So, I think we will get started with our session now. So, what we will do is now we will take standard flow over a flat plate. So, in most of the cases when we take flow over a flat plate what are the let us just write what are the let me just take flow over a flat plate that is this is my flat plate and let this be u infinity comma t infinity. Now, what is the continuity I am taking two dimensional incompressible steady flow two dimensional incompressible steady flow that means velocity is varying only with x and y in x it is going to be u and in y it is going to be v. So, what is continuity equation for incompressible 2D flow del u by del x plus del v by del y equal to 0 is that right. Now, what is the momentum equation x momentum equation there is no z momentum equation because w is u. So, I have u del u by del x plus v del u by del y del u by del t is not there because my flow is steady equal to minus of 1 by rho del p by del x plus mu by rho into del squared u by del x squared plus del squared u by del y squared. If I write y momentum equation what is that u into del v by del x plus v into del v by del y equal to minus of 1 by rho del p by del y plus mu by rho del squared v by del x squared plus del squared v by del y. Professor Arun has already told I will not write energy equation right now I will go ahead and write that also what is there energy equation is u del t by del x plus v del t by del y is equal to 1 by alpha into del squared t by del x squared plus del squared t by del y squared let me check whether I have written it rightly or not I have made a mistake that is k by rho c p actually this is this is alpha not 1 by alpha. So, now what happens to my y momentum equation what happens to my y momentum equation we know very well that v is very small compared to u v is very small compared to u. So, that makes my all this term will be vanishing this term will vanish and this term will vanish then what is what does this mean implies that del p by del y is equal to 0 that means pressure is a function of pressure is a function of x only x only. So, what does this my x momentum equation reduced now here let us go to the next term what happens to flow over a flat plate for del p by del x yes let us d p by d x, but that is not the answer see let us write let us write the x momentum equation continuity equation is del u by del x plus del v by del y equal to 0 u del u by del x plus v del u by del y equal to minus of 1 by rho d p by d x plus mu by rho into del square d u by del x square plus del square d u by del y. Now, first thing is for a flow over a flat plate for flow over a flat plate if you see the equation what I have written here I have not written d p by d x and I have also not written del square d u by del x square, but I need to now show why they can be neglected. So, now for flow over a flat plate for flow over a flat plate what is u infinity all over u infinity is constant it does not vary. So, now if I apply as professor was telling I have two portion whenever a flow takes place over a plate there are two portions one is viscid portion another one is inviscid portion that means in the inviscid portion viscosity effects are not there. If I apply this equation for the inviscid portion in the inviscid portion what is u infinity? u infinity is constant u infinity is constant if you ask for proof for that I would say that if you put a put a tube and measure very much outside the boundary layer u infinity is constant you can check it that way. Now of course, from potential flow solutions also one can show that u infinity is constant because inviscid flows can always be handled by potential flow.   So, this term is 0 again here u infinity is constant this term is 0 that means what what did I end up getting d p infinity by d x is equal to 0 this is only applicable for whom for all those flows where in which u infinity is constant. But now if I take u infinity is constant for flow over a flat plate if you take flow around a cylinder if you take flow around a cylinder u infinity is not constant if you take this is u infinity flow in the potential core that is very much away you will get u by u infinity as some sin theta or something sin theta no I would say some function of theta let me put it like that I do not recollect the function exactly u by u infinity is a function of theta that means u is not a function of u is not constant anymore it is a function of location where I am then in that case d p by d x is not equal to 0 is that ok. So, that is why for flow over a flat plate d p infinity by d x. So, what is this d p infinity by d x this is d p infinity by d x is the pressure with which is the pressure with which the inviscid portion is impressing upon impressing upon whom impressing upon upon boundary layer see because we said that there is inviscid portion this is viscid portion and this is inviscid portion this d p infinity by d x is acting on the boundary that is how the inviscid portion and the viscid portion get connected with each other. So, one solution is required for the inviscid portion to get d p by d x in the boundary layer equation that is for this equation d p by d x in the boundary layer for getting this equation I need to get d p by d p infinity by d x using the inviscid solution using the potential core solution of course we are not taught you how to get the potential core solution it becomes too much I would be ending up teaching only fluid mechanics, but not heat transfer. So, this comes from potential flow solution. So, this means to say that inviscid solutions are also as important as the solution because that is how both the viscid and the inviscid portion get connected with each other usually this point is not told by most of the text books directly this is something which comes out of inside of course, Schlichting tells this and various specialized book tell, but usually undergraduate books do not tell this kind of insights. So, this is what we need to impress upon the students otherwise he will be asking us always why for a flow or a flat plate d p by d x equal to 0 I will not be getting convincing answer. Next let me tell what is this del squared u by I have neglected del squared u by del x squared equal to 0 why now let me do that for doing that I will be using the recourse of scale analysis. If you see here I have neglected del squared u by del x squared and similarly del squared t by del x squared I will just answer this in a little while, but I am going to start off something very interesting what is called as scale analysis or order of magnitude analysis order of magnitude sis or it is also called as scale analysis this analysis what I am teaching you can get it from Professor Bejan's book either heat transfer book or convective heat transfer book John Wiley publications. So, what is my continuity equation del u by del x what am I up to what am I up to I am going to take only the continuity equation and the x momentum equation and I am going to get the velocity boundary layer thickness and skin friction coefficient without solving the equation. Let us see how do we do that it appears like a magic, but yes indeed it is a magic let us see how do we apply that magical idea of scale analysis del u by del x plus del v by del y equal to 0. Now, what am I doing here I am going to take if I take flat plate again and I have u infinity and this is y direction and this is x direction u is of the order of note this I am using the notation tilde u is of the order of u is of the order of what u is of the order of what or x is of the order of what it is of the order of length of the plate means if it is 1 meter I can say that anywhere I take at least it will be 0.1 meter, 0.2 meter, 0.2 meter it is of the order of meter. So, x is of the order of L x is of the order of L u would be of the order of u infinity that means if u infinity is meters per second let us say 10 meters per second throughout the boundary layer u infinity will be of the order of 1, 2, 3, 4, 5 meters per second. So, u is of the order of u infinity what is the order of y we are interested in boundary layer thickness. So, I am interested not outside the boundary layer I am interested only within the boundary layer thickness. So, y is of the order of delta. Now, let us find the order of v from continuity equation what do I get what is the order of del u u infinity minus 0 that is u infinity upon delta x L minus 0 that is L these two of the same order know they are equal to another. So, this is of the order of v upon what is the order of delta y delta del y is of the order of delta. So, v is of the order of what is the v what is the order of v u infinity delta by L remember it is not equal to if you have to make equal to you have to put some constant number that is equal to u infinity delta by L. But remember I am not putting equal to here I am putting tilde that means it says that it is of the order of that is v infinity what does this say what do I understand from this how does delta compare with L delta by L is always less than 1 is not it. So, that means v is always less than u infinity how much times delta by L times is that ok. So, now, I have got the scale of v now let us take x momentum equation that is u del u by del x plus v del u by del y equal to p is not there for flow over a flat plate only I am doing d p by d x we have made it as 0 I have explained you just a little while ago why because it is u infinity is constant. So, 1 upon or mu upon rho into del squared u by del x squared plus del squared u by del y square. Now what is the scale of del squared u by del x squared mu by rho into u infinity upon upon x x squared is L squared.  So, which of these two terms are important now which of these two terms are important anything divided by delta squared delta is very small 0.1 meter. So, it will get amplified L is 1 meter or 2 meter. So, this will be of the order of meters this will be at least 10 times larger than this. So, that is why I can neglect what I can neglect del squared u by del x squared after neglecting that let me see what do I get that is u del u by del x plus v del u by del y equal to mu by rho del squared u by del y squared. I have answered the question why del squared u by del x squared is neglected is neglected here and the same arguments you can do this here also, but let us do that little later on.  Now let us see this is important or not important because we get a doubt v del u by del y v is very small why am I not throwing it out why am I not throwing it out now you see what do I get delta delta cancels out what do I get here u infinity squared by L which is same as u infinity squared by L that means what this term is of the same order as the second term that is why although v is small which one is large del u by del y is large that is why I cannot neglect. So, the overall scale of the left hand side is what u infinity squared by L which is equal to mu by rho u infinity by delta square what does this imply what does this imply delta squared by L is of the order of mu upon rho u infinity one u infinity has got cancelled out is that ok if I write now delta squared by L squared that means both sides I am dividing it by L.  So, delta by L whole square delta by L is of the order of R e L to the power of minus half is that ok. So, now this answers the question delta is of the order of L R e L to the power of minus half. So, now I can tell boundary layer increases with the increase of the length you can see here delta by L if you see if I from here if I write directly delta is of the order or delta squared is of the order of what mu L by rho u infinity delta increases with the increase of viscosity it increases with the increase of length it decreases with the increase of velocity and it decreases with the increase of ok. So, yesterday and day before yesterday we had so many questions how does boundary layer thickness depend on velocity thermo physical properties location this is the answer for that remember I have not even solved if I solve this equation I am going to get the constant I am just going to get the constant if I have to show you that constant I am going to get the constant little later in the day that point it is going to be 5 delta by x equal to 5 times that is delta by L equal to 5 times R e x to the power of minus half that 5 is the only thing which is missing, but otherwise I have captured all the parameters now let us see how do I get c of x that is shear stress what is c of x c f x equal to tau wall upon half rho u infinity square is that right what is tau wall what is tau wall tau wall equal to mu del u by del y upon half rho u infinity square ok. So, now c f x is of the order of equal to I cannot write mu is of the order of means I do not have to worry about this half ok mu what is the order of del u by del y u infinity by delta upon rho u infinity square. What will I get mu by rho into 1 upon u infinity into delta, but I have just now found what is delta what is delta delta is c f x equal to mu by rho upon u infinity what is delta L R e L to the power of minus half just now we have found is that right what will this give me u infinity L upon mu what is that 1 by no I think I have made a mistake here somewhere rho u infinity square yeah it is right nothing is wrong nothing is wrong nothing is wrong yeah that is right. So, what do I get 1 upon R e L into R e L to the power of minus half so, what do I get c f x is of the order of R e L to the power of R e L to the power of minus half. So, what happens to my shear stress with the increase of the Reynolds number it decreases shear stress decreases with the increase of the increase of Reynolds number is that ok. Now, let us see you see so nicely we have found all that you are going to do is in the derivation that this constant is going to be 0.332 after you derived without deriving we have got the functional form the beauty is we have got the functional form that is the beauty let us see can we do this for boundary layer thermal boundary layer thickness and also nusselt number. So, there I have to specify this is what I have said delta by L is R e L to the power of minus half and c f is of the order of R e L to the power of minus half is that ok. So, now, let us see what do we do if in case of boundary layer thermal boundary layer I am taking Prandtl number less than 1 if I take Prandtl number less than 1 what will I get Prandtl number less than 1 is always results in thick thermal boundary layer Prandtl number less than 1 gives me thick thermal boundary layer ok. So, what is my energy equation first let me write the thick boundary layer. So, that is I am taking Prandtl number less than 1 where does this occur it occurs in liquid metals like lead mercury. So, these are the indium tin these are the ones which will typically have Prandtl number of 0.02, 0.003 and things like that. So, very less Prandtl number if you have such a low Prandtl number what will happen we have said that already delta by delta t is of the order of Prandtl number to the power of n where n is a positive number right.  If I take thick thermal boundary layer how does that boundary layer look like actually boundary layer thickness will be this is delta and this is delta t ok. This is hydrodynamic boundary layer this is thermal boundary layer tbl and hbl or vbl velocity boundary layer.  So, that is this is my velocity boundary layer or the velocity profile within the boundary layer and also outside and this thickness is what is this thickness hydrodynamic boundary layer delta this is u infinity and remember I am maintaining this at constant temperature ok this is my plate. Now how does the thermal boundary layer look like thermal boundary layer we have realized already it is thicker than hydrodynamic boundary layer. So, I will get a temperature gradient this is t infinity and this is what I am going to get as temperature and this boundary layer is delta t thermal boundary layer thickness ok. So, now if you go to the document back and see here what is my energy equation u del t by del x plus v del t by del y is equal to alpha del square t by del y square. I think I can write this directly now on the same arguments the way I wrote for del square u by del x square plus del square u by del y square. If you see del square u by del x square what did I do I neglected that because I end up getting u infinity by l square. So, that is going to be smaller than this. So, similar argument here if I take for if I take for del square t by del x square I will get delta t by l square as opposed to delta t by delta t square. So, delta t square is very much small which is sitting in the denominator which will blow to my complete number ok. So, now what is the scale of u what is the scale of u yes within the thermal boundary layer what am I going to do is that within the thermal boundary layer because my thermal boundary layer thickness is much larger than the hydrodynamic boundary layer. So, within the thermal boundary layer I am going ahead and making the velocity is same and that velocity is equal to u infinity. In fact, this point was told by professor Arun when he had touched upon the energy equation earlier in the yesterday yesterday he had told he said it is little early, but this is what he was meaning when he told when he told that. So, that is u infinity. So, what is the scale of u u infinity what is the scale of del t delta t what is this delta t this delta t is nothing, but t s minus t infinity it is of that order what is the scale of delta x del x l what is the scale of v we have done already u infinity delta by l remember this is hydrodynamic boundary layer then del t is delta t what is the scale of del y only delta we are in the thermal boundary layer the energy equation is for thermal boundary layer y here means for us delta t thermal boundary layer thickness is of that order of alpha delta t upon again delta t. Now, which of these terms are important and which of these terms are not important let us just have a look at it see you see here what is that I am saying. So, you see here what do I get delta by delta t how does this delta by delta t compare is delta by delta t larger small number it is a small number because parental number is less than 1. So, I can neglect this term compare to u infinity delta by l is that right delta by delta t is a small. So, anything which I multiply with that has to be a smaller number. So, what is there left hand side I have u infinity delta t by l is of the order of alpha delta t by delta t square. So, what do I get for delta t square delta t square is of the order of alpha l by u infinity is that right now what is this alpha l k by rho c p into l upon u infinity. Let me multiply and divide by mu multiply and divide by mu there is l here what is mu c p by k given by mu c p by k is what parental number Reynolds number Reynolds number is u infinity l by rho infinite mu infinity by mu. So, let me rewrite this I am going to rewrite this whole thing in the next board that is delta t square equal is of the order of delta t square is of the order of k by rho c p l by rho c p l by rho mu upon u infinity mu. Now, let me divide both sides by l square. So, delta t square by l square is of the order of if I do rearrangement mu c p by k into u infinity l rho by mu this is what I get is that right what is this by the way 1 upon parental number what is this 1 upon r e l. So, delta t by l is of the order of r e l to the power of minus half p r to the power of minus half. Now, what is Nusselt number let us go to heat transfer coefficient quickly I am I am not explaining this here much because I have already explained how you can look at all that I am saying is this constant is missing this constant is missing. So, h equal to minus k del t by del y at y equal to 0 upon t s minus t infinity what is the scale I get for this h k into del t by what is the scale of del y delta t upon delta t t s minus t infinity of the scale of delta t. So, this delta t this delta t cancels out. So, what do I get h is of the order of k into what is 1 upon k upon delta t that is h is of the order of k upon delta t is what delta t is what l r e l to the power of minus half p r to the power of minus half this means what this implies that h l by k h l by k is of the order of r e to the power of half and p r to the power of half we have reached where we are supposed to reach all that is missing is here constant which we are going to derive in couple of minutes from now. This is the beauty of scale analysis we are skipping the Prandtl number greater than 1 and I would like you to ask plenty of questions I would like to take this tomorrow I would like all of you to work on this and derive and show that delta t by l is of the order of p r to the power of minus 1 by 3 r e to the power of minus half for thin thermal boundary layer that is p r greater than 1 and nusselt number is r e to the power of half and p r to the power of 1 by 3. Let us see how many of you can do it. Now I am handing over the charge to professor Arun who will teach us the closed form solution of flow over a flat plate. So, good afternoon we will quickly now that we have understood professor Prabhu has very elaborately explained why pressure gradient over a flat plate necessarily has to be equal to 0. So, d p by d x is equal to 0 for flow over a flat plate and therefore, in case of in case of flow around a cylinder or a sphere d p by d x is not equal to 0, but d p d p infinity by d x pressure with which the inviscid portion is impressing upon the boundary layer I am using whatever he has written that basically is related to the potential flow solution. Now he has done the order of magnitude or scale analysis and this is something which unfortunately we do not tell our students at undergraduate level which is ok, but the students usually have this doubt when to use what why is it r e to the power half p r to the power half r e to the power one third p r to the power one third so on and so forth. And all those questions I think have been answered by the scale analysis which was done so elegantly professor Bejan's book on convective heat transfer has this scale analysis done in detail with this scale analysis we do not have to teach this to our students, but at least those who are interested where people come and ask why how do I remember how do I know that it is r e to the power half p r to the power one third means we say that yeah it is given like that it is an empirical correlation this is what we give an answer, but it has come primarily from scale analysis and this I think if this two or three sets of transparencies which have been flashed these things are the heart of the scaling thing ok. So, you can with confidence hopefully answer questions related to Nusselt number being a function of r e p r that we showed when we showed the non dimensional solution now we are showing that it is r e to the power half p r to the power one third for one of the case and half and half to the other case. Now this dimensionless similarity solution and all these things these are essentially the mathematics related to flow over a flat plate fluid mechanics has basically taught us all this I do not I am not going to spend more than 5 minutes on this it is essentially what is the similarity variable we have done similarity analysis when we did this semi infinite medium essentially eta was defined to take care of the x and the t dependency x and the t dependency was there in a semi infinite medium because our governing equation was d square t by dx square is equal to 1 by alpha dt by dt and then we said let me define a variable eta which was something related to under root alpha t x by under root 4 alpha t or something like that which then took care of the variation in both x and t such that a PDE was converted to a ordinary differential equation and that ordinary differential equation we could get a solution analytically in the form of some error function or conjugate error function we could do that very easily and then obtain a close form solution. Now what we are doing going to do here for this laminar flat plate solution which most fluid mechanics books will have if you take fluid mechanics by Frank White or Fox and McDonald's, Engels, Engels and Simbala any of the standard good books they will have this Blasius solution. So, I am not going to spend time on it we are just going to see the similarity variable let eta be defined in terms of this y under root u infinity by nu x and that u by u infinity is a function f prime of t. It is given as f prime for convenience f prime of eta therefore, you get y over delta is essentially of this form because of the scale analysis delta over x is of the order of Reynolds number to the power minus half you can write this in this form and then the stream function definition you use u is equal to d psi by d y v is equal to minus d psi by d x continuity equation you can get this psi is equal to integral u d y substitute for u from here etcetera. So, this maths has been every step here has been given.  Now, my thing is I have u as a function of x comma y v as a function of x comma y this eta will convert this p d e into a o d e such that my d u by d x therefore, gets defined in this form u infinity eta by 2 x f double prime and this functional form is f double prime etcetera we will see what it is. So, therefore, I will get v if upon integration I will get v and this was one of the things which we had asked in the previous tutorial and if you do the maths you will get d v by d y as eta by 2 u infinity by x f double prime d u by d x you will get continuity equation is automatically satisfied therefore, u by u prime is a function of eta is a correct assumption why are we doing this we will just understand a bit. So, with this assuming that this f prime of n is a correct so called similarity variable that is a variable which transposes or which converges p d e to a o d e my momentum equation therefore, on applying this whole thing this becomes d u by d y is equal to d by d y of u infinity times f prime this is the definition of u. So, I want to expand this. So, summation rule I will get u infinity is a constant so this is just going to be u infinity d by d y of f double prime y, but then I do not want x is n y that is the basic idea of the similarity solution all the x is n y's have to go away that means I have to write let me just go back here see eta is a function of y and x. So, this when I want to write something as a derivative with respect to y it is essentially a derivative with respect to eta, but it includes this aspect also so d u infinity f double prime this d eta by d y this d eta by d y comes because of the change in the coordinate system. So, x and y gets transposed to eta so because of that we are going to get d eta by d y I am not consciously spending any time on this you can do all the max every step is outlined here. Therefore, I will get an expression for d u by d y d square u by d y square everything is boxed in red u by u infinity is known. So, d u by d x can be obtained. So, this x momentum equation basically got transformed to a ODE third order ordinary differential equation f triple prime plus f f double prime by 2 equal to 0. So, when I do the transposition my boundary conditions also get transposed get converted. So, this is what is given here.  And the boundary condition this is third order. So, I need 3 boundary conditions in eta when eta equal to 0 f is equal to 0 what is this how did we get that when eta equal to 0 f it equal to 0 came from the gradient at the what is f here f of eta is nothing but psi divided by u infinity under root nu x by u infinity.  So, eta going to 0 refers to y equal to 0 that is the wall surface u equal to 0 no slip condition eta tending to infinity refers to y tending to infinity which is far away from the boundary layer. So, u is equal to u infinity f prime equal to 1. So, with this idea I will say this ODE with the three boundary condition was first solved by Blashe's in 1908 and this solution all of us know gives us this nice elegant form d f by d eta is equal to u by u infinity is equal to 0.992 and that eta equal to 5 therefore, delta which comes out to be 5 divided by u infinity by mu x and this is how all of us have seen this delta over x is equal to 5 over Reynolds number that we have seen that is this Blashe's solution delta over x is equal to 5 over under root Reynolds number where x is the distance from the leading edge of the plate leading edge means if I have a flat plate going from left to right this is the leading edge the starting. So, that is why there were questions on Moodle which asked why should a boundary layer grow how far does a boundary layer grow with respect to x the answer is given here delta is directly related to x to the power one half x divided by square root of x is x to the power one half. So, delta grows as x increases. So, it will grow as long as there is a solid flat plate associated as long as the viscous effects are going to be there as long as a no slip condition is imposed at a solid surface this as long as there is a flat plate which is there and that boundary layer essentially is seeing the effect of the flat plate in a y direction as long as that is there we are going to have the growth of the boundary layer. I know I have gone fast, but this whole this 4 5 transparencies are essentially algebra there is hardly any physics to be explained. So, start with this and just proceed in this direction what are we checking first is we are defining this and defining this and then we are saying is this going to satisfy as a similarity variable a similarity variable basically we have to see whether it is going to satisfy this equation. So, that once it is satisfied we said it is then we substitute for these d u by d x.  So, continuity equation will give me v like this.  So, please I we urge you to go through this I know there is lot of go through this at home kind of things today, but most of these things are algebra. So, what are you what are the three things that you will do at home? This one the scale analysis for case which Prabhu outline where for the Prandtl number is much greater than 1 that you are going to and do then you are going to do the energy equation non-dimensionalization. So, we told in the morning that you will be tested on that thing also. So, energy equation non-dimensionalization scale analysis for P r much greater than 1 and this Blasius solution Blasius solution blindly you can take from any of the good fluid mechanics book there is nothing to teach here. So, this basically gives a shear stress which is mu d u by d y at the wall and that shear stress go grows like this R e to the power minus half thus unlike boundary layer thickness wall shear stress and the skin friction coefficient decrease along the plate to the in the form x to the power half. So, boundary layer thickness increase. So, if I want to draw it like this whiteboard boundary layer thickness for a laminar boundary layer definitely we are going to have this kind of a functional form. How did this come? This came from fundamentals what are the fundamentals solution of the governing equation? Solution of the governing equation how could I solve this? I had a bad looking equation momentum equation which was cast in a convenient form by the definition of a similarity variable basically what it did was the following I will just summarize this I had u d u by d x plus v d u by d y is equal to mu d square u by d y square I had this form this with the associated boundary conditions was there. We what happened to the y momentum equation this is the x momentum equation y momentum equation professor showed us that it gave us only the following d p by d y equal to 0. So, we have already exhausted it. So, only this one remains now this equation I cannot solve directly. So, what I do is I have go to this so called similarity analysis where I say I define f of eta which is nothing, but f prime of eta is u by u infinity and then I define eta is equal to y u infinity mu x under root. So, y is here x under root and then I go through the whole algebra.  Once I substitute into the x momentum equation I will get a equation of this form f triple prime or 2 f triple prime plus f double prime equal to 0. This with the appropriate transformed boundary conditions will give me a solution what will this give me it will give me f as a function of eta correct because this is my independent variable and this is my dependent variable. So, this is going to give me f as a function of eta. So, this f as a function of eta when why did I start this I forgot to tell this has come from fundamental solution. So, after having obtained this one after having obtained this it is not come from magic it is a solution only thing is that we are not understanding the solution because it looks more different because you have f eta so on and so forth, but it is essentially the same equation. So, once I get this solution I recast this in the form of u d u by d y so on and so forth and see here we have d u by d y in terms of d square f by d eta square at eta equal to 0 d square f by d eta square has been tabulated here this is f prime this is f double prime this is the header of the table at eta equal to 0 is 0.332. So, this 0.332 is the second derivative of f with respect to eta. So, second derivative of f with respect to eta d u by d y first derivative of u with respect to y has become second derivative of f with respect to eta, but this was partial this was ordinary this value is 0.332 which has come from here and that is why I will get tau wall is of this functional form. So, why is this 0.332 why is it not something else that question is answered directly by looking at this table of solution where what is this table of solution this table of solution is a solution for this differential equation with the boundary condition and what is this let us not lose lose site this is the transformed bound governing equation of the x momentum equation along with the boundary condition. So, I have solved for the x momentum equation only closed form laminar flow solution, but I cannot directly see it because it is in terms of f and eta and that is what I am getting here. So, solution of the x momentum equation gave me the velocity distribution which in turn gave me the wall shear stress this is the functional form of the wall shear stress and that is why you get 0.332 and not some other constant and therefore, c of x varies as x to the minus one half. So, I just want to take over and do this part and then we will quickly finish this and go to internal. So, now actually I just want to give a historical perspective of this in fact, what we whatever taught us just little while ago was actually the Blasius solution Blasius did this solution as a part of his PhD with Prandtl. So, in fact, the solution is called as Blasius solution. In fact, the solution does not it did not come the way it has been taught now. In fact, this whatever we gave you the solution directly is actually based on numerical methods that is there is something called shooting method technique, but let us not get into that the point is at that time Blasius did not have that time means what it was in may be around 1920s 1908 1908 there is no question of calculator or computer at the most you have logarithmic table. So, what did it do is actually the solution is obtained by power series solution. So, you just take f as a function of eta that is a polynomial function f dash is a function of eta f double prime as a function of eta. So, let me just write just I will write one equation.   So, I have this equation just solve now differential equation has got reduced to d squared f by d eta square. So, this becomes f double prime now differential equation has got reduced to just an algebraic equation which we need to solve and remember all the coefficients of this equation for each of 1 eta, eta square, eta cube, eta to the power of 4, eta to the power of 5 all of those coefficients should be such that I should be getting this back to 0. So, if I equate that I will be getting a 1 a 2 a 3. So, that is how he got so intelligent is not it we do not have computer also what we got this solution. So, that is how bless has gave even today that 0.3302 which we have got it from computer he has given it from simple power series solution he has just done what is this whatever I wrote this is what is called as which we study any in mathematics actually what is called as power series solution beautiful it is what beauty also cannot contain what the meaning we mean actually. So, this is how we get this 0.332 and if I integrate if I put that I am going to get 0.332 here 2 is where that gets multiplied and I get 0.664 now another important thing we need to realize is how does one get eta equal to y square root of u infinity by new x I would remind you what did we do for semi infinite medium, semi infinite medium also we took the scale analysis and derived this similarly we have already done the scale analysis what is my scale analysis telling y upon delta what is delta delta is x re x to the power of minus half that is what I am substituting x upon square root of u infinity x of new so I get y into square root of u infinity upon this is essentially if I put that is not like a magic it has come from scale analysis without scale analysis the point is actually this there is a one important point I we want to emphasize here is always a question is whether we should teach to teach scale analysis or not to teach scale analysis I would say for under graduates please do not teach scale analysis it will be little too much for them but if you can teach and spend time and make them understand we teach here in UG in IIT we do teach scale analysis myself and professor Arun do teach scale analysis why because of this simple reason otherwise I cannot explain the student how did I come upon eta equal to y into square root of u infinity upon new it is just otherwise impossible to convince my student how do I know that I have to take u infinity upon square root of u infinity upon new so that is the reason why I say that scale analysis is very important order of magnitude analysis is very important in fact why only here for a post graduate approach I would say that for natural convection for forced convection for all cases one can derive boundary layer thicknesses and nusselt numbers from simple bound scale analysis okay okay so now with this we will move on to energy equation at the same pace what we did it for and definitely we do we have not covered it in we are not solved what we are not done is we are not solved this equation for you we have set up the equation and we have given the boundary condition similarly we are going to do for energy equation so in fact this can be solved by if you are if you are very particular about numerical method I am sure many of you are very good in numerical method this can be solved by rangay kutta method by what is called a shooting technique that is I will assume f prime and then go on iterating so this is combination of rangay kutta method and shooting technique if it is a post graduate class I would ask the student to solve this and submit this as an assignment if it is UG class I think it is little too much for ask them but if someone is interested do not discourage them for them to solve okay so that is about the shear stress it is very logical now bless us solve the temperature profile and now sorry velocity profile by solving the momentum equation now it is very logical for us to get to energy equation so I have energy equation U del t by del x plus V del t by del y equal to alpha del square t by del eta del y square there is nothing difficult here all that I have to do is I have to differentiate or I have to express t in terms of theta this is not new for us we have done this already in transient conduction so theta of x comma y is t of x comma y minus t s upon t infinity minus t s if I do that if I do this express this t in terms of theta and I anyway have eta already eta is y square root of U infinity by nu x if I do the transformation I have already have U V if I do the transformation for del t by del x and del t by del y and del square t by del y square I know I am going very fast what I am not doing is I am not differentiating all that you need to do is del t by del x can be written as del theta by del eta into del eta by del x so differentiate this with respect to eta theta with respect to eta and eta with respect to x you end up with this if you do similarly del t by del y you end up with this again you differentiate the same thing with respect to y you get del square t by del y square if you put that and simplify you are going to get what is as the way it is looking like this that is theta double prime theta double prime plus Prandtl by 2 into f theta prime f theta prime equal to 0 this is my equation and again I have boundary condition I am going to solve only for constant wall temperature that is if I take constant wall temperature case and substitute those boundary conditions I am going to get delta t again I am skipping all the mathematics I am skipping all the mathematics I intentionally I am doing please go back home and do this so delta t I get 5 I want to write this because delta t I get as equal to 5 re x to the power of half we are to the power of minus 1 by 3 does this equation look familiar with us does this equation look familiar by scale analysis we had found that I think I have made a mistake here it is delta t by x that is delta t by x equal to okay so that is we had found already delta t by L is of the order of re l to the power of minus half pr to the power of minus 1 by 3 so now you can write a cell number equal to just we need to flip that is 5 re to the power of re x to the power of half and pr to the power of 1 by 3 that is what we are going to see in a cell number you see here you are going to get 0.332 re x to the power of 0.5 and pr to the power of 1 by 3 why because I need to substitute minus k del t by del y k by delta t yeah k by delta t if I put delta t 1 upon 5 okay 1 upon 5 I should be getting yeah but k by delta I should get so if you substitute that you are going to get let me come back to you how exactly I get that that is I get 0.332 re x to the power of 0.5 and pr to the power of 1 by 3 so here what is missing is yeah yeah yeah yes 0.332 I get here because I am from this this is what so I get directly 0.332 and from that I am inferring delta t I think there is a small problem here no no there is no problem everything is fine 0.332 is what we had got from f prime at eta equal to 0 f prime eta equal to 0 that is what we have substituted and if I do that I am going to get the nusselt number the point is we have demonstrated through this that nusselt number is going to be a function of Reynolds and Prandtl we have shown that it works for it works for flat plate case any other configuration we take as long as the flow is incompressible we can represent the nusselt number as a function of Reynolds and Prandtl number we should be able to come up that is how we see hundreds of correlations put in data handbook after understanding this now you use any correlation left and right absolutely there is no problem I think I will stop at this point.


Exponential Ansatz The exponential ansatz is suitable for ordinary differential equations of arbitrary order that are linear and have constant coefficients. Of course, the method quickly becomes complex for higher orders. The method is best suited for second-order differential equations. The general form of a second-order linear differential equation looks like this. y prime prime plus k1 times y prime plus k0 times y is equal to s. Here we assume that the coefficients k1 and k0 as well as the perturbation function s are independent of x. They are a constant. If the perturbation function is not 0, then you must first ask a mathematician. Then he will tell you that the general solution y of an inhomogeneous linear differential equation is composed of two parts, a homogeneous solution yh of the homogeneous differential equation and of a particular solution which we denote by yp. The homogeneous solution yh solves the differential equation if you set the perturbation function s equal to 0. In the method of the exponential ansatz, as the name suggests, we make the guess that the homogeneous solution yh has the form of an exponential function. yh is equal to c times e to the power of lambda times x. Since the first and second derivatives of yh occur in the general formula of the differential equation, we must differentiate our exponential ansatz twice. The first derivative is yh prime is equal to c times lambda times e to the power of lambda times x. And the second derivative is c times lambda squared times e to the power of lambda x. Now we can insert the exponential ansatz and the corresponding derivatives into the differential equation. Let us factor out c times e to the lambda x. If we divide by this factor, then we get the so-called characteristic equation for lambda. When we solve this equation, we find the unknown lambda. Since it is a quadratic equation for lambda, we get two solutions, lambda 1 and lambda 2. We have to consider both of them. Basically, you can set up the characteristic equation directly by looking at your differential equation without having to do all these steps. Compare your differential equation with a characteristic equation.  In this case, the coefficient is 1. The coefficient of lambda is in front of the first derivative of y, in this case, k1. And the coefficient k0 in front of the function y itself stands alone in the characteristic equation. By the way, if you had a homogeneous third-order differential equation, then the characteristic equation would start with the cubic term and so on. A quadratic equation has two solutions, lambda 1 and lambda 2, and you can determine these, for example, with a quadratic formula. Since you get two lambda values, we have to consider both. To do this, you have to extend the exponential ansatz by another term in which the second lambda value is in the exponent. With the corresponding lambda values, this is the solution of the homogeneous differential equation of second order. Depending on the values of the coefficients k1 and k0, the solutions may show different behavior, because if k0 is greater than k1 squared over 4, then you are taking the square root of a negative number. In this case, you get a solution that describes oscillations. I will show you this in an example. If the differential equation to be solved is inhomogeneous, that is, the perturbation function is not 0, then we still have to add the particular solution, yp, to the homogeneous one to find a general solution. For the particular solution, we have to choose an appropriate approach, depending on what form the perturbation function s has. Here we look at the simplest possible form, namely when the perturbation function s is constant. Then the particular solution is given by a perturbation function divided by the coefficient k0. yp is equal to s over k0. Now, to get a general solution of such an inhomogeneous differential equation, we have to add the homogeneous solution and the particular solution. The two unknown constants, c1 and c2, are determined by the constraints, as you know. Also note that this method of the exponential ansatz is a gas method. This is the case if the actual solution of the differential equation is not of exponential form. Therefore, if you use a solving method that contains the word ansatz in its name, be sure to check your solution. You do this by substituting the solution you have found in the differential equation and checking if both sides are equal. Let's make an example. Do you remember the differential equation for the oscillating mass? This is a second-order differential equation with constant coefficients. The perturbation function is zero. That means we only have to find out the homogeneous solution and we do that with the exponential ansatz we just learned. Let's take the fastest way and directly write down the characteristic equation. We expect a quadratic equation because we have a second-order differential equation. The second derivative is preceded by the coefficient 1, so we just write lambda squared. Then the coefficient in front of the first derivative. Since the first derivative is missing in the differential equation, the lambda term is absent as well.  This coefficient stands alone in the characteristic equation. All together, the characteristic equation reads lambda squared plus d over m is equal to zero. For this equation, we don't even need the quadratic formula. We get the solution directly if we first bring d over m to the other side and then take the square root. Consider that the inverse of squaring gives two solutions, a positive and a negative square root. Also, we have an interesting case here when the square root of a negative number is taken. Square root of a negative number is not a real number, but an imaginary number. Do you remember what that means? We expect that the system must oscillate. Even if you don't know imaginary or complex numbers yet, you can split the term inside the square root into a product of minus 1 and d over m. According to the square root laws, you can split this product into two square roots. Square root of minus 1 is defined as the imaginary unit, a number which we denote by the letter I. That's all you need to know about imaginary numbers. Let us denote square root of d over m shortly as omega. If we insert the lambda values we have just found into the exponential ansatz, we get the general solution for the considered differential equation. This solution seems very abstract, but I will show you that this solution corresponds to an oscillation. Before that, let us determine the unknown constants c1 and c2 with initial conditions for our problem. For example, we could have observed that at the time t equals 0, the displacement of the spring was 1. The spring was displaced to the maximum, so the initial condition is y of 0 is equal to 1. Insert this condition into the solution to determine c1. e to the power of 0 is 1, rearrange for c1 and you get c1 is equal to 1 minus c2. Next step is to determine the unknown constant c2 using the second initial condition. For this, we use the fact that at t equals 0, the velocity of the mass was 0. In physics, you learn that velocity corresponds to the first-time derivative of the displacement. So our second initial condition is given by y prime of 0 is equal to 0. So let's differentiate our general solution with respect to time t.  And then we insert the second initial condition into the derivative. The exponential functions become 1 and the factor i omega cancels out, rearranging for c2. We find out that c2 is equal to c1. So we know that c2 must be equal to c1. Nice. Let's replace c2 with c1 to determine concrete value for the constants. The equation results in c1 is equal to 1 half. Since c1 and c2 are equal, c2 must be also equal to 1 half. Insert 1 half into our general solution. Now let's find out what this abstract solution has to do with oscillations. For this, we get our friend Euler to help us who tells us his famous Euler formula. e to the power of i omega t is equal to cosine of omega t plus i times sine of omega t. This relation tells us how the complex exponential function is related to cosine and sine functions. So the first complex exponential in our solution becomes cosine and sine with positive omega t and the second complex exponential becomes cosine and sine with negative omega t. We can omit the minus sign in the argument of the cosine function because cosine is symmetric. That means it has the same value for arguments x and minus x. The sine function, on the other hand, is anti-symmetric. Therefore, we cannot omit the minus sign in the argument, but rather pull it out in front of the sine function. Very nice. The complex sine function drops out and the cosine can be summed up. And this is our final solution. As you can see, the displacement y changes periodically with time. The mass attached to the spring oscillates back and forth and the oscillation is described by the cosine function.


Matt here for Leighton Sports Cards ripping one box of 2022 Panini Select football h2. This is for alexandros s. You had luck. Okay, you'll them quay walker. Theos jones jr. Rookie disco. Julio Jones. Ballinger. Trevor Lawrence. Minka Fitzpatrick. And gold disco Alvin Camara. That is nine of 10. Nice Alvin Camara. Gold disco. Jurgans. Guys, I'm going to jump off the stream almost immediately after this box is done. Appreciate y'all hanging out, Philip. They're keeping me busy. Chattin. Kenneth Walker. Rookie. Justin Fields. Tyler Lockett. Silver. Terry McLauren. Red disco. That is to 49. Ryan Asamoa. Sean Watson. All right, here we go. Penning. Donovan Wilson. Jonathan Taylor and Derek Carr. Blue disco. That is to 25. Lamjack and Chris Olave. Rookie on the back. That'll do it for your box, alexandros. Appreciate it. We'll get these right out to you.


My name is Melissa Armo with the Stock Swoosh and tonight I wanted to lecture you a little bit about one of my favorite topics, it's perseverance. It's coming into the end of the calendar year 2016 and if you have not achieved or reached your calendar goals this year for your trading, you sometimes may ask yourself what am I going to do in 2017? What should I do? Well just because you haven't made your goals this year doesn't mean you're not going to make your goals next year and it doesn't have to take you that long. I've found that we really create a lot of the obstacles for ourselves in our mind that prevent us from doing the things we need to do. Part of life and achieving your goals is having perseverance. You will go along if things will be going great then something will happen to you and you'll have to get over the hump but by all means don't work against yourself by creating your own humps you will have humps and bumps in the roads of life that will be things that come into your life that make you go out of control for a period of time and this could be in your trading too, I'm using this for your life and also your trading that are beyond your control and you have to persevere to get over them. So don't also create those same obstacles for yourself in your own mind. Anything that you want to achieve you can. The amount of time it takes will vary.   You might lose the internet, you might not be able to short a stop that you want in the day, maybe the platform is doing that strategy, maybe that's why you're listening to me. Maybe you've been trying to trade the market for the last 10 years and you haven't been able to do it and you've lost money or paid for classes and didn't learn a thing and now all of a sudden you've found me and you are interested maybe in what I do or teach yourself your own strategy you're going to have to learn from someone else like me or someone else. So what are you going to do? If the chips are down and you've had a shitty experience in the market you're going to have to persevere through taking another class like mine or someone else's or you're going to have to teach yourself that part of the perseverance. Many times I talk to people and they're in this cycle like I can hear it in their voice when I'm talking to people or even reading their emails, you've got to make your decision. Do you want to do this or not? If you want to take my class, yes, it will cost you money from the information. If you want to go out and teach yourself your own strategy and create it, go ahead and do it. But guess what? That's going to cost you money too. And there's an unlimited amount of money that's going to cost you. My class is a fixed amount. My class is $49.99, it's $5,000, that's a fixed amount. You go out and teach yourself how to trade the market yourself. And there's an unlimited amount of money that the market can take from you until you teach yourself your own strategy. That's pretty much what happened to me. So I'm lucky that I made it. It took me three years to figure out what I know, which I now teach people if you want to come learn it. You will have these roads in your training and also in life. You will have to persevere. You can be strong and you can do it. Wining like a baby or crying all day long isn't going to get you where you need to be. It's okay to be sad and depressed for a couple of hours or a day, but that's it people. You got to buck up, you got to learn how to do it, you got to make up your mind. It is a conscious decision. It's a conscious decision in your mind to train the market at all. It's a conscious decision to train to take the risk to take a class like mine. You have to be willing to do it. You have to be committed and you have to be willing to persevere when the hard times comes. I don't know why this is true, but it is in your training. You will get to this point and you may be there now before you come to me and you're like right there. You're almost going to make it and then you meet someone like me and I can take you to the next level. That's really what I've done with many people that I've met and hopefully I can do that for you if you would like to become one of my students. If you'd like more information on my Golden Gap course, email me at Melissa at thestockswish.com. Don't forget the things that I said tonight and hopefully something that I said will help assist you in your training and in your overall life. Thanks everyone for watching.


consistent to what state use and the city had used as commodity codes. Based upon those codes and key words, our system would do a search of every government database. If those words or codes come up, they will send you an email notification. They do the work for you. You don't have to go online to see what opportunity exists.  As I mentioned, there's certifications for small disadvantaged businesses, women owned businesses, hubs owned businesses, where there are set asides and sole sources that the federal government have, whereas they have goals they must meet, annual goals to do business with small business owners. And if you do meet the requirements and you are eligible and the government could seek your company directly and hopefully award you a contract. And we don't just limit the federal government, we also include state and local government. Part of our process is making sure our businesses understand that there are opportunities, the rewarding opportunities for you that also are available to states and local businesses. Some think because of Hurricane Ida, let me just do work with FEMA. No, the counties, the cities, and even local businesses may be a need of your services. But the call we had lately was FEMA, FEMA, FEMA. So the goal is to educate our businesses on all opportunities that are available. Our goal is also to assist you in marketing your business to federal, state, and local agencies. So that's the premise of the PTAC. This opportunity has been great for me because in my previous capacities and procurement, oftentimes there are requests for debriefing. And one of the conversations, the conversations normally include the fact that you didn't do a very good job of highlighting your skill set or your abilities. We're gonna assist you in that regard, making sure that you showcase what makes you different from every other company to the government. And also we are going to create a website whereas it's gonna be a one-stop shop. So government has small purchasing procedures. So if they're trying to, they're seeking quotes, they could kill two birds with one stone, but that better way of saying it, whereas they could acquire a contract opportunity with a business that also could meet their needs, they could meet their needs, but also be a small business. So they could meet both of those requirements. That's our plans.  We have objectives and we have goals and we are confident we will meet them. So ptechsc.org, www.ptechsc.org has a lot of information to the services we provide. That also has a resource page in terms of if you need assistance with sound registration or anything else, we have a lot of information there. And this is the first page of the website, we are gonna enhance it to make it more robust. Contact us if you are in need of any type of assistance in terms of you doing business with any level of government, local, state or federal, we could assist you. I wanna thank you so very much, Ron. We understand that procurement, it really is the key to us getting large state contracts and understanding of procurement and knowing what to do and when to do it is highly important and it's really reassuring to know that your office exists to help us double check our work to make sure that we're in line and ready to do what we need to do. We wanna thank Ron and the South Carolina Procurement Technical Assistance Center for being with us today. We also wanna thank the folks from the city, Shannon and Jewel from the city of Columbia's Procurement and Contracts Office. Thank you for the valuable information you share and from Richland County Procurement, Jennifer, thank you for all that you do as well and thank you for being willing to share with our audiences today because oftentimes when we're talking about getting business, it's important that we realize there's an older space where we can get business and we should know about doing business with the government, so thank you all very, very much. It is now time to be transitioned to our next topic. Before we do that, I wanna pause and see if there's anything that Melissa wants to add before we move forward. Yeah, I think Kalina had something to add regarding the city of Columbia and commodity that we procure. Kalina, would you state that briefly for us? Thank you, Ayesha. Just to kind of clarify what we all know, in the Minority Business Opportunity Directory, we do have a listing of all commodities that all of our city departments, including procurement and contracts and our community partners are listed to purchase. And I'm gonna stop the directory again in the chat, but if you miss it or you'd like to just have questions about it, my contact information is on the cover page of the chat. If you have any questions regarding the directory. Thank you. Thank you very much, Ms. Gignard. That's very, very helpful. Besides using examples of the businesses that are doing so well, we are very appreciative for the resources that you're bringing to the table and helping us to understand how to access them and then walking us through the processes to make sure that we have a good chance of getting access to those resources. And I gotta tell you, that's why you guys are meeting and exceeding your DBE goals consistently because of all the efforts that you're doing to make sure we know about these opportunities. So thank you so very, very much. And thank you to all the resources that you are bringing to us today in the procurement offices in Richland County, City of Columbia, and also the assistance that is offered on the state level. As we transition now to talk about our small business resource partners, I'm delighted to be able to introduce these powerhouses. Benedict College's Women's Business Center, Mrs. Cheryl Sallie, director is with us today. Richland Library, Ms. Diane Lucy, the business and careers manager, Mrs. Andrina King, workforce development specialist is with us, Mrs. Angela Brewer, business opportunity specialist with the U.S. Small Business Administration. We thank you each for being with us today and we'll start with you, Ms. Shat, Ms. Sallie. Thank you, thank you, thank you so much, Ms. Hardy. And thank you to the City of Columbia and all the leaders, Ms. Lindler, the staff. This has been really, really, really great greetings on behalf of Benedict College and the Women's Business Center. I've taken so, so many notes. I've covered up the notes that I had to speak from because it's so, so wonderful. Next slide, Aisha. But thank you so much. It's a delight and a pleasure for us to be here. Cheryl Sallie, I serve as the director at the Women's Business Center here at Benedict College. And we, like so many small businesses, we launched in the middle of the pandemic, we're a year old as of our official launch, September 2nd of last year.  Our proposal, grant proposal was the initial plan. We took it, we had to immediately implement it, hit the ground running, develop the infrastructure, market, engage and market to the clients, serve the clients and so on and so forth. We have to wait, we're all the hats. We have to do what we have to do, but we're happy to be here and happy to serve. We are one of three HBCUs in the country that operate at WBC. We are an economic development organization serving our small minority women-owned businesses and a part of a network of nationwide community business and economic development programs. And I'll talk about our professional and well-trained business advisors. Next slide. And to summarize, this is a lot of narrative here, but just to summarize our purpose, our mission, our vision, in alignment with Benedict College's mission as a catalyst for economic development, the WBC serves as the first gender-focused statewide entrepreneurial initiative for socially and economically disadvantaged small and minority-owned businesses throughout the state of South Carolina. Our vision here is to be a gateway out of poverty for our women-owned, particularly our women of color entrepreneurs. And to help them to start their businesses, grow their businesses, take that leap of faith into and on this business journey. So we want to make sure that we do everything we can to provide the tools, the information, and the education, the training that they need. Next slide, Aisha. And a part of that is to, at the end of the day, leave a legacy for their families. Leave a legacy in this community. So we're here to serve and have impact. This is just an overview of our staff. We have a great staff and collectively, it's probably 60 years of experience. We are, as Chris mentioned, early on, probably serial entrepreneurs. We haven't been afraid to start businesses. So we share all of our knowledge, all of our experience and our skills. So we're just happy to have the staff. We're looking for two more full-time positions to a full-time business manager and another business resiliency advisor for our newly awarded grant. And I'll talk about that a little later. Next slide. Who we serve, start-up businesses as well as existing businesses, for-profit organizations, women, women of color. But while we do focus on women, we serve everybody. We don't discriminate based on race or gender. So we welcome you to come on. We serve all business industries, business segments throughout the state of South Carolina. Next slide. These are some of the services we provide, everything from business planning, so you have the idea. All right, so let's put that idea on paper. Let's put it on paper, let's plan. Let's see where you are now, where you wanna go, three months, six months, six years, 60 years and what are the steps that you need to take to get there?  So how do you assess where you are? What do you need to do? And don't be afraid to make that adjustment. So we provide those services, resiliency, recovery, financial management. I am a stickler for making sure that you understand business systems and infrastructure. I always say, if you leave your business seven days, 14 days, 30 days, if your business miss a beat, you have a high-end hobby because businesses continue to run. So how do you duplicate yourself and make that transition to work on your business versus working in your business? But we provide the services, government contracting, networking, show up. Show up, learn every day is an opportunity, but you have to stay engaged in the community and network and collaborate. Don't be afraid to collaborate. Strategic planning, that's where we are now at the end of a year, taking a look at the previous year and that's what every business has to do. What do we do well? What did we do not so good? What are the weaknesses? And what do we need to do to strengthen those weaknesses? What do we need to pivot on? What do we need to embrace? And so we're doing what we advise businesses to do and how do we measure the outcomes and the results? And everything we do is at no cost. All the 101 consulting and all of the webinars and trainings that we've provided over the past year has been at no cost. So it's just a wealth of information and knowledge that you have access to at no cost. And I'm gonna tell you about some more of the services as well that comes through our partners. Next slide. And yeah, I know you can't read this, but this is just a snapshot of performance since we kind of hit the ground running probably over the last six months. We've touched at least 1,200, over 1,200 contacts and businesses through training, serving about over 200 clients and 14 entities, I think we have helped them to invest over $700,000 in their business, which may not be a lot on the big scheme of things, but it's a lot for a business that's trying to launch and grow. So we're very proud of that and to be able to say we haven't seen any of the businesses close. So that is really, really good. Over 300 jobs supported through our efforts. So we're really happy about that. Next slide. This is a sample of some of the workshops and training and tentative workshops that we are going to look to launch over the next several months. We have a partnership with Nexon Pruitt Law Firm and it's just a great partnership and opportunity. They provide legal services to our OWBC clients at no cost, no charge to them. So we're happy about that and we're gonna be doing some programming on business law, contract law, intellectual property, marketing, and so on and so forth, social media law. There are things that you're not supposed to be doing on social media and we want you to know. So we're happy about that partnership as well as the other partnerships, finance and banking, how do you build a bankable relationship? Business and everything we do is all about relationships but your financial system, financial planning and banking. We want to do some workshops and programming on that. This is our upcoming mobile unit. We're gonna take the Business Development Center, Women's Business Center on the road so that we can make sure we're reaching our entrepreneurs in the rural areas and we thank Boeing for sponsoring our business unit that will be ready to launch in January. Next slide. And the other things that we couldn't do it without our partners in South Carolina, the city of Columbia, as well as all the other partners that we collaborate with and we network with to make sure that we are serving and engaging with the clients. We are looking to also launch a mentorship program. I think Miss Goodwin is on the line, Roslyn Goodwin and we're so happy about that mentorship program. We want to connect our seasoned women business owners with our businesses who are ready to scale their business. So we're looking at women in business, women in different industries, women in media, Ms. Hardy, women in government contracting, women in medicine. So we want to really have a huge collaboration and community of women who can reach back and help our women business owners. So thank you so much for this opportunity. We'll have our, this is our contact information. Please connect with us. We're here to serve you. Thank you so much. Well, thank you very much, Ms. Cheryl Salas. Cheryl has just a long distinguished career in assisting small businesses to reach their full potential. And we want to let Cheryl know how much we appreciate her sharing the resources today. Give our regards to Dr. Roslyn Clark, artist, Kim Hunter, our very good friend, communications over there and all the other good folks over at Benedict College and the Benedict College Women's Business Center. So thank you very much. Our friends at Richland Library are waiting in the cut to give us their presentation.  Richland Library, you guys, our library doesn't operate like a typical library. They're truly a community service and a community resource, a community gym. Diane Lucy and Andrina King are here to talk to us about the library today over the next six minutes. Ladies. Thank you, Cynthia. First of all, I'd like to begin by thanking the city of Columbia for the invitation today. And I'd also like to congratulate Cheryl Sally on that mobile unit. What a fantastic resource for our community. So congratulations on that new service that's coming soon. I also wanted to tell you that we do have a special guest that we're gonna be introducing during our presentation. So stay tuned for that. So next slide, please. We're going to very briefly touch on four areas of free small business resources available through Richland Library. Our goal is always to help remove barriers for access to these resources. One of them is the Entrepreneur-in-Residence Program. The other is our Library of Things. Third is Small Business Programming. And the fourth area is Other Library Small Business Resources. So, which brings me to our special guest, Richland Library's newest Entrepreneur-in-Residence, Verschanda Pratt. Ro will be with us. Her residency started yesterday, so she's been our resident for all of one day. It'll run through January 15th of 2022. Ro, I know you're on the call, so I'd like everyone to join me in giving Ro a huge welcome. And I think she's gonna share a little bit about how she sees her role as her newest Entrepreneur-in-Residence for Richland Library. Okay, Ro. Hi, everyone. I am really excited to be here. And I think this is a fascinating opportunity. I often joke with the folks at the library that I am their biggest fan. It's one of my favorite places in the Midlands. So I am so honored to be the new Entrepreneur-in-Residence following behind some pretty dynamic people. I'm excited to bring my television news experience. I'm a former television news producer of 20 years. I decided to bet on myself and left television news to become a full-time entrepreneur. And my zone of genius is focusing on everything from visibility, because your visibility is power, storytelling, taking your idea to market, and everything it takes to help you become a successful small business and entrepreneur. So I'm very excited. As Diane said, my calendar is now officially open. This is my first speaking opportunity as my new title enrolls. So I'm very excited, but my calendar is open. And so I would love an opportunity to hop on. There's the information right there on your screen on how you can book some time on my calendar so we can help you be successful this year, even in the midst of a pandemic. I'm rooting for you. I'm excited. Thank you, Diane. Thank you, Ro. Yeah, we are just thrilled to have Ro with us. And just to let you know, her one-on-one small business coaching appointments will begin on the week of September 20th, which is next week. She'll be meeting with five individuals virtually each week. And she'll also be doing some exciting programming. So stay tuned for that as well. So let's talk for just a minute about the Library of Things Aboriginal Library. So originally our studio services department started out with four kits that customers could check out. And these were entrepreneurial items, such as a light kit, might be a podcasting kit, might be a camera kit. And this service just quickly took off. And we realized that because of the pandemic, we needed to add many, many other items that you may not think of as an entrepreneur. This could be something like a power washer, a lawn mower, even games for your children, because we realized that lots of folks had their hours cut back, some people lost their jobs. So what we did is we created this resource, removing a barrier of access. So to look at the collection, you would go to richelinlibrary.com forward slash library of things. The guidelines are pretty straightforward. You can check out an item for up to one week with one renewal. And the card holder is responsible for all the materials they check out from library of things. If the item is lost or not returned, those charges go to your library card and we can actually reserve the right to cancel if you don't pick up the item within 24 hours of your reservation because we have in some instances, many people waiting in line to check out these items. Next slide, please. So, okay, if you're 18 or older, you are able to check out equipment. You have to have a Richelin Library card in good standing which means you're less than $10 in fines. And if you have a computer card or a courtesy card, those particular types of cards do not have borrowing privileges for library of things. You would sign a waiver and all you need is your government issued ID to check out this equipment. Next slide, please. The other thing I wanted to quickly mention is that Richelin Library facilitates a monthly community partners meeting which is held on Zoom. We started this service back in March of 2020 and we were meeting weekly because as we all know, the pandemic really threw us a curve ball and I know Cheryl referred to pivoting. So this was one way that we could still get together and support one another, talk about status updates, upcoming programming resources that we provided for the small business community. Now we meet monthly and our next meeting is on Monday, October 4th at 2 p.m. So if you are a small business agency helping small business owners in our community and are not currently participating, please send me an email to request the Zoom link. We'd love to have you. So now, next slide, please. I'm gonna turn things over to Andrina King, our workforce development specialist who's gonna share information about the library's small business programming. Thank you, Andrina. You have three minutes. Thank you. Richelin Library's small business programming includes various collaborations with community partners that cover topics on marketing, 10 steps of starting a small business and financing to name a few. In addition to our artists and our entrepreneur and residents, our three previous EIRs from the library's field business branch were Karen Ari Jenkins, Kevin Williams and Shanice Cleckley. You may have also heard of One Million Cups, Columbia, which was formerly held here at Richelin Library. Due to the pandemic, the location has been moved to Noma Warehouse every Wednesday at 9 a.m. Next slide. We're excited to have an entrepreneur and small business Facebook group. The Richelin Library, entrepreneur and small business Facebook group was created last year when the pandemic first started. We wanted to create a space where businesses could bounce ideas off of each other, ask questions and find the tools and resources they needed to keep their businesses running during these uncertain times. We're excited to see that the group currently has over 2,000 members. Next slide. Richelin Library's small business databases. Richelin Library has several small business databases that are great resources for research and learning new skills. LinkedIn Learning is a popular database that provides over 16,000 web-based video courses, step-by-step tutorials covering creative techniques, technical skills, business strategies, and more. And what's unique about the database is a certificate of completion is awarded after successfully completing each course. Next slide. For Entrepreneurs, is a dedicated page on our website that provides resources to support emergent businesses at every stage of their development. Richelin Library is committed to offering a risk-free entrepreneur hub to support Midlands area dreamers, doers, and makers. Our library residencies. The library's button residency program connects the community with local working artists, entrepreneurs, and writers. It provides educational opportunities to the community that supports cultural and artistic exchange. In addition to building an ecosystem that sustains the development and growth of future entrepreneurs in our region. Does anybody have any questions? I don't know if y'all are taking questions at this time or not, but thank you so much for the time. Thank you so very, very much. We appreciate all of the resources that the library brings to our community. And you just seem to get better and better every year in terms of meeting the needs of the community. We appreciate what you guys do in the small business space. And certainly congratulations to Rishanda Pratt for being the artist in residence or the business in residence for this month. So thank you guys very, very much for being here today. We're going to hear from Ms. Angela Brewer with the U.S. Small Business Administration. She's a business opportunity specialist and you're our last speaker for today. Angela Brewer, thank you so very much for all that you've done in the small business space. And thank you for being with us today. Thank you so much, Ms. Hardy.  It's one of the last or it is the last program that SBA has right now. We've gone through the PPP that has expired. We've gone through the restaurant revitalization that has expired. So the only thing left now is the EIDL that expires on December 31st of this year. Next slide. There has been some changes. Well, let me talk a little bit about the EIDL. The EIDL is a 3.75% interest on a 30-year loan, a two-year deferment. That means if you are applicable and get the loan this year is deferred for two years, you don't have to make a payment for two years. If you have had a loan that got the loan last year, it's still on a two-year deferment. But those of you who have not applied or have gotten denied for some reason, there is still a chance. There has been a few changes that has gone on. We have the target EIDL and we have the supplemental EIDL. They're not on my slide presentation, but they are still available in order to get the target EIDL. You must be in a low-income, socially-economic area in order to apply. You must, I mean, in order to get that grant. It is a grant, it's not a loan. Your business must be in existence as of January of 2020. You have to have had a 35% decrease from 2019. It doesn't matter which quarter or which month, but you have to have a decrease. And for the supplemental, which is $5,000, you have to have had a 50% decrease. And we got, I'm the one that manage our general mailbox. And those are the questions coming in to us that I answered and I sent back to them that you have to have a decrease. If you did not, I got denied for the EIDL or the target EIDL, the supplement. That could be one of the reasons you did not have a decrease in your business income over that timeframe. Next slide. These are some policies, some changes that just took place September the 8th. I'm not gonna read over them. Aisha's gonna send out a PDF file to everyone who registered on the new changes for that happened over the year, I mean, September the 8th, but this is a list of the changes that just happened. Next slide. We always get the questions of how long it's gonna take. You know, what's the timeframe? If you are, if you're trying to get between one to $25,000, you don't have to put down a collateral. If you over 25 from $25,000 to $500,000, you have to put something up for collateral. And if they just increased it from $500,000 to $2 million, so depending on your revenue that you generated, your small business could be eligible for a $2 million loan at a 3.75% for business and a 2.75% non-profit. And as you see, you have to have at least a minimum of a 570 credit score for up to $500,000. You have to have a 625 credit score in order to be approved for those loans. Next slide. Here are the changes. I'm not gonna read over all this cause I know we're pressed for time, we've gone over time, but here are some of the changes that just happened, like I said on September the 8th, there has been a few changes that was made. Next slide. These are the NAICS code that the changes that was affected by the new changes. So just take a look, Aisha's gonna send this out to you. So when you get a chance, you can narrow it down to see if your business industry fought in the NAICS code change. Next slide. Next slide, I'm trying to get to one particular slide. Here we are. A lot of applicants call or they wanna know I applied for my loan, I applied for the loan, I have not heard back. If you're applying for a loan and it's been more than three weeks and you're eligible for less than $500,000, it should only take three weeks. If it's more than $5,000, it takes six weeks. But if you're applying for the $2 million, it's gonna take three weeks plus the six week, which is nine weeks. And as you know, we're now in the middle of September. So if you think that your business is eligible for the $2 million, you need to get on it right now because the deadline is fastly approaching. And if you don't get it done and don't get in the system and get looked at by a loan officer, it will not be approved because, and then money is slowly running out also, so we're fastly running out. So make sure if you're gonna apply, this the only thing the SBA has right now for the disaster, the pandemic. So make sure you get your application in now. If you need the application and go to COVID, the number 19relief.sba.gov, and that's how you apply. Make sure you track your application number. Here at the district office level, we don't have access to the system to approve any loans, but we can look in the system and find out what's going on, figure out. If you can't figure out, you can't get documents, we can upload documents, we can look in there and see what happened, what stayed your application and just call our office or send us an email and someone can look into it for you. I've been getting a lot of emails about unverifiable information. One thing that you all who don't know, the 4506T is the tax document that the IRS, that the SBA sent to the IRS, and they want to see what you submitted on your taxes. So they say there's unverifiable information because we get that a lot. That means that what you put on the application and what the IRS have from your 4506T, there's a request for your tax transcript does not match. So if you're getting that error message or getting denied for that reason, that is the reason. So make sure you don't make a mistake or you get a correct copy of your taxes and submit that because that's one of the main reason why a lot of people are getting denied for the EID alone. Last slide. This is my contact, you can go to the last slide. This is my contact information. My name is Angela Brewer. I am in business opportunity business. I'm working mostly with a small business trying to expand into the federal contract arena. I give classes on how to get certified. I go over the SAM woman on ADA and all the different programs, but I can't answer questions about the financial side or the EIDL. If you call our office or you email me or call me and have questions and I will get back with you. Thank you so much for having me. We just want to thank Angela Brewer for all that she has meant in our community over the years and thank you so much for the information that you're providing on the SBA and opportunities that are there. The wealth of information at the Office of Business Opportunities this year today is more than impressive. Thank you Sheryl Sally, Diane Lucy, Andrina King, Roschanda Pratt, Angela Brewer for headlining our segment today on meeting small business resource, meet our small business resource partners. You guys are invaluable partners and it's been a wonderful experience for me today. I know we've gone over a little bit. I'm going to turn it back over now to the officials at the Office of Business Opportunities. You'll be now hearing from Aisha Driggers and then of course, Melissa Linda Aisha. Thank you, Cynthia. This has been wonderful. You've done a great job of keeping us moving and it's a great conversation that we've had with all of our presenters and our panelists that we appreciate everyone for taking the time to join us today. We know for you to be here means you're not doing something else so we really appreciate your common effort. There are so many thank yous that we have, our OBO staff, Ms. Hardy. Thank you so much. It's always, it's such an honor to work with you so we appreciate that. Our keynote presenter, Chris Hively. He gave us a wealth of information. I like Sheryl have a page full of notes from that conversation so we appreciate it. Our small businesses, thank you so much for joining us. Thank you also for providing some of our door prizes. We really appreciate that and we'll be getting to that in just a moment. Also our procurement representatives from the city, Richland County and procurement technical assistant center. Thank you so much for joining us today. As always, we appreciate our business partner, Richland Library, Benedict College Women's Business Center and SBA. We work on a daily basis together to figure out how we can meet the needs of our small businesses. So we appreciate you taking the time today. Thank you, Rick Fitz with Fitz Production for our video. We really appreciate that. We are gonna get that out to everyone that attended today so you can watch that as well as a YouTube link for today's event. Also have the slides provided to you as well. In a survey, we wanted you to tell us how, how you think things went today and what can we can do to improve as we continue to provide these webinars to our communities. We wanna meet the needs of the community so it's important that we get the feedback from you. Also the City of Columbia Public Relations staff always helps us to make sure that we can live stream these events on our Facebook page and on the city YouTube. So we appreciate that as well. So as Melissa's gonna do some closing remarks and I am doing a number generator for our door prizes, it's an actual app that I'm using to generate numbers based on attendance. So you have to be in attendance to win. So I'm gonna just check all of these and see this here while Melissa does the closing remarks. Thank you, Aisha. Really just ditto. Cynthia, prior Hardy on Point Media. Jim Hardy, you all have done a fantastic job with this. Thank you for all of your help and your partnership to get information out into our community and to make sure that our resources are shared and people know about the free resources that we have right here in Columbia. Again, thank all of our partners. Thank our departments that have joined us today. For curing it, we do work very closely with Ms. Wright, Sandra Wright, Middleton, Shannon, Lizzynski. Also, I think I did pretty good with that, right? I did okay. I was the color Shannon, but we have a great team here at the city of Columbia and we're always willing and open to work with you. If you need any resources or direction, we've got some information or we've got some questions about business licensing. I will say this, if you're doing business within the city of Columbia, you do this in business license. You definitely need a Richland County license, but please make sure you go through Richland County to get that information. Great resource. Again, the county's always there willing to help. I do want to acknowledge Erica, I think she's still on. Erica Wei, yes, she is still on, she's needed, but I wanted to acknowledge her. She's with the Richland County Office of Small Business Opportunity. And again, they're a great resource partner as well. Pam Green is also part of her team. She's on the call as well. And I don't want to leave out our Columbia Chamber. And she actually put some information in the chat box regarding Accelerator Program, Small Business Minority Business Accelerator Program, and with a link to that information and also some upcoming network opportunities for you. She's on the call as well. I don't want to leave anybody out, but again, thank you to everybody for your partnership. Thank you to our small business community for your partnership. Because we know that as a city, we can't do what we do if it was not for you, your employer, your provider, your supplier. We really do appreciate you. And thank you for everything. And I'll turn it back over to Aisha for our great prizes. Now, I want to say our prizes from our small business community, they didn't have to, because we were willing to pay for it. But I don't mind spending money, but Tanya does mind spending money in our office. I love putting money into the community, right, Tanya? But they actually donated these prizes. And the size of the prizes are, I was just like, what? But she'll share that with you. And again, thank you all for everything, Andrina, Diane, oh gosh, thank you. Just our resource partners. I'm so appreciative of everyone and our small businesses that we highlighted. Thank you so much for your innovation. Thank you so much for your willingness to share strategies and tips that you're utilizing. And thank you so much for filling those gaps and finding opportunities out of crisis. Thank you so much, Aisha. All right. So our first prize is a one-on-one consultation with Chris Hively, the MapQuest co-founder, which I think will be a great wealth of information where you can talk to him one-on-one. And the winner of that one is Brie Maxwell. And I just look like Brie is still going here. Congratulations, Brie. We will reach out to you to get you that information to partner with Chris. Next we have our Sunrise Artisan Bath & Body gift box. Alyssa, I think you said that might be worth over $100 worth of products in there. And that has been really helpful. Yeah, I was going to say, I was trying to sneak some products out of there, just full disclosure. Ms. Brown, because he might not let me, so I just bought my own. Oh. Yeah, it's full. It's full. It's a full box, full box. So the winner of that is Carolyn Sawyer. So congratulations on that. I'm sure you'll enjoy those. I don't. Next we have a $25 gift certificate for Uptown 1 Main Street. They have lots of different products. And the winner of that is Lizzie Clemon. Make sure she's still on. Yes, Lizzie Clemon is still on here. So we'll make sure to get you that. And then next, Uptown also donated a $50 gift certificate. And that will go to Shamika Huff. Shamika's still on here. Good job. Congratulations, Shamika. Next we have Bang Back Pinball. And so that is a pinball kind of arcade over in the five points area. And that will go to Rodana Scott, so $25 gift card to Bang Back. Congratulations, Rodana. You're still here. Yes. And then they also donate another $25 gift card. And that will go to Linda Harvey. Linda's still here. Congratulations, Linda. Ron might want to share that with you. So congratulations on that. Next we have Kola Kicks. They provided a $100 store credit there. But I don't know. They talked about those $30,000 sneakers, but I'm sure they'll have something that'll cover this amount. And that will go to James Travis. Yeah, I'm here. Oh, good job, James. We'll be in contact with you. And congratulations on that. All right. Well, thank you so much. I'm excited. The deal. Okay, next, Moda Enterprises has provided Spanish lessons. So she said that's worth over $500. And the first, we have two of those. The first one will go to Roslyn Goodwin. Roslyn Goodwin? Yes, you are. Good job. So we'll get that over to you, Roslyn. And then we have another one. Angela. What? Carnita Wilson. You still here? I think I see her name on here. Angela. We'll be in touch with you for that Moda Enterprises. Yes, good job. Thank you for responding in the chat. And lastly, we have a $100 gift card from Hotel Trundle. And that is going to go to Ms. Tiffany Floyd. Congratulations. Hotel Trundle is a great boutique hotel in downtown Columbia.  Congratulations, Tiffany, on that. Everyone, you'll receive an email from us so that we can coordinate getting those items delivered to you. Thank you again for joining us. We know we ran over a little bit, but we got some free prizes out of it. So that's great. We appreciate your time and your effort. If there's anything off of the business opportunity can do for you, we are here. We are a resource. If we don't know the answer, we will connect you with someone who has the right answer. Melissa just made a comment against Cynthia, you've done an amazing job. Thank you so much for making sure we flowed well today. It was a great conversation and we appreciate it. Do you have any final words, Cynthia? Just that this has been tremendous. I've learned a lot myself. It's always really great to join our colleagues in small business for these empowering kinds of events. And thank you for giving Unpoint Media an opportunity to participate. All right, everyone you'll receive. If you registered, you will receive an email with the survey, the presentation and all this great information that we've shared. So I hope everyone has a great rest of your day. Please stay safe and take care of each other. I do want Cynthia to stick her tagline for us. Okay. But I think it's very, very critical here. And I think everything we talked about, the information that is shared, it means a lot for Unpoint Media. Thank you. We just appreciate it. We know how important information is for empowering ourselves, our families, our communities because without the information, we're just not as sure-footed in our efforts. So we thank you for the opportunity to inform and empower. Thank you, Cynthia. All right, have a great day and thank you again. Thank you guys. Thank you. Bye-bye.