Good morning to all of you, I welcome all of you to the last day of workshop on competition fluid dynamics. In the yesterday's lecture I was in the middle of a topic which is solution of new stoke equation on a staggered grid. I had started with an introduction and completed finite volume method on a staggered grid where I had shown you that let me start the lecture on whiteboard and because I have as I said that in my last lecture that this is a this is probably the most difficult and challenging part of teaching CFD. And I had discussed the issue of pressure velocity decoupling and the role of staggering of grid on resolving this problem of pressure velocity decoupling in the lecture slides. But let me do the same thing slowly in whiteboard so that each one of you are able to understand and appreciate this issue of pressure velocity decoupling and how staggered grid is a remedy to that. Before that I would like to make some general statement saying that right now you may be thinking that we are discussing staggered grid so when staggered grid is used and when collocated grid is used. Let me tell you that this staggered grid was the first remedy to the problem of pressure velocity decoupling. However, when people as I said that the way if you go to the development of CFD initially people were not able to solve the new stoke equation due to this pressure velocity decoupling problem. Then the first remedy which came up was a staggered grid. Now when this staggered grid was good as far as standard coordinate system or problems are concerned like if some CFD problem is in standard cylindrical, spherical or Cartesian coordinate system then this formulation of staggered grid works well. However, when you move into more challenging industrial problem in those situations the geometry becomes much more complicated and you have to go for complex geometry formulations using finite volume method. Now when you go for complex geometry formulation it is very difficult to or it becomes almost impossible to use this staggered grid approach. So with that problem people came up with a second remedy to the pressure velocity decoupling which is called as a collocated grid method. Due to limitation of time I am not taking that topic here. However, we teach the collocated grid method also in our CFD course here, but I would just like to mention there the grid point same grid point is in collocated grid same grid point is used for pressure, velocity, temperature. So that is for all the variables which is sitting at the centeroids of the control volume. However, to resolve the pressure velocity decoupling problem instead of linear interpolation to calculate the mass flux at the face center of the control volume there is another type of interpolation which is done which is called as momentum interpolation. So let me start with explaining what is pressure velocity decoupling in a whiteboard. So when you discretize the continuity equation you get terms like if you take a control volume let us say this is the east face center, this is the west face center. So when you want to calculate the total mass transfer across the control volume let us say in the x direction it becomes rho u e delta surface area in the east face is delta y minus rho u w into delta y this is the total mass transfer in the x direction. Similarly, you can do in y direction I am showing you here only one of the direction. So what you get is rho delta y u e minus u w now if you do linear interpolation where this is the node p this is the node capital E and this is the node capital W then you get u capital E plus u capital P divided by 2 minus u capital P plus u capital W divided by 2. Now when you do the subtraction this u p cancels down and you end up with an expression in mass conservation equation which is this. Now what you realize is that to calculate to do the mass balance your equation is such similarly when you do in y direction you get rho delta x v capital N minus v capital S divided by 2. So what you realize is that to let us say this is the node N and this is the node S. So to do mass balance the nodes which are involved are this the u velocity of the east and west neighbor and v velocity of the north and south neighbor. Similarly when you can when you apply the momentum conservation so this is for mass conservation similarly if you apply a momentum conservation then at the same point so this is the u velocity grade point.  Now when you do the linear interpolation it becomes p capital W plus p capital P divided by 2 minus p capital E plus p capital P divided by 2 into delta y and this p p again cancels down. So the pressure force which is acting for when you sorry this will be capital W and this will be capital E. So the pressure force which acts in the x direction comes from not node p but neighboring nodes capital E and capital W. Similarly when you calculate the pressure force in the y direction when you apply the law of conservation of y momentum the pressure of the north and south comes into the expression. So the point is to calculate the velocity at a particular grid point the pressure of neighboring grid points are used. So this is what is called as pressure velocity decoupling because to calculate pressure at a particular sorry velocity at this node the pressure of the same node is not being used. So there is a breakage in communication between the pressure and velocity of the same grid points which results in what we call as pressure velocity decoupling. And why this is happening this is happening because of this interpolations. I will write down this is the first interpolation here I am showing this is the second one this is the third one and this is the fourth one. So first and second are the interpolations for the normal velocity at the phase center. Third and fourth are interpolations for the pressure at the phase center. So the here the issue is this decoupling is occurring due to this interpolation. So if you want to resolve this problem of pressure velocity decoupling we have to somehow do some other interpolation which is done in a collocated grid method called as momentum interpolation for normal velocity. Otherwise the best thing is that we avoid interpolations and that is what is being done in a staggered grid. How we avoid interpolations I will show you in the next page is the way to avoid interpolation is I say that this is the grid point for pressure this is the grid point for u velocity this is the grid point for v velocity. Now this if this is p subscript p then there are two vertical phase centers. So what I am doing is here is note that if I want to avoid this interpolation which is of normal velocity at the control volume of pressure I am prescribing proposing the u velocity on the vertical phase center and v velocity on the horizontal phase center. Now the convention which we are following the running indices of this corresponds to the running indices of the cell center at which it is lying on the east phase. So this will become up similarly this become vp this become uw and this become vs. So when you apply mass conservation now you can see it becomes rho into delta bar up minus uw if you want to calculate net out flux of mass plus rho into delta x vp minus vw. So in this you see that we are not doing any interpolation for the normal velocity. So we have eliminated that now when you go to the note that this is the control volume for mass conservation. Now control volume for x momentum conservation will be a control volume where now this should be sitting at the centroid of the control volume. So now if you want this square to be sitting at the centroid of the control volume it is staggered in the x direction by delta x by 2. Now this becomes up from here you can see this is pp then this becomes pe. Now v velocity sitting at the top this is vp this will be vs this will be vse this will be ve. E means i plus 1 s means j minus 1 se means s is j minus 1 e will be i plus 1. So now if you see this control volume for u velocity in which you apply law of conservation of x momentum now you see that what will be the net pressure force in the x direction p e divided by delta y this is for x momentum. Similarly if you generate control volume for y momentum then it will be let us say this one. So for y momentum control volume you get pressure now you can see for this control volume pp is here p capital N will be here and the net pressure force for y momentum will be p capital P minus p capital N into delta x. So now you see that this values at the phase centers we are avoiding any interpolations which we are doing in previous slides. Now with this it is very clear step by step how the interpolations for pressures at the phase center in case of momentum equation and in linear interpolations of normal velocity in the mass conservation is avoided. Now let me go to a case when I said that this is good in standard coordinate system but this is very difficult in a complex geometry problem. In a standard coordinate system the control volume we have let us say this is the control volume in Cartesian. Let us say this is the control volume in cylindrical. In a standard coordinate system you can easily find the locate this because there is a coordinate along which you have let us say in this case green square are along horizontal direction. In this case the green or the square here not green there is no color here this square here are along the angular directions. Now this red inverted this inverted triangles which corresponds to the v velocity here in Cartesian they are in y direction whereas in this case they are in radial directions. This is a r phi case but when you have a complex geometry what is meant by complex geometry I will discuss. In that in a complex geometry what happens is the control volumes are not aligned along the any standard coordinate directions you may have coordinates like may have control volumes like this. Let us suppose this is P this is capital E this is capital W. Now in this you can see that this is x coordinate direction this is y coordinate direction this is phi coordinate direction this is on a radial coordinate direction. Now here when you join let us say this points what you realize is that they are not in any standard coordinate direction and in this case if I want to define velocities here which velocity I should define that is the question because the norm actually we want we do when we want to define grid point it is for normal velocity.  So face to face the normal velocity direction is varying in a standard coordinate system note that there are only two directions of the normal velocity horizontal and vertical in carton and angular and radial in cylindrical. So when you want to use this staggered grid formulation for a complex geometry problem then you cannot use this staggered grid. So then people came up with a formulation which is called as a collocated grid method where instead of using linear momentum sorry linear interpolation for normal velocity in the mass conservation they used an interpolation which is called as momentum interpolation. I had written one chapter on finite volume method in a book computational fluid flow and heat transfer edited by professors Murli there from IIT Kanpur and professor Sundar Rajan from IIT Madras and in that chapter that is one chapter which is added in the revised edition and in that chapter this collocated grid method for complex geometry is discussed we teach this topic complex geometry formulation using collocated grid method to our students here however due to limitation of time I am unable to cover that topic here. So I had mentioned that if you use a fully explicit method then to get u velocity you can take x momentum equation and when you take x momentum equation and use fully explicit method where you take not only u velocities and v velocities of previous time level you take pressures of the previous time level also but whenever you take pressures of the previous time level you can solve u momentum equation explicitly such that if there are 25 yellow circle 25 interior points you will get 25 equation and in each equation there will be only one unknown. So you can solve for u velocity and this u velocity will be we are using a staggered grid formulation. So in a staggered grid formulation the u velocity when you go for mass conservation is the normal velocity for the pressure control volume. Then if you use bi-momentum equation and obtain the v velocity explicitly then you get that v velocity x is the normal velocity for pressure control volume. So let us go back to this window and I will show you an animation. So using the pressures of the previous time level velocities of the previous time level by solving x momentum equation let us suppose you have obtained the u velocity v velocity of the next time level but they are not obeying mass balance. As they are not obeying mass balance this velocities are called as predicted velocity. Note that here the methodology which is being discussed it is like predictor-corrector method and it is iterative image. Why it becomes iterative? So I had mentioned there is a issue that we do not have explicit equation for pressure. We do not have any conservation law where we have del p by del t. There is no explicit term for temporal variation of pressure and we want to capture temporal variation of pressure. Let us say to create a movie of how pressure changes with respect to time. So when you go to solve mass conservation equation continuity equation in fact we do not have any pressure term and we want to use del u by del x plus del v by del y for an equation as an equation for pressure. So mathematically it is converted into an equation for pressure but here I am showing you, here I am discussing the same formulation philosophically and later on so I am starting with some physical discussion and then I will go to the mathematical expression. So using the predicted velocities which acts less, acts as normal velocities on the face center of the pressure control volume we end up with this velocity which are predicted velocity they are not obeying the mass balance. So we need to tune the pressure. So let us suppose you have a knob, so right now more mass is going out as less mass is coming in. So to do a mass balance you want to increase the inflow. If you want to increase the inflow and if you have a control knob you would, what you would like to do? Increase the pressure, what is the effect of increasing or decreasing the pressure as far as inflow is concerned? If you are increasing the pressure of this control volume it will resist the inflow. If you decrease the pressure of this control volume it will create a suction. So more mass will come in. So in this case what do you want? You want more mass to come in.  So one atmosphere is the pressure of the previous time level you have a pressure correction which is minus 0.5. So p plus p prime 1 minus 0.5 which is 0.5. This completes your first iteration then you again predict. So with this change in the pressure you again predict the mass flux and let us suppose with then updated pressure of the cell center you got updated or let us say corrected mass fluxes, mass flow rates not mass flux sorry. This is the second iteration. Now what is happening? More mass is coming in. Now we should create suction. You should decrease the pressure or you should increase the pressure. Now you should increase the pressure of this control volume. So let us say the tuning which you have done is 1.5. So you are increasing the pressure by an amount 1.5 atmosphere. Then the total pressure changes from 0.5 plus 1.5 which is 2 atmosphere. Then what will happen? Third picture will come which will be the corrected value. So you start with the prediction then tune the pressure you get a corrected value of mass fluxes. From that mass flow rates you calculate the mass imbalance. From the mass imbalance you tune the pressure you have to see whether the mass imbalance is positive or negative. So this way it goes on and on. And this is the philosophy with which we convert the continuity equation as an equation for pressure or pressure correction. I will show you symbolically about the equations. So the first equation is in a functional form is prediction of mass fluxes as a function of velocities and pressures in the previous time level. And what is this mass source? Mass source is net mass going out. So in the first picture what is the net mass going out? 4 minus 2 kg per second which is 2 kg per second. In the second iteration or second picture what is the net mass out flux? 2 kg minus 6 kg minus 4 kg. So mass source is net mass out flow. Note that when the mass out flow is positive when the mass source is positive pressure correction is negative. What is the pressure correction in the first figure? I have shown through that yellow arrow minus 0.5. What is the mass source in the first picture? 2 kg per second. So when it is plus 2 kg per second pressure correction is minus 0.5. Now let us go to the second picture. Here mass imbalance is mass source is 2 minus 6 minus 4. So when mass source is negative pressure correction is plus 1.5 atmosphere. So note the relationship and later on I will show you mathematical expression which follows this relationship. So meet equation which I am showing you. First I had shown you the first for mass flux prediction. Here I am showing in a symbolic form. Later I will show you detailed expression. Second to calculate the mass source. So the prediction of mass flux which I had just written as in the first picture 1 kg, 2 kg all these are calculated by one equation. The mass imbalance there is an equation. Right now I have discussed philosophically that let us say you have a knob by which you can tune the pressure and by certain amount which we call as pressure correction this tuning is done by an equation. And that equation philosophically you can appreciate and understand that it should be function of mass imbalance. So later on I will write out symbolic here. Later on I will show you detailed expression. Once you tune the pressure let us say on the top phase in the first picture it is 2 kg per second. In the bottom phase in the bottom picture at the top phase it is 1 kg per second. So there is a mass flux correction minus 1. And on the left surface between the two picture the mass flux correction is plus 2 kg per second. So this mass flux correction which is plus 2 or minus 1 shown in the fourth step in this slide. I will show you that this mass flux correction I will show you equation as a function of pressure correction. Because this correction occurs when you tune the pressure. Then you as I said that you start with a prediction then do a correction then get an updated value of. What is updated value? Like on the left phase on the first picture it is 1 kg and 3 kg per second in the second picture is an updated value. And the correction is plus 2 kg per second. So you get an updated value of the mass fluxes as in the second picture. And with this you get a new mass source that is step 2 new pressure correction new mass flux correction as a function of pressure correction. So this is this way the picture changes always. So it goes in an iterative mode. This is right now shown philosophically. Let us go to the slide and discuss the detailed expressions. Right now here the equations are symbolic in nature and I have discussed with figures and animation the philosophy. Now let us discuss the equations. Here as I am not giving you the lecture notes so whatever I have discussed that is written in detail. I will not discuss this because it is the same thing which I had discussed just now. Now let us go to the expressions. Here let me go I would like to mention that the continuity equation should always be at a time level and plus 1. So whether you are following an explicit method let me tell you in neo stokes equation there is no fully explicit method. Why? Because it does not work. In a neo stokes solver either you have what is called as semi explicit method or semi implicit method method. What you may know more is that semi implicit pressure linked equations which is called as simple algorithm, simpler algorithm, simple algorithm. As a semi explicit method is more towards explicit but there is some reason one term which is made implicit so that is why it is called as semi explicit. Semi implicit method is more close towards implicit. The more will be clear as this lecture proceeds but in both the methods note that the pressure is taken implicitly because we want to make sure that the velocities which we have predicted for the next time level it should obey the mass balance. So if we say that we have predicted from x momentum the u velocities and v velocities from the y momentum for the next time step then it should obey this mass imbalance for the next time. Now in an explicit method as I said why it is semi explicit because pressure is taken implicitly. Note that advection and diffusion are taken explicit in a semi implicit manner even this advection and diffusion are taken at a new time level. So the difference between the original proposition in semi explicit and semi implicit is in a semi explicit advection as well as diffusion terms are explicit. In a semi implicit these terms are implicit that is they are calculated from new time level values. So we come up with an original proposition and follow up predictor-character approach which I had discussed philosophically in animation through an animation philosophy of pressure correction. So this is the original proposition but this has a starting trouble. What is the starting trouble? Which term is causing the starting problem? Whether this term will given any starting problem? This is the rate at which x momentum is lost by the fluid in the control volume using the velocities of the previous time level which we already know. So this is there is no problem in calculating this. There is no problem in calculating total viscous force acting in the x direction using the velocity from the previous time level.  No. So this is the term which causes the starting trouble. So how we can overcome this starting trouble? What we know pressure? Pressure what we know is pressures of the previous time level which is this but if you use this you get a velocity which you have to check whether it is obeying the mass balance or not. So the pressure of the time it is not. So this velocity is called as predicted velocity. Now here I have shown you through arrows. The difference of the velocity actual velocity of the or updated velocities of the next iteration un plus 1 minus u star actually let me tell you that this un plus although I have written as un plus 1 but actually this becomes un plus 1 only if you get a picture where mass balance is obeyed because it is not that you do one calculation for prediction and one calculation for correction and you are done. You are not done in first iteration. You have to keep doing it. So actually this un plus 1 is an upgraded value of u star unless you get a u star value in a particular iteration at which the mass balance is obeyed then that u star becomes the un plus 1 which is the velocity u velocity for the next time. However the difference of the u velocity between two consecutive iterations is what is called as velocity correction. In my previous slide as I said that the mass flux is changed from 1 kg per second to 3 kg per second on the left wall. So that 1 kg per second is let us say u star. 3 kg per second is not un plus 1 it is an updated value of u star but symbolically here we are writing as un plus 1 and the difference between the two which is 2 kg per second is u prime. I can show you this is u star divided by rho into delta y because it is a mass flow rate. This is updated value of u star this is not un plus 1. So this is basically 3 divided by rho into delta y. This is u star this is an updated value and what is the correction that difference between the two. So this is the velocity correction but if you want to convert into mass flux correction you have to multiply by density and surface area which is delta y and that mass flux correction was 2 kg per second on that left wall. Here again pn plus 1 is an updated value of pressure. I will go back and show you this you started with this pressure of the old time level and you got an updated value of pressure by this pressure correction. Now this updated value of pressure becomes the correct pressure for the next time step that is pn plus 1 if and only if in the next picture it is obeying the mass value. Otherwise this is an updated value and the difference between this and this is pressure correction which is shown here. So this note that I am showing you which equation right now it is velocity correction is a function of pressure correction but you just need to multiply by density and area to convert into mass flux correction as a function of pressure correction. Now let me go back and show where is symbolically represent mass flux correction as a function of pressure correction where is that symbolically represented in which step, third step mass flux correction as a function of pressure correction. Here I had shown symbolically here I am showing you mathematically it is just that you have to multiply by density and surface area but this is the mass flux correction in the x direction for vertical faces. Then this is the slide for horizontal faces same thing is done the only difference is now we are doing this for y momentum equation note that for x momentum equation the control volume which is used is different from what is used in the y momentum equation. So in this slide we calculate the mass flux correction in the vertical direction. So that way we have completed the equation for mass flux correction as a function of pressure correction. So if you tune the pressure it causes this must change in the mass flow rate at the faces by tuning the pressure the consequence of the tuning the pressure. If you create suction mass flux correction will be positive if you increase the pressure mass flux correction will be negative. So as there is no specific question for calculation of pressure there are actually two class of methods for calculation of pressure one which is called as pressure correction method which is most commonly used simple method simpler simple r pressure correction method. There is another method where instead of using pressure correction pressure is used as a variable called as projection method. So how what is the expression between the velocity predictions and the correction these are the expressions. But note that there is a statement which I had written at the bottom which I had emphasized earlier in earlier slides. The pressure and velocity correction is continued iteratively till a divergence free velocity field is obtained then only the right hand side of the above equation becomes the velocity and the pressure for the next time step otherwise it is just an updated value. So this does not happen in one iteration with that we had we had done with the mathematical expression for mass flux correction as a function of pressure correction. We need other expression also for that so we have already completed the discretization of x momentum y momentum. Now we are left with continuity because now what is the objective here for from x momentum and y momentum we have been successful in getting an expression for mass flux correction as a function of pressure correction. Now we want to get an expression for pressure correction which is pressure tuning as a function of mass imbalance correct which is let me go back to the slide what I am saying is that from x and y momentum equation we have been successful in getting detailed expression not symbolic expression shown here for this. Now we want to get an expression for pressure correction as a function of mass imbalance. So this we have got from momentum equation this we will obtain from continuity equation this equation is very simple mass imbalance is just doing a balance. So you know this expression there is nothing difficult about it this I had already shown this is you predict u velocity using the u velocity of the neck you predict u velocity for the next iteration using the velocities and pressures for the of the previous time step when it is a first iteration. So and this mass flux correction I had already shown you the expression. So there are in fact two main expression mass flux correction as a function of pressure correction which comes from momentum equation and the second pressure correction as a function of mass imbalance which comes from continuity equation mass conservation equation which I will show you now okay. So this is the discrete form of mass conservation equation note that the normal velocities are sitting at the places of the control volume in which we apply the mass conservation and we need not do any interpolations. If u p n plus 1 in previous slide for x momentum I had shown you this expression what I am showing you here is basically this is p p prime minus p e prime into delta y.  Similarly you can do for v p n plus 1 that is what I am showing you okay and this is the predicted velocity and this whole term is called as the velocity correction from which you can calculate the mass flux correction. Now the objective is that we obtain expression for u p n plus 1 v p n plus 1 which I am showing you here.  Now how is the nature of this equation let me ask you what are the terms in this equation what are the terms in this equation predicted velocities and pressure correction. Now these predicted velocities are which velocities they are normal velocity they contribute the mass flux mass flow rates. So we will get pressure correction as a function of mass imbalance from this equation. However what is the nature of the equation do you find any similarity of this linear algebraic equation with heat conduction equation with volumetric heat generation okay. Now let us suppose let us look into this term do not you think that this term has analogy with k dT by dx on east phase center do not you think this is analogy with not plus k minus k dT by dx what is the discrete form of minus k dT by dx it is minus k p capital E minus t capital P divided by delta x as it is minus k when you make it plus k it becomes t at capital P minus t at capital E divided by delta x. So instead of temperature here we are having pressure correction and instead of conductivity we are having delta t actually this delta t is analogous to minus of k okay. In fact this equation although right now it is an algebraic form but it can be converted into a differential form also and it is a Poisson equation it is like a steady state conduction equation note that it is analogous to steady state conduction equation you can see there is no del by del t term here there is no unsteady term time derivative is not there there is no delta t in this equation there is delta t is there but the delta t is analogous to minus of k. So this is analogous to if you convert into a differential form it is basically del square p is equals to some source term which is of Poisson equation and it is analogous to steady state heat conduction with volumetric heat generation now what is that volumetric heat generation analogy or source here is it is mass imbalance because this up star note that you have to multiply by rho delta y and what is up star into rho delta y this gives you I am showing you here all this star u star and v star it can be expressed into this now what is this term this is the discrete form of the continuity equation using the predicted velocity and in fact this is the negative of mass imbalance note that it is negative it is negative of mass imbalance and I would like to remind you that I told you that we should have expression pressure correction as negative of mass imbalance let me go back and show to you pressure correction is negative of mass imbalance if the mass imbalance is positive 4 minus 2 which is plus 2 pressure correction is negative when the mass imbalance is here 6 minus 2 sorry 2 minus 6 because we are talking of outflow minus inflow 2 minus 6 which is minus 4 then the pressure correction is plus 1.5. So the pressure correction mass imbalance should be related with the negative sign exactly similar relationship is shown here pressure correction as a function of negative of mass imbalance this is mass imbalance coming from the predicted normal velocities. So with this I had shown you all the equations I started with the philosophical discussions and showed you the equations showed you the equations for velocity correction mass flux correction as a function of pressure correction as a function of mass imbalance. So this is called a semi explicit method this is the formulation for the semi explicit method where I had given you shown you expressions. So this is what is called as formulation now let us go to the formulation of semi implicit before moving on to solution methodology that this equation because we do not have more we have more than one equation what should be the methodology what should be the procedure to solve this equation one by one. So before discussing solution methodology let us discuss what is semi implicit method what I am discussing here is basically simple method the pressure is taken fully implicitly as in case of explicit method and this is the original proposition to obtain u velocities and v velocities for the next time level. Now in this original proposition what is the difference between this original proposition in semi implicit method with that of semi difference is how we treat the advection term and diffusion term. Note that the diffusion term and advection term have a superscript n plus 1 which indicates that to calculate net y momentum or x momentum lost by the fluid in the control volume and net viscous forces acting in the x direction and y direction is calculated using the velocity of the next time level and you know that in implicit you get a system of algebraic equation where there are more than one unknown. So at each time step when you want to solve this equation you have to solve iteratively now here again there is a starting trouble you may say that I do not know the pressure of the next time level I do not know the velocities of the next time level. So let us use the pressures of the previous time level and let us predict the velocity and when we are predicting that velocity let us use the predicted velocity of the neighbors although they are not known because that is what happen in a implicit method because in this case note that you will get a linear algebraic equation for up star u w star u capital E star u capital N star u capital S star and in each equation there will be 5 unknowns if it is an interior point. So there are 25 equations there will be there are 25 interior points you will have 25 equations and in each equation there will be not more than 5 unknowns. So if you want to predict this velocities you have to solve this equation iteratively similarly for v velocities this is in semi implicit method note in that in semi explicit method this advection and diffusion were calculated using the velocities of previous time level. So in this in a semi explicit method you get there are 25 grid points you get 25 equation for u velocities 25 equations for v velocities and in each equation there is only one unknown. Now if you want to get an equation for mass flux correction as a function of pressure correction that is what I had shown earlier for from the momentum equation. So you just subtract the original proposition with this predictor step you know what is predictor corrector method prediction plus correction is equal to the final solution. So this is the final this is the original proposition this is equals to the prediction plus correction if you want to get an expression for correction you have to subtract this 2 equation. So when you subtract this 2 equation note that in pure explicit sorry semi explicit method when you subtract the original proposition let me show you note that in semi explicit method when you subtract this 2 to get a corrector step this an is same in both original proposition as well as prediction. So it cancels down completely diffusion also cancels down but in a semi implicit method when you subtract original proposition and the predicted equation for predicted step whether this 2 will cancel down this is a p at n plus 1 this is a p at star which uses predicted velocity in a coupled sense. So when you subtract this 2 this is non-zero and in fact this involves velocity corrections of the neighbor and you get an algebraic equation like this actually in CFD we want to get an algebraic equation and we want to get an algebraic equation which is sparse in nature. If there are million points you will get million equations but in an implicit method in each equation there will be not more than 5 unknowns. So million minus 5 in each row will be 0 so there are many zeros so that is what is called as sparse linear algebraic equation. If you want to solve this equation it will be like million equation having million unknowns for each point there will be almost no zeros. So your coefficient matrix will be completely filled with numbers till this step let me tell you all the steps which we are doing is fully implicit. So why it is called a semi implicit because at this step we neglect let us say it is like z is equals to x plus y and then we are saying z is approximately equal to y. So we are neglecting the this term let us say x so when you neglect the velocity correction of the neighbors and why we are doing there is a purpose because in CFD we want to have sparse matrix and we solve iteratively. Note that at this step it becomes semi implicit. Before this step it is fully implicit this we did for u velocity we have to do this for v velocity control volume also shown here. Here again we consider the velocity corrections of the neighbors as negligible note that there is an equal sign here and there is an approximate sign here and at this step it becomes semi implicit. So when you substitute the so this is the equation for normal velocity correction as a function of pressure correction when you substitute into the continuity equation the way I had shown earlier you get a linear algebraic equation this is basically the discretization of Poisson equation del square p prime is equals to minus of mass imbalance. So this results into pressure correction equation pressure correction equation is an equation to obtain pressure correction is a function of mass imbalance. Here I can note the negative sign negative of mass imbalance. So this completes the formulation before I go into the solution methodology let us go into the implementation details. Now actually we have three different control volumes. Now when we want to predict u velocity so what is the idea actually if you want to know how to proceed go back to that animation which I had shown for pressure what was the first thing which came in the animation we take a control volume we use the pressures of the previous time level. Then what is the first thing which came into that animation the predicted mass. What is the second thing mass imbalance what was the third thing pressure correction is a function of mass imbalance. What is the fourth thing mass flux correction is a function of pressure correction or a weighted value of mass flux to be proceed in a solution algorithm or our procedures that way. So let me start with the first step that is prediction of u velocity. Just to connect to you we are taking an example problem which was given to you in yesterday lab session lead driven cavity flow which is just a square cavity whose three walls are stationary left right and bottom top wall is like a conveyor belt which is moving from left to right with a constant velocity u naught. So what will happen as this is moving the fluid which is close to it will try to moves along with it at the corner it cannot go out. So it has to turn this will create a clockwise vortex the flow will generate a clockwise vortex near to that corner there will be an anticlockwise vortex small anticlockwise vortex.  So I will not show you an animation here what I will do is with this slide I want to show you the how to predict the u velocity for the u in the u control volume. Same thing I will show you to an animation it will be your feel better feel of what is happening. So let us come to the animation this is the square domain which we are using for lead driven cavity flow problem this is the lead this is the velocity with which lead is moving. Now you see that this vertical lines are not equi-space near to the boundary why because here I want to show you control volume for u velocity and you know that the control volume for u velocity are staggered in horizontal direction by delta x by 2 whereas the horizontal line will be equi-space because u velocity is not staggered in the y direction and where are your u control volumes this are the running indices note that we are not using different running indices for pressure velocities and we have used a convention where we are making sure that we use a single running indices. In fact in the formulations which I had given you in the slides although there are certain subscript in denoting the neighbors or the cell center but when you program you have to convert into appropriate ij. So this is the interior grid point for u control volume how many are they 4 in the x direction 5 in the y direction this are the boundary grid points left bottom right and top note that on the left and right boundary there is a half control volume but do you need to solve in that half control volume do you apply a conservation laws in that half control volume what is the grid point coming in that half control volume boundary grid point and I had emphasized in my earlier lecture that for boundary grid points we do not apply conservation laws what we apply is boundary condition we apply conservation laws in the full control volume and this is the typical computational stencils this is the interior control volume I am also showing you the running indices for the interior control volume. Now what I will show you is that as you know when you apply law of conservation of mass in this control volume you will have to calculate the fluxes mass flux x momentum flux viscous stresses at the faces. Here I am showing you the face centers at which you will calculate the fluxes in the x direction and I am showing you the running indices in the i direction which is which is different from the interior node but in the j direction it is same. So I am not repeating that note that this yellow squares are 5 in the x direction and 5 in the y direction. Now what are this squares this are the this pink square corresponds to the points at which we calculate the fluxes in the y direction mass flux in the y direction x momentum flux in the y direction viscous stresses here we will calculate always in the x direction whether it is a vertical surface or a horizontal surface for vertical surface it will be normal stress and for horizontal surface it will be a shear stress. These are the boundary points and these are the running indices. Note that for vertical and horizontal I am showing only one of for vertical I am showing you i running indices for horizontal I am showing j. The other running indices is same as what is given for interior control and this is the control volume where we are trying to conserve the momentum. Note that I will again repeat that when you want to calculate x momentum flux in this control volume x momentum flux also involved mass flux and note that although in continuity equation mass flux grid point was sitting at the faces of the pressure control volume. But in this control in case for to calculate mass flux you have to do linear interpolation.  This is the convention which we are following. So, at this points we calculate the mass flux momentum flux and viscous stresses in the x direction and on this face again we calculate the mass flux in the y direction x momentum flux in the y direction and the viscous stress which is the shear stress in the x direction. Similar thing we do for y momentum now here you see that vertical lines are equi-spaced, but horizontal lines near to the boundary there is half control volume and these are the control volumes for v velocity interior control volume. In for you it was 4 in the x direction 5 in the y direction here it is 5 in the x direction 4 in the y direction. I would like to mention here that this pictures are very good although they seem to be that there are very few points, but if you develop a program to begin with it is just that you have to use some loop.  I will be showing you the pseudo codes also this is to help you as far as possible through equations, figures, animations, pseudo codes. So, that you do not struggle and start teaching the CFD course. This is the boundary grid point in the half control volume where we do not use the law of conservation of wire momentum. We have used boundary conditions this is at the top this is on the left boundary this is on the right boundary. Next the flux calculations. So, this are the interior control volumes this are the running indices or this yellow inverted triangle is the these are the points where we calculate the flux in the x direction. This is the boundary at the boundary on this green inverted triangle we calculate the mass flux in the y direction momentum flux in the y direction discuss stresses in the y direction and the running indices is this. This is a convention which we are following as far as running indices is concerned. So, x momentum y momentum now I am showing you mass conservation. Now, the control volume here is equi-spaced now here in this case there will be no half control volume. These are the pressure grid points these are the phases where you calculate mass flux in the x direction. These are the phases where you will calculate mass flux in the y direction. Note that this mass flux is only used in the mass conservation this is not used in the momentum conservation. There we have to do linear interpolations. If you look into the temperature the temperature control volume is same as that of a scalar control volume which is pressure. So, these are the grid point for temperature these are the running indices. You calculate heat flux and enthalpy flux in the x direction at this phase center. You calculate conduction heat flux and enthalpy flux in the y direction in this red circles. So, with this I had completed the showing you animation of the different control volumes and the different phase centers which are involved.


Irregular migration and the dangers thereof. For obvious reasons, bordering on economic justice, human rights, climate, etc. Nigerians, youths especially, are indeed ready to leave the country at any cost and this is not surprising. However, one must ask, is the grass truly greener on the other side? In examining this issue, research and statistics show that for Africans, moving to Europe has become increasingly difficult. It is indeed a very high risk situation. Thousands are now discovering this reality. Today, I wish to highlight a few dangers facing irregular migrants. They do say knowledge is power and information shared may help you or someone you know. Some of the issues include exploitation by smugglers. Smugglers are criminals who pretend that both the journey and settling in Europe are easy in order to get money. Smugglers often lie about the safety of the route and modes of transport. The United Nations has sanctioned human traffickers in Libya for many crimes. An example of issues around this is Angela, a 25-year-old Nigerian woman, who was abandoned by her smuggler along with 50 others in the Sahara Desert and left to die. And only six of them survived. Smugglers are also known to sell migrants en route. They may also demand a ransom from family members. Being violent to the point of killing migrants are seen recently by smugglers who reportedly shot migrants on the beach in Libya for refusing to get on the boat. Kidnap and human trafficking. There are many stories of migrants being kidnapped when crossing Mali, Mauritania and Niger, as well as trying to leave Libya. This often happens in the Sahara Desert and is sometimes planned in advance with drivers selling the migrants to kidnappers for large sums of money. This horrific experience goes hand in hand with the severe torture of the migrants. And we have the case of Azam, a 30-year-old Nigerian male, who recounted how masked kidnappers stomped the truck he was traveling in with 30 other irregular migrants, shot and killed the driver and three passengers as a means of instilling fear into the rest of them. This kidnappers proceeded to torture Azam and his companions daily until their families paid huge ransom for their release. There are also regularly reported cases of migrants being robbed and dispossessed of their belongings by armed bandits as they travel through the desert. We have abused and exploitation of children and adolescents. Migrant children are the most at risk of abuse. Three in four children and young adults face various forms of exploitation on their journey to Europe through the Mediterranean.  More than 1,200 migrant children died from 2014 to 2018 as recorded by the International Organization for Migration, otherwise known as IOM. And nearly half of them perished while attempting to cross the Mediterranean. The question arises. Given the somewhat despondent nature of things in Nigeria and most African countries today how do we encourage young people to refuse irregular migration options? As the question always arises, will you feed me and my family if you ask me not to leave? These are the issues and government and citizens alike must rise up and put in place social programs that empower the people. For this is the most sustainable panacea to irregular migration. I mean that's just the truth. That's the final question, your songs of everything. You're telling me not to go, will you feed me and my family? You say, crazy question, yes I know. But the truth is, as much as we say will you feed me and my family let's look at the amounts that are used in this migration. These amounts can set up small and medium enterprises in Nigeria but love for the stability dreamt of abroad with no balance or understanding of what you're going to meet there. We seem to think to, let me use the language. Japa, Nadiwe Ford. When I was a banker there, I had the first quarter in shock. People, families will sell their homes, sell their properties to send their children abroad and they don't know what the child is going to do there or what the future is going to be like. They don't care, all they know is that just go out there, it is better, the grass is greener on the other side. So they're willing because after applying for visas over and over again and being rejected, to them that's the option. And you don't have, if you don't have here, here's a family, they don't have any sustainable means of income, they are living below the poverty line, there's no other way. They don't see any other means of escape. So to them, there's nothing you can tell them. And of course they have the son of Papachi Nadiwe. Nadiwe, let's go out of the conversation. Exactly, I'm just going to say that the thing is that it's a game of chance. There's always that person who has done it and who has successfully played it. Like Sholah, right? Absolutely. So you have a trash record. Sholah is a good example of her. Sholah is good. I am somewhere that really encourages a lot of people to migrate if they've got the necessary skills and means to survive or get jobs legally in the U.S. and Canada. The reason why I say so is that I always say that if you have children you've got to be in that second experience. You can't let your children go to New Zealand, so this is normal. However, I've always said that if the government, look, I know what, for somebody to sell their house, to sell their family house and go and rent the property so that one child of two children can attempt to cross and go abroad without knowing whether it will be successful, without knowing whether they will make it and even remember it, it shows you how desperate people are. Are we going to say that? People are just that desperate to think that, you know, it's worth taking the chance. Even if one of these children makes it, we're set up for life. And that shows you how sometimes hopeless situation can be for some families in Nigeria. Especially the middle income, you know, the old middle income families who are civil servants and such. Things have become so, but they can't get their pension. There's no means of livelihood. And somebody talked about in business. I will say what I'm talking about now. And I will tell you that I have invested so much money, more money than I have with you two. But my business is you can do always do better in Nigeria because in Nigeria some government policies come and wife structures and start all over again. You know, they never end in cycle and people are kind of tired of people. And even the exchange rate that always flows. So I mean, yeah, to learn my voice to this, having been in this space for a few years now, I think that it's really a situation of a chicken and egg approach. It's an issue where we have to look at different sides. And as Shola has rightly said, when hope is totally lost, believe me, you resort to any means possible. That does not mean I am condoning the actions as declared by Eno. What I'm saying here is that as the last line of my advocacy said, how do we actually find the balance, civil society, government and even the people themselves? What are we likely to do? Fundamentally, migration is a human right. As a human being, you have the right to move from place to place. But then one thing I said in Loiza, my own right stops where your own begins. So while I may have the right to move to your country, I cannot come to your country without the established methodology and the prescriptions I said, this is how you come into my country. So what we're advocating for and what we are pushing for and one thing that I have said all the while is that if you're going to migrate or leave your country to another place, do not go through the irregular routes. Do not lose a limb livelihood and everything for a dream that you have no understanding or idea about. Find out what the real options are, find out how you should go and then also understand the place of developing your society. We are going to still need people who have got in the experience and the way we thought to come back. And there's no forget that. Remi, please say hi in Nigeria's economy. Okay, so the local government and its importance is what Cully wants to tell us about after the break.


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Amen Mark chapter 16. So of course the story of the ladies going to the tomb and seeing that Jesus was risen. Happy resurrection day and that's the day that is marked down in the Bible here in Mark chapter 16 when they discovered the empty tomb. So of course they are there on Sunday morning the first day of the week. I'm not preaching on when Jesus was crucified or the timeline of that event this morning. I believe Jesus was crucified on Wednesday. The Bible is pretty clear about that. Many people think, you know, Good Friday. Good Friday makes no sense at all, you know, for three days and three nights in the tomb. There's no way to basically figure out that Jesus, you know, one does not equal three basically. So I believe that Wednesday is the best explanation. Thursday has got some credibility as well, but it's definitely not Friday. So they go to the tomb early Sunday morning and they see that Jesus is in the tomb. In Matthew 12 verse 40 the Bible says this for his Jonas was three days and three nights in the Wales belly. So shall the son of man be three days and three nights in the heart of the earth. So he was put in the tomb early Thursday morning. You have Thursday, Thursday night, Friday, Friday night, Saturday, Saturday night. That's three, okay? That's three days and three nights. But look down at Mark chapter 16 and we come to early Sunday morning. So early Sunday morning, like the early AM hours of Sunday morning. And it says when the Sabbath was passed, which is Saturday. Remember there was two Sabbaths because of the Passover, but Saturday was the regular weekly Sabbath. And when the Sabbath was passed, Mary Magdalene and Mary the mother of James and Solomon brought sweet spices that they might come and anoint him. And very early in the morning, the first day of the week, they came unto the sepulchre at the rising of the sun. And they said amongst themselves, who shall roll away the stone from the door of the sepulchre? So this morning we celebrate the resurrection of Jesus Christ, our Savior, the Savior of the world. And what does that mean for us? And what I want to do this morning is kind of have a little bit different take on the resurrection. You say, you know, it's kind of hard as a pastor. Like you preach, you know, on Easter, it's the same message or the same. But there's always a different way of looking at things in the Bible. There's always a different way. The Bible is infinite. So it's really not that difficult to come up with different Easter messages. But this morning I want to look at a phrase that is in the Bible that's not in Mark chapter 16. Go to Jeremiah chapter 10 in verse number 20. So we're celebrating the risen Savior, the risen Jesus Christ this morning. And there's a phrase that is used to describe God in the Bible. And it is numerous times in the Bible, dozens of times in the Bible. You will see this phrase describing God. And I think that fits, this phrase fits perfectly for a resurrection Sunday, a resurrection, a day where we celebrate the risen Jesus Christ, the source of our salvation. Look down at Jeremiah chapter 10 and look at verse number 20. Let me just give you one instance of this phrase used to describe God. The Bible says this in verse number 10, or verse number, I'm sorry, Jeremiah chapter 10, verse number 10. The Bible says, but the Lord is the true God, He is the living God and an everlasting King at His wrath the earth shall tremble and the nations shall not be able to abide His indignation. So as we celebrate the resurrection this morning, I really want to focus in on this phrase, the living God. The living God. Over and over and over in the Bible we see God described as what? Not just God as the living God. Like he's the only one. There's not another one. He's not a living God. He is the living God. It's one of the ultimate demonstrations of this attribute of God through the resurrection of Jesus Christ. So I'm going to give you three points this morning on this idea that we serve a living God this morning. The first one is this. The first one is this and is demonstrated through the resurrection of Jesus Christ. But the first point is very simple, and I won't spend a lot on this first point, but the living God, the living God is the only way we can live. The only way that we can live spiritually is through the living God, and that is not an accident. It is only through the death, burial, and resurrection of Jesus Christ that we can live. It is only through that. Turn to John chapter 14 in verse number 6. Look, you said that seems pretty simple, pastor. Well, it's not that simple to people today. It's not that simple to people today. I had somebody I talked to yesterday. My wife went and we did some visits yesterday, and I had somebody tell me yesterday that I had never met before and never talked to. Well, you know God, he's got a lot of different names, and everybody, certain people call God different names, but we all worship the same God. Wrong! We do not all worship the same God. Look down to John chapter 14 in verse number 6. Jesus sayeth unto him, I am the way. Notice, I mean, look, you should read this verse with the proper emphasis. I am the way, the truth, and the life, meaning there's only one way. No man cometh unto the Father, but by me. When you see the little bumper stickers, they always leave that part of the verse off. When you see the little refrigerator magnets, they always leave that part off. I am the way, the truth, and the life, end statement. That's not the whole verse. I am the way, the truth, and the life, and no man comes to the Father but by me. So I told that man yesterday. A very nice way. I stood in his yard. I told him, no, that's not correct. There's only one way. There's only one name. I mean, the Bible literally says this. There's none other name under heaven given among men. Whereby we must be saved. There's only one way to be saved. So when somebody comes up to you and says, you know, you ask somebody how you get to heaven, and they say, we'll just have faith in God. Just have faith in God. What's God's name? Well, you know, His name is Jesus. That's who it is. That's the only name. That is the only name that you can trust in that will actually lead you to salvation. It's not just God. He has a lot of names. The living God has only one only begotten Son, and that is the name that we must trust on for salvation. So it is only through the living God. The first point is very simple. The living God, turn to Matthew chapter 16. Turn to Matthew chapter 16. The living God is the only way that man can live. Look at Matthew chapter 16 in verse number 16. Jesus asked Peter this question. Jesus asked Peter, who do you think that I am? Who do people say that I am? Who do you think that I am? And look what Peter says in verse number 16 of Matthew 16. And Simon Peter asked him and said, Thou art the Christ, the what? The Son of the living God. And Jesus answered and said in the name of blessed art thou, Simon, for flesh and blood had not revealed unto thee, but my Father which is in heaven. Jesus Christ is the Son of the living God. Without the Son of the living God and without this event that we celebrate this morning, there is no chance of life. Jesus Christ is the only way. The world doesn't want to say that today. Christian churches don't want to say that today. Billy Graham didn't even say that. But the Bible is very clear that Jesus Christ is the only way. If we care about people that's what we will tell them. We are not here to lie to people. Some pastor that will get up and tell you that God has many names and all these different things and he doesn't really care about anything is a liar and a fraud and does not love the people that he is talking to. Without Jesus Christ and the resurrection there is no chance of life. It's not complicated. But people don't want to hear these truths today. Here's the second point. Turn to Daniel chapter 6. There's too many examples in the Bible of this one, but I just chose Daniel chapter 6. I really like Daniel in the Bible. Not just because it's my middle name, but Daniel is just such a great character in the Bible. Look at Daniel chapter 6 and the second point is this. The second point is this. The living God is active. The living God is active. The Bible does not teach a God that is a God of deism. This deist idea of this God that just kind of made everything, set everything in motion and then doesn't get involved. The death burial and resurrection of Jesus Christ is the biggest example in the Bible of God being active, of God being involved, of God getting involved in the lives of men. Look at Daniel chapter 6 for another example. Daniel chapter 6, Daniel finds himself in trouble here under Persian rule. Remember Daniel, he was a prophet during Babylonian captivity when the lower kingdom of Judah was taken into captivity by the Babylonian Empire. Just a few years later, the Babylonian Empire, Daniel quickly rose to second in command of the Babylonian Empire under King Nebuchadnezzar and then they were taken over by the Persians just a few years after the captivity. So one empire replaced by another empire, but Daniel was so valuable that they're like, hey, we're overthrowing this entire empire and just wiping everybody out, but we're keeping this guy. And we're not only keeping this guy, but we're putting him second in charge too. Basically, the prince of Persia, the king of Persia puts Daniel in charge of everything. Look at verse number one. It, please, Darius is set over the kingdom in 120 princes. So he's obviously got princes that he's delegating his power to, which should be over the whole kingdom, and of these three presidents of whom Daniel was first. So there was three, there was 120 princes, there was three presidents over those princes, and then Daniel was top of those three that the princes might give accounts under them and that the king should have no damage. So this king delegates his authority well. Then Daniel was preferred above the presidents and princes because an excellent spirit was in him and the king thought to set him over the whole realm. Then the presidents and princes sought to find occasion against Daniel concerning the kingdom. All the other rulers didn't like this. They didn't like that Daniel was liked and favored amongst the king. But look, this is a perfect example that is God that promotes. It is God that promotes.    No, it is God that promotes. And Daniel is a perfect example of this. Then the presidents and princes, they're conspiring against him. It's a real life conspiracy here. Then said these men, verse 5, we should not find any occasion against this Daniel except we find it against him concerning the law of his God. They're like, he doesn't do anything illegal. What are we going to do? We're going to have to make something up here that goes against his God. All the presidents and princes assembled together the king and said unto him, thus unto him, King Darius lived forever, flatterers. All the presidents of the kingdom, the governors and the princes and the counselors and the captains have consulted together to establish a royal statute to make a firm decree that whosoever shall ask a petition of any God or man for 30 days save of the king, he shall be cast into the dead of lions. So they make this rule, make this rule that you can't worship any God or even consult any other authority. They know Daniel praise. What was it, three times a day? No matter what? They're like, king for a whole month we're going to have anyone, they're only allowed to consult you. No matter what, the king, you know, probably a prideful person falls for this flattery and he says, well, that sounds pretty good. Now, oh king, establish a decree and sign the writing that it be not changed according to the law of the Medes and the Persians. So the Medes and the Persians, they had a rule that once you sign the law, even the king himself couldn't go against it. Wherefore, King Darius signed the writing and the decree. Skip down to verse number 13. Keep in mind that the king loves Daniel. That's why he favors him. Then answered they and said before the king, that Daniel, which is of the children of the captivity of Judah, regardless not the oh king nor the decree that thou hast signed, but make it his petition three times a day. He's still praying three times a day they're saying. Then the king, when he heard these words, was sore, displeased with himself. He's like, oh, they tricked me, these people. And he set his heart on Daniel to deliver him. So he tried to figure out a way to get around this law. And he labored till the going down of the sun to deliver him. Then these men assembled unto the king and he said unto the king, now oh king, that the law of the Medes and the Persians is that no decree nor statute with the king established may be changed. So they got Daniel in this case. Very foolish men, by the way. Do they think that the king is happy with them at this point? That's another story in itself. Let's skip down to verse number 20. So the king had to throw Daniel into the lion's den. And he did it. But then right away, after the sentence is carried out, after he was in the lion's den for a night, he rushes down to the lion's den. Look what he says in verse number 20. The king goes to Daniel and he came to the den and he cried with a lamentable voice unto Daniel. And he spake unto Daniel. He said, oh Daniel, servant of what? Servant of the living God. Is thy God whom thou service continually able to deliver thee from the lion's? What was the answer? The answer was yes. The living God intervenes. The living God is not some God that sets some, our God is not some God that set things in motion and does not intervene. And as he intervened for Daniel, look at Hebrews chapter 11. As he intervened for Daniel, he intervenes today. He intervenes for people both saved and unsaved today. Look at Hebrews chapter 11. Look at verse number 33. I mean, how in the world did all these normal, uneducated, weak men throughout the Bible and women throughout the Bible, how did they achieve the things that they achieved? Why did God use the weakest? Why did he use? Why did he use a bunch of fishermen? Why did he use people that were just normal people to, you know, use the Holy Spirit through them to witness for him, to start this movement that started after the resurrection? Look at Hebrews chapter 11 verse 33. Look at this. Look at this. Stopped the mouth of lions, quenched the violence of fire. Remember Shadrach, Meshach, and Abednego, a very similar situation. Literally thrown into a furnace that was so hot that the people that threw them in were burned up. And the living God intervened. The living God saved them, escaped the edge of the sword. Out of weakness were made strong, weak people, subdued nations through what? Through the power of the living God who intervenes. Waxed valiant in fight. People that weren't valiant became strong in fight, turned to fight the armies of the aliens. How in the world did they enter the Promised Land with Joshua and defeat these nations that were stronger than them? And defeat all these armies that were bigger than them? There's no military explanation to it. There's no military explanation. I was just telling some of the guys. I'm reading a book, an eyewitness account of the last two years of the Civil War. One of the most interesting things about the book is that it's a lot of military strategy and numbers on how many people the Union lost versus the Confederacy. The Union lost more people than the Confederacy a long ways because typically an offensive operation is more costly for an army. These are just military strategies.  Then there's things like Thermopylae where there's some new military thing introduced or theory or strategy introduced. But in general, an army that constantly is on the offensive will just simply lose more people. It is really the army that wins is the bigger army, the nation with more resources that can continue to fund and just replace and build more weapons and build more equipment and throw more men at the situation. That's typical of military strategy and you'll see that throughout military history. However, the nation of Israel, they cross the river and they just defied all of that. Why? Because they had on their side the living God. That's why. And that's why God used weaker armies and weaker men because he wanted people to be able to look back and say, that doesn't make sense. Something else must have been going on there. Something else must have been intervening there because this goes against any strategy or any theory that men could put together. This is why the spies came back and 10 of the spies, before they even went to battle, 10 of the spies, they said, this doesn't make any sense that we would go to war against these people. There's more of them, they're bigger, they're stronger, we're going to lose. But what do they lack? Through faith they subdued kingdoms. We'll get to that one in just a second. Back to the point I'm trying to make here. The living God intervenes. The living God is active with mankind. From the beginning it will be like that until the end.  You worship some fairy tale. The mistake the unsaved make when they say things like that, they assume that we worship a God that is make-believe. But no, we worship an active living God. And it makes no sense that he's not active because he hasn't been intervening because we've been seeing it happen throughout history, we see it happen in our lives. But just because they think that because they have never personally seen him or seen him, they think that they've never seen him or they think that they've never seen him intervene in their lives. This is what people think today. But the truth is they have seen him and he has intervened in their lives. Some people, many people, I don't like reporting this, but probably the majority of people have simply hardened their hearts to be able to ignore it. I can give you examples of this from just a few days ago. If you're a soul winner, you know this. I was talking to a person earlier this week. I was talking to this person at their door and this young man was about to go to jail. He was telling me how he's about to go to jail and he got in this situation where he assaulted somebody and he's going to go to jail for, he's said for maybe a year or so. And he said that, he got into some altercation and he said, you know, I've just really been pressed lately. He says, funny, you're here. Because I've really been pressed lately that, you know, on all these questions like where am I going to go? Because what did I ask him? I asked him if he knew he was going to go to heaven. And he said, boy, I've really been thinking a lot about that lately. Because of this situation that is happening to me, and you can tell he was worried about it. He was worried about it. He did it. He did it.  I should have done it. I should have done it. I got an altercation with somebody and I just, I just, you know, I should have never gotten physical. And I just, but since this, all these questions have been coming into my mind. I've just been thinking about all these things that you're bringing up to me. I've been thinking about them a lot in the last couple of weeks, he says to me. And he says, I know. And he says, I know I shouldn't have done what I did. I said, how did you know that? He's not saved. He doesn't know a word of the Bible. How did you know it was wrong? He's the one that told me that. I didn't tell him that. I'm not lecturing him on what he did. He says, I know it was wrong. I said, how did you know that it was wrong? I told him. I said, you have a living God that wrote his law in your heart. That's how you know. Because when you were born into this world, God wrote that law down in your heart. And that's how you're standing there today. You know nothing of what the Bible says, and you're telling me what you did was wrong. Tell me that's not God intervening for you. You have a living God. You have to ignore him.  You think it's an accident? Some Baptist pastor walks into your driveway, and here you stand. Just yesterday, Benjamin ran into the same type of people. This man and his wife, maybe we've been talking about this for the last two days, non-stop. You think it's an accident? That we're here? Answering the questions that you've been asking each other for two days? You just want to, like, what in the world?  People ignore him, though. He is active. He is there. He did write the law. He wrote the law for a reason. So you would have moments like that. Turn to Romans chapter one. It's good to tell people when you get in situations like that. It's good to tell people, like, do you think it's an accident that I'm here with the Bible? Look, I don't know when he's special, but I'm standing here with the Word of God. And I'm willing to explain it to you. I'm no more special than you are. I'm just here with the answers in my hand. God is real. He's alive. He's active. Look at Romans chapter one. People have just gotten very good at ignoring him. I mean, look, because of this, the fact that people are breathing, the fact that they are thinking and contemplating is proof of this living God. Look down in Romans chapter one of verse number 20. The Bible says this, for the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made. Everybody is walking around with this law written in their heart, and they are walking around in the creation that God has made. And the fact that they're conscious, that they're consciousness, the fact that these questions keep coming into their mind, it's proof that God intervenes. But you can ignore them. You can ignore them. And if you read the rest of Romans one, you can turn against him. It's possible. But it's the world around us. And it's the cares. I'm talking about man's world around us. And it's the cares of man's world around us. It's the lusts of the flesh around us that suppress the natural search for the living God that is in every man. But he's there. And there's evidence of it for every single person who has ever lived. Turn to Hebrews chapter 10. So the living God is the only way that we can achieve life or that any man can achieve life. The living God is active.  The third one is this. And this isn't one you're going to hear a lot today. But the third one is this. Because we have a living God and the only way to life is through the living God and the only way that living God is active, the living God should be feared. Look at Hebrews chapter 10 and look at verse number 26. You say by who? By everyone. By both the unsaved and the saved. Look at Hebrews chapter 10 and verse number 26. The Bible says this. This is for somebody who is saved and is just abusing God's grace. Because look, if you're saved this morning, nothing's going to make you unsaved. You are saved as a gift. You are saved by trusting on the risen Jesus Christ that we celebrate this morning. You're not saved by 0.1% of anything that you've done. If you think that, you're not saved. You're saved by trusting in Jesus Christ. You're saved by trusting in the Son of the living God. But you go ahead and abuse that there's going to be chastisement coming your way. Look at Hebrews chapter 10 and verse 26. If we sin willfully after we've received the knowledge of the truth, the remaineth no more sacrifice for sins. It's like God's like, I don't know what I'm going to do with you. I've saved you. I've sealed you with the Holy Spirit. There's only one thing I can do with you.  I'm not just going to have Jesus come and die every five minutes, which is what you would need. It's like there's no more sacrifice for sins. Instead, a certain fearful looking for judgment and fiery indignation which shall devour the adversaries. He that despised Moses' law died without mercy under two or three witnesses. Look, everybody that died under God's judgment under Moses, they're not in hell.  They physically died. Oh, how much sore punishment suppose ye shall ye be thought worthy who had trodden underfoot the Son of God and have counted the blood of the covenant where he was sanctified an unholy thing and had done despite unto the Spirit of grace. This is the Christian who's like, I'm saved. I know I'm saved. Hey, great. I'm glad you know that. And then they just despise God.  He says we know that he has said vengeance belong to me. I will recompense saith the Lord and again the Lord shall judge who? His people. It is a fearful thing. He's not going to save people here. He's not going to save people trying to wake them up on the chastisement of God. Look, the chastisement of God on the believer, on your life on this earth is nothing to shake a stick at. That's what he's trying to do. He's trying to wake people up. You will find people. I even asked this question myself before I was saved and somebody gave me the gospel. I even asked somebody myself they're telling me the gospel's free. All I have to do is just trust on Jesus.  As long as I trust in Jesus. And the answer is yes. But God will beat me. God will beat me possibly to death is what Hebrews 10 is saying. The chastisement of God is nothing to joke about.  Great. But don't abuse that because God will chastise His people severely. And look, as far as the unsaved goes turn to 1 Samuel chapter 17. As far as the unsaved goes they should fear the Lord as well. But they won't as much as they should just because they don't take Him seriously. But they should. And people in the Bible knew this. David knew this. Look at 1 Samuel chapter 17 and look at verse 26. Goliath was coming out and he was challenging the army of Israel. He was challenging the army of Israel and he was just making fun of them and they were all just terrified and in fear. And David was just like, look at his reaction here. Look what he says in verse 26. David knew that Goliath should fear God. Goliath knew that even these pagan people even though they didn't, they should. Look at verse 26. It says, And David spake unto the men that stood by him saying, What shall be done to the man that killed this Philistine? And take it the way the reproach from Israel. And he's saying, look, it's an embarrassing thing. Israel has reproach upon it because they're allowing somebody like this. And look at this. Look what he says. For who is this uncircumcised Philistine? This unbeliever is what he's saying that he should defy the armies of what? Of the living God. You know what David is saying? Doesn't this guy know that God intervenes? Doesn't this guy know that God is active? Doesn't this guy know? I mean, he hears what Goliath is saying. He's like, he's afraid for him. He's like, he's defying the armies. He's not of Israel. He's defying the armies of the living God. David was insulted for God. Look at 2 Kings 19. Same thing with the armies of Syria. And Rab Shaki, who comes and just starts insulting God. 2 Kings 19, look at verse number 4. Everybody, saved and unsaved, should fear the living God. Look at 2 Kings 19, verse number 4. The Bible says that maybe the Lord God will hear all the words of Rab Shaki, whom the king of Assyria, his master, has sent to what? To reproach the living God. And he will reproach the words which the Lord God has heard. Wherefore, lift up thy prayer unto the remnant that are left. They're saying like, we should pray that God intervenes here. He intervened. Nations that reproach the living God are systematically destroyed. If you don't get a theme from the Bible, look, nations are not individuals. Nations are judged on this earth. You are not saved as a nation. You are saved as an individual belief or not belief. Nations are judged now. And you just see that again and again and again in the Old Testament. Why do you think it's there? Nations that, no, think about this. Nations that reproach the living God are judged harshly on this earth. Do you believe me? Our president a couple days ago declared Easter Sunday as Trans Recognition Day. That's real. I had to read it twice. That's a real thing. That really happened. That will not go unanswered. Why? Because everybody should fear the living God. Why? Because God intervenes. Not because of anything I will do or you will do, but because God is active and he's a living God. And people that reproach him, nations that reproach him will be destroyed. I actually think that there's a silver lining here and I'm going to talk about that tonight. I'm going to get into that this evening. Turn to 1 Timothy chapter this morning. Turn to 1 Timothy chapter number four. I mean, everyone should fear the living God.  Churches that are removing all of God's wrath so they can be nicer.   Hello, what are people doing in these churches? Hello, what are people doing in these churches? What are people doing listening to this? What are Christians who are saved and believe the Bible doing listening to this garbage? The living God is active and the living God should be feared by everyone. Look at 1 Timothy chapter four. Look at 1 Timothy chapter four. My last point is this. My last point is this and this is really for us this morning. They're all for us this morning, but this one really I want you to focus on. The living God should be trusted. The living God should be trusted. Look at 1 Timothy chapter four and verse number ten. The Bible says in 1 Timothy chapter four and verse number ten, it says, therefore we both labor and suffer reproach. Guess what? As you live amongst a nation that at least has leaders that reproach the living God at this time anyway, you will be reproached because why? Because we trust in the living God who is the Savior of all men. This is such a great verse here. It totally disproves Calvinism in this one verse. It says this living God, we trust in the living God who is the Savior of all men, especially of those that believe. You know what those last two points of that verse say? They say that God is that, it proves God wants all to be saved. It calls God the Savior of all men. God wants to be. It is His will that all men should be saved. That is what God wants. It is not God's fault that people ignore Him. It is not God's fault that people turn against Him. It is not God's fault that people decide to hate Him. God wants all, it is His will that all would be saved, especially of those that believe, but only those that believe will be saved. See what's saying there? Doesn't that make perfect sense? So look, I pray that you're all saved this morning and if you're not, ask somebody after service. But you are saved through trust in the living God. You are saved through the trust in the death, burial, and resurrection of God's Son, the Son of the living God. You are not saved by following Him. You are not saved by turning from your sins or repenting of sins or whatever you want to call it. You were only saved by trusting in Him. But guess what? You had at least enough faith to trust in Jesus for your salvation, which guess what? You got something from. You were given eternal life personally for that. So my question for you this morning is this. Why can't you then trust in the rest of His Word? Why can't you trust Him going forward in your Christian life? Why would somebody get saved and then never even open the Bible to see what God has for their life going forward? Why would they not do that? Why can't we trust the Word of the living God? Because God doesn't just give us this path to salvation. He tells us, hey, now here's what I want you to do. I mean, wasn't that Mark 16? The end of Mark 16? Jesus literally saying, hey, there you go. Now go tell everybody else. He tells you what to do now. He didn't just save you. He tells you what to do now.  I mean, personally at this point, it's harder and harder for me to believe every single day that people would choose the way of this world over the Word of God. I mean, who in the world, this is just from my viewpoint, who in the world could look at this country? What believer could look at this country and the things that are happening and conclude that the Bible was the problem? I mean, what in the world? Who could do that in South Dakota? And I guess Georgia, they passed some new law, like literally limits free speech. Isn't South Dakota, this governor of South Dakota, like all the conservatives like, oh, pass a law that literally makes it illegal to preach that, look, it's to preach that it was the Jews that killed Jesus. She makes like Matthew 24 illegal. Look, I'm not mad at people that are of the Jewish religion, but they need to hear the gospel. But to pass laws that are against preaching the truth, look folks, you cannot choose Jesus Christ and the religion responsible for killing Him. I mean, how backwards much to be to believe that the Bible was the problem today?  I preached a sermon at first worked called Honky Tonk Hip Hop. And I compared the themes of hip hop music versus country music. And you know what my first point was in the sermon? My first point was hip hop especially, but they both kind of do. Hip hop especially promotes the abuse of women. And somebody said to me a couple of days ago, I'm like man, that sermon was prophetic.  He's been singing about it for 30 years. He's been singing music and promoting people that sing music that's like, hey, we abuse women for 30 years. And I'm some prophet. What in the world?  They're telling you, we worship the devil. And there was like, wow, they worship the devil? They're telling you, we want your kids. We want to grab your kids. We want to pervert their minds. We want to destroy them. People are like, they're after our kids? Can that be true? They're telling you. They're telling you. They're coming out and they're saying it. How backwards must you be? Or how far? How far into the sand must your head be? To be like, I think the Bible, the problem. Facebook now is graying out Bible verse posts. We have to click an extra button where it says, this is sensitive. I don't even know what it says. Sensitive or graphic or violent content. And then it's like a King James Bible verse. Because the Bible is the problem. We're done with Facebook, by the way. Church announcement. I told the online manager, blow us out of this thing. I don't want anything to do with it. Why? Because it's bad. It's completely all bad. I'll preach a whole sermon on it, or I'll tell you after church or whatever you want. But it's all bad. There's nothing good there. There's nothing good there. But look, I'll talk about the craziness of a lot of this tonight and how there's a big silver lining here. And I do believe that there is a silver lining here. But the point is this. Back to the point. The clowns of clown world. And the less separated from it you become, some of the things in the Bible might even seem to be radical to you. If you don't know what's in the Bible.  I mean, don't hold illogical positions. As a Christian. You can't, don't think that you can do what everybody else does. Don't think that you can go where they go. Don't think that you can see what they see. Don't think that you can engage in what they engage in. Don't think that you can speak like they speak. Think like they think. And suddenly you won't reap what they reap. You won't become unsaved. But you will reap. Turn to Proverbs chapter 3. If you don't want those results, then you need to trust the living God.  But people, they'll get so entrenched in worldly things, then maybe even the Bible seems extreme to even a person that is saved. There's less and less and less knowledge of the Bible out there, folks. It's not getting any better. I meet more and more people every single month, every single year that haven't even heard of Jesus. Especially young people. Typically people would have heard of Jesus five years ago. Now you're meeting a lot of young people, never even heard of him. Look at Proverbs chapter 3 and verse number 5. We must, as saved believers, we must still trust the living God. We must trust his word. Look at Proverbs chapter 3 and verse number 5. The Bible says this, what? It says, trust in the Lord with all thine heart. And lean not onto thine own understanding. As things get crazier and crazier, thine own understanding is going to get worse and worse. The temptations of the flesh are going to get worse and worse.  Satan can't make you unsaved, but he can make you unprofitable. Just trust in the Lord. In all thy ways acknowledge him, and he shall direct thy past. He'll tell you everything what you need to do. You don't even need to know anything. You know what it means? Look, if you just knew, it wouldn't be trust. If you just knew, it wouldn't be trust. Even if you read something in the Bible, you just have to accept this. Like, hey, you know what? If you read it, do it. If you hear it preached, just implement it. That's what trust means. Like it doesn't make sense to me. Lean not onto thine own understanding. You've been corrupted. You've been corrupted. Look at Matthew 24. Look at Matthew 24, verse number 35. Matthew 24, look at verse number 35. See, Jesus is the Word. Jesus is the Word. And the Bible says in Matthew 24, verse 35, Heaven and earth shall pass away, but my words shall not pass away. We have the Word of God. We have the answer. It's just a willingness to trust in it. Notice at the beginning of Hebrews chapter 11, when it talked about all those wonderful things that those people were able to accomplish. All those weak men and women were able to accomplish. How does the chapter start out? It says through faith. Why? Because they trusted. Daniel's like, whatever, I guess. I'm not stopping praying. I'm just going to trust what God says. And you know what, Shadrach, Meshach, and Abednego? I love their statement. I'm going to paraphrase it, but they're like, if God saves us, he saves us. If not, he doesn't. Whatever. We're not going against his Word. They trust his Word. Like I'm following the Lord, kill me, whatever. I'm following the Lord. You can't go to church anymore. Hey, whatever. Do your worst. I'm following the Lord. I'll follow the Lord. Because you know what? I'm way more afraid of him than I am of any man. I get it. There's people that can put fear in you. And we shouldn't have that fear. But look, we're men. We're women. We're flesh. You just fear the Lord. Fear the Lord more. I'm way more afraid of God than I am of you or anybody in this world. Look, folks. Jesus is the Word. And he's risen. And his Word will not pass away. He's the eternal King. That means it's just like forever. Forever, he's the King. We have a living God this Easter morning. We have a living God that is active on this earth with us. That is not... Look, Jesus was here and he lived what you're living. He lived the temptations that you can't get away from. He lived all these things and he succeeded against them all. Yet without sin. He lived amongst... He lived amongst a nation that was way worse than ours. There was way sicker and anti-God than ours. This Roman Empire that was of Jesus' time. But guess what? He's here and he's available to you.   And he gave you instructions to do so. So look, trust in this. Trust in this. This Easter morning. Say, what did you learn Easter morning? You already knew that Jesus rose from the dead when he came to church this morning. But you learned that you should trust him in everything. Just trust him in everything and you know what? The more you trust him in everything, the more sense. You may not like it, but it makes perfect sense. I'm not confused about anything that's happening today. I don't like seeing it. I don't like people reproaching the living God. But I'm not confused about anything. There's a lot of people that are confused. People tell me that all the time. Older lady yesterday. Beautiful, wonderful older lady. My wife sat down and visited with yesterday. And this gave us just like, it's crazy out here. It's crazy. And she's pointing to her front door. And I'm like, agree, 100%. But here's the thing, folks. Take advantage of this. Take advantage of this because who's going to hurt you when you have that? Who can do anything to you when you trust in the living God in everything that you do? When you just have Jesus directing your past. I mean, there will be no excuse. Just look at the front of your bulletin. I'll read for you Philippians 2. There will be no excuse for anyone. There's so many people, pastor, that have no interest in the Bible. They're not going to get saved. They don't seem to want to care if they're going to heaven or not. But at the name of Jesus, every knee. At the name of Jesus, every knee will bow. And every tongue will confess that Jesus Christ is Lord. Many people, when they do that, it will be too late.  But that tongue will confess that Jesus Christ is Lord. Right before they're thrown into the lake of fire. We have a living God this morning. Jesus really rose from the dead. He was seen by hundreds of people. It was not a hidden resurrection. That's why I put this as the verse of the week. He was seen above 500 brethren at once. The resurrecting of Jesus Christ was not hidden from anybody. Neither will his second coming be hidden from anybody.  That wouldn't make any sense. Everyone will see Jesus coming back. It's not some fantasy book where it's going to be hidden. Because we have a living God who is real. He's active. He's involved. He should be feared and he should be followed. By us especially. So happy resurrection day. Happy living God day. Let's bow our heads and have a word of prayer.