Good morning everyone. Let us continue with what we started yesterday which is differential analysis. We had already started the discussion and more or less completed it also for the differential mass conservation equation which is what we call a continuity equation. So, let me just briefly look at that. So, what we have done here is that we have employed our balance statement for an elemental control volume which is shown on the slide right now as the rectangular control volume with elemental length delta x and delta y. And then we identified the mass flow rates coming in to this control volume, mass flow rates going out from the control volume which were expressed using the first order Taylor series. And then completed the balance statement by realizing that the mass flow rate coming in that is a total mass flow rate coming in which will be m dot x plus m dot y minus the total mass flow rate going out would be then the mass flow rate that will be sorry the mass that will be getting accumulated. So, this is how the balance statement was written and by substituting the expressions for the mass flow rates m dot x and m dot y we simplified it to the form that is at the bottom of slide number 3 here. So, this is what we call a conservative form several questions have been asked already about the conservative form and the non-conservative form. So, let me take a minute to say here that when you employ a balance statement such as what we have done here inherently you will generate a governing equation whether it is a governing equation for mass or whether it is for linear momentum or whether it is for energy in the conservative form. So, if you see what I mean by conservative form you just go to the top line here the top box and this is essentially a conservation law employed for the control volume as you can see the first term on the right hand side is the net mass flow rate going out rather coming in to the control volume. Similarly, the second term on the right hand side with the minus sign that is is the net mass flow rate coming into the control volume and essentially the total net mass flow rate coming into the control volume will accumulate at the rate given by the left hand side. So, this is what we call a conservation statement or a conservation law and if you employ a conservation law through a balance statement like what we have done here you will always generate the governing equation in the so called conservative form. Using a mathematical manipulation here we converted the conservative form which is given by equation number 1 into another form which is dealing with a substantial derivative and this substantial derivative form is actually what is called as a non-conservative form. So, let me go ahead and talk about that in a second. Here again we obtained the same conservative form which is at the bottom of the screen, but the way it was obtained you can imagine again that we started from the integral mass balance equation. So, again fundamentally it is a balance equation written for an integral control volume we chose a stationary control volume and performing certain mathematical manipulations which are listed here using the Leibniz rule and the divergence theorem what we did was we converted the area integral into the volume integral. We brought the differentiation with respect to time inside the integral sign and made it partial differentiation and then essentially what you obtain is the same conservative form on a per unit volume basis because we said that the control volume is arbitrary and if a quantity integrated over an arbitrary control volume is giving you a result of 0 then that quantity itself must be identically equal to 0. So, what we mean by that quantity is the essentially the integrand. So, the integrand is nothing but this partial derivative of rho with respect to time plus divergence of rho v and this has to be equal to 0 was the conclusion and which is nothing but the mass conservation on a per unit volume basis or again the continuity equation expressed in a conservative form. So, again the point to emphasize here is we start from a balance statement and we obtain the expression in a conservative form. Again doing some mathematical manipulations which by the way here in this slide number 6 at the top where my highlighter is the mathematical manipulation is essentially opening up this divergence of rho times v and opening up of this divergence of rho time v has been achieved through the mathematical background that was discussed on day number 1. Essentially it is the same operation that was done here on slide number 4. On slide number 4 it has been done explicitly for Cartesian coordinates whereas, in slide number 6 it has been shown in the general vector form. So, please do not get confused about whether we are doing something different here than what we did earlier here. They are exactly the same here it is explicitly written out in Cartesian form whereas, on slide number 6 what we are doing is we are using the vector form directly. The purpose of doing all this is to make you familiar with different ways of manipulating the equations and primarily because different books will follow different approaches and you should know these approaches so that you do not get stuck while going through any particular book. Anyhow, so when you open this divergence of rho v you get a gradient of rho dotted with v plus rho times divergence of velocity and the first two terms which are which are underlined are essentially combined to obtain the substantial derivative of density and we obtain our non-conservative form. So, in this manner the non-conservative form has been obtained from the conservative form and using some mathematical manipulation. Exactly the same way it was obtained here we started with the conservative form in the Cartesian coordinates and opened up these differentiations and then collected certain terms together to obtain the non-conservative form. One last thing that we discussed here was that the non-conservative form can be obtained directly by realizing that we can employ our Lagrangian approach for mass conservation. When we employ Lagrangian approach what we say is that we will identify a fluid particle a given fluid particle which will essentially imply or mean that we are always following the same mass. A fluid particle will necessarily mean that we are following the fixed amount of matter or mass as it moves from one location to the other and therefore by definition since it is the same mass that we are following and we are following it as a Lagrangian point of view the rate of change of that mass itself has to be identically 0. This is from the definition and then rho times delta v is essentially the mass content of the fluid particle where delta v is the volume element for the fluid particle rho is the density and in general everything is assumed to be variable. So that this substantial derivative operator remember we talked about this substantial derivative as an operator also simply operates on a product of rho multiplied by delta v using the standard product rule of differentiation I have written out the next step which is substantial derivative of rho times delta v plus rho times substantial derivative of delta v equal to 0 then we divide the entire equation by rho multiplied by delta v and realize that this 1 over delta v times the substantial derivative of delta t from the discussion on kinematics is nothing but the volumetric rate of strain which was found in kinematics to be equal to the divergence of velocity where my highlighter is right now and therefore we obtain again the non-conservative form of the mass conservation equation directly. So let me just take a minute again and use the whiteboard to write that the mass conservation equation on a differential basis this form is called as the conservative form and if I write it in the form of a substantial derivative this will be the non-conservative form. Keep in mind that the conservative form is obtained by employing a balance equation on a control volume whereas the non-conservative form can be directly obtained by employing the conservation of mass statement to a fluid particle which means that we are necessarily employing Lagrangian approach. So let me continue now and try to wrap up this mass conservation and go ahead with the remaining part. Remember that if we have incompressible flow from kinematics we had obtained the condition that the divergence of the velocity field is 0. If you go back and look at our non-conservative form of the mass conservation equation if the divergence of velocity is equal to 0 the second term will be identically equal to 0 and therefore an alternative statement for incompressible flow is going to be that the substantial derivative of density is also equal to 0. Now let us look at a special situation where we are dealing with a two dimensional incompressible flow in Cartesian coordinates. So in Cartesian coordinates this divergence of velocity equal to 0 in two dimensions will simply mean that partial derivative of u with respect to x plus partial derivative of v with respect to y will be equal to 0. In two dimensions for incompressible flow we many times use a concept called a stream function which is usually denoted by psi such that the way this stream function is defined as the u velocity which is essentially the x component of the velocity will be given by y derivative of psi. So psi remember then is a function of both x as well as y. So the partial derivative of psi will be giving you the x component of the velocity and negative of the partial derivative of psi with respect to x will give you a v velocity. So what will be the utilization of this definition if you take this definition and substitute it in the continuity equation let us see what happens. So let me go to the whiteboard again. So what we end up doing is that we realize that the continuity equation is satisfied automatically assuming that then the psi function is a well behaved function where whether you calculate the cross derivative as first the x derivative and the y derivative or first the y derivative and the x derivative does not really matter. So automatically the continuity equation is satisfied with this usage of stream function psi. So many times what happens is in some classes of flow many times the continuity equation is replaced in terms of an appropriate equation involving psi. So that is the utility of this stream function. Now let us see what else this stream function signifies physically. So if you recall from our kinematics discussion yesterday that we had talked about stream lines and in stream lines we realize that the differential equation which gives us the equation of the stream line is given by dx over the x component of the velocity equal to dy over v component or sorry the y component which is v. This is in two dimensions of course that we are talking about. So if you cross multiply and rearrange the equation you will see that it turns out to be u times dy minus v times dx equal to 0. Now here if you substitute u equal to partial derivative of psi with y and v equal to minus partial derivative of psi with respect to x what you will get is d psi dy dy that is partial derivative of psi with y times dy plus because this minus and this minus will cancel each other. Partial derivative of psi with respect to x times dx equal to 0 but then what will be left out on the left hand side is essentially the total derivative of psi equal to 0 and therefore this will imply that we are talking about d psi equal to 0 or if d of psi is 0 that is the total differential of psi is 0 then psi must be a constant and that is what the conclusion is that for a given streamline the value of the stream function is a constant. So physically if you want to imagine what is going on let me let me draw a sketch. Let us say that we have a streamline field or a flow field which will be represented with a bunch of streamlines. This could be in general some flow field and I have shown four different streamlines. Each of these streamlines will have a constant value of psi associated with it. So this could be psi 1 which is a constant for this entire streamline number 1 this would be psi 2 which would be constant for the entire streamline number 2. This will be psi 3 which will be constant for entire streamline number 3 and so on. So many times when we plot streamlines as a result of the CFD computations, in effect you can say that we are generating this stream function value and then we are plotting these stream function values as a contour plot which will generate these streamline patterns. So this is something that is special for a two-dimensional incompressible flow. Please keep in mind that the stream function is an entity which is defined only for two-dimensional incompressible flow and it is defined in the following fashion. With the use that the continuity equation gets automatically satisfied therefore you can replace the continuity equation in terms of an equation involving psi. I will not talk about that right now. I will want you to think about it a little bit that what will be the equation of psi that certain types of flows will satisfy. Importantly right now what we wanted to point out was psi equal to a constant is an alternative representation for the equation of a streamline and as I drew those few streamlines what we realize is that each streamline will have a value of psi associated with it which will be a constant for that streamline. I think this is more or less here. This is more or less what we have to discuss as far as the differential mass conservation statement is concerned. So the continuity equation again going back non-conservative form and conservative form. These are the two important expressions that you would like to know. So let us move on and start talking about the linear momentum principle on a differential basis. So the ideas are now going to be repeated and therefore in order to avoid boredom in some sense what I will say is that let us see what we are doing here and let us see if you understand the overall idea as to how things are getting computed. The specific expressions for each of these quantities I will not again go in serious detail because they are very very straightforward to understand once you know what is going on and also anyway in each slide everything has been written out. So if you want you can always go back to a given slide and see how it is getting computed. So let us see what is happening. We go back to our elemental control volume which is shown on the top of the slide. The x dimension is again delta x, y dimension is delta y as usual these are our elemental lengths and for the purpose of later connecting the CFD terminology what I have shown is I am marking the bottom face as the south face, I am marking the top face as the north face, marking the left face as the west face and marking the right face as the east face. So this south north west east terminology is quite common in finite volume methods which will be covered in the next week and therefore, let us start getting used to that terminology alright. So what is happening now is we have already discussed mass flow rates coming in and going out. So I have just shown for reference m dot x coming in from the west face and m dot y coming in from the south face. Obviously again we know that using Taylor series we can calculate the mass flow rate going out of north and mass flow rate going out of east face that is not been shown because we have just completed it. What happens is now because the flow is considered to be two dimensional here. The entire derivation is done in two dimensions in reality everything is three dimensions but the ideas are similar two dimensional derivations are easy to handle from the algebra point of view and that is why we will stick to two dimensions. Generalizing to 3D is not really a big deal at all. So since the flow is two dimensional the mass flow rate that is coming in from the west face in particular let us say will carry with it in two components of velocity x as well as y and therefore this mass flow rate m dot x will bring in a linear momentum both in the x direction where my highlighter is and a linear momentum in the y direction as well. So just for the showing of these two linear momenta that are coming in at the west face I am showing the x momentum entering the control volume by a horizontal line and y momentum entering the control volume from the west face using some sort of a L shaped arrow. I hope the idea is clear mass flow rate is coming in but it carries with it two components of velocity and therefore it brings in both x linear momentum as well as y linear momentum. And the same idea is repeated for the south face mass flow rate m dot y will come in two dimensional flow hence it will bring in y momentum as well and an x momentum as well. So the terminology used here on the picture is p is our linear momentum dot is the rate of linear momentum x is the x direction linear momentum and finally I am also representing which face we are talking about. So either it is south or west or north or east alright. So let us look at one expression and hopefully you will understand what is going on. Let us look at the west face y momentum. So what we are looking at is the second line here in the list of expressions y direction momentum coming in from the west face. So fundamentally you should realize that it is the mass flow rate which is entering from the west face that is bringing in this momentum right. So therefore the mass flow rate is m dot x for the west face and that brings in a y direction momentum which will be given by the v velocity or the y velocity and therefore the product of these two will give me the rate of y momentum coming in from the west face. Now what is the expression for the mass flow rate m dot x? Remember that it is the normal component of the flow with respect to area element that carries the mass flow rate. So what is our area element here? It is delta y multiplied by one which is the depth into the depth of the paper. So that delta y shown here multiply that by the normal component of the velocity which is u or the x component multiply that by the density. So rho multiplied by u multiplied by delta y is the m dot x multiply that by the y component of velocity which is v and what you get is the linear momentum coming in from the west face. So putting it together what we realize is that this p dot y from the west face is rho times u times v times delta y this is simply rearranging the previous expression. Same thing you can do for the remaining momenta coming in. So remember that there are two momenta coming in in x and y direction from the west face. Similarly there will be two momenta coming in from the south face again in the x direction sorry x direction and y direction and those four expressions have been written out here. I hope that with the explanation that was provided just now you will easily get to understand how these four expressions have been obtained. So for example if you look at the last line absolute last line on the slide it talks about the y momentum coming in from the south face which is simply the mass flow rate coming in from the south face multiplied by the y direction velocity itself because that is what we are looking at. Now the mass flow rate coming in from the south face is delta x which is the area element multiplied by 1 of course multiplied by the normal component of the velocity which is v times rho and that gives you the mass flow rate. So rho v delta x is the mass flow rate for the south face and multiply that by the y component of the velocity and you will get the y direction linear momentum coming in. So with this we know how to obtain the four linear momenta coming in what we will do on the next slide and I will not do this in detail because it is very very obvious now what we are going to do. We will express the linear momenta going out of the control volume from the east face and the north face in terms of a first order Taylor series expansion about the linear momenta coming in from the west face and from the south face and this is precisely what has been done here. I am repeating the figure for convenience on the left top here but if you see I will look at one expression let us look at the very first expression. I am interested in knowing the linear momentum in the x direction leaving the east face so p dot x east. I will express that in terms of the p dot x west which is coming in plus the first order Taylor series expansion term. So partial derivative of p dot x west with respect to x times the delta x distance between the west face and the east face. We already have the expressions for p dot x w which was rho times u times u times delta y from the previous slide the very first expression here and you substitute it then in these two places and realize that you will generate the volume element delta x delta y as a constant factor multiplying the second term and that is about it. So what you have is the linear momentum leaving the x in the x direction from the east face is simply rho times u times u times delta y plus partial derivative with respect to x of rho times u times u times delta y delta x which is simply the volume element in the two dimension. In exactly the same fashion the remaining four linear momenta remaining three linear momenta are figured out which three would be those y linear momentum leaving from east face x linear momentum leaving from north face and y linear momentum leaving from north face and those have been listed out here. Then we go back to our balance statement which says that here is our standard balance statement now we will employ it for linear momentum rate of accumulation equal to rate of inflow minus rate of outflow plus the source we have done this before for the integral we will now do this for differential mass content within the control volume multiplied by the x direction velocity will give me the linear momentum content within the control volume and a time rate of change of that will give me the rate of accumulation of x linear momentum within the control volume. Now here what is the mass within the control volume mass content within the control volume so let me just go quickly to the whiteboard it is simply the product of the density and the volume element of the control volume. The volume element is delta x delta y multiply that by the density and therefore m suffix c v which is the mass content of or within the c v is simply rho times delta x times delta y which you substitute in here what you will realize is that delta x multiplied by delta y will then be appearing in all three terms. So, on purpose I did not write it on the first term on the left hand side or the term on the left hand side I want you to substitute that expression for the mass content within the c v in here and you will realize that this delta x times delta y delta x times delta y will come in in all terms the first term on the right hand side and first term second term on the right hand side will be corresponding to rate of inflow minus rate of outflow terms. So, remember we are still talking about x direction linear momentum. So, because each because there are four x direction linear momentum that we are talking about one here, one here, one here and one here you will generate two terms. So, please make sure that you take these expressions and incorporate those in the balance statement to realize that these two terms will come in for the x momentum equation and then finally the source term for the linear momentum equation as we had seen in the case of integral analysis is the net force acting in the x direction on the material contained within the elemental c v in this case. So, for convenience because this delta x delta y term is appearing in all terms we choose to express this net force acting on the material in the c v as some sort of a intensity of the force that is per unit volume. So, that is small f of x x multiplied by delta x delta y. So, this is just for the convenience of formulating the equation in a compact form. If you substitute for capital F x in this form you will realize that then the delta x delta y terms will cancel from all terms and what you are left with is something what is shown in the box. So, this is what you will call a differential x momentum equation. The first term here is the rate of accumulation of linear momentum on a per unit volume basis now. So, the rate of accumulation of x linear momentum within the control volume or a per unit volume basis and the two terms next to the left hand side term the remaining two terms on the left hand side will combine combinedly say that this is the net rate of x momentum leaving the control volume. So, the reason two terms come about is because as we said there are four linear momentum in the x direction two coming in and two going out and that is how we end up generating the momentum equation. Finally, on the right hand side we have this small f suffix x which is from our usage is the net x force acting on the material contained in the elemental CV on a per unit volume basis. So, this is something that you may note down right away that all differential equations of motion that we will talk about whether it was earlier the continuity equation or whether now it is this momentum equation whenever we are talking about these differential equations they will be necessarily on a per unit volume basis. Each term here whether it is the accumulation term or whether it is the net rate of x momentum leaving the control volume term signified by these two terms on the left hand side will be all always on a per unit volume basis. So, this was worked out completely for the x direction linear momentum if you go back I already have listed all y direction momenta as well. So, y direction momentum coming in from the west phase, y direction momentum coming in from the south phase, y direction momentum leaving from east phase, y direction momentum leaving from north phase. So, similar to this x direction momentum you can go ahead and formulate this balance statement for the y momentum and what you will realize is you obtain what will be given in the box at the bottom. So, again let me go to the wide board if this is the form of momentum equation that we are generating this is essentially a y direction linear momentum equation where what we have is on the right hand side small f suffix y is the net y force acting on the material within our elemental control volume on a per unit volume basis. So, each term here there are four terms let us say all four terms are on a per unit volume basis. So, the first term which I will now circle signifies the rate of accumulation of y momentum within our elemental control volume on a per unit volume basis. The next two terms together will signify the net rate of y momentum leaving the control volume on a per unit volume basis and the reason we again get two terms is because we are dealing with four linear momentum in the y direction two coming in from the west and south phase and two leaving from the east and north phase. So, keep this in mind that each term in a differential equation of motion is going to be on a per unit volume basis. Again we obtain these two equations in terms of employing a balance statement for our control volume. So, inherently we are following the Eulerian approach where we are using the control volume and then identifying mass flow rates in and momentum flow rates in and going out and so on. So, what we are generating here is the momentum equations in their conservative forms. So, keep that in mind again just like when we did our mass balance equation using an Eulerian format like what we are doing we generated a conservative form. So, so far we have obtained this on a conservative form. One last point is that we still have not talked about in detail the nature of these forces f suffix x and f suffix y the net x force on a per unit volume basis etc. We are going to talk about that little later. So, this was so far done from a an Eulerian point of view where we utilized our balance equation for the elemental control volume. Now, let us see what we can do if we want to employ our integral equation. So, just as we obtained the differential mass conservation statement in the continuity equation from an integral equation we can do this here as well. So, we start with our integral form of the momentum equation which if you go back two days on the first day we had we had discussed this. So, I am directly writing the vector form which is available in your in your notes from the first day for a stationary CV exactly the same procedure as was employed for the mass balance situation namely use the Leibniz rule for first term to bring in this time derivative inside the integral use the divergence theorem to convert the area integral over the control volume integral or volume integral. And just because we want now all terms as volume integrals we for convenience express the net force acting on the material in this control volume in terms of an integral representation where the small f vector is again simply the net force acting on the material inside the control volume on a per unit volume basis because that is what is getting then integrated over the entire control volume. So, now what we have achieved is that all integrals are on the volume basis. So, that we can combine the integrants together and we combine the integrants in the following form the partial derivative with respect to time of rho times v plus divergence of rho times v times v all vectors v minus this vector f which is on a per unit volume basis integrated over the entire control volume is giving us 0 again using the same argument as before arbitrary control volume will necessarily mean that the integrand itself is identically equal to 0. So, again writing that as writing that with the f small f the per unit volume force onto the right hand side we obtain the momentum equation on a per unit volume basis. Now, remember what we have done here we started from employing our balance statement in a Cartesian frame of reference. So, we explicitly written wrote down linear momentum x direction linear momentum in y direction coming in from both west and south faces etcetera and went ahead and then obtained something like this for the x direction something like this for the y direction. So, in some sense I will say that even though may be the procedure is easy to understand it usually involves more algebra. However, if you look at this integral equation to begin with and then employing our Leibniz ruler and divergence theorem we convert that into a differential form you directly obtain within a few steps the same equation as what we have obtained using the Cartesian framework. Only thing is that here it is a vector equation directly, but that is no problem and we can always generate the corresponding Cartesian components by replacing this vector v with either u or small v or small w. So, again since we have obtained the momentum equation here which is again on a per unit volume basis using a balance statement because this was the balance statement for the integral form we have converted that into a per unit volume balance statement which is essentially the differential momentum equation. This again is in the conservative form alright. So, let us see how we can now obtain the component equation again from our vector equation. Just one point I would like to make here that the term that you see here inside the divergence rho times vector v times vector v. So, this is a special product of two vectors it is what is called as a tensor product of two vectors. Let us not get into the details of this right now usually the properties of these vector products of rather tensor product of two vectors etc. We deal with in an advanced fluid mechanics course. At the moment you do not have to worry about it make sure that this is neither a dot product nor a cross product it is what is called as a tensor product of two vectors. We are not going to be bothered about this much because we will usually deal with the component forms. What happens is when we are obtaining the differential equation from the integral form directly this product will come in without really you doing anything when we employ the divergence theorem. So, that is the way it has appeared but do not worry too much about it because immediately we are going to start writing the component equations. So, what I said was if I want from the vector equation the corresponding x direction equation I will simply replace the vector velocity v here and one of the vector velocities v there with the corresponding directional velocity which is in this case u. So, let us just look at this carefully on the whiteboard. So, this is the form of the equation let me go back and make sure that is what it is. This is right here I have not bothered to write the brackets around this rho times small u times the vector v but that is about it. So, now let us see what we how we can expand this into our standard Cartesian form. So, remember that our grad operator is unit vector i times partial derivative with respect to x plus unit vector j times partial derivative with respect to y and unit vector z k rather times partial derivative with respect to x and the velocity vector v is i times u plus j times v plus k times w as we know. So, all that we do is we will substitute in here for the velocity sorry for the gradient operator and for the vector velocity v here what will end up happening is we will get d dt of rho u as it is the first term on the left hand side plus the very first term on the right hand the first term when we expand this divergence of rho u v will be d dx partial of rho times u times u plus d dy partial of rho times u which is common in here. The second term will be essentially j dotted with j and therefore, you will obtain a v the third term will be d dz partial rho times u which is common at the at the top multiplied by w and this will be equal to f of xx which is the net force acting in the x direction on a per unit volume basis. So, forget about right now the third dimension the z dimension. So, let us get rid of this for the time being because earlier we were discussing two dimensional derivations. So, let us get rid of this let us get rid of this. So, what is left is partial derivative of rho u with respect to t plus partial derivative with respect to x of rho u u plus partial derivative with respect to y of rho u v equal to f suffix x on a per unit volume basis and let us make sure that this was exactly the statement that we had obtained through our initial Cartesian derivation and it is right here ok. So, you do not get anything different is just that the different approaches to obtain the same equation and that is that is what you will obtain right. So, coming back to this then now I have actually written this with all three directions. So, therefore, we have this partial derivative with respect to z also term which is fine in exactly the same fashion you can repeat this for the y direction and the z direction and you will realize that this is how things will turn out. So, my suggestion then is just like what we worked out right now for the x direction you start with the vector form and obtain the corresponding expressions for the y direction and the z direction by replacing this vector v with small v and small w and one of these vectors v with small v and small w and you will essentially generate this set of equations. So, this is to sort of get familiar with the way these equations are to be manipulated. What we will start talking about now is the nature of this net force that is acting on the material within the control volume and we said that the way we have expressed this is on a per unit volume basis. So, in general the vector small f will be considered as the net force on a per unit volume basis acting on the control volume the material contained within the control volume and during the integral analysis we had noted that in general the forces are surface forces and body forces as well. So, both surface forces and body forces will contribute to the net force. So, let us start talking about the nature of these forces. But before that there is one another manipulation which I want you to carefully see. See what we had done let me go back to the mass conservation equation. Remember we employed our balance equation and obtained the mass conservation equation. This was obtained in the conservative form because we started with our Eulerian balance approach for the control volume. Then what we did was we did some manipulation and obtained the non-conservative form from the conservative form. Essentially all that was done was opened these derivatives that is about it. Now here in case of momentum equations so far we have been obtaining the momentum equations on a conservative form. So, the one that you see right now here on the board projected or the vector form that you see here or all these forms written on slide number 12 they are all in conservative form because essentially they have been obtained using balance statements. Just the same way as we did some manipulations on the continuity equation to obtain a non-conservative form from the conservative form we can do that here as well and that is what is written on slide number 13. So, we start with the conservative form of the momentum equation which is at the top and subtract from it u multiplied by the conservative form of the continuity equation. So, please realize that inside here what we have is partial derivative of rho with respect to t plus d dx of rho u plus d dy of rho v plus d dz of rho w equal to 0 is nothing but the continuity equation that we derived few minutes earlier on the conservative basis. So, the entire continuity equation which is equal to 0 by the way is multiplied by u and subtracted from the conservative form of the momentum equation. So, because the entire continuity equation is equal to 0 you multiply anything to 0 and add or subtract to the equation at the top it does not really change the value of the equation so to say. So, this is perfectly fine. So, let us see what happens. I will write this only for two dimensions just to save my writing a little bit. Remember that this entire continuity equation is equal to 0 and therefore adding or subtracting any multiple of the 0 to our top equation is not really going to change anything. So, let us just see what will happen. The first term is just let us just look at the time derivative I have d dt of rho u as the first term on the top equation minus u times partial derivative of rho with respect to time what will this give us. Essentially what you need to do is you open this first derivative using product rule. So, when you open you will get u multiplied by d rho dt which is going to cancel with this term and what will be left is rho multiplied by d u dt on a partial basis. I hope you understand this.  Let me explicitly write this one term and you will see what is happening. This will imply that for the purpose of now convenience what I am going to do is I am going to split this as two terms inside the product rho u and u and then rho u will be treated as one term u will be treated as another term and then you expand this derivative using the product rule.  Is that fine? Let me ask some of the audience here if they are able to follow it minus u times d dx of rho u and as you can see the first term and the third term will cancel and what you will end up with is rho times u times d u dx. In exactly the same fashion you can manipulate the third term as well and I hope now you can complete this yourself. So, let me go back to my slide and as you can see the first term here where my highlighter is standing is the first term that we had obtained on our whiteboard rho times partial derivative of u with respect to t. The second term where my highlighter is standing was what we obtained on the whiteboard a minute ago which was rho times u times partial derivative of u with respect to x. Similar two terms you can now obtain on your own by subtracting from the third term here u multiplied by third term here and so on. What you realize is that rho is essentially a constant entirely on the left hand side.  So, I will write this in the form that rho times partial derivative with respect to time plus u times partial derivative with respect to x plus v times partial derivative with respect to y plus w times partial derivative with respect to z operating on the x component of the velocity u and that is equal to the x direction net force acting on the material within the control volume on a per unit volume basis. So, that has been carried the way it is the reason is because we were essentially adding 0. So, nothing was changing in the first equation, but realize what this is whatever is incorporated inside the brackets is nothing but the substantial derivative operator. So, I can replace this entire bracket with capital DDT which is nothing but partial derivative with respect to time plus u dot grad as they say or u DDX plus v DDY and w DDZ. And therefore, an alternative way in which the momentum equation has been now obtained in the x direction is that it is rho times substantial derivative of the x direction velocity is equal to the net force acting on the control volume on a per unit volume basis. So, the way to interpret this is as follows.  So, imagine that a fluid particle which has the same volume as the volume of our elemental CV is occupying that elemental CV at the instant of our interest and let the volume element of that fluid particle be delta v. So, let me multiply this entire equation by that volume element delta v which in our case of Cartesian control volume is essentially delta x multiplied by delta y, but let me not write that explicitly I will simply write. So, now we are suddenly seeing something different in the sense that rho multiplied by delta v can be essentially interpreted as the mass content of the fluid particle which is occupying the same position as the control volume that we have been talking about. So, mass multiplied by an acceleration equal to net force in the x direction is nothing but the statement of Newton's second law of motion. And therefore, what we are saying is that the momentum equation which we started out as a balance statement for a control volume we have been able to do some manipulations and we have been able to interpret this as essentially a statement of the Newton's second law of motion for a fluid particle which is having the same volume element as our elemental control volume and that has been essentially occupying the space where our control volume is. So, in that sense what I have written out is rho multiplied by the substantial derivative of u equal to a net force in the x direction per unit volume basis is essentially a statement of Newton's second law of motion for an elemental fluid particle which is supposed to be coinciding with the elemental CV at the time of our interest where everything is getting analyzed or calculated. And so, this entire thing is expressed as a per unit volume statement. The point here to be noted is that since we are talking about now a fluid particle per se we are now suddenly talking about a Lagrangian point of view. And therefore, the last statement which is boxed at the bottom of the slide can be interpreted as a non-conservative statement or form of the momentum equation because now we are talking about this as a Newton's second law of motion in terms of f equal to ma applied for a given elemental fluid particle just so happens that by the design of our derivation we will essentially say that that elemental fluid particle is occupying the same position where our elemental control volume was when we began our discussion. So, just like we obtained a non-conservative statement of continuity equation earlier this way you are able to obtain a non-conservative statement for the momentum equation as well. This was of course done for an x direction similar treatment can be performed for the y direction as well and one can even generalize the statement in a vector form. In fact, let me do that on the whiteboard. So, at the top we have x direction similarly we can write for y direction and finally to generalize it we can write it as rho times substantial derivative of the velocity vector equal to force net force vector on a per unit volume basis. So, this all these statements that have been written out here are all on per volume. So, keep this in mind that same equation you do some manipulations and you can interpret that from a Lagrangian point of view and then you are in a business of saying that this equation now is in the non-conservation form alright. So, this is what we wanted to point out in this case. So, having done this discussion now we are in a position to start talking about the forces that are acting on the fluid particle. So, now what we will do now is because here on this slide number 13 we established that whether we start with a control volume and do a balance on the control volume to obtain the momentum equation or equivalently we consider a fluid particle which is coinciding with the control volume at the instant of interest and choose to employ Newton's second law of motion to that particular fluid element which is coinciding with the control volume we essentially get the same result ok. So, therefore, for the purpose of further discussions what we will say is that we will now start talking about that particular fluid particle for which this non-conservative form of the momentum equation has been derived, but keep in mind that they are interchangeable these two forms. When we talk about the conservative form it is essentially to deal with the control volume formulation when we talk about the non-conservative form it is essentially to deal with a fluid particle which has been identified as a fixed amount of mass where we are employing our Newton's second law of motion. One way or the other we are essentially saying that the fluid particle will be coincident with the elemental CV at the instant of interest. So, from that point of view the net force which has been shown here can be considered to be acting either on the control volume or equivalently on the fluid particle which is exactly coinciding with the control volume. So, please keep this in mind.


A fiery horse with a speed of light, a cloud of dust, and a hearty high old silver, the lone ranger. With his faithful Indian companion, Tonto, the daring and resourceful, masked rider of the planes led the fight for law and order in the early western United States. Nowhere in the pages of history can one find a greater champion of justice. Return with us now to those thrilling days of yesteryear, from out of the past from the thundering hoof beats of the great horse, Silver, the lone ranger rides again. The lone ranger and Tonto had watered their horses in the wooded stream and were ready to continue on their journey. Tonto, there are those bloodhounds again. They seem to be heading in this direction. Ah, if prisons escape, then come this way maybe. Yes, they would head for the woods. Still, how long can they remain at large, I wonder? Not long, Kimusabi. Too many clout. Quiet, Tonto. In the reeds, there. Ah, let me see him, Kimusabi. All right, you in the water, hiding in the reeds. No use, we see you. Come out of there. Kimusabi, Kimusabi, escape prison, it's sure. Yes. Hold on, you guns. I give up. I couldn't breathe any more.  But on the side of the law, you're going to turn you in. No, you can't. You can't turn me in, please. Don't send me back to prison, not now. I'll go back. Sure, I'll go back, but first... First, you want to return to Cureville and get revenge on Jim Keppel. Your name is Fred Tracy. You were in jail for killing your uncle, Mike Fahey.  And I don't want revenge if you mean killing Jim Keppel. And how did you know who I was? I recognize you. Your father was a friend of mine, Tracy, a good friend. I never thought I'd see the day when his son would go to prison for murder.  I was framed by Jim Keppel. It's true. I think he couldn't even prove my uncle was dead. The evidence was planted. That's an old cry of people who get caught. I'm telling you the truth. I swear it. And why haven't you appealed your case? You've been in prison for more than a year. Yes, but it's useless trying to fight from there. You hurt yourself by making escape. Why are you run away now? Because tomorrow I have one last chance to clear myself. How? I don't know. But I do know I want to keep Jim Keppel from taking every bit of property left by my uncle Mike. He can do that? Yes. I've learned he's going to appear in court tomorrow. To present papers that'll give my uncle's entire estate to him if they're accepted. I see. You want the estate. I don't. I don't want a penny of it. Whatever the papers are, though, they're fake. They'd have to be. You have anything to prove what you say? Nothing. Only my story and the knowledge that I'm innocent. Listen to those bloodhounds. They're getting nearer. Even if you would approve your innocence, I'd have to go back to prison. You know that. Of course I do. I want to be free only long enough to do whatever I can. If I'm successful, I'll go back to prison knowing they'll let me out soon. If I'm not successful, then, well, anything after that doesn't matter. I'd like to hear your story, Tracy. It's too long. The guards will be here before I finish. Yes, I realize that. I realize that if you are innocent, I owe it to your father to try and prove it. You mean you'll help me? For one day. You've escaped from prison and are going back. I don't imagine it matters whether you're away one day or not. Don't, guards. Plant in here, Kimosabi. Yes, I know, Toto. I'll have to get away from here at once and use all the back trails we know. We could start a mixture of ground, Kimosabi. High trail, maybe. Yes, do that, Toto.  While Toto tries to get rid of your trail around here. I'll never be able to thank you for this. Don't you two. When we get to my camp near Cureville, I just want you to tell me the complete story of why you were sent to prison. The lone ranger, Tonto and Fred Tracy, had arrived at a secluded spot on the Altskirts of Cureville late that evening. Their journey had been without incident and Tonto was now arranging a tent. Let me pick some tent with panty cover. You sleep. Why are we gone? But, first, Tracy, I want your story. You know some of it, I believe. I know that your uncle, Mike Fahey and Jim Keppel were partners in various enterprises. That's right. And I work for them as sort of manager, bookkeeper and general everything. I'll see. One thing Uncle Mike didn't know was that Jim Keppel owned the White Yucca Cafe. He wasn't a partner in that. Is that important? No, except that the cafe was used as sort of a private headquarters by Keppel and the many associated with away from business. I see. Now, what I know the case is mere hearsay. But as I understand it, your uncle went to Frontier Town to get a large sum of cash from the bank there. Yes. That was a little more than a year ago. Huh? Uncle Mike and Keppel used the Frontier Town bank for all the big transactions. He got tent all fixed now. Good, Tonto. Go on, Tracy. Well, on that particular morning, I was the one who was supposed to go to Frontier Town and pick up the money. Supposed to? You didn't go? No. At the last minute, Keppel said my uncle was gone. He sent me out to his ranch. It's just a few miles from here and gave me some instructions for Chad Durrigan. As I've heard of him, he's Keppel's sidekick, isn't he? Yes, and was long before Keppel and Uncle Mike joined up. Anyway, at that time, Berrigan was running Keppel's ranch. Well, something happened at ranch when you go there? Nothing. The place was deserted and no sign of Berrigan around. Yet at the trial, Berrigan testified he'd been there all the time and produced witnesses who swore they'd been there with him. And Keppel swore he'd never sent me on the air. Now, you tell the truth. That frame... It is the truth. I tried to convince the judge and jury, but they didn't believe my story. Your uncle was gone when you got back to Cureville? Yes. He'd gone to Frontier Town. Oh.  But no one ever saw him after that. And find no trace of him, though. Not to this day. They take it for granted he's dead, but there's nothing to prove it. Didn't they find part of the money from the Frontier Town bank among your belongings? Part of it, yes. A very small part. And there was blood on it, too. Uncle Mike's blood, they said. It was Keppel who suggested the search. Yet what they found was enough to convict you. On a trumped-up robbery charge. The jury wanted to convict me of murder using Keppel's and Berrigan's testimony as a basis. There was no body, though. That's what the judge told him. Not a trace of the body. So he dropped their proposal and made the robbery sentence as stiff as he could. I see. Tracy, what do you know of this paper Keppel is going to produce in court tomorrow? I've been told it's supposed to be an agreement signed by both Uncle Mike and Keppel. In it, one agrees that his complete possessions be turned over to the other in the event he dies, or deserts the business for a year. There's something like that, anyway. It's very bad, isn't it? Look, Tracy, on the day your uncle disappeared, whatever happened, would have had to take place between here and Frontier Town, right? I think so. But you'll never find twos after a year. And if you did find the body after all this time, I don't want to think of it. Identification would be the important thing. How would you identify your uncle forgetting his physical setup? Well, he's closed for one thing.  And then, well, the most outstanding thing was the beaded belt. A beautiful, elaborate belt given him by Chief Manna Wanta. Chief Manna Wanta? You know him. Wanny prized so much, he always wore it. Uh, come here, Tunnel. Oh, come with me. Tracy, look at Tunnel's belt, will you? All right. Uh, get him to the fire, Tunnel. Uh-huh. Yeah, that's it. Now, Tracy, was your uncle's belt anything like that? Well, yes. I didn't think there was another. Yes, like that, except that, well, his name was on the belt. Not in beads. It seems that the beads had been left off where the name was written. You could do that to your belt, couldn't you, Tunnel? Uh-huh. Hey, Miss Hubby, take a few hours, maybe, but me do it. What are you going to do, stranger? I'm not quite sure yet. Maybe there's a way of making Keppel break so that he'll expose himself. We'll see. Well, we'll get going to Cureville, Tunnel. If Keppel's not at the White Yucca Cafe, he'll be at his reins most likely. It'll be late when you get to Cureville. We'll make it by midnight, but before we visit Keppel, I think we'll see the sheriff first. Keppel can wait. Jim Keppel was waiting in his office at the White Yucca playing solitaire and occasionally looking through the open door into the crowded room outside. He'd been playing listlessly most of the evening when suddenly he looked up and saw his right-hand man, Chad Berrigan, pushing his way recklessly through the customers, heading directly to the office. Now, what's eating, Chad? Hey, Keppel, I got to see you alone, Keppel. What's the excitement, Chad? Keppel. Keppel, Tracy's escaped. Is that on the level? When did that happen? He broke prison this morning. I just heard it at the sheriff's office. They're getting a posse together. They think he's heading this way. How do they know? Just guessing, I imagine. They say he's coming back after you. Yeah? It'd be a good excuse to wipe him out if he does show up, wouldn't it? Keppel, do you think maybe he learned something? About what? About Tamara, maybe. Suppose he heard you were going to court to claim my claim. Suppose? There's no supposing about it. He knows. What? You heard me. If he stayed in jail, he'd know it in a couple of days and go on living. It happens, he knew it a couple of days ahead and he didn't stay in jail. He's still living, maybe, but... You mean you wanted him to break on? I wasn't sure he'd try, but it's what I wanted. No matter where he is, now he's a Clay Pigeon. We couldn't get him hanged a year ago, so we, uh... He'll get himself shot now, here or somewhere else. It works out all around, see? It was nearing midnight and Sheriff Wert Campion was about to close his office for the evening. Then he saw a shadow flip past his front window. Without hesitating a moment, he rushed to the door and jerked it open, gun in hand. All right, you up with your hands. One move towards your gun and that's it. I'll step inside here. Well, an Indian sneaking around outside, huh? We'll soon find out why. Me tell you why.  Thank you, Sheriff. What the...  That's the way. I think he's got another one. Huh, me get it. Nice man together, huh? A trick. Yes, a trick, Sheriff. It worked well, didn't it? What do you want? Whatever it is, you won't get it. My posse... Your posse is out searching for Fred Tracy. How do you know that? Because I happen to have Fred Tracy. You have? Say, what is this? A trade, if you'll listen, Sheriff. I'll trade you, Fred Tracy, for 12 hours. Now you're talking nonsense. What do you mean? Just what I said. I can produce Tracy if you'll cooperate with me in the cause of justice. I won't cooperate with you in anything. You're nothing but... Say, what is that you've got in your hand there?  In frontier time? That's got nothing to do... Thank you, Gully. That's a silver bullet. Sure, I'm a friend of Sheriff Taylor's, and he's told me, but... A masked man and an Indian. Gully, you must be... That's right.  Now, I know you turn over Fred Tracy, like you said. Gully thinking of meeting you. Here, now, put down that gun and let's talk. Come on, come on in. What is it you want me to do? Just a few things. The first center's around Jim Keppel. But I want you to understand that I have no way of... The curtain falls on the first act of our Lone Ranger story. Before the next exciting scenes, please permit us to pause for just a few moments. Continue our story. It was after midnight when Jim Keppel and Chad Berrigan finished another conference at Keppel's ranch house. Keppel looked with pride at the papers he was presenting caught in the morning. Ah, this is a nice job, Chad. Great. Right down to that fake notary signature. Good thing the case comes up before Kingsley ended. Hey, what's that? Is it the window? Is someone wrapping? Sounded like that. I'll see who it is. Oh, that's the door. Someone's there. I've got a cover, Chad. Come on in. I said come on in. What is this? Get over and open that door, Chad. Get the door.  There's no one here. Hey, the window again. Leave that door and come on over here, Chad. Somebody's being wise. Hey, look that way. I'll open the window. Hey, the door again. No, you don't pull it. Someone shut that door, Chad. It's me outside. I can't see anyone near the window, Kepp. It's a trick. I'll keep the window covered. You go outside and see who's around. Be careful now, Chad. Watch out. Something's funny here. You wait there, Kepp. Go around the side and look. Well, Jim Kepp will face the window, his gun pointing at the opening. Chad Berrigan hurried out the front door, gun in hand and walked into the night. He peered about him in the inky darkness and saw no one. Then he turned a corner. Don't move other inch, Berrigan. You're covered. Who are you? Where'd you come from? Never mind. Turn around and start walking again. I've got your horse ready for you. They're taking a little ride to Cureville. Why are you going? While Jim Kepp will watch the window in his ranch house, his gun arm tensed, his whole being alert, the Lone Ranger and Tonto rode back to Cureville with the reluctant Chad Berrigan between them. It was shortly before dawn when a masked man and Tonto appeared once more while the sheriff had been waiting most of the night in his office. Sheriff, we have Berrigan tied up. He's out in your stable. Is that all right with you? Oh, I'm afraid this is all too high-handed. I'm using my office to... To find the truth. Just wait till this is over, Sheriff. Now, that window tap and a door pound of nonsense didn't work on Kepple, did it? I bet he's back there fighting with Phantoms right now. Yeah, he's on edge. And unless I miss my guess, he'll take off and dawn breaks and ride here. He'll want your protection in case Fred Tracy should be around. Sheriff, you look at belt now. What do you think, huh? Well, I'll be dawned. It looks just like Mike Sayhe's. Good. And the rest depends on how we use it. Yep. Jim Kepple will take to the bait. That's our one hope. Meanwhile, Tottenham and I'll take Berrigan out a bit on the frontier town road and get ready for our entrance. We'll keep an eye on Kepple, Sheriff. Put those ropes on Berrigan. Remove that gag from his mouth. We do it. Thanks, Tottenham. All right, come along, Berrigan. We're taking you on another ride. What are you trying to do? Where are we going? Just a short distance down the frontier town road this time so we can ride back as if we'd been someplace. Where's that? Don't ask questions. Just do as you're told. Here, take this belt, Berrigan. Whoa, you look startled. It does look like Mike Sayhe's belt doesn't it? A lone ranger had guessed rightly. Jim Kepple, perturbed by Berrigan's disappearance and fearful of Tracy's presence, waited till dawn before taking off. Sheriff Wurt Campion was waiting in his office when he heard Kepple ride up outside. When Kepple opened the door and entered, Campion chuckled. All right, Kepple. Court doesn't open for quite a spell. Sort of expected you, though. You expected me? Why? You wouldn't know why. You've got something on your mind, haven't you? Huh? What are you trying to get at, Campion? The truth about Fred Tracy, for one thing. All right, here he escaped. They got him yet? Well, forget about that part. You go to court this morning to try and collect Mike Sayhe's money, don't you? What's that got to do with Tracy? I might know the answer to that in a minute. It might mean a lot, depending on what those horsemen outside picked up. Excuse me, I'll keep them. I want to see them before they come barging in here. I'll be back in a minute. Hey, Chad Berrigan's with them. Hang to that now. As Sheriff Campion bustled out the door, Jim Kepple, suspicious and ill at ease, seemed to understand the Sheriff's allusions. He walked to the window and saw a sight that made the blood drain from his face. The Sheriff told the truth. Berrigan was outside, a stride of horse and sitting afraid-like between a masked man and an Indian. What was that Berrigan was handing to Sheriff Campion? It looked like... but no. That couldn't be. Mike Sayhe's belt? No! He had no time to debate the question with himself. As the three horsemen took off again, Kepple hurried to the center of the room. Sheriff Campion entered. I just learned a lot of things, Kepple. Yes, sir. Would you like to have a look at this belt? Why should I? I thought you might find it interesting. Just take a look at this belt, will you? Strangest belt I ever did see. It's exactly like the one Mike Sayhe used to wear. The room began to spin around as Kepple eyed the beaded belt the sheriff placed on the desk. A beautiful belt, indeed. The like of which he had seen only once before. And yes, there was the name worked into it. Mike Sayhe. This was a trap. Campion think Kepple would react to it. Berrigan revealed the truth. He mustn't show it a panic, he felt. He had to be cool in a spot like this. Cool. Where are you going, Kepple? I thought you came to see me. I've changed my mind, Campion. See you at court later. Berrigan, you will. I want to talk to you more about this belt next time. After all, Sayhe was your partner, isn't he? Sheriff's office, Jim Kepple walked slowly to give an air of composure. Just in case Campion was watching him from the window. But he walked slowly more because his legs were wobbly, as was his mind. He was running his horse, and with no one in sight, he talked to himself. But Chad wouldn't squeal. Why should he? Maybe Tracy's come back with some information. No, that's foolish. Where would he get it? That beaded belt will say his name. Must have found the body, sure enough. Why did Campion let me walk out if they did? He was acting funny and smart like. Maybe he's just playing a cat and mouse game. I'd better right away from here, Proto. Here I am, bullet. Steady now. All right, get a bullet. What Campion expect me to do when I saw those clothes in that belt? Break down? I think like that, I bet. Sure. Those clothes weren't phased, the bell either. Everybody knows he wore things like that. It's just a plant to make me talk to the court today. They found the body. I'd better get out of here in a hurry. If they didn't, I cannot bluff them. I'm gonna see if that body's still there. Get a bullet.  Jim Keppel made his way to the cave where Mike Fahey's body had been left more than a year ago. A cave deep in the hills and hidden from view. He made his way stealthily so that no one could see, hear, or follow him. He did, and the far reaches now walking steadily, his heart beating louder than the steps that seemed to echo around him from all sides. And now he was the turn that would give him the answer he sought. He closed his eyes for a moment, then opened them as perspiration felt in his brow. Now it's... Ha! Ah, dear! That body's still here. They didn't find it. The body or the rest of the money. They didn't find the body. They have now, Keppel. No, they'll never find your body either. Who's low, Keppel? Who's low, mentally, too? Didn't know you were followed. Let yourself be fooled by the belt we've fixed for you. My shoulder! That's only a wound, Keppel. It's going to be much worse than that. I'll take you right on to Sheriff Campion's office. Father should have arrived there with Fred Tracy by now. Judge Kingsley entered his courtroom and looked with amazement at the crowd assembled before the bench and in the spectator's seats. Oh, I see. Rightly, or is that a man in a mask back there? Your Honor, he just brought in these men I had before you. Sheriff, who are the... Now look, according to my calendar, the first case is that of Jim Keppel in the estate of Michael Fahey. Now that's half right, Judge. Only it's the state asking an indictment against Jim Keppel for the murder of Michael Fahey and the appropriation of his estate in part. And also the state against Chad Barrick and this other man for Aden and the Betten Keppel. But wait a minute. This is too much. This other man. What in Heaven's name is it? Why, he's wearing a convict's uniform. For only a few days more, Your Honor. You'll take the steps to right the wrong that's been done him. That's Fred Tracy and he's... Now say it, Sheriff. You're acting as prosecutor, grand jury, defense attorney, adviser to the court and general confuser. Now let's start at the beginning. The first question I asked was about that mask man back there. Well, say it. Where is he? He's done his job, Your Honor. He's gone. Done his job? What nonsense is this? Job? Now look, Sheriff. Who is he? Well, didn't you recognize him, Your Honor? That was the Lone Ranger. I was... Tremble production directed by Charles D. Livingston, edited by Fran Stryker and the part of the Lone Ranger is played by Brace Beamer.


You have to pay attention to detail and you have to give it your maximum effort. Navigate! Let's play! Let's play! Welcome to the Only Easy Day was Yesterday, the official Navy SEAL podcast. Mental and physical health is essential to a successful trip through Navy Boot Camp. Even more so for NSW candidates. I'm Daniel Fletcher. As we continue our Boot Camp series from Great Lakes, Illinois, we sit down with medical liaison for Crew Training Command, Chief Hospital corpsman, Jeff Ramirez. We answer some common questions about recruit medical history, mental health, medications, and preventative care. Listen up. Thanks for sitting down with us for one and if you could just briefly talk about what you do here, that would be a great start. So I deal with all medical related issues in terms of recruit appointments, any injuries that we have here going to network hospitals, to the federal health care center, outsourced down to Chicago, any questionable areas that the doctors have that they need to liaison with the RDCs here in terms of missed appointments or recruits not eating enough, or even if they feel like they're getting too much exercise, because there's instances where we start breaking some recruits down that are couch potatoes and then they get over here and they're learning right away that it's a little bit different here. What types of tests or any type of screening do you administer or is that not part of your position? So that's not part of my position here. So I deal with the docs and it's going to range from mental health to your physical therapy, your preventive meds, and then general sick call. But it's every illness or injury or anything medical related between RTC and the providers. Okay. Are there any ailments or injuries that you see specifically for the 800 guys that are coming through the pipeline here? 800 guys, I would say the biggest injuries that I see would be shin splints, stress fractures, and not getting enough nutrients. Really? Rabdo, rhabdomyolysis. We'll see that. They didn't train for the pipeline before they got here. Okay. And so when they're doing the DiMO PT, their body's breaking down. Pretty severe. The muscles are breaking down pretty severe, yes. I guess that's a kind of form of a failed test, so to speak, if someone's put in position physically where their body's not holding up. Are there any other specific medical tests that are given periodically or on a routine basis that you see NSW candidates having issues with? Special physicals. What are special physicals? They'll answer the questionnaires there, go through the overall history, see if there's anything that raises any red flags. In terms of anything periodic, that doesn't happen unless they choose to go to sick call. If they're having some issues with RTC, they see any 800, any recruit really walking around with limbs or looking distressed or sick, we're going to send them to sick call regardless. You get your labs drawn over there, or if you've got to go to bone density scan or X-rays, a lot of 800s, they really want to be here most of them, so it's kind of hard to get them to go to medical sometimes. So it's our responsibility, and I get a lot of phone calls about that, that I have a recruit that's been limping around, he says he's okay, but however, most body sensors telling me he probably needs to get seen, then we'll go ahead and send them in there and usually find out something, something else. If there's somebody who gets to this point in the process and then a medical screening turns up not not good enough or a fail, whatever you want to call it, are there chances for a candidate to kind of retake a test? Can you tell me a little bit about that landscape of that kind of situation? So if we're going to use vision for an example, so if you came here and your MEP says, you know, the MEP's doctor says your vision's a certain score and you come here and it's not or you're colorblind, they're going to reissue the actual test again. So you're going to do one MEP's, you're going to do one here. If you fail that, you're going to do it again. And if, you know, if all the scores end up just the same here, then they're going to either request a waiver if you're eligible. The special phys docs, well, they're going to determine who's waiverable for the Bumet instruction. So they'll go in there and make sure it is a waiverable condition, whether it's vision hearing. If it is waiverable, we'll keep them in the pipeline here, and then the waivers usually come back by the time they get out of here. If it's not, since most of them are contracted, then they can opt to either pick a different rate or pretty much get separated in their contract. So your basic recruits, they're coming in, they're not contracted like the A-hundreds. If they don't meet specific requirements for a certain job, then they'll get put in another job before the A-hundreds. If they don't want to fulfill their contract, because they can't do something medically related, then they can either opt to go or stay with another rate. Our primary audience is, I guess I could say the layman or layperson or civilian, so I'll do a little bit of interpreting for them. So someone who comes into the special warfare pipeline is kind of quasi hired by the Navy and they can then choose to choose a different direction to take their career. Is that accurate? Yes, that's exactly what I'm saying. And if they got their heart set on being a seal and you don't meet the medical qualifications to fulfill that contract, then yes, you can drop on request and get separated from the Navy altogether. So you decide to make a different decision. How often do you see that misalignment of MEPs decision versus something onsite here? There's those that do fall through the cracks. There's always human error. It's not as much as we think. Most of our drops are usually drop on request, really, after a dive mo. In terms of medical, it's not as much as we actually think. Okay, well, that's good to hear. What types of tests can a recruit fail to have them say, you're not going to be able to be in the Navy at all? Hearing's on a bigger scale right now than vision is. Why is that? Any kind of permanent hearing loss, the deeper decibels, it's an automatic disqualifier and some of them are even waverable. So you're saying that it's starting to become a bigger issue than it used to be or it's more noticeable? I think it's probably more noticeable. All the loud music. I think it, I would say that the earbuds and the working out, I'll be hypocritified and say I do it myself. I would imagine they have something to do with it. So if you listen to something loud right away, you'll have the minimum hearing loss, the temporary, the lower frequencies, but the higher frequencies and that would be, you can consecutively listen to your earphones really loud while you're in the gym 24-7 or in your cars nowadays. No good. I'm guessing that there's sometimes injuries that happen during basic training that can kind of put someone in that same situation. Is that something that people should be concerned with or is that pretty rare in terms of shin splints that are so severe or whatever it may be that they're not able to continue? Do they have a chance to maybe have a few weeks to kind of heal up? So we have the recoup convalescent unit that's actually here in chip four. The reason for that is if you do get shin splints or stress fractures, you actually get a rehab time.  So that's 21 days of rehab. So doing that 21 days, they're gonna go to a chip four and their job is to get better. They'll be putting a hold status until they are fit for full duty. Once they're fit for full duty, they'll incorporate with another 800, to keep the training going. Does it happen often enough that you think that's a successful way to deal with that? I don't think it happens as much for the 800s. For the regular recruit divisions, yes. For the 800s, no. And that's the same process for both of them in terms of the time that they're given. Yes, if you look at the recoup convalescent unit, you'll have for every 30, 40 recruits, you might have one 800 in there. Usually the one 800s get injured is because that individual didn't prep. Right, right. So he came over here and did a little more running than what he or she was doing at the time. Right, right, right. From your perspective, what type of advice would you give to someone coming into the pipeline to avoid types of medical issues that we're speaking of? The biggest thing is prep. So don't do the bare minimum before you actually get here. Going over to Dymo PT, they're gonna give you a workout. Right. So if you weren't prep before you got over here, it's gonna show and it's gonna show quick. Dymo is really where they start falling out. It's not regular PT. That's not the PT that all the other recruits get here because you're contracted. You're gonna do something more strenuous. So it's definitely prep is huge. Is there part of the process that you think people should be maybe more aware of? You have to prep. Prep's the biggest thing. If you're taking a lot of protein powders and all this stuff that you can get from GNC or bodybuilding.com, whatever you're taking, just remember you're not gonna have it here. Okay. You know, the stuff that gets you over a good workout because sometimes you're pushing your body so much that you need the extra protein or you need a little bit extra. Or whether it's pre-workout. You're not gonna have it here. That's gonna be cut straight out of your whole diet. Now you need to know how to eat correctly. And we do our classes here and teach you how to eat. However, it's a lot better if you have the history of doing it and you know your greens, you know your fruits, you know what you're doing. Cause you can't just take that simple protein powder and call it a day. It's not gonna happen. Right. If there's anything that you see on a common basis that you think that people should be more aware of, it'll be great if you kind of cover some of that as well. One big thing is the psych issue. And when I say that would be more geared towards the mental health part of it. Mental health's a big part here and it's a great, great tool for recruit training command. Some of these, the 800 guys that we have will get them and they are watching videos and everything since they were young, they're real motivated, can't wait to do it, which is outstanding. However, they come to boot camp and they figure out maybe this is not what I wanted to do. So it's a general hype that they get theirself going through for however long that they get theirself hyped up for, but they get here and they kind of shut down a little bit. And once they shut down, it's kind of hard to pick them back up. So then they end up going to mental health and talking to them. Sometimes they can say things that may disqualify them. Does it happen? Yes, and it's not on a huge scale, but however it does happen. Almost like they pretty much psych themselves out. The first couple of dive mode days they kind of psych themselves out and they're like, okay, maybe this is not what I want to do. Cause it's strenuous, it hurts. At the end of the day, this is what you wanted and they're getting trained by the best people we have over there. But sometimes they can psych themselves out and they put themselves in a bad position. I think you brought up a good point. So much focus on the prep and even further along in the buds process and then throughout there's a huge emphasis on physical preparedness, physical capabilities, but the mental aspects seem to be coming to the forefront a lot more than it used to be in the past. And even awareness of mental health issues speak to that a little bit. Are there underlying issues that people should be aware of? If they want to come through basic training or maybe in the past, they had issues with depression or anxiety, what's the Navy's kind of opinion on that in NSW and the big Navy? How does that kind of fit into your job? I work with mental health really, really closely here. And the reason being is because it is a different kind of day and age. We have kids getting prescribed meds from early age. However, that's not all disqualifying factors. Yes, there's certain diagnosis where it's going to be a disqualifying factor and that's just kind of what it is if you bipolar type two or something like that. It's going to be a disqualifying factor. However, if you struggle a little bit through school or even your first year of college and you show progression and you have those notes by a doc saying that you're good to go, you've been on meds for a little bit or something bad happened in the family, these are all things that they are waiverable. You just got to show the actual documentation. So in terms of someone prepping to come here, it's good to know that if you're going to get your civilian medical records and it's going to say that you were put on a certain medication, don't just stop. Don't just stop and say, okay, well, I'm going to stop taking this because I plan to go to boot camp in a year. So if you stop when your doctor tells you to stop and then you show the progression for the one or two years, preferably two, the doc can sign off on that and say, hey, this person fell into a kind of a slump, did what he had to do, she had to do and recovered fully, we're good. Where we have the issue is when someone goes to a hard part in their life, gets prescribed some medications and then decides this is what they want to do, they want to go to boot camp and then just stops it. So when you get some civilian medical records, it shows that you were getting treated and then there's a blink. So there's really nothing to go off of and now you got to get reevaluated here and then there's a huge, it's huge. So there's nothing wrong with mental health, it's a part of all of us, whether we like it or not. However, it's the way we go about it. Well, I think that's a good thing to point out. I think a lot of people might say, whatever this existing condition, whether it's physical, mental or whatever, disqualifies me. I think reading between the lines is, talk to your doctor and you guys want to see as long of a spread between whether it's a prescription or a diagnosis and have some evidence to say, like, hey, this is where I am now. I think that would be helpful for a lot of people to hear because a lot of people might just say, well, I can't do that now. And I mean, you know what I'm wrong, in mental health there's a lot of different areas, a lot of different diagnoses and this doesn't speak for every single one of them. But there's a big chunk out there that you can still be in the Navy, not wrong with it. They just need to see the full treatment plan. They can't just cut it off in the middle because you never went through treatment then. There seems to be maybe some confusion in the aspect of medical records and how that's integrated into the Navy from civilian into integrating into the sailor life. What's that process look like? Or does the Navy get all, does they scoop up everyone's medical records when you come in? Well, any diagnosis, anything that you, legally, that you have to put on your medical history form and that's a medical history form that's legally used by the whole Department of Defense. You do have to provide that documentation if you put a diagnosis on there. So if you put a diagnosis on there, you have to provide that actual information because the doc's gonna know what the treatment plan was for and how that's gonna affect you. In terms of receiving all the civilian medical records, what we'll get with you put in there. Do we get them all? Probably not. With that being said, we probably don't get it because they still got the human error or you got humans lying as well. Right, you see that as an issue normally or I guess not normally, but. I think it is an issue.  Past injury acts up here and we find out that it's something that was pre-existing. So now if they wanna stay in, we're gonna request those civilian medical records so we can see if you can actually stay in. But in terms of the Navy just reaching out to civilian hospitals and get medical records, that's not even legal. Well, obviously it seems like being honest on your forms is probably not only legal but the better thing to do for your own success. Yes, absolutely. So like I said, any medical history form you get in the Navy, you have to list legally. That's why they're asking you. We gotta know what's wrong with you. And if you don't provide that actual information and you can get in trouble for that. But in terms of if you don't provide it and even if we think something, it's not like we could reach out to where you got seen as a kid and say, hey, I need those records. It probably doesn't work like that. Are there any kind of medical issues that can develop here that can disqualify someone completely from the Navy? Recurring stress fractures or reoccurring shin splints. You're contracted to be here. However, you can't get past dive mode because you just keep on breaking unfortunately. That's not your fault. Right. However, that's something that would definitely, you want to make it much further than that because you're pretty fragile. And after this pipeline, you're going on to something even more aggressive. Right, more aggressive. I appreciate you taking the time to sit down and talk with us. I know this might not be the most glamorous topic, but it's just as important as every other part of this pipeline. You know, crawl, walk, run, medical is a big part of that. Thank you for your service and thank you for the time today. Absolutely. Find out more at Sealswick.com and join us again for the next NSW podcast.


It's 2022. The perfect time to announce the sequel to 1988's Beetlejuice. What are we doing anymore? It was just reported recently that Brad Pitt's production company Plan B Entertainment just said Beetlejuice's name three times, which gets him to appear. That's a reference to the film. Subscribe for movie references. Michael Keaton and Wynonna Ryder are back, baby! Reprising their role from the 80s hit film. Is Tim Burton coming back to direct this one? I don't know. What about Catherine O'Hara or Gina Davis? No idea. Is Alec Baldwin returning? I'm gonna guess not. He's been...  All we know for certain is this thing could get greenlit any day now and production can begin as early as this coming summer. How exciting.  These sequels come out decades after the original. They try to recapture the lightning in a bottle, the magic that you saw on display. The writing is usually hacky at best, full of nostalgic references. It doesn't have half the cast back, and the ones they do get are 40 years older. They're tired. They're not as exciting as they were. Granted, Michael Keaton is still at the top of his game. He is hilarious. He can put on all these different costumes and sell us. He's gonna be Batman again. So I do have confidence he can do it. He's really getting this, like, renaissance lately, and it's great. I'm all for it, but it just... I don't have any faith in any other part of this. Now, not all these sequels are bad. We've had some really good ones that came out years and years after the original. Look at Blade Runner recently. Solid film. Very solid. Mad Max Fury Road blew me away and a lot of others. Although, that felt less like a sequel and more like almost a reboot of Swords. And so it was really its own thing. Some people really like Tron Legacy as well, so there are definitely examples. But then you have the Coming to America sequel or Dumb and Dumber 2, the abysmal Zoolander 2. There's a ton you can look at also and say, man, what, why? Why? I am a fan of the original Beetlejuice. Not a super fan by any means. I did really enjoy the movie. I recently rewatched it with my kids at Halloween. They both really liked it too. And they're 9 and 13. So there's definitely an audience. The new generation, I think, as long as their parents showed them the original film, probably would eat a sequel up. I've just been burned an awful lot by movie studios trying to bring back classics for a new generation. Look at the recent Home Alone. What an abysmal waste of everyone's time. So yeah, just a quick video reporting on this news. Would love to hear what you think in the comments below. Let me know if you're excited about Beetlejuice 2 or if you're kind of like me and just thinking, I don't care. I don't need it. There's the possibility it could be great. There's also a much larger possibility it won't. So maybe we should just let Sleeping Dogs lie. I think that's the expression. I don't know. Subscribe to the channel if you like movie commentary, reviews, rants. I do them all here. And hopefully I'll see you next time. Hey, since you're still here, maybe you're one of the people that've been watching me for a while and thinking about ways you can give back. Well, I'm on Patreon at patreon.com slash adamdoesmovies. There's a $1 tier even or a $5, $10, depending on what you want to give and how much you appreciate what I'm doing. Or you can become a YouTube join member right here. Hit that join button and you get access to exclusive videos and a lot of the same perks the Patreons do. If you're a poor college student or you just have to save that money for a Starbucks every day on the way to work, I get it. In that case, maybe just share the video around with your friends and family, post it on Twitter, Facebook, all that crap. Every little bit helps. I appreciate it.


Borrow. Borrow. To borrow means to take and use something that belongs to someone else for a period of time before returning it. Bring back. Bring back. To bring back means to return with something or someone. Celebrate. Celebrate. To celebrate means to do something special or enjoyable for an important event, occasion, or holiday. Gift. Gift. A gift is something that is given to another person. Lend. Lend. To lend means to give something to someone, to be used for a period of time, and then returned. Loan. Loan. To loan means to give something to someone for a period of time. Mention. Mention. Mention means to talk or write about something or someone in a brief way. Don't mention it. Don't mention it. Don't mention it is an expression used to answer someone who has just thanked you for something. Paycheck. Paycheck. A paycheck is the money that you regularly earn. Sharp. Sharp means having a thin edge that is able to cut things. Supplies. Supplies. Supplies are things such as food, equipment, fuel, and so on that are needed for a particular purpose. Use. Use. To use means to do something with an object, machine, person, or method in order to accomplish a task or do an activity. When Americans ask for permission, they can choose to use either may or can. In this lesson, Anna asks her coworkers to borrow some things using may. May is the more polite way to ask. Can is the more familiar way. English teachers in the past told students to use only may to be polite when asking for permission. Now, grammar teachers tell us it is fine to use can to ask permission. Listen to how Anna asks Jonathan for permission to use his scissors. Can I borrow your scissors? Now you try it. Ask for permission from a friend. May I borrow your book? Anna asked Amelia and Jonathan if she could borrow things. When they answered, they used the word lend or loan. Sure. I can lend you my stapler, Anna. Yes, I can lend them to you, but must return them. English learners have trouble with these two verbs. Here's one way to remember the difference. Borrow starts with the letter B and so does the phrase bring it back. When you borrow something, you must bring it back. Lend and loan start with the letter L and so does let. When you lend or loan something, you let someone use it. Now listen to Anna say to Jonathan. Wow. Those are sharp. I will be very careful and I'll bring them back tomorrow. Now you try it. A friend asks you. Can I borrow your book? You answer.