 Good morning, we will continue our discussion of multi-phase flows and see one possible route by which we can model these sprays. Before we go to modeling multi-phase flows, what I want to do is try to see if we can recap some of the information we learnt earlier and concepts we learnt earlier with regards to probability density functions. So our old definition of probability pdf, if I define an f I will use r for the radius of the drop and we will write I will write down the nomenclature. Define such that this is the probability of finding a drop in the range r to r plus dr and velocity v to v plus dv and in the space in the spatial location x to x plus dx. So that is let us just quickly understand what this means. If I take a spray and I will plant my coordinate system just to since we are now talking about a real physical system and modeling a real physical system. If I take a tiny volume element and imagine there is a bunch of these drops inside here then if I take f this f times this probability density f of r, v, x, t is time of course here is basically given by the probability of finding a drop in the radius range r to r plus dr and velocity range v to v plus dv. So we should be clear about what this vector v to v plus dv means. So I will write that out v to v plus dv is the same as saying or rather let us be it is easier to define the dv vector. dv is equal to dv1, dv2, dv3 which is the velocity components in the three directions. So it is like a little tiny volume in the velocity coordinate. So if I take a drop of radius r and I can easily understand what the meaning of r to r plus dr is. So it is like 50 to 51 microns. What the volume element in the velocity coordinate this dv vector given by the product of dv1, dv2, dv3 means finding a drop of velocity let us say 10 to 10.1 meters per second in the x direction some 1 to 1.1 meters per second in the y direction and let us say 5 to 5.1 meters per second in the z direction. So a drop satisfying all three conditions is what will fall inside this dv element. Likewise dx is easier to understand we have been trained to understand this since high school. I will refer to these by x1, x2, x3 it makes more sense. This is dx1, dx2 and dx3. So this is easier to understand because it is physical space. It is the space in the spray like a little tiny volume and I am looking at all the drops that fall inside this volume that is my starting point of all the drops that fall inside this volume. I want to look at the drops that have size r to r plus dr and have velocities between v to v plus d1, v1 to v1 plus dv1, v2 to v2 plus dv2 and v3 to v3 plus dv3. That is like a tiny if you can think of this now as a volume element in the velocity coordinate. So this is I want to make sure we are all on the same page. We are used to spatial coordinate and volume in the spatial coordinate. We are now defining a velocity coordinate and defining a volume in that velocity coordinate. So I can now define any other coordinate just like this and I can define a volume in that coordinate. The reason I have a volume in the velocity coordinate is because velocity is a vector. So I require three components to describe the velocity vector and it naturally forms a volume in the velocity coordinate. Whereas if I took radius, radius is a scalar and I just need r to r plus dr. So essentially I am still defining a volume but it is volume in a one dimensional space. So r to r plus dr defines a volume in 1d space of the scalar quantity radius of the drop. dv defines a volume in the velocity coordinate in the three dimensional velocity vector space dv1 dv2 dv3. dx defines a volume in the spatial coordinates in the spatial coordinate vector space dx1 dx2 dx3. So what we have done is we are used to space as having coordinates. We are now introducing velocity as also being a coordinate. That is a fundamental shift in the way we are going to look at these models. I think there is a reason we do that. We will come to that maybe later on. So the moment I have made this, I have defined this multivariate probability density function f with r, v and x as the independent coordinates. So this is in 7 coordinates and time as being a parameter. So we are not defining, I can define an instantaneous probability density function. So at this current instant of time, if I gather all the drops that are in the spatial location dx, x to x plus dx and parse them into radius bins and velocity bins, this gives me the multivariate probability density function. Now we looked at this even when we defined multivariate probability density functions. The fact that the fact that the f at this point is clearly not the same as the f at another spatial location. Now f is a function defined in this 7 dimensional space of 3 velocity coordinates, 3 spatial coordinates and the radius of the drop. So typically we will use the word external coordinate to define the spatial coordinates. Those are like obvious to us. We have learned that from Cartesian geometry days. These velocity and radius are internal coordinates. Now I can continue on and define many more internal coordinates as many as I need to completely determine the state of a single drop. Say for example, we talked about this in our earlier lecture. If I have a mixture of let us say methanol and ethanol that I am spraying, then the fraction of ethanol in a given drop is an internal coordinate of that drop. I can define a probability density in that internal scalar coordinate. So I can now define concentration in a general sense of some n dimensional mixture as being an n vector and I can continue this argument forward. So I just want to bring the distinction between external coordinates and internal coordinates in the way we are going to look at space. So now, so we want to write an equation that describes the evolution of f in space time and the velocity coordinate. So the velocity is now an independent coordinate just like space. We are not going to make any distinction between velocity and space. It is like at this point I have a certain drop at this location moving at this velocity. This position and velocity are both characteristics of that drop. They are independent variables that there could be other dependent variables that fall out of this. We will look at those. So now let us go back to this idea of simple co-moving derivative and see what we are used to this say if I have some function f or some function g which is a function of x and time. We write a co-moving derivative of g. This is what we very often refer to as capital dg dt where u is the velocity, scalar velocity in the x direction. This comes from the basically defining g as a field in x and time. So essentially if I have this g is a property everywhere in x and time and u defines the transport velocity at every point in space and time. So u is also a property of x and time. That is generally how we understand fluid mechanics. And this comes from basically defining velocity as a field and not velocity as a property of a particle or property of the material. So if I define velocity as being a field variable depending on the spatial coordinate and time. Whatever material particle passes through that point in space and time acquires that velocity. That is our basic assumption on which our hydrodynamics and Navier-Stokes equations are built. So if I now take that forward and if I have x and t as my independent variables. This is my co-moving derivative. If I continue this forward if I define f like we have here is a function of r, v, x and t. So for this general probability density function how do we write the co-moving derivative and what does that mean? This part is fairly straight forward. So since f depends on since f is a function of the spatial location x, y, z or x1, x2, x3. I could have an advection of this property f. f is just like our old g. Any property which is a field which is defined as a field which is defined as a function of x and time can be advected through that in that spatial coordinates and time by their respective velocities v1, v2, v3. So if I have v1, v2, v3 as the velocities at a given point of this material f of this property f then the partial derivative of f with respect to time plus v1 df dx1 plus v2 df dx2 plus v3 df dx3 gives me the rate of advection in space and time. But we now have additional coordinates in terms of v and r which are now defined as our independent coordinates. So how do we account for those? You require additional terms. What does this mean? If f1, f2, f3 is a force vector per unit mass acting on the population of drops. So just like I require velocity which has units of meters per second. Velocity has units of meters per second to advect the probability density function in the spatial coordinate which has units of meter. So if the spatial coordinate has units of meter the corresponding velocity with which the property is advected has units of meters per second. If the internal coordinate has units of meters per second the corresponding property the corresponding let us say vector by along which this property is advected in that the probability density function is advected in the velocity space will have to have units of meters per second per second. So essentially this force is I mean f1, f2, f3 is the force vector per unit mass which is what we know as acceleration. So if there is a force per unit mass acting on this population of drops that is going to cause movement of this particle in the velocity coordinate. I want to make sure we understand this and also let us take a very simple example one we are familiar with from high school. If I take a mass let us say a block of mass m being pushed by a force f let us say it is initially moving at some velocity v. So the acceleration we know is f over m and we are only going to look at one dimension one spatial coordinate. So one way to look at say for example it is currently at this location which is x equal to 1 let us say or x equal right. So if I look at the probability density function at this x equal to 1 or the probability density function in space it is a direct delta function at x equal to 1 that is I have a mass at x equal to 1 and everywhere else there is no mass. So we will just looking at a point mass let us say okay. So I have if I look at the probability of finding particles or number density of particles we are used to number pdf and we will define this f also as a number pdf. The number pdf at every location other than x equal to 1 is 0 and or rather the number probability at every other location is 0. So if I take a location from 0 to 0 plus dx whatever dx may be the number probability in that spatial location 0 to 0 plus dx is 0. The number probability in 10 to 10 plus 10 plus dx is 0. The number probability from 0.95 to 0.1.05 is 1. So I have one particle and there is so there is a probability of finding a particle around 1. So in the limit of that dx shrinking to 0 we get our probability density function and that probability density function when plotted in the spatial coordinate takes a form of a direct delta function at 1 and basically it is like a very sharp peak going towards infinity at 1 with the area underneath this curve being bounded being equal to 1. That is the meaning of a direct delta function. And so this is the probability in this is a pdf in the spatial coordinate. If I plot the same pdf in the velocity coordinate if it is initially moving at some velocity v0. So I am now disregarding space. I mean it is a little trivial to talk of pdf of 1 particle but I thought this example should illustrate at least the idea of an internal coordinate as being not very different from an external coordinate. That is my objective here. I can look at if I have let us say a bunch of particles I can I we all understand how to create a pdf in that velocity coordinate. If I do a similar pdf of this 1 particle there is a probability of finding a particle at v0 and no other velocity. So I can at some velocity v0 there is a direct delta function that says essentially says all my particles have 1 velocity v0 all in this sense is 1 but that is a material to the pdf. So all my particles in this in the system have 1 velocity v0 and you know I can actually make the argument that this is a mass containing you know many particles and they are all moving at the same velocity. They are all at nearly the same physical location x. So I mean these things can also apply physically to that system. So now how do I move this pdf in the spatial coordinate by the spatial velocity. So if the particles have a spatial velocity if this is the pdf at time t at some other time t plus delta t this pdf shifts to 1 plus v0 dt. So at some future time t plus delta t t plus dt this same direct delta function has moved to a new location x1. So the probability density function in x coordinate is now given by delta. So if I define f of x at time t this is given by the direct delta function this is given by the direct delta function at x minus 1. The probability density function at t plus delta t plus dt is given by this x minus x1. So this t plus so all we have done is velocity in the spatial coordinate has advected this pdf from being p at x equal to 1 to some x equal to x1. Similarly if I have a force acting on this system and the acceleration is f over m f of v at some time t the f of v we are only writing probability density function in a single coordinate just to sort of understand what is happening and in this case the multivariate or the bivariate probability density function of x and v is the product of these two whatever we are writing here we will show that also. So I know that at some time t I had a direct delta function defining my probability density function in the velocity coordinate and the direct delta function was a peak at v0 the value v0. At some future time what would this look like? So just as we said we updated its position due to the velocity we say it we update the velocity due to the acceleration. So we are in this formulation the velocity is also a coordinate along which motion is possible due to acceleration. I want to take one more last little bit of complication let us say this block was growing. So let us say initially this is of some mass m or we will say for example of some size l. So cubic block or some size l let us say and I have a little mass accumulating on here at some l. or some v. some volume is being added to this block and as a result its size is also changing. So this is just like a condensation or an evaporation process in a droplet right. So what do you I can now say I can do something exactly similar so let us say initially it was some size l0. l0 is the size of the particle at time t and if I now take at the next time the same probability density function in the size coordinate if l. is the rate at which that size coordinate is changing then l1 given by l0 plus l.dt defines the new peak in my pdf. So notice how size velocity and spatial coordinates are practically indistinguishable except when a drop grows by let us say condensation the we talk of not the drop growing but the drop pdf being advected in the size coordinate space. Likewise if the particles are accelerated we talk of the particle pdf being the particle pdf being advected in the velocity coordinate. Likewise in space we are used to the idea of advection in space we have now introduced the idea of advection in the velocity coordinate and advection in the size coordinate. Now I can have advection in any other internal coordinate. I have methanol ethanol and ethanol is preferentially condensing on the drop at certain rate on the drops ensemble. So I am taking a whole population of drops to really talk of pdf right. So if I have some ethanol condensing on the population of drops then the rate of condensation of ethanol is going to advect this population of drops in the ethanol fraction coordinate. Now in the granular material literature so when I am dealing with let us say crystallization or when I am dealing with powder material flow if I take a single powder particle I in order to completely define the geometry of a single powder particle I may need to define more than its size. I may need to define like number of facets or some polygonality or some I can come up with measures that characterize the single particle and let us say I take a whole population of this powder and I found it. If I am taking water what might be perfectly cubic particles and creating particles with more facets. So this if I define a coordinate like average surface area per unit volume then pounding the cubic particles down to finer and finer particles which are non cubic amounts to an advection in this area per unit volume coordinate. So whatever force I am imparting in this case I am not talking the physical force but the act of pulverizing this powder is creating an effective advection force in the area per unit volume coordinate. So these are the this is a different way of looking at looking at transport of a pdf in a given space. In this case we are looking at x v space x v r space. So we still have one more last term to define which is x v r so here r dot is the rate of growth in the r coordinate. So if I now look at this all these possibilities of advection in the different coordinates let me take a very simple situation first if I take just a bunch of drops that are all stationary but are of different sizes. So I have an initial f as a function of r but they are all stationary and in both stationary in position and not moving. So I mean clearly they are stationary. So if I now allow some sort of a condensation process to be initiated or an evaporation process does not matter it is a matter of just a sign right. So r dot is a function of the r current radius of the drop. Then d dr of r dot times f defines the rate of advection of this of this pdf and df dt which is the rate of change of the pdf explicitly due to time. So this df dt the partial derivative of f with respect to time is where the pdf is explicitly evolving in time is due to r dot. So if everything else in here the black and the red terms going to 0 the only the explicit variation of f with respect to time is due to plus the rate of change of rate of advection of the pdf in the r coordinate together define the rate of change of this pdf in a system sense. So for you to completely understand this I would suggest you read an undergraduate textbook and understand this Eulerian Lagrangian frames of reference and where we use the Reynolds transport theorem to go back and forth between the system and control volume view of a material right. So if you do that df dt this total derivative is our systemic view of the material all of this right hand side is a control volume view of the material that is find out define a control volume in x1 x2 x3 v1 v2 v3 r space this is the rate of motion in and out of that control volume. Now this alone is not enough so if I now follow a certain group of drops and follow them follow the exact same group of drops without paying reference to their velocities their spatial locations and their size. So I have marked a certain set of particles or drops and I am only going to follow those there are we can write a law a balance law for that group of drops this is just like our systemic balance law of force and force and momentum where we say the rate of change of momentum is equal to the force acting on the system. So likewise this left hand side the right hand side that I have is our control volume view and we have to replace the left hand side which is our systemic view of the fixed set of particles or drops with an equivalent balance law. So what could be happening to these drops that could cause the probability density function in a systemic view to change so I will take a I will just post this in some sort of a physics sense and we will come back write the mathematics next time. I will just again go back to a very simple view of two particles let us say undergoing an inelastic collision so I have two particles moving towards each other with the velocity v and v in opposite directions with velocity plus v and minus v speeds v and v of the same mass coming towards each other and colliding. If the collision is perfectly inelastic the velocity of this resultant entity is 0. So these two particles come collide they let us say coalesce into one drop which is going to become stationary. So or I am going to treat this as you know two drops that are attached to each other so the probability density function used to have two peaks at plus v and minus v now has become one peak at 0 and there is no external force acting on these particles right the forces are all internal to these two particle system. So everything on this right hand side barring of course if I take that instant where they were all at the same physical location so I am going to ignore the black terms for a moment the black spatial advection terms for a moment when you can imagine this happening in a very small neighborhood before and after the collision so to say. The advection in all the coordinates is not is absent you only have a collision driven change in the pdf. So from a systemic view this is what is responsible for the change in the pdf from having two peaks at plus v and minus v to having one peak at v equal to 0. So collisions between the particles of the system so I have a system of particles collisions between the particles of the system is response could be one force that is responsible for change in this pdf ok. We will look at how what that means to the left hand side of this equation in the next class.