 So, far in this particular course, we have emphasized more strongly on incompressible flows, but compressible flows are also very important and the scope of this particular course does not contain too much of an opportunity of discussing the details of the compressible flows, but we will try to identify some of the important conceptual paradigms which are associated with compressible flows and we will see that how they may be strikingly different from that of incompressible flows. So, first of all let us revisit the requirements of compressibility or incompressibility. To do that, let us consider that we have a one-dimensional unsteady flow so that if you write the continuity equation, you are having this particular term, terms. Now, let us try to have a sort of feel of what would be the corresponding form if it is an incompressible flow. So, to do that, let us say that we divide this term into two parts. One is if it were an incompressible flow then obviously the only term that would have mattered for that case is this term, whereas if it is a compressible flow, we expect that this particular term would also turn out to be important. So, if this term does not turn out to be important then it is as good as treating the flow like an incompressible one. That means we are interested to investigate a situation when you may have a sort of effect which gives rise to some kind of compressibility but the compressibility is not substantial. That compressibility will not be substantial when you have this term much, much less than this particular term. So that means if you just write in terms of order of magnitude, u into delta rho is much, much less than rho into delta u. Also just to get a feel of the relationship between delta u, rho and delta p, we may refer to the Euler's equation. That is dp by rho plus u du equal to 0, assuming negligible changes in potential energy. So this we are just treating it in an invisible limit and try to figure out that what could be the important parameters that may dictate the extent of compressibility in the fluid. So this is as good as writing that delta p by rho is of the order of u into delta u. So our requirement is delta rho by rho is much, much less than rho, much, much less than delta u by u and in place of delta u, you can write this as delta p by rho u square is delta u by u. So from here what we can say is that delta rho by rho is less than equal to or we can simply rewrite it by writing it in terms of delta p by delta rho is much, much greater than u square, right. So what is delta p by delta rho? We will see very soon that this delta p by delta rho is an indicator of the sonic speed within the medium. So let us say that c is the sonic speed within the medium. So this in fact may be written as c square where c is the sonic speed that is the velocity, the speed at which disturbance propagates through a medium. So since u by c that is the fluid speed by the sonic speed is known as the Mach number. So we can say that this is as good as Mach number square is much, much less than 1 in terms of the order of magnitude. So from this what follows is that if it is a very low Mach number case then the situation may be treated as if it is like an incompressible fluid. So from this very simple understanding what we can conclude is that one of the very important roles that is being played is by the sonic speed and therefore it is important to know that what are the factors or what are the parameters on which the sonic speed depends. To do that we will consider a very simple experiment. Let us say that you have piston cylinder arrangement and there is a container and there is a piston. So the container is like a cylinder. What is being done is that let us say this is a confinement and this piston say is moving towards the right with a particular speed. So when the piston is moving towards the right with a particular speed it is forcing the molecules which are adhering to that to be disturbed and that means it creates a disturbance in the fluid medium that is there in the space between the piston and the cylinder. Question is that what is this medium? Let us say that this medium is a gas and let us try to see that what would be the difference if in place of a gas if it would have been some kind of a liquid. Now what we may say is that if it is a sort of a liquid or a highly compact system then if you create a disturbance now the question is here if you create a disturbance and if it is a liquid with a high compactness the disturbance whatever it is you may not be able to compress this further because of the very poor compressibility of the medium. But what you may get assured is as and when you try to impose this disturbance this disturbance gets propagated from one point to another in the medium at infinitely large speed. So it is like all the molecules or all the fluid particles if they are very compact they respond almost instantaneously towards this disturbance. On the other hand if the medium is not that compact then what happens then if there are certain molecules which are in contact with this particular piston then those molecules are first displaced from their mean position. So it is a sort of a pressure pulsation. So that pressure pulsation then is propagated to some other molecules in the surroundings and then from that to another set of molecules in the surroundings to maintain the continuity. But all the molecules in the medium are not simultaneously disturbed by that pressure perturbation. So what happens is that there is a sort of disturbance the disturbance propagates through the medium at a finite speed and if it is a sort of a weak pressure wave that propagates through the medium at a speed which is a function of some of the properties of the medium and one of our objectives will be to identify that how it depends on some of the properties of the medium. But before that we may appreciate that the effect of this pressure perturbation it may have reached at certain location and beyond that location the effect is not felt at a given time as time progresses may be this location up to which the effect of pressure perturbation is felt propagates further and further. So we may call this as sort of a wave front remember that it is not exactly like a flat line that we have shown in the diagram. So the wave front may be typically a part of a very large sphere with a particular radius but we are considering only a small part of it so that it is almost like a flat surface and this is propagating relative to the medium at a certain speed say C which we call as the sonic speed that is the speed at which a disturbance in terms of a pressure perturbation traverses relative to the medium in form of a weak wave weak pressure wave. Now we need to identify that what happens upstream and downstream of this so if we just consider this particular line which is say propagating towards the right at a particular speed C and because this line is very very thin almost all properties are continuous across it. We will see that in not in all cases it may be possible that even if you have a a limited thin interface across which properties are continuous but here we will assume that the properties are continuous across this because this is just a normal wave front but if it is a shock wave front then that may not be possible. So let us say that the pressure on this side is P or let us say that the pressure on the right side is P here the pressure is P plus dP. In the right side the pressure is the density is rho here the density is rho plus d rho. In the right side the velocity is 0 and in the left side the velocity is say du. So this is undisturbed and this is disturbed. Now remember that this front is propagating towards the right with a velocity u to make an analysis it may be important to transform our reference to a front which is stationary. To do that what we will do as if we will add a speed of C but opposite to the positive x direction so that the speed of the front relative to that reference would be 0 and then all other quantities will be measured with respect to that. So then the situation will be that here you have made a transformation so as if you are sitting on a reference frame that moves with a velocity same as this one. So with respect to that moving reference frame moving with a velocity C what would be the velocity of flow here remember what we are doing we are adding as if to make it stationary we are adding a velocity of minus C. So this is in this direction C and here plus du is along positive x and minus C superimposed on that so here it will be C minus du in the negative x direction to maintain it so this is a transform reference. Now what we will do we will write equations of mass balance and momentum balance for a control volume that surrounds this particular front first of all mass balance. So conservation of mass now one of the important assumption that we are going to make here and possibly for most of the important fundamental theoretical derivations for compressible flows at least within the scope of this particular course is that the flow is like a one dimensional flow that means u is not varying with y. So then conservation of mass what we can write that rho into A into C is equal to rho plus d rho into A into C minus du where A is the area of cross section of the control volume that is drawn. If the velocities were non uniform one would have taken the average velocity in place of this. Now from here what we can write is that rho into C is equal to rho into C minus rho du plus C d rho minus d rho du rho into C gets cancelled out d rho du is a product of two differentially small quantities which may be neglected in comparison to the other terms. So we can write that d rho is equal to rho by C du this is what we get from conservation of mass let us say this is equation number one. Next conservation of linear momentum so conservation of linear momentum what we get the resultant force that acts on the control volume we will just be using the Reynolds transport theorem. So the resultant force that acts on the control volume is what? So from the left side it is p plus dp into A from the right side it is in the opposite direction p into A. So dp into A is equal to m dot into v2 minus v1. So since all the flows we have shown in the negative x direction maybe let us write it conservation of linear momentum along the negative x direction to make the algebraic sign simple. So m dot into C minus du minus C right in place of m dot we can write rho into A into C. So this becomes minus rho AC du so dp you can write rho C du. Now if you divide 2 divide 1 by 2 so 1 divided by 2 what it gives dp d rho is equal to C square right. So from here we may get an expression for the sonic speed which is square root of dp d rho. So we get an important property which sort of is related to the elasticity of the medium because what it shows? It shows the effect of compressibility directly that is in other words it is inversely related to what is the rate of change of density with respect to change in pressure. So if you say pressurize the system whether how strongly the density will change or not. So if you require a large pressure to create a small change of density then C is very large that means if when you require a very large pressure to create a change in density when it is not a very compact when it is a very compact system when it is not a very compact system you require a very small pressure to create whatever change in density you want. But more and more compact the system is you require more and more pressure that means more and more compact the system higher and higher will be the sonic speed. So the fact that this kind of sonic speed finite sonic speed is existing it shows that it is not a perfectly compact system that we are talking about. So analogy here with mechanics or basic or classical mechanics is like between rigid bodies and deformable bodies or say rigid bodies and elastic bodies. So if you have a rigid body and say you have a disturbance all points equally feel the disturbance. If you have an elastic body there is an elastic response to it so all points do not respond equally to the disturbance. So it is a compressible medium is like an elastic body and in fact this is related to the elasticity property of the medium. The next important question is that when you write dp d rho it depends on the thermodynamic process by which it is occurring. So to have a further simplification of that we must refer to certain processes. So let us try to see that try to have a feel of what kind of thermodynamic process that we are talking about. In most of the examples that we will be discussing in this particular chapter we will be referring to the special type of flow known as isentropic flow or the process thermodynamically is an isentropic process. Isentropic process is just equivalent to something known as reversible and adiabatic process. Of course this is not a course of thermodynamics so we will not go into the details of what is the reversible process, what is the adiabatic process is straightforward. You know that a process where heat transfer is 0 is an adiabatic process that is relatively clear but what is the reversible process? So reversible process is defined as such a process which if it has taken place it may be reversed and while it is being reversed at the end it leaves no change in the system or in the surroundings. So how it is possible? It may be possible only in a hypothetical environment that does not exist in reality because a reversible process has to be a very slow process. So as an example if you have a piston and a cylinder like this now say that you are moving the cylinder or moving the piston in the cylinder upwards. So how you may move it upwards? One of the possibilities is that let us say that you want to move it from the initial level to the final level dictated by these tops where you want it to stop. So one of the possibilities is say initially it was restrained by a huge weight which was pressing it with the bottom stops. Now you take this weight away so when you take this weight away it will suddenly jump from this location and reach the top. It is a very fast process. Now if you want to revert it back to its original state what will happen? Say it has come to this one now you put the load back and it will come back here. When it comes back here it does not mean that it will come back to the same thermodynamic state. Its temperature pressure all the states would be different from what it was initially and only way in which you can make sure that you also return it back to the same thermodynamic state is by making the arrangement in this way making this as a summation of very small slices, very small weights. You remove the weights one after the other so that it will move only slightly from its bottom position to the top position. So all the loads are removed say the loads are so designed that it has just reached the top most position. If you want to revert it back then put the loads back one by one slowly and then it will come back to its original position. So if that is done then what will happen is this will be back to its original state spontaneously. On the other hand if it moves very fast then when it has its forward motion and then its reverse motion after that it would be found that inside its temperature is higher say as an example than with what it started. To bring it back to its original state some transfer of heat to the surroundings will be required then and then in that process the system will come back to its original state but the surroundings are no more at the same state because of the net heat transfer. So that is not a reversible process. To have a reversible process one must have a very slow process. So in reality such a slow process such a slow process is known as a quasi equilibrium or a quasi static process. So in reality such a slow process may not be achievable and therefore a reversible process in reality is something which is sort of it is an imagination. It may be only achieved by thought experiments but real processes may only approach that in a limit but it will not be really reversible because if it is really reversible it will take infinite time to complete the process. Now if you are thinking of a reversible process there are many factors actually which are against it in reality which makes real processes irreversible. One of the important factors is friction. So friction is such a factor which will make a real process irreversible in a way that we will not have enough scope here to discuss but just keep on important thing in mind that if you want to have a reversible flow or a reversible thermodynamic process it should be devoid of friction because friction is one thing that makes the real process deviated from a reversible one. So when we say a reversible and adiabatic process we mean basically a sort of frictionless and adiabatic process and a very slow process. It will be striking to some of you who have first learnt an adiabatic process as a very fast process because that is how adiabatic processes are introduced in high school physics that you have a piston cylinder arrangement and you make the process so fast that there is insufficient time for heat transfer to take place and then you call it adiabatic yes that is one of the ways of achieving an adiabatic process approximately but it is not a reversible adiabatic process. To have a reversible adiabatic process it has to be adiabatic it has to be slow for example in this particular system as a thought experiment you could have a reversible adiabatic process by putting an insulation around the cylinder everywhere so that there is no scope of heat transfer to take place and insulating material you put and then make a slow change so that will make a sort of reversible and adiabatic process. So we are thinking of to get a fair idea first that this process by which the pressure and the density they change and they are related to each other is a reversible adiabatic process. It may be shown that if it is a reversible and adiabatic process of an ideal gas then you have p by rho to the power gamma is equal to constant where gamma is Cp by Cv. Remember it is a misconception that this is valid for any adiabatic process of an ideal gas it is only reversible adiabatic process of an ideal gas so if it is adiabatic but irreversible this is not valid. The other important thing is that when you say that this is a constant now what are the factors on which Cp and Cv depend? For ideal gases it may be shown that Cp and Cv are functions of temperature only such that Cp-Cv is always equal to R but it does not mean Cp and Cv are constants these may be functions of temperature but when you have the difference the function of temperature gets cancelled out but certain special types of ideal gases will have constant Cp and Cv and those are known as calorically perfect gases these are different between perfect gas and ideal gas. So for a calorically perfect gas you have Cp and Cv as constants not function of temperature ideal gas in general Cp and Cv may be functions of temperature. So here we are discussing about perfect gases so we will not explicitly tell all the time but in this particular chapter whatever substance we will be considering is a perfect gas calorically perfect gas that means an ideal gas with constant Cp and Cv. So for that particular substance we are having this p by rho to the power gamma as constant as the process. So for that let us then find out what is dp d rho. So if we say that reversible and adiabatic process why it is called an isentropic process because if it is a reversible and adiabatic process it may be shown that there is no change in entropy during the process that is why it is called as an isentropic process. So if we want to find out that what is the value of C corresponding to an isentropic flow then it is better to write this in a more formal way that it is partial derivative of p with respect to rho by fixing the entropy. It ensures that it is a reversible adiabatic process because you could have many different types of thermodynamic processes by which you could have a change in pressure and a change in density. So if you have p by rho to the power gamma as constant you may take log of both sides and then differentiate. So you will get dp by p-gamma d rho by rho is equal to 0. So from here you can write dp d rho is equal to gamma p by rho. So this is root over gamma p by rho for an reversible adiabatic process and we know from the equation of state that p by rho equal to RT. This R keep in mind this is not the universal constant this is the universal constant divided by the molecular weight of the substance. So this is the specific gas specific constant. So the universal constant usually is given a symbol of R bar in the common literature that is 8.314 joule per mole Kelvin. So kilo joule per kilo mole Kelvin or joule per mole Kelvin all the same and this R is this divided by the molecular weight. So it is for air this turns out to be roughly 287 joule per kg Kelvin. Of course molecular weight of air is not a very simple calculation but you may roughly consider it as some proportion of nitrogen and oxygen and like 79 is to 21 roughly and then calculate the equivalent average molecular weight. So if you do that this will be the value of R. So keeping that in mind we may replace p by rho as RT so this becomes root over gamma RT. So we can see that what are the factors on which this sonic speed depends. It of course depends on the gas constant it depends on the Cp by Cv and the absolute temperature. But we have to keep in mind that this is not the most general expression the most general expression will be this one because then depending on the nature of the process dp d rho will have different expressions this is for an isentropic process. So for a normal temperature condition this will be close to 340 meter per second and since sound wave also propagates in form of a weak pressure wave within a medium. So it is a natural coincidence that this is also same as the speed of sound in a medium. Sonic speed is not the speed of sound in a medium by a fundamental definition. But because sound also propagates in the form of a weak pressure wave it is coincidental and scientifically coincidental that they are the same. Now let us work out a very simple small problem to just illustrate some of the facts that we have discussed till now. So the problem is like this a weak pressure wave which is a sound wave a weak pressure wave with a pressure change of delta P equal to 40 Pascal delta P equal to 40 Pascal propagates through steel air at 20 degree centigrade propagates through steel air at 20 degree centigrade and 1 atmosphere. Estimate number 1 the density change and number 2 the velocity change across the wave. So the density change and velocity change across the wave it is a very straight forward problem it is the use of the formula expressions 1 and 2 for calculating the density change and the pressure change. So if you calculate the density change and the pressure change so first of all you see here the dp is equal to rho cdu just look into equation number 2. So you have dp that is given that is delta P 40 Pascal. So from here you have to find out what is delta u the rho of the medium you can find out given the temperature and the pressure as 1 atmosphere you can use the ideal gas equation to find out the density. So let me give you the answer so that you may verify this but it is a straight forward use of this one. So the answers are delta u equal to 0.097 meter per second and delta rho is equal to 0.0034 kg per meter cube okay. So for this expression you assume that the r as 287 temperature remember the value of the temperature should be absolute temperature. So not 20, 20 plus 273 because all calculations here where second law of thermodynamics is involved will require absolute temperatures. So that is the only place where you have to be careful otherwise it is a straight forward use of the formula that we have derived okay. Now the next thing that we will try to see is that we have till now seen that what is the sonic speed what is its physical implication and what are the parameters on which the sonic speed depends. The next question will be that we have seen that there is a important non-dimensional number called as a Mach number which is the ratio of the fluid speed or flow speed to the sonic speed and therefore it might have its important consequence so far as the extent of compressibility is concerned that also we have seen. So how that Mach number influences the nature of compressibility or the nature of propagation of disturbance within a particular medium let us try to look into that bit more carefully. So we will consider some important limiting cases. So this is just to have a physical understanding on the implications of Mach number. So as a first example let us take say Mach number less than 1 just let us say Mach number equal to half. Let us say that there is a source of disturbance which we are approximating as a point source. The source is moving in the medium with a certain speed and let us say that the speed is u. So the source so the medium is stationary and the source is moving with a speed u is as good as relative to the source the fluid is flowing with a speed of u but in the opposite direction. So there is a source which is at time equal to 0 here it is moving along this direction say at time equal to some delta t it is there at this position 1 which is given by u into delta t say it is moving towards the left that means relative to the source flow is moving towards the right then at time 2 delta t it is there at the point 2 and so on. So we consider another times at time 3 delta t it is at the point 3 which is again another distance u delta t. So in this way the source is moving and from each of these points through which the source passed waves are emitted and when waves are emitted see from the point so let us say that we are now considering the current instant of time. So at the current instant of time how much time has elapsed 3 into delta t. So in time 3 into delta t if there was a wave that was initiated or that was propagated from the point o then how much the wave will travel in the time 3 delta t see is the disturbance of propagation of the wave. So c into 3 delta t would be the distance at which distance by which the wave would travels. So you have Mach number equal to half that means u by c equal to half that means in time 3 delta t the wave will propagate by a distance of c will be 2u so that means double of this one. So by taking this as a center and 6u delta t as the radius if you draw a circle in all sides it will be a very big circle then that will be the wave front in the plane of the figure the radius of this circle is c into 3 delta t where c is 2u. Thus from the point 1 when the source was at point 1 after that how much time has elapsed 2 delta t. So by this time now this has emitted something which will go by a distance of c into 2 delta t. So by taking this as a center and c into 2 delta t as radius this will be the wave front corresponding to the wave that was emitted from 1. Similarly when the source was at 2 after that delta t time has passed so it is emitting a wave that is now having a front like this where this is c is 2u. So you can see that the disturbance is always ahead of the source and the disturbance propagates in all directions just because the figure will become very clumsy I have not drawn the other part of the figure it will be too big to draw here but just imagine that these are full circles. So the disturbance is propagated in all in all the directions and it moves ahead of the source that means at this point if you think of this point this point knows the message that there is a disturbance before the source of disturbance comes there this is very very important. Next let us consider the second limiting case say mach number equal to 1. So when you consider mach number equal to 1 again let us take the same example but just the different mach number. So 0, 1, 2, 3 like that. So here mach number is equal to 1 that means u equal to c. So if you construct the similar things for just as the wave fronts that you have drawn for the previous figure c and u are the same. So we will see that these fronts are still circles but having a sort of common tangent at the point 3 right. So this is a limiting case. Let us take a third example mach number greater than 1. Again we draw the similar figures all these intervals are u delta t. So when a wave is emitted from 0 in time say let us take an example of mach number equal to 2 that means u by c equal to 2. So when one wave is emitted from 0 in a time of 3 delta t it will have its zone of influence as c into 3 delta t and c is u by 2 that means it will be a circle with half of the radius of the distance as between 0 to 3. So this will be c into 3 delta t. When it is at 1 by the time 2 delta t it will emit a wave with radius c into 2 delta t which is same as u into delta t. When it was at 2 similarly this is c into delta t. So in this way it is possible to draw envelopes of these particular figures and it will give something which is very very interesting. So if you draw an envelope as a common tangent to all these circles so what does this line represent? See all the waves are confined within this zone. So this is like a cone in a 3 dimensional space. So these green colored lines are the bounding lines edge views of that bounding lines of the cone. So all the waves are confined within that and the disturbance can be understood only if wave has propagated at a particular location. So outside this at any point the disturbance is not failed and within this cone the disturbance is failed. So this is known as within this zone it is known as zone of action and this is known as zone of silence and this cone which is generated this is known as mac cone and the semi vertical angle of this cone may be obtained very easily. Let us say that mu is the semi vertical angle of this cone. So you can write so if you draw a tangent to the circle so this distance is C into 3 delta t and this distance is u into 3 delta t. So sin mu is C by u right from the right angle triangle that is 1 by mac number. These bounding lines are very very interesting in terms of mathematical theory of the compressible flows. So if you see these bounding lines any location which is just on one side of this bounding line and just on the other side of this bounding line these 2 locations are somewhat different. One location will feel the effect of the propagation of the disturbance other location just on the other side of the bounding line will not feel it. Therefore there is a strong discontinuity across these bounding lines and as if the disturbances are propagated following these bounding lines. So on one side disturbance is felt on another side the disturbance is not felt and there is a sharp discontinuity across it. So these types of lines are known as characteristics of the physical system. So here if we think about the corresponding partial differential equations these are known as hyperbolic partial differential equations and these lines are known as the characteristic lines of the corresponding hyperbolic partial differential equations. We will not go into the mathematical theory of these things but the interesting observation if you see that say if a aircraft is moving with such a high speed. So roughly if you see mac number less than 1 is a relatively low speed flow and these are known as subsonic flow. So mac number less than 1 is subsonic. Mac number equal to 1 is sonic. Mac number greater than 1 is called as supersonic just some important names. So it revolves around sonic which is mac number equal to 1. Why sonic? It is as the sonic speed and the flow speed are the same at that particular condition. Now close to mac number equal to 1 there are certain names these are known as transonic. So it is like close to mac number equal to 1 but one may have a slight variation that means some mac number say within the range 0.8 to 1.2 or whatever like near sonic regime is like a transonic regime and very high mac number are known as hypersonic flows. So very very high. So anything greater than 1 is supersonic. So very high value of mac number typically greater than 3 is known as a hypersonic flow. So if you have a supersonic flow then what is an important thing that you observe say you have a supersonic jet say you observe a jet plane moving in the sky. So you will see that once the jet plane has moved after some part of its movement then only you realize that its sound that you are getting and you will see that the smoke that is emitted from that will form such a cone. If you observe jet planes moving in the sky you will see that it will form this type of envelope of disturbance. Once you are within this envelope of disturbance then only you feel that it is there. That is why when it is there at some distance such that you are within this mac cone then only you feel that yes you have heard the sound of that. Otherwise when if you are outside this mac cone you are not hearing that. So let us work out a problem to illustrate this. So the problem statement is like this a supersonic aircraft flies horizontally at 1500 meter altitude. So there is a supersonic aircraft which is moving at an altitude of h equal to 1500 meter with a constant speed of 750 meter per second. The aircraft passes directly over a stationary ground observer. The aircraft passes directly over a stationary ground observer. How much time elapses after it has passed over the observer before the observer hears the aircraft? How much time elapses after it has passed over the observer before the observer hears the aircraft? Assume the sonic speed as 335 meter per second. And the aircraft creates a small disturbance that may be treated as a sound wave. And the aircraft creates a small disturbance that may be treated as a sound wave. So it is if you schematically consider it there is a mac cone like this and consider that there is this is the ground. The observer is standing here and it is 1500 meter height at which the aircraft is moving. In a time interval of delta t the aircraft has moved by v into delta t where v is the or u into delta t where u is the speed of the aircraft. You have to find out what is the delta t so that the observer now comes within the mac cone and so this is a limiting configuration when the observer comes within the mac cone. So this angle is given by tan mu is equal to h by u into delta t and we know that mu is equal to sin inverse of 1 by mac number from the previous diagram and mac number is u by c both are given. Therefore tan mu may be obtained h is given u is given so from here you can find out what is the delta t and the answer to this is 4.006 second. So, we have seen that what is the implication of different ranges of mac number and in terms of the propagation of disturbance and how that may be related to the sonic speed. And in the next lecture we will go ahead with this and we will move on into some more interesting and important properties of compressible flows. So that we will take up in the next lecture. Thank you.