic isentropic solution. So, in which case what ends up happening is that that curve will simply come down to something like this. So, if you maintain a back pressure equal to this value this corresponds to the supersonic solution for this A E over A star value in which case the pressure continuously decreases from P naught reaches a value of P star at the throat will keep on continuously decreasing and finally will smoothly reach the value of P B which you are maintaining at the back pressure. So, this case is what is popularly called as the design condition. So, the design condition is when the convergent divergent nozzle is operating in a completely isentropic fashion with a supersonic solution in the divergent part. So, the pressure then that you reach at the exit plane here is what we call a design pressure design exit plane pressure and if you go back to your A over A star value whatever this is you go and you read off from the isentropic flow table that corresponding to this A exit over A star value there is one unique value of the supersonic mach number and if the pressure in the back side is maintained at this value what you will do is to realize that the exit mach number will be equal to that value from the isentropic flow table. So, that let me just draw it here M exit design is what you will achieve if the back pressure is maintained at this value corresponding to which the throat pressure is P star entirely the flow is isentropic and supersonic completely through the divergent part. So, these are the two limiting cases that will exist corresponding to the value of A E over A star one is what we call the subsonic solution the other one is what we call a supersonic solution. So, now the question is that what happens if we maintain the back pressure in between these two value. So, let me try to to draw a new figure for this. So, this is the throat location again. So, let me draw for the purpose of reference this limiting subsonic case the supersonic solution this is subsonic solution and this is the supersonic solution. So, this is what we called the design condition the supersonic solution in both cases the pressure at the throat is equal to P star. Now, let us say that this first limiting case corresponded to some specific value of P b let me call it P b 1 and the second one corresponds to some other value P b 2 the design value here in between if you want to maintain the back pressure somewhere between P b 1 and P b 2 what is the situation. So, let us go back to the limiting subsonic case here. Now, let us assume that you are operating with the limiting subsonic case and now you want to reduce the back pressure a little bit from the P b 1 value that you are maintaining to achieve the limiting subsonic case. What ends up happening now is at the instant when you reached this P b 1 value you reached P star value at the throat and therefore, the mach number at the throat is going to be equal to 1. So, therefore, if now you reduce the value of P b 1 below the value of or back pressure below the value of P b 1 there will again be the generation of those acoustic disturbances which will try to move up the nozzle and while doing so will try to change the conditions within the nozzle. However, when it comes to the throat location it will experience the same problem as it had when we were discussing the purely convergent nozzle namely that the flow itself has reached the sonic velocity at the throat and therefore, this acoustic disturbance is not able to go anywhere and change any conditions up the location of the throat. So, therefore, beyond the throat nothing changes this remains completely frozen here. On the other hand what ends up happening is that because let us say you are trying to maintain a pressure like this here which is P b 3 let us say that is neither low enough equal to P b 2 nor high enough equal to P b 1 which are the two limiting isentropic solutions. What ends up happening is that because the mass flow rate is chosen the nozzle is essentially forced to flow the frozen mass flow rate and the only way this situation is resolved is by generating a little bit of supersonic flow in the divergent part a shock situation occurs and post shock the flow becomes subsonic and will slowly reach back the value of P b 3. So, here is a situation where a shock occurs within the divergent portion of the nozzle corresponding to a value of the back pressure that is in between this limiting value of P b 1 and P b 2. It so happens that if you are going close to P b 1, but just below P b 1 this shock will be closer to the throat ok. If you go away from P b 1 below toward P b 2 the shock will actually occur at a later cross section. The location of the shock simply depends upon the back pressure that you are maintaining and the phenomenon that involves in formation of the shock is exactly the same as what we were discussing yesterday namely coalescence of individual acoustic waves etcetera. It is just that here we are dealing with a inherently steady flow situation, but physically there is absolutely no difference between the shock formation that we talked about yesterday and what we are discussing here today. As we are reducing the back pressure some transient flow take place and until the steady state is reached and within this transient flow what ends up happening is that there is that coalescence of individual acoustic waves to form a shock at a certain location within the divergent section. So, going back again to the board what will end up happening is that if I want to redraw the plot. So, I will now draw only that situation which we are talking about. So, here is where we have reached P star that is not going to change further there is a supersonic flow developed which ends in a shock. Post shock this is subsonic as we saw in the yesterday discussion this is from here to here this is supersonic in the throat it is exactly m equal to 1 and this is the value of the back pressure that you are maintaining which is achieved at the end of the exit plane. If you keep on reducing this back pressure what ends up happening is that this shock will keep on moving and at certain value of the back pressure it will actually stand exactly in the exit plane. Let me call this P b 4. So, if you are maintaining this P b 4 the shock actually stands in the exit plane the value of the Mach number here is exactly equal to the design value. However, the back pressure is not low enough. So, that a completely isentropic flow can be achieved, but there is a normal shock that is standing exactly at the exit plane. Now, if you want to reduce the back pressure further what ends up happening is that there are two possibilities. So, let me draw this in a new figure let us say that this situation is such that the normal shock is standing here corresponding to which the back pressure is P b 4 this is your design condition. So, anywhere in between this if you employ the back pressure what you generate is what is called as an oblique shock which is outside the nozzle which will eventually reach the value of the pressure that you are maintaining. So, this situation is what is called as an oblique shock that is outside the nozzle as against that under expanded situation which we saw in the case of a purely convergent nozzle. This is what is called as an over expanded nozzle the reason why it is called over expanded is because the exit plane pressure here is below the back pressure that you are maintaining and the back pressure is achieved by increasing the value through the oblique shock outside the nozzle. Again the oblique shock situation is something that we have not studied within our quasi one-dimensional framework we cannot really study those for that you have to go to multi-dimensional flow analysis. So, again when it comes to explaining to the students it is probably sufficient at this level to explain that there is this oblique shock phenomenon that will exist in case of these over expanded nozzle and the oblique shock is actually acting outside the nozzle and it is raising the pressure from this design value in the exit plane to a higher value which you are maintaining outside the nozzle. If you keep on lowering the back pressure you will achieve the design value again there is absolutely no waves generation everything is isentropic and this is where you want to typically operate the nozzle in it is not always possible. So, if you end up lowering the back pressure below the design value of the back pressure what will happen is what we have already seen namely an under expanded nozzle as was the case in the case of a purely convergent nozzle. So, whenever you encounter the words under expanded nozzle what we can understand is that the the nozzle is such the nozzle is operating such that there is an expansion from the exit plane pressure to the lower value of the back pressure that is occurring outside the nozzle through expansion wave. If on the other hand the nozzle is over expanded we we understand that there is an increase in the pressure from the exit plane value to whatever value that you are mentioning at the back back side through a series of oblique shock which are again outside the nozzle. So, this is how usually the the nozzle operations are are described and with the help of our isentropic flow tables and the normal shock tables it is possible to analyze more or less all these situations that we have we have discussed. So, what we will do is in the afternoon we will go through a couple of example problems where I will try to outline a few more features of how to solve these some of these problems and post that we will have our discussion session, but I will really request that once we have gone through our afternoon session today you please go back and try to solve the remaining the problem from the assignment sheet or the exercise sheet because many of these ideas which are described here you know cannot be fully understood unless and until you attempt to solve those those problems. So, with this I think I will I will stop this was more or less what I wanted to discuss as a part of introduction to compressible flow in case this topic is introduced in the thermodynamics course. Remember that we have already discussed far more things than what are typically available in the thermodynamics books as compressible flow chapter and the reason is because if you want to explain more details as to what is happening in these operating nozzles and so on you have to utilize ideas from fluid mechanics and it may not be possible for the thermodynamics book to really include all those fluid mechanics ideas and therefore the way at least I would look at it is that if this topic is included in the thermodynamics course my suggestion would be to stick to the minimum possible requirement and to provide these additional explanations through the kind of discussion that we have already gone through. So, with this I will I will stop.