So, welcome to the lecture 16 of multi phase flows. We were looking at this particular equation last class where this equation arises in the context of a pulsatile flow in a circular channel okay. And what we did was we concentrated on the limit of rw much lower than 1 was the limit when the frequency is very low okay. And so now what I want to talk about a little bit before I go on to the next problem is talk about the other limit of rw being much greater than 1 because you know you have 2 timescales and we looked at the one limit wherein we said when rw is very much lower than 1 then the inertial terms are not significant. What you have is a balance between the viscous forces and the pressure term and we found that things were in phase, the velocity was going to be in phase with the pressure. So, now the question is what happens when the frequency is very large, when rw is much greater than 1 the other extreme and is it possible for us to take a look at this system. So, clearly when rw is very large the one thing which we need to include we need to include the pressure term because the pressure is the one which is going to be driving the flow and remember the pressure term is coming from here okay. The other term this comes from the inertial term the derivative the time derivative term and this is your viscous term. So now in this limit we need to retain the pressure term the inertial term is also going to be important because the inertial term is negligible in the limit of rw being very much lower than 1. So, when rw is much greater than 1 this has to be important. So, essentially what we want is that we want to be able to simplify this equation so that these 2 terms become dominant and this becomes negligible okay. But the way this equation is written it looks like when rw is much greater than 1 you do not have a balance between the inertial forces and the pressure forces okay. So, how can you possibly get a balance between these 2 terms? One way in which you can get a balance is if we write h as let us say h bar divided by rw that is as rw increases h becomes smaller then h rw becomes h bar okay and this would have a magnitude of which is comparable with this whereas that is h as I increase rw h has to decrease and if it decreases inversely as I have written then this will have a magnitude which is going to be comparable with this term and that I get the balance between the inertial term and the pressure term and what is going to happen when I substitute this here I get h bar divided by rw okay. So, then I have a 1 by rw here which makes this term very negligible okay. So, when I substitute this here what I get is multiplied by 1 by rw I need to put this bar here minus ih bar equals minus 1. So, now I have a situation what I have done is I am basically hypothesizing that in the limit of rw being much greater than 1 h is going to be very small h per se and the way I am going to capture that is by seeking h as h bar by rw okay. Now whether it is inversely with rw or to the power 2 or something is going to be decided by the form of the equation. So, here this gives me this equation and what this means is in the limit of rw being much greater than 1 this guy is now 0 I can knock off this term the viscous term and now I have a balance between the inertial term and the pressure term okay. So, here in this limit the viscous term is negligible we have a balance between the inertial pressure forces okay.  So, that is my h bar and once I know h bar at least to some approximation of rw turning to infinity this is h bar I can go back and calculate h I can go back in substitute and get back my velocity by taking the imaginary part okay. The thing which I want to mention here is that we have neglected the viscous terms remember this is therefore going to be valid as long as you are away from the walls of the channel. Near the wall of the channel this business of neglecting the viscous term is not a good idea physically because you have the no slip boundary condition the liquid is going to be at rest at the wall and immediately next to it you have the liquid moving. So, you are going to have velocity gradients viscosity is going to be an effect. So, point I am trying to make here is that this is valid away from the walls. Near the walls the viscous effect is dominant and need a different approach. So, basically this particular problem of this pulsatile flow in a circular channel is discussed in Gary Lee's book on advanced transport phenomena. I pretty much try to stick to the notation which he has used okay. So, that when you guys go and read the book you will be easy for you to follow okay. So, I am suggesting that you go and look at the reference to this problem is Gary Lee advanced transport phenomena okay and I believe you need to look at chapters 3 towards the end and chapter 4 towards the beginning. So, chapter 3 basically talks about the complete solution which we spoke about and chapter 4 talks about the asymptotic solution okay which we have discussed. So, that is a place and here he will also talk about how to take into account this viscous effect and where this is. So, if time permits towards the end we may come and revisit this problem of how to do mass asymptotic expansions. Right now I just want to talk about this perturbation theory and then move on okay. So, this is one problem which we have solved. The next problem that I am going to talk about is also a solved problem is a workload problem in Gary Lee okay. So, again that is from chapter 4 and this problem has to do with the taking into account the effect of viscous heating in a shear flow which is let us say one dimensional. So, our objective is just to illustrate the idea. So, we are going to keep the math as simple as possible okay. So, we just illustrating our analysis focus to a one dimensional problem and the applications of this are in basically in viscometry okay. So, for example how do you measure viscosity of a fluid? You put it in a viscometer then you have a wall at the bottom which is stationary and usually you have a cylindrical surface of the top which is actually rotating and then you know the shear rate you measure the shear stress and you find the viscosity of the fluid right. So, there are many applications where you have one wall which is stationary another wall which is moving and there is a liquid below. So, here the motion is going to be induced by the motion of the one of the walls. So, one of the walls moves and that induces the motion in the liquid. So, rather than worry right now about the circular geometry what I am going to do is worry about the simplified things at least geometrically which is just talk about two flat parallel plates which are infinitely long okay. So, keep life simple which is take the x direction going to infinity keep the y direction going to infinity and have the z direction also going to infinity okay. So, y and z directions go to infinity oops y and z do not go to infinity x and z go to infinity and y is bounded between 0 and h and y is bounded between 0 and h. In applications like viscometry where we are focusing on determining viscosity of fluids okay one wall is moving relative to the other wall. So, the idea is the same. So, I am just using the same idea to try and explain a concept to you tomorrow if you really want to go and do a cylindrical geometry you should be able to go and do it okay. We move one wall relative to the other and find the viscosity that is the idea okay. The objective is to find viscosity and then we do this by doing the simple experiment of moving one wall relative to the other. Now, so what I am going to do is I am going to keep this lower wall stationary u is 0 here and I am moving it in the x direction and I am going to move this with the velocity capital U okay and just to make sure that there is a difference between the small u and the capital U I am going to put a subscript 0 there. Now, you know this particular classical problem what is this called? This is the Kuwait flow right and you know that if you have a liquid which is between 2 plates and if the upper plate is moving you have a flow profile velocity which is actually linear varying from 0 to u0 okay and what is this profile going to be given by? This profile is going to be given by u equals u0 multiplied by y divided by capital H okay and y equal to 0 it is 0 and y equals capital S it becomes u0 and it satisfies your momentum equation that is no externally imposed pressure gradient. What we want to do now is supposing your fluid happens to be very viscous you are trying to measure the viscosity of oil let us say and very viscous viscosity being something like friction is going to end up generating a lot of heat okay. So, when you are doing this movement there is going to be heat which is generated and this heat is going to be generated in the form of a viscous dissipation term. Normally what we have done in the past is we would end up neglecting this viscous dissipation term as a result of this viscous dissipation term heat is generated and although the wall of the lower the temperature of the lower wall and the temperature of the upper wall would be constant let us say at room temperature because of the heat generation there is going to be a temperature profile which is going to be induced inside okay. Now if your properties your viscosity and your thermal conductivity were actually dependent on temperature okay then what you need to do is you need to include this effect if you say I am not worried about the dependence of the viscosity on temperature because the actual temperature prevailing inside your viscometer is going to be different from what you think it is because and therefore what you would be measuring is possibly an inaccurate value of viscosity okay. So, what is the point I am trying to make here is there can be liquids when you can have a significant amount of heat generation if there is a significant amount of heat generation temperature is going to rise if the temperature rises you would actually be measuring some kind of apparent viscosity or an average viscosity over the temperature range okay because viscosity is a function of temperature thermal conductivity is a function of temperature okay. So, what we want to do is see how we can get an estimate of the velocity field and the temperature field by in this problem where viscous dissipation is the cause of the heat generation okay. So, that is the idea so and clearly you can already see what I am going to do I am going to in the limit of the viscous dissipation not existing I know the solution I know this linear profile temperature is going to be uniform and I know the velocity is linear. So, what I am going to do is include viscous dissipation and treat the dimensionless parameter which is going to be associated with it as my small parameter and do a perturbation series to find how the actual velocity profile and temperature profile is. So, that is the strategy that is the plan okay. So, now let us go back and write some equations and proceed. So, the momentum equation look I am going to keep things one dimensional remember the actual momentum equation will have d by dy of mu du by dy equal to 0 okay. I am keeping the viscosity inside here because normally what you are used to is saying viscosity is a constant and you take it outside the derivative but in general remember what I am interested in is viscosity can be a function of y implicitly because viscosity is a function of temperature and temperature can be a function of y and therefore viscosity comes inside. So, for my energy balance my differential equation is going to be d by dy of k dt by dy okay plus mu times du by dy the whole square equals 0. So, this is my double conductivity and this is my viscous dissipation term. Remember the viscous dissipation term is a source of heat therefore the square there is a positive it is always positive always going to be generating heat okay. Whenever you are going to have flow there is going to be only thing is depending upon the value of the viscosity this can be small or high. Now, I need to have some boundary conditions for both velocity and this and you already know the boundary conditions at y equal to 0 u is 0 and temperature equals t0 and at y equals capital H u equals u0 and t equals t0. So, since we want to talk in terms of doing a perturbation analysis what I want to do is rather in talking terms of the actual magnitudes of k and mu we need to make things dimensionless and then talk in terms of the relative role of time scales etc. So, when you make things dimensionless you get dimensionless numbers which tells you the magnitude of inertial over viscous or this time scale over the other time scale. So, it is a relative magnitude is what we are interested in. So, next thing to do is to make this dimensionless right. So, I am going to make this dimensionless by choosing my new characteristic as u0 because that is going to decide my velocity y characteristic is clearly H okay and rather than talk in terms of t0 I am going to define temperature scale will be t0 but I want to do it in terms of getting my boundary conditions homogeneous. So, what I am going to do is I am going to define theta as t-t0 divided by t0 clearly t0 is my reference temperature whatever is the temperature of the walls okay but rather than talk about the absolute temperature I want to talk about the thing in a relative way. So, I am just going to define this as t-t0 by t0 this will basically give me an idea of how much is the temperature rise inside my viscometer this indicates the temperature rise inside okay. So, let us make this equation dimensionless. What do I get? Let tilde denote dimensionless variables and clearly you would get okay then I am also going to define mu tilde or mu average as being mu divided by mu0 and k as being equal to divided by k0. What is mu0? Mu0 is nothing but the viscosity evaluated at t0 k0 is the thermal conductivity evaluated at t0 okay. So, mu0 is mu of t0 k0 is k of t0 okay.  So, this basically this is a constant and this cannot be 0 therefore this has to be 0 okay. This implies d by dy tilde of mu bar du tilde by dy equals 0. We need to make the energy balance dimensionless okay and let us do that I am going to get k0 by h squared times t I am going to replace in terms of theta I am going to get t0 here d by dy tilde of k bar. So, that is basically what I have and this clearly must equal 0. I am going to make the coefficient of this one move these terms to the next to the second term and basically what I will get is a dimensionless number. This dimensionless number is called the Brinkman number okay. So, I will just write that final equation maybe here I am going to write that equation here and I am going to drop the tilde as but for convenience okay. But remember my equations are dimensionless just to keep life simple. So, what do I do I am going to write this as d by dy of k d theta by dy plus mu du by dy the whole square this is going to be multiplied by this Brinkman number equals 0 and let me define what this Brinkman number is it is going to be mu 0 times u0 squared the h squared does not show up and I have a k0 and I have a t0 okay. So, all I have done is move though that coefficient that to the second term and this is my dimensionless group okay. So, I have a one dimensional problem because I just want to keep my life simple and I have this equation which relates the temperature to the viscous dissipation of the velocity gradient. I also have the momentum equation which is this one which is d by dy of mu du by dy equal to 0. I want to tell you a couple of things here that these 2 equations are actually coupled to each other okay. There is how is the coupling? The coupling is occurring because clearly the theta depends upon u and this equation is non-linear mu depends upon theta the temperature. So, you need to actually solve both these equations simultaneously. It is not that I can solve this equation for velocity and I can substitute it here and solve for temperature. The 2 equations have to be solved together if you actually. So, one way to do this is to do a numerical solution do a finite different scheme and you try to get a solution. But the other thing is what we are going to do which is exploit the fact that for small values of Brinkman number we can possibly get an analytical solution. What motivates me to do that is when Brinkman number is 0 when there is no viscous dissipation I know the solution. I know that the temperature is uniform. There is no temperature gradient. The velocity is linear. So, when the Brinkman number is going to be small I expect only a small deviation from these 2 solutions some small correction and what is that correction that is what perturbation theory helps me get to okay. So, that is the idea. Remember these 2 equations are coupled and non-linear. We can do a numerical solution or we can do an approximate analytical solution okay. So, the solution is obtained numerically or approximately by using some perturbation method or we can get a perturbation series solution okay. Why? Because for Brinkman number equal to 0 we know u will be equal to y. When I have scaled everything u will be y is going to be going from 0 to 1 okay and theta will be 0 because temperature will be equal to T0 everywhere. In terms of the dimensionless numbers theta will be 0 because t is equal to T0. So, for Brinkman number equal to 0 this is the solution. So, for small amounts of Brinkman number it is going to be different from this and so we need to find out what is going to happen right okay. A couple of things but before I proceed I need to make an assumption of how does the thermal conductivity and how does the viscosity actually change with temperature. So, clearly the easiest thing to do is to assume a linear variation depending upon how crazy you are you want to make a higher order approximation maybe for second order, third order but we will keep things simple not be too crazy today. Then we will just keep k as k0 times 1 plus just trying to think beta theta. So, basically the idea is or maybe I do not want k0 here because it is already normalized right. So, it is just 1 plus because this is remember k by k0 which is my k tilde. So, this is my k tilde although not killed in the dimensionless equation now but this is the linear form okay. When theta is 0 when T is equal to T0 k tilde is 1 but k tilde is k divided by k0 remember that. Similarly, I am going to do something here which is slightly different rather than say the viscosity is a linear function I am going to say the reciprocal of the viscosity is a linear function because it helps me with the algebra okay. So, I am just going to assume that the reciprocal of the viscosity is 1 plus alpha times theta. No, I mean you can always say mu tilde as 1 by 1 plus alpha theta which is 1 minus alpha theta in a binomial expansion. So, I mean I think it does not really matter because alpha the only thing which is going to decide is whether it is increasing or decreasing function that is going to be decided by the sine of alpha okay. So, the sine of alpha will tell you if it is increasing or decreasing you can choose mu tilde as 1 plus alpha theta as well but my point is the alpha star let us say that alpha star will be negative of this alpha. So, that is the only thing. So, this is only to make my life simple when I substituting this back inside the differential equation okay.  So, so we are all set.  So, I mean you can go further as a power series plus higher order terms but we are going to stop only at Brinkman number to the power 1 today okay and as far as theta is concerned that is also a function of y and Brinkman number and that is going to be equal to theta 0 plus Brinkman number theta 1 plus Brinkman number squared theta 2 okay. Restrict v focus up to order of Brinkman number to the power 1 that is I am interested in getting only the corrections u1 and theta 1. You already know what is u0 and theta 0 because u0 and theta 0 correspond to Brinkman number equal to 0 okay. So, I mean and that should come out when you substitute this inside your differential equation okay. So, u0 is clearly equal to y and theta 0 is clearly equal to 0 okay. I am going to since this is the solution for Brinkman number equal to 0 okay. So, now we are all set we will look at the momentum equation. The momentum equation tells me that d by dy of mu du by dy equals 0 and I am going to say this is therefore d mu by dy which is nothing but d mu by d theta sorry. Let me write it this way let me do this step one step at a time d mu by dy times du by dy plus mu times d square u by dy square equals 0 that is just expanding this out okay. All I have done is just expanded this out and now I am going to write substitute for u in terms of u0 and Brinkman number times u1 and I am going to write d mu by dy as d mu by d theta times d theta by dy because mu is a function of theta okay and this I am going to leave it as it is and I am going to use the fact that there is a linear dependency of mu and things like that. So, let us do that.  What is d mu by d theta? Now, we of course remember this is all in dimensional terms. So, somebody has to tell me what is d mu by d theta. So, this is the only one place where I have to do this is 1 plus alpha theta.  Is this right? So, d mu by d theta is that theta squared. Maybe I will do something beyond this. Since I am only interested in powers of Brinkman number to the power 1, what I want to do is I am going to write this using a binomial form minus alpha times 1 plus alpha theta to the power minus 2 okay and I am going to substitute this as minus alpha times 1 plus alpha times theta remember is theta 0 plus Brinkman number times theta 1. Theta 0 is 0. So, I am going to write this as 1 plus alpha times Brinkman number to the power theta 1 to the power minus 2 okay. So, what I am going to do is I am going to, if I did a binomial series expansion I am going to, this is going to be minus alpha times 1 minus 2 alpha Brinkman number theta 1 plus higher order terms which I am not going to worry about because that is going to involve Brinkman number squared and I am not interested in Brinkman number squared.  What is d theta by dy? d theta by dy is, when I substitute in terms of theta 0, it is going to be Brinkman number times d theta 1 by dy okay. That is Brinkman number times d theta 1 by dy and what is d u by dy? d u by dy will have 2 terms which is the u0 term and the Brinkman number term. d u0 by dy is 1, we know that and plus Brinkman number times du1 by dy. That is, this is this term here plus the viscosity term which is now 1 by 1 plus alpha theta times d square u0 by dy square is 0 because we already know u0 is y and what I am left with is Brinkman number times d square u1 by dy square equals 0 okay. What I am interested in is taking the terms which have Brinkman number to the power 1 right, that is the thing I am interested in. So, what are the terms which are Brinkman number to the power 1? I have this term here and here this, I have Brinkman number here. So, when I multiply this with this and this, I get Brinkman number squared. This with this is going to give me Brinkman number squared. So, the only thing I need to worry about is the contribution of this term and this term because there is already a Brinkman number outside okay. So, this is at order Brinkman number to the power 1. What do I have? This gives me minus alpha d theta 1 by dy plus 1 plus 1 plus alpha theta and I should actually substitute this here again in terms of which is important. I should do okay. Give me a minute. Let me write this later.  By now I am in a series. Remember theta is theta 0 plus Brinkman number theta 1. So, this becomes 1 minus alpha times theta 0 is 0. So, I am just going to directly write Brinkman number theta 1 okay. So, theta I am just going to substitute in terms of this. Clearly, the only term which is going to contribute now when I substitute this here, the only term which is going to contribute is going to be the 1 multiplied by this term because this already has a Brinkman number. So, this multiplied by this is Brinkman number squared. So, I am not interested in that okay.  Do you guys think you agree with me on this? Yeah, the alpha is going to cancel off in both the terms. Yeah, it is 1 multiplied by I think you are right. You are right. There is no alpha here. Yeah okay. So, I believe this is what I am getting d y square or plus this equals 0. I need to go back to my energy equation. Yeah, both are plus right. Yeah, that should be plus.  So, now this is minus okay. So, that is what we get okay. So, now I need to go back to my energy equation and work on that. Yeah. In both the terms, yeah, you could have possibly done it. After taking the Br out. I should do, I mean if we just remove the Br, then we are left with order 0 and order 1 also. Okay, the reason the order, see the order 0 has disappeared because I am directly substituting the solution which I have. See, I am not trying to find out u 0 and theta 0. If I had left the u 0 and theta 0 as it is, then the Brinkman number, to the power 0 would have contributed, I would have gotten u 0 and theta 0 okay. That would have corresponded to Brinkman number to the power 0, that is order 1. Order Brinkman number will be the first order term which is what I have got. So, what I have done is basically use the find that I already know the solution for the isothermal system okay. So, what I am getting is the first order. This is not the 0th order. You are suspecting this is 0th order. Yeah, if Br is taken up. No, this is not 0th order. They have already substituted the solution okay. Yeah, because it is just going to minimize my algebra. So, that is the reason I wanted to get rid of the theta 0. Otherwise, I keep it and then at the end I have to throw it out. So, since we already knew the solution, I just wanted to use that directly. Let us look at the, you know, I thought we would be done today but I guess we would not be done today. So, let us look at the energy balance and this is possibly a good time to stop because it gives me an opportunity to ask you people to do the energy balance and I will do it in the class next time and then we can verify if indeed it is right okay. So, the energy balance is d by dy of k d theta by dy plus mu du by dy the whole square equals 0 okay. And remember this is actually mu bar mu by mu 0 and this is k by k 0 and what I want to do now is substitute this as I had alpha there. So, 1 plus beta theta multiplied by d theta by dy plus 1 plus alpha theta multiplied by du by dy whole square equals 0 okay. So, all I am doing is substituting the assumed dependency on temperature and now I am going to substitute things in terms of my Brinkman number the expansion that I had and you should be able to equate terms of order Brinkman number to the power 1 that gives you your first order term okay. Brinkman number to the power 0 is implicitly been already built in because I assumed u0 is y and theta 0 is 0. So, when you do that you should be able to so find terms of order Brinkman number to the power 1.  So, you guys were asking me you know saying why is this guy choosing the reciprocal I am choosing the reciprocal because now mu is the reciprocal of that when I bring it to this side it becomes the numerator it simplifies my algebra a little bit okay of course I would differentiate over there a little bit. So, anyway you cannot have everything your way. So, anyway I want you to find the solution to this equation Brinkman number to the power 1 in fact you would find that you will get something for theta 1 you solve for that and then you can substitute this and then get the velocity. So, you should be able to actually decouple. So, what you will find out is that the 2 equations are getting decoupled okay and then you will be able to find the solution.


Meantime, the reconstituted COVID-19 Management Centre will hold its inaugural meeting on Thursday, November 11, 2021. The meeting, which will take place at the Financial Administrative Centre at Point Serafin, will be addressed by Minister for Health, Wellness and Elderly Affairs, Hon. Moses Jabatist. Members of the COVID-19 Management Centre will receive a report by the Chief Medical Officer, Dr. Sharon Belmar George, on lessons learned from St. Lucia's experience during the fourth wave of the pandemic. It is expected that representatives of the private sector on the COVID-19 Management Centre will share their perspectives on a policy and strategic framework for handling the pandemic in a manner that will protect lives and livelihoods and promote a healthy economy. The meeting will be telecast live on the National Television Network from 10 a.m. The Ireland's Minister for Sustainable Development has been assessing the existing systems and capacity for waste management. Minister for Education, Sustainable Development, Innovation, Science, Technology and Vocational Training, Hon. Sean Edward, accompanied by management of the St. Lucia Solid Waste Management Authority, recently conducted a tour of the Declos Sanitary landfill. Minister Hon. Edward indicated that the government, or St. Lucia, to achieve its goal of a landfill-free St. Lucia by 2030 must overcome the many challenges that currently exist. One such challenge is that the St. Lucia Solid Waste Management Authority is currently without a general manager. The Sustainable Development Minister explained that the government of St. Lucia will be looking to appoint a general manager to lead such a critical agency. Hon. The ultimate objective is for us to have a landfill-free St. Lucia by the year 2030. Moving forward, there is also a very robust education plan and campaign that will be rolled out by the Solid Waste Management Authority. Generally speaking, I think St. Lucia's have adapted well to Solid Waste disposal. And as I said, it is just a matter of ensuring that whatever we do, it is then in conformity with the best practices that exist in other parts of the world. As we speak, there's a situation where we are hauling waste from the south of the island all the way to the Declos landfill. I do not believe that this is the best practice and this is something that we have to revisit as a new administration. It has cost the government of St. Lucia somewhere in the region of two million dollars just to hauled waste from view for two Declos and in an environment where the government doesn't have too much fiscal space in which to manoeuvre and to roll out programs with the rapidity that we would have liked as an incoming administration. This is something that we have to review and we believe that the two million dollars that has been expended so far in trucking garbage from view for two Declos is money that could be used in other areas of national development to provide relief for the citizens of this country. The government of St. Lucia, in addressing the issue of hauling waste from view for two castries, will be reviewing its agreement with Desert Star Holding, DSH, to allow for changes where possible and necessary in the best interest of citizens. The Sustainable Development Minister also explained that efforts will be channeled towards improving the work conditions at the landfill. When I came here a couple days ago, well, probably I should say that from time to time I make an impromptu stop at the landfill just to get an appreciation for myself in terms of what the workers go through when the cameras are not around and there's no oversight being provided by management and you get a much truer picture of what obtains and I must tell you that when I came here two Sundays ago I was not particularly thrilled with some of the conditions that I noticed the workers had to endure. I have brought up to the attention of management. They too were aware and it wasn't a deliberate action on the part to have the workers enjoying the conditions that are obviously not favorable but moving forward we have all recognized that there's need for improvement and I can assure you that the next time you come here you will have the workers at the landfill and just workers at solid waste generally speaking much more favorably in terms of the conditions under which they have to work. Acting General Manager of the St. Lucia Solid Waste Management Authority, Lorianus Les Flores explained that the landfill has been faced with the issue of waste not being covered resulting in increased pests. He noted however that this would be addressed over the next two months. The Deiglo Sanitary Landfill was commissioned in 2003 with a 20-year lifespan. Operation of the landfill comes at a monthly cost of $45,000 for equipment rental and wages as the authority does not own any equipment. Based on the current situation the authority is looking at an additional 9 to 12 years lifespan for the landfill, providing that a number of provisions are made including the reduction of waste coming into the landfill. If we reduce the volume of waste coming into the landfill, organic waste make up over 50% of the waste as deposited there. Now the authority have a plan in terms of doing composting at the landfill and I will show you the site that we have prepared for that. In the coming months we will be encouraging households and businesses to undertake composting. So our objective is to have very little compostable waste coming into the landfill. With regards to the other waste streams, example cans, plastic bottles, metal, we have a plan to do some segregation of that waste at the household level but that's in the long term. So those measures would increase the life of the landfill considerably so that hopefully we could achieve, like the minister said, zero landfill by the year 2040. The St. Lucia Solid Waste Management Authority also explained that some 20 incinerators were procured, however it has been facing a number of challenges with their operationalisation. Four of the incinerators were commissioned in February 2021, however as of now only one remains operational. As such a significant effort will be made towards rectifying existing issues with the incinerators. From the Government Information Service, I am Janelle Norville.


It was only a few times. Did I mention, you're the best provider. If you like true revenge stories, you found the best place for your vengeful needs. The first story will test your boundaries. Would you, count it as cheating? Second story. As a team grew out of her awkward phase, into a beautiful woman. She didn't grew out, of her ex cheating on her. This revenge act is served cold, literally. The last story, will grind your gears. It's featured as an experiment as I wanna know, what your thoughts are. OP chose to ignore all the red flags, to his own detriment. He stayed loyal to the wrong person and learn it the hardest way. Tell the like button, his presence is more disappointing, than an unsalted pretzel. Warning, these cheating revenge stories might be upsetting to some audiences, especially cheaters. This happened years ago, I believe I was around 9-10. My mom was dating this personification, of a pure disappointment. We'll call him Jay. He was never the best to me and my little brother. He chose to treat us bad, and we had to endure it physically and emotionally. There were more things that happened behind the scenes, when my poor mother wasn't around. I can't name it specifically here. If you think of something horrible, you most probably guessed it right. Jay had a problem with watching inappropriate, stuff on the internet, which led to numerous viruses on our computer. We shared one computer within the household. My brother and I, tried our best to get rid of them, considering how young we were. But one day, we just came back home from school.  My mom heard our reaction and marched over. She took control of the computer and looked through the search history. She found several sites, that would make most grown people blush. She got rid of the search history and cookies, and let us continue playing on the computer. I guess she was waiting for Jay to come home from work or the liquor store, to chew a piece off of him. We heard my mother screaming at him, saying she would pack up his stuff and send it off on a greyhound to his hometown, if she sees it one more time. Me, being a brave 9-year-old thought. Why not give myself the best birthday present? And quickly plotted a devious plan. I would sneak onto the computer before school, and simply go onto the forbidden area of the web. Then close the tab but left the search in history, and went on whistling on my way to school. I was eagerly waiting to get home. I couldn't concentrate at all, all I could think about, was that my mom would eventually use the computer, and the browser would. Pop up, full screen. It went a bit different from my imagination. As it was a half day at school, my mom would get off of work earlier to pick us up first. When we got home, it was like clockwork. Favorite music on, and hit the computer to play some Roblox. I checked the search history, and called my mom over. She saw the search, and her face, went red. She looked at us, and told us to help her pack Jay's stuff. A few hours later it was all packed up, and dropped off at the greyhound station. It was sent to one of his relatives' place. We returned to the house and gave him his ticket, before we dropped him off at the greyhound too. My mom was so happy. She never knew what happened until two years ago, when she was asking about a certain thing that happened during her and Jay's relationship. When I confessed, she looked at me and smiled. She thanked me, and we both ended that topic with, he's the hole, where the sun don't shine. Clarification from OP. Thank you for reaching out royal.  She sort of knew some of the stuff that was going on when she wasn't home, but it was compared to a small sliver of everything that was actually going on, until she left him. My little brother and I were finally able to tell her without fear of this man. I'm 18 now, so the entire thing took place 9 years ago. Also, I checked your channel out, you've managed to find some super good stories, I must admit. I hope you have a wonderful day. This story is told, from a female's perspective. My story takes place across a decade. When I was a teen girl, I got my first boyfriend, and his name was Roy. We were both in high school. Roy was popular and cute, so I was very happy to have him as my first boyfriend. We went to the high school dance together. I had my first kiss with him there. We were in the same school clubs, which meant we got to go on the same last school excursion, together. The location of the excursion was about an hour away, so it was a long bus ride. I sat with Roy in the back of the bus, where we were kissing like teens do. But he was pressuring me to do more than I was comfortable doing. He kept pushing, but I didn't want him to. I told him no, and he would stop for a few minutes, and then try again. I heard one of my best friends start to cry.  I was worried for her and wanted out of this uncomfortable situation, so I jumped at the chance to move seats, and go check on her. I comforted my friend for 15 to 20 minutes. I made sure she was feeling better, and then I was going to move back to sit with Roy. When I looked back at Roy's seat, he wasn't alone. Helen was sitting with him. But she wasn't only sitting. I saw his euphoric expression on his face and I saw Helen's head bobbing up and down, so I knew what was happening. Roy, the reason why the middle finger was created, didn't even care that I saw. Obviously, it broke the teen love fairy tale and I broke up with him. I didn't speak to him again for several months. We lived in a very small town, approximately 500 people, and like I said, we were in the same school and clubs, so I would see him around. He apologized to me several times. I did not forget what happened, but we got to where we could be in the same group, with our mutual friends, and I could stand to be around him again. Nonetheless, we moved on with our lives and it was fine. I ended up moving away from that small town a year later when I was older, flash forward over 10 years. I graduated from university and still in my late 20s. I've always been a skinny girl with no curves, no figure, and had the awkward preteen look. I had clearly grown out of that. It sounds weird to say this about myself, but I will say that I blossomed into a beautiful young woman with a nice figure I was proud of. I maintained beautiful long wavy hair and learned what style rocked me best. So when I went to visit best friend Tara, in the same small town where we had grown up, I made sure I looked my best. When I was visiting Tara, her cousin, Jeremy, and his roommate also came over to visit. What a surprise, the roommate was Roy. Roy had not seen me since I was a teen and still awkward looking. Now I was a size two and had double Ds with long wavy hair. I was wearing jeans and a low-cut shirt, so I know I looked my best.  His eyes lit up like Christmas came early, and he said it was good to see me. Like I said, I had moved on from what happened all those years ago. But, I had not forgotten. Roy was flirting with me, and it was clear that he was interested. I told him I still thought he was cute, and I shamelessly flirted back. I would get close, and then I would move away. I would also whisper flirtatious words into his ear. Then I stood up and asked Tara if we could have privacy in a bedroom. She uncomfortably said yes, looking at me confused as it to ask, why? Tara had been with me all those years ago and knew what happened between me and Roy. Why would I give him a second chance? I gave her a knowing look shared between best friends, and she knew. But Roy's face looked like he could do the happy dance. I whispered to him to go into the bedroom, uncover fully, and wait for me. I was going to run to the restroom, and then meet him there. He sprinted to Tara's bedroom as I went to the restroom, touched up my makeup, and waited enough time, so I knew he would be dressed down waiting for me. I opened the bedroom door, and Roy was lying there as expected. His unfolded clothes piled on the nightstand. I said some flirtatious things, as I slowly walked closer. I told him to close his eyes. He didn't want to. So I told him I would only approach if he closed his eyes, so he complied, at least for a few seconds. But a few seconds was all I needed. While walking towards the nightstand, I grabbed his clothes, and I bolted out of the room. I ran through the house to get outside and when I did, I just kept on running. It was snowing, it was freezing. But I was warm with adrenaline. So I threw some of his clothes somewhere in the snow, but when I was about to throw his pants, I could feel his wallet and car keys in them. So I took these separately, and threw them somewhere else in the snow. His wallet in the opposite direction, because I'm petty like that. But just when I threw his wallet, I could hear commotion coming from the living room, with Roy screaming in his birthday suit, from the door. What are you doing? I told him, this is for high school. This is payback, for your cheating with some troll, that was more important than our relationship. Maybe you'll think twice about being a cheating frik-boy in the future. Then lastly, to give more power to my words, I swung his pants and threw them on the Salty Street. My best friend Tara and her cousin were standing there, watching us while laughing. Both Tara and Jeremy had been on the excursion that day over a decade ago, so they both knew how much it hurt me when, he did that. Neither Tara nor Jeremy helped Roy find his wallet or keys. While Roy was looking for his things, Tara and I locked her house and left Roy in the snow, while we went out to eat dinner. It was the best revenge I've ever gotten, and it all came spontaneously. For the ones who care about Roy, Jeremy stayed with Roy when we left him in the snow, so we knew he wouldn't be in any danger. Jeremy had his own car keys and turned the heater on in the car for them. I knew he would be physically fine. I'm petty, but not that cruel. Small side note, I absolutely did not think about doing any of this during the decade before being there. I just thought of it on the spot when I saw him walk through the door with Tara's cousin, and I could tell he was interested in me. I didn't think about him or expect to see him ever again. He wasn't living rent-free in my head for over a decade. I want to know what happened to Helen. Do you have any information on that? She's still a woman of questionable morals and virtues. I definitely didn't keep in touch with her. I heard through the grapevine a few years ago that she has four kids from four different guys. She still lives in that same small town and doesn't work. The following story is added as an experiment. Let me know what you think in the end. Your opinion matters. My current life is nothing short of a nightmare. Here is my story, my first real Reddit post.  We were high school sweethearts, married for 12 years and together for 15. We have three kids together, wonderful healthy kids. She grew up in an extreme Christian household, where sexuality wasn't a topic of discussion, but more of a taboo. You save yourself for marriage or your future husband and figure out the rest later. So she did, her first time was on our wedding night. We had a healthy marriage, or so I thought at the time. Always made time to talk and listen to each other. I work away from home, so I'm gone every other week. We both have great jobs and are doing well financially. After we decided we were done having kids and my wife's body changed so much for giving us our children. We talked about breast implants and decided to go ahead with surgery. Very nice indeed, certainly helped my wife gain her confidence back about herself. I've always been into fitness and going to the gym, so we bought some workout equipment at home and she began to use them excitedly. She made great progress. So great that she wanted to share that with others. At this point she talked to me about wanting to make an Instagram fitness page to show her journey. I shared my concerns with that and the negative attention that will bring. But as a husband normally does, I ultimately decided to support my wife's new hobby. A few months later, I get home on International Women's Day. I came home to my wife crying and telling me she needs to show me something serious. Someone had sent her a picture of herself. It was a bikini photo she had uploaded to Instagram, but someone had used an app to remove bikini. The sender told her it was found on a dirty site. They just wanted to let her know. This killed me inside, seeing the pain it caused my wife. I suggested she removed her Instagram since the initial picture was taken from there. But from that moment, the incident wasn't bad enough. Over the next few months, her posts changed from helping people and documenting her fitness journey to showing her body off with less clothing. When August 2021 comes around, I start having my own health concerns. I had left testicle pain that lasted for two months. After many doctor trips and having ultrasound, I get cleared of cancer, which was a relief. In the meantime, my wife starts becoming emotionally distant and I'm trying to figure out why. One month later, I'm still struggling with my libido and drive big time. I get tested for low-testosterone and it comes back well below low normal average. Doctor tells me they need a second test before starting on hormone replacement. The second test comes back shortly after at the low normal cut-off limits, and they tell me they can't proceed with treatment and that I'm doing fine. But they recommend antidepressants, which I decline. That week, I get another blood test from a hormone replacement clinic and book in for consultation in early January. Early December 2021 rolls around and my wife loses her uncle and her grandmother, seven days apart. A major loss for the family. I come home from work early and take extra time off work to support my wife because she isn't doing well at all. It killed me inside, having to go back to work after two weeks, knowing my wife is still grieving such loss and needed my support. January 2022, I get approved for my hormone replacement therapy and take my first shot of testosterone on the 13th, that month. Two weeks later, I'm already feeling like I haven't in a very long time.  My wife tells me that she can't believe the changes I've made in literally three weeks and that I'm a 10 out of a 10 dad and husband. A few weeks later, I'm at work again and I'm reading articles on emotionally distant spouses and midlife crisis. I sent her the links and she agrees that does sound like her, but she's not sure why. I then come across an article that read, the top 25 signs, your wife is having an affair. I'm reading this and I'm blown away that my wife scores 22 out of the 25. I decide I'm going to confront her, but not until I'm home. D-Day arrives in March. We go out on a date night and I genuinely had a great time. I sit her down and tell her I'm going to talk for 15 minutes and want her to listen and she'll have plenty of time to reply afterwards. I read her the top 25 signs and afterwards she says. Wow, I'm shocked. I can't believe you think some of those of me. I reply by telling her this is the time to clear your soul and whatever has been bothering you needs to come out. I was serious in my tone as I thought about a specific shady incident and that there was a drunken one night stand from the summer. The tears start to pour and the long silence develops before she tells me she has been having an online affair through Instagram. Within the previous mentioned months, she showed herself to another man who lives overseas and he had a party of one with those pictures. I'm in disbelief that she could do this, but I think it's something we can reconcile over pretty quickly. I then tell her that if there's anything else that needs to come out, she now has the opportunity. She only tells me. This is going to kill us. And cries even harder now, followed by minutes of silence. I'm preparing for the worst. She breaks the silence and tells me she is having an affair. She met him on Instagram and it all started a week after her grandmother's funeral and only a few days after it took extra time off work to be home. She made sure to add. It was only a few times and it ended in January. As I come to find out that only half truths come out and only as information as to limit the damage and fallout effect of the affair. A few days of seriously talking all day and night with no sleep, all the details start to come out. The truth was that the affair was alive and very much active and sparkling on the same night I confronted her. Thousands of text messages and averaging four hours of phone calls a day. They would meet in random parking lots in the back of a truck, hotels, a fair partner's home and the last time in my own home. You thought it was bad? No, it's getting way worse. The guy spent the night while our three children were sleeping upstairs while I'm away at work. She let him stay the night, sleep in my bed while my children were sleeping in the same house. She confessed to 11 confirmed times in 2.5 months but it could be as high as 15 to 20 times as she can't remember all the dates. She would tell me she was headed to the gym but in reality meeting him in a parking lot. She's on birth control, so they didn't use protection. I asked her to pack her stuff and move out, the kids had the worst week of their lives. Here we are today, both going to individual counseling and have a first MC scheduled for next week. She basically told me she had written me off as we were going down the path of divorce which justified the cheating but I think it can never be justified. Nonetheless, it changed when I made a miraculous turnaround thanks to my hormone replacement. In her eyes, she got the man she actually wanted and by still cheating on me, it drove the guilt and shame even harder. She says there's been no contact with her affair partner since the 18th of March. She actually started her personal counseling while she was in the middle of the affair. She's regretful, remorseful and wants to do anything to make this work. You might think I'm weak and she uses me as a doormat but when it's really happening to you and you have children, it's much harder than perceived if you're on the outside. I'll just say it, I let her back in the house. Her living with me and the children in trying to repair what is broken. She's been tested for STIS slash HIV and pregnancy. She's reluctant to have sexy time or even kiss me and through open discussion has admitted she is having a very hard time getting over the affair feelings and the withdrawal from it. The guy she was with is 33 years old.  What about my needs? Obviously not wanting to be a doormat here. I'm not wanting this to work just for the children either. Questions, comments, concerns are all welcome.  Even worse, while your children slept upstairs. There is no coming back from that. Imagine the level of disrespect. That's a big F you. No affair fog excuse can justify that. That's designed humiliation, probably laughing at you in your own house. No man, she's got to go. I think it is a common theme in cheating cases that show that it's not a thing of being weak-willed or having a lapse in judgment but all about having a selfish personality. When the cheating partner is having a hard time, the loyal partner does his her best to be there and help them recover. When the loyal partner is having any kind of problem, no matter how mild, they jump ship. You were there for her as you should when those loses happened. When you are going through your testicular issue and low testosterone counts, she starts thinking, well, this is going to end in divorce. I should just start looking for something new instead of trying to fix the relationship. She would rather throw 15 years and three kids together to the trash rather than put in some work. However, your health issues manifested in your relationship causing problems. They didn't came out of nowhere. You both knew why it was happening and you were getting treatment for it and yet she didn't have your back. She was just looking after herself. Luckily, it wasn't cancer, but what if it had been? Or if you were to lose someone in your family and get depressed for a while, will she take time and make efforts to be with you and help you be better? Or will she ditch you and seek to have her needs met by any random follower of her insta-page? She didn't just fail her promise to be faithful but also being there in sickness and in health too. Want to bet what would happen if you tested the other ones? If you lost your job and couldn't get another one, would she uphold the for richer or for poorer one? This woman won't have your back when things go bad. Pretty much my thoughts there as well. Much has transpired since my last post. Looking back, I wish I took the advice of people on here who reached out and genuinely cared. The hardest part is accepting the reality that all is lost. Unfortunately, I found out the hard way by my own doing. Reconcile was attempted after I let her back in the house as I mentioned in previous post. Sue many red flags as if it was a communist parade.  That's not all. The list of red flags went on and on. After a period of trying, she pushes me away on purpose, saying this is too hard and that she needs some time to figure her life out. I agreed. Not seeing each other would give us both some breathing room and it could be healthy. She ends up renting an Airbnb but was reluctant to give me the address of the place but ultimately does. One morning I got curious and drove by the place. I see a truck parked a block away. I know this truck, I've seen it before as it's the truck her affair partner uses. Seriously? Shouldn't be a surprise at this point. I'm driving by and I see her and the guy walking out of the alley towards his truck. I turn around, drive towards them and pull up door to door. He's inside the truck. She didn't have time to get inside so she is ducking down behind the tailgate, trying to hide from my sight. I rolled my window down and said, real impressive, you lasted one day on your own. I pulled to the back of the truck to get a better view and she runs to the front and hides by the hood. So I left and it all was clear to me. I made sure to take the necessary steps like I should have done way earlier. I canceled her cell phone that was on my plan and then just waited for her head to pop up. Two days later she did. She shows up at the house with her bags packed. She says in a defeating tone. I want to come home. I'm sick of living this shady lifestyle. I said that's a good decision but you're not welcome to stay here.  She seemed understanding and left. Mother's Day rolls around. She wants to do something with a family. I disagree. She calls a few hours later and invites me to a movie for the kids. This time I agree. We had a great night as a family and when we get back home, I sent the kids to bed. She stays for a bit but when I'm done for the night, I ask her to leave. She refuses saying. No, I rather sleep. I'm just tired. I'll sleep on the couch. I go upstairs and see her old cell phone on the table. Don't judge me, but I decide to go through it. I found evidence in pictures and messages of the affair. Before this point, I hadn't actually seen it so blunt in my face. Whatever she told me was either changed in a way to make it seem less bad or when I saw her with the guy, she would refuse to take ownership of what it really was. I'm tired of being the guy. She keeps walking over. I have the proof in my hand and walk towards her. I wake her up and confront her. She goes ballistic straight away. Long story short, police show up and she's arrested and charged with assault at 1 a.m. Three weeks of no contact and she's not allowed to visit the home but after some time the charges were dropped. Shortly after we meet up again, as she wants to meet, she wants to work on things, I'm hesitant, but she's a mess and I take that as finally being remorseful. I rent my own place for five days and leave her at home with the kids. When I'm back, things seem to go a bit better as communicating has improved drastically but there's no intimacy at all. She won't even kiss me and her phone still has a password I don't know about. We are in bed one morning and at 6 a.m. her phone rings. She looks at the phone and then looks at me telling me it's the guy she had an affair with before I can say anything she adds. Yes, I had blocked him, but I unblocked him again.  She didn't know, I already have my lawyer paperwork prepared for separation at this point. I know it's over and I'm prepared. I have a fire in the backyard one night. She's at a neighbor's for a glass of wine. She comes back a few hours later and we talk. We dance a bit outside for the first time in years. I try to kiss her, she pulls away. I'm surprised and make a fool of myself later when trying to initiate again shortly after. It felt so bad I had to sleep downstairs after. The next day, she forgets her phone again and I slide the screen down but can't get into it. Nonetheless, it shows the email notification from the guy again. Another dagger to the back, I confront her again. She lies again saying she didn't message him. He must have gotten her email somehow. I go downstairs, pull up her email and read the entire conversation. It says they've been going on for days behind my back. I show her, she breaks down crying. I said I'm filing for divorce the next day and leave. She calls me 15 times over the next hour. I don't answer and she texts me to come home that she's having a breakdown. I return home hours later. She calls her a fare partner while I'm there on the phone and tells him it's over. He asks for my blessing to pursue a relationship with her if we don't work out. I laugh, she hangs up. After some time, she made some changes, was very positive, reassuring me she is in this for the long haul and that it could take years to get to the place we wanna be at. I quit my job on the road and took a new position to be home every night with my family. When I start my new job, she pushes me away again. We go for a walk, she says it isn't working and doesn't think we can get there but never directly says she wants to separate or divorce. We had a friend's wedding to attend that weekend but I got my new work schedule and couldn't make it. She reaches out to her a fare partner again and spent the next three days with him after the wedding. I finally grew a spine and filed my paperwork. In August, she was served. I listed our home and it was sold in one month including possession date. Separation agreement was signed the same week. I'm in a much better place over the last few months being free from this madness which ultimately I created for myself. Trying to do everything possible to help this woman and save my family but she isn't the same one I married 12 years ago obviously and unfortunately all at my expense. Everyone said she would do it again. I didn't wanna believe it but deep down I knew it would happen. Kids are adjusting well to their new home with me, shared custody and I focus my time and attention to them at all times. She's not with her a fare partner anymore and he is apparently heartbroken. Recently I saw her and she told me you're a great man, a great provider, husband, father all the qualities I'm looking for. I know I won't find another man with even 50% of what you have to offer but I'm not attracted to you but I haven't ruled out getting back together someday. Going forward, I can live with knowing I tried everything possible to work on our family unit. She did not and that's something she will carry to her grave. Stop talking to her about anything but the children. Seriously, just get away from her. She's toxic to your future happiness. Already done. I love how she just assumed she can always come back to you. She's completely delusional. She won't understand until you have fully moved on and start seeing someone else. Then it will hit her like a ton of bricks how badly she messed up. Be prepared for when it happens. A whole new set of drama will take place. I know this is coming. The story isn't over. You need to make a copy of all the bad things she has done to you. Let it be the venom you need as motivation.   To then be treated like a doormat and told you aren't good enough. Write it all down. Then cry a bunch.  You have never really seen what else is out there. You need to start dating ASAP so there is another woman pushing your cheating ex-wife out while filling up your heart. Don't fall in love with the first girl or two you date but just stop wearing your ring and let your friends know you are single now.  It seems because you think she is the only one you can be intimate with and you crave that like a drug. Get a new supply by realizing other women want you because as you said, you work out, earn well and put your loved ones first. Other women will see this and chase you down. This is also to battle your cheating ex-wife telling you she isn't attracted to you. That is a huge blow to your self-confidence. I have a new girlfriend now. She treats me well and I'm finally happy again. Thank you for your support. I really appreciate you and show your vengeful devotion by tickling the like button without mercy. How did you experience the last story? Was it infuriating like you were expecting? Let me know your thoughts. I'm really curious and will reply to you. That's a promise. I'll be seeing you in the next one. Remember that these stories are shared for your entertainment. This content is to be taken as such and nothing else. Royal AI rejects application or instigation of illegal actions.