 Thank you everybody. Thanks a lot for attending this talk. And I want to thank the CTP for giving me this opportunity to be here and to present this talk. Entitled phase filled model for immiscible to phase a flow in a micro-fill with a junction when we put two immiscible fill with in a junction, what are the kind of pattern that can happen. This work actually is done in collaboration between the National Engineering School of Sparks and the Indian Institute of Technology Europe in India. And it will be presented by me, Zahir Dil-Hafsi, and that was provided by both Professor Mananjan Mishra from the Indian side and Professor Samuel Oud from the Tunisian side. So quickly to the outlines of this presentation, we will start by an introduction talking about the micro-fill with X, and then we will try to talk about application of two phase flow in micro-fill with X, why we are talking about these kind of things. Then we'll present our mathematical model, the two-phase flow phase filled model, which is built by the Navier-Stokes equations and the Kahn-Helier equations which is the basic equation for the phase filled model. Then we'll give some numerical simulations result. We will start by the T-junction to validate the work with the previous works, and then we will present some results also related to an H-junction. Of course we will conclude. As an introduction, junction shape may have different and different topologies. We can talk about Y-shaped, T-shaped, C-shaped, H-shaped, and in this study we will focus on T-shaped or T-junction shape and H-junction shape. The objective of this study actually is to follow up the flow of two immiscible fill with when you put two immiscible fill with. Of course a micro-fill with the junction, what are the possible paterns that can happen and what are the phenomenon that can be seen? And we'll highlight the capability of the phase filled method to capture the interface of a lotion compared to some previous method. And we'll discuss the different observed pattern and we'll compare with the previous find. The two-phase flow in micro-fill with X and its different applications you can encounter this kind of flow in encapsulation industry for foods, drugs, chemical products. Incomical reaction and biochemical assays, separation chemistry like chromatography, droplet generation and manipulation and we'll see these kinds of droplets, biomedicine and material synthesis. Also there are different other applications. For the mathematical model we will talk about immiscible fill with flow. What does it mean immiscible fill with the flow? It means that we can see a limiting interface between the fill with. It will be observed along the simulations. And to flow up the flow pattern we have to focus on the variation or the geometrical variation of this interface. We'll give us an idea about the flow pattern that happens. So what we'll use actually is a laminar fill with the flow. Laminar fill with the flow can be solved using Navier-Stokes equation for momentum and mass conservation. That will give us an idea about the pressure and the velocity of the fill with. But we want to track the interface. That means we want to see the different patterns that happen. That means we need an interface tracking method as well. Like volume fluid method, level set our phase fill. For our case we'll use the phase fill method. So our model is built. Reposal run, Navier-Stokes equation, the two equations of momentum and mass conservation. We'll not talk about energy equation because we'll assume an isotermal flow. And we'll use the phase fill model to build our numerical scheme. This equation will be solved using the Kamsor multi-physics 5.2 software. It's a finite element based software. And there is a model inside this Kamsor called laminar two-phase flow phase fill model and we'll use it. Well, quickly the equation is for immiscible incompressible isotermal liquid-liquid flow. We have Navier-Stokes equations, the conservation of mass and the conservation of momentum. And then we'll add the interface tracking method, the phase fill method, which is based on the Kandelier equation for time evolution of diffuse interface profile. Actually, the coupling between these, the set of equation of Navier-Stokes and the phase fill model is doing the other surface tension force, the FST. And the FST is expressed as function of the phi, the order parameter in the chemical potential times, the derivative of the order parameter phi. So what is the order parameter? This order parameter actually is the phase fill parameter or the order parameter phi is a parameter that we want to use to describe the transition between fluid one and the fluid two. Actually, phase fill method, it is diffuse interface method. Why it is diffuse interface method? Because in reality, when you put two immiscible fluid, fluid one and fluid two, a sharp interface is seen between them. That means if we will go from fluid one to fluid two, the transition will be abrupt from row one to row two. But this is not very useful numerically. So we assume a diffuse interface model. We assume that there is a small interface between the two fluid, at which there is a little bit of diffusing in order to ensure the smoothness of the transition between fluid one to fluid two. So we are assumed that to go from fluid one to fluid two, we are going smoothly. So using this representation, we'll be able to solve our model. And instead of using row one and row two, we are using just the phase fill parameter phi. It's a global parameter that gives a description of the whole system. If phi is equal to minus one, we are in fluid one. If phi is equal to one, we are in fluid two. And between them, all values are between minus one and one, are describing the diffuse interface. And we are giving a small thickness epsilon to the interface. We assume between the two fluid. Actually, the diffuse interface approach allows us to do what? Allows us to analyze the two fluid system as one fluid flow. As we are having just one fluid flow with a density row and a viscosity mu. And they are given by these relations. They see the full description. Actually, if we will put phi equal to minus one, we'll be back to row equal to one. And if we will put phi equal to one, we'll be back to row equal to row two. So these expressions are giving a general description to the system via the phase fill method. We are not talking about fluid one and fluid two. We are talking about a system of as we have one fluid flow. And this is very, very useful numerically. Actually, for the numerical simulation, what will be done is- The following. We will focus on a reference model first. This is the experimental numerical works of Tiva Comeridi-Cherlou and his co-authors in 2010. What they did actually, they are putting two invisible fluid. They are putting water and kerosene. They are injecting water from the top and kerosene from the bottom in a T-junction. And the pattern they have find are they find some bubbles and some slugs and stratified flow and a little bit wavy also. And here are some slugs. This is experimentally and this is done numerically via the volume of fluid method. This is the previous work of Cherlou et al. Actually, what Cherlou has done is the following. They are having a T-junction. Initially, the T-junction is full of air. Then they are injecting water from the top, kerosene from the bottom, water and kerosene are flowing. They are pushing out of the air, out of the junction. There there will be some patterns that are formed between water and kerosene. These are the patterns we have seen. This kind of pattern, sorry. This is the red one in the water, the blue one in the kerosene, and we have some slugs here and here are slugs detachment and these kind of things. But what we will do is a little bit different because Cherlou model actually is a three phase flow system model because we have the air, water and kerosene. What we assume is that because of kerosene and water will push out the air. So we start by filling the domain by water and kerosene. That means at T equal to zero for our model, the initially the domain is half filled by, filled with one water in the top and filled with two kerosene in the bottom at rest. Then we will apply a velocity to the water and the velocity to the kerosene. This is the 3D model. We are exemplifying the 3D model to 3D model. Actually what we have here, water inlet, kerosene inlet and the outlet. For the simulation parameter at the first stage, we are taking the same value of Cherlou at all. And for the boundary condition, we have a constant inlet velocity and a constant pressure outlet, the atmospheric pressure. Wet world condition also is assumed in the boundary wall. Well, this is a snapshot of the laminar two-phase flow model of COMSOL multi-physics. This is just a snapshot, it's not very important. So what we did actually is a free triangular fine mesh and this is our base state. This is at T equal to zero. We are filling the junction half filled with water, the blue one and kerosene, the red one. Then we will apply a velocity from this side and the same velocity value from this side. The simulation will be done or will be run in one second. What are the different patterns that we have observed? The disturbance, there is an onset of disturbance that starts at T equal to zero zero five seconds. What we see actually is stratified flow plus onset of instability. There is some stratification and there is kind of instability that occur here. This is the starting of wavy flow. Here we can see the wavy flow and there is starting of slag detachment. The slag will be, there will be slag detachment here. Here we can see. And then these are the slags. T equal to 25 seconds. We can see the slags here. See the transition between the wavy and slag flow and then fully slag flow is seen as Shirley has found also. So starting from T equal to zero zero three second, the transition between wavy and slag flow is completed and the flow is fully slagged. This is the permanent established regime. It's a permanent slag flow. This is the first simulation result but we'll try to go a little bit further and we'll try to see the effect of a grand little velocity if both fluid are not having the same velocity. So here is the same velocity that means the previous result. Here is the case when we are giving water, let's say, a greater velocity than kerosene. So we can see that the slags of water are much more larger than that of kerosene. And inversely, of course, if we will give velocity of kerosene greater than that of water, the slags of kerosene are much more late. So we can see, we can say that slag dimension is proportional to the inlet velocity. This is a fast observation that we can take. So here we have an animation. I don't know if it will work or not. Just, sorry. Anyway, this animation is showing the slag detachment and how the slag formations can be seen. You are not able to see this. Right. For the H-junction, so we did the T-junction model just to compare with Sherlock's work. And we will try to see what can happen in the H-junction. Actually, I assume an H-junction. The same parameter on most of the T-junction, just we have here, sorry, two outlets. We are injecting water from the top here and from the bottom, and we have two outlets. That means the flow has the possibility to go here and there. The same velocity here, free triangular fine mesh, but the pattern that we can observe. Here we will focus on the effect of the surface tension. That means we run our model with surface tension and without surface tension. With surface tension where we can see the permanent regime is the slag flow. Slag slag there, then in the outlet or in the outlet of the horizontal challenge, a subdivision of slag see the seed. This is similar to the previous work in the T-junction, but without surface tension, a new pattern is observed. We are observing bubbles. Bubbles are there. We have a stratified flow and a bubbling phenomena. We will try to focus on these bubbles. Let's say how these bubbles are formed. Actually, giving a velocity to k-rosion and the velocity to the water, sorry, red, and k-rosion in the blue. At initial contact between the two fill-with, bubbles occur and the bubbles are pushing out, to the channel, let's see here, are pushing out by the stratified flow. That means the bubbling phenomena is just a transition phenomena. So it's pushing out, then it's pushing out. So we have seen this stratified flow, bubbly flow, and some vertical slag flow. If the bubbles are going out, they are coming. They become some slags and they go up. Actually, and then finally, here we have the disappears of the disturbance. That means the bubbling phenomena is just transition and we will finally, we will find the stratified flow, establish it again, and you see the final shape, this one, at zero three, three one. Water is pushing from the top, going out from the top, k-rosion from the bottom, going out from the bottom, and it is the permanent state for the H-junction without surface tension effect. Let's see, just to discuss the bubbling phenomena. Finally, just, we have, we will see the effect of the gradient of velocity, same as we did in T-junction. Yes, it's almost the last one. If we are injecting water with a greater velocity than that of k-rosion, here, due to the mass conservation equation normally, water will, will exit from the top and also from the bottom of junction. We are injecting water from the top, but you can see that it, due to the gradient of velocity and because of what is very fast, or twice faster than k-rosion cell, it allows it to exit also from the bottom of, of the channel. Actually, this is almost all just to conclude. Well, we are seeing that the phase field method is capable to capture some, the flow regime in micro fluid injection. Actually, the accuracy of the results are well depending on the interface thickness parameter epsilon that we are, that we assume between the two fluid. The surface tension effect between the two, the two immiscible fluid is an onset of, is responsible for the onset of instability. What are they saying? Actually, no, no important thing to say. Just for consultancy physics, there is another model called a ternary phase field model. We are not using it. Maybe you can use it there, in coming up. Some references are there. Just to thank Professor Manurajan Mishra from Indian Institute of Technology, Europa and Professor Samarit from the Mechanical Engineering Department in National Engineering School of Science, Tunisia, and they have to acknowledge the National Engineering School of Science, the Indian Institute of Technology, Delhi AM Center for Fellowship, I have it. Obtained in New Delhi, India, and of course, the ICTP for this opportunity. Thanks to you. Thanks to you, Paul. Thank you. I wish to be Dr. Gasson. It's been defined by capital forces. So, when you see your slats, it should be something like rounded hips. Like it is something. Due to, due to the capillary forces. Sometimes, also, the viscosity is the bubble. Actually, what we have seen is that the bubble in here, because we are assuming the wetability. How do you think your surface tension is? Yes, the smearing part of it. The surface tension? Surface tension. Well, it's almost included in the model itself, here. Because this model, all these sets of equations, and this one, the cannular equations, that gives the relationship between the surface tension force and the phi parameter. The phi parameter is the parameter that distinguishes the two phases. Actually, seeing the bubble means that there is a difference of phase. Phase one is bubbling here, and the phase two is, there is no bubble. That means the surface tension force, and this is the key equation, I can say. Because it's coupling the model, the Navier-Stokes equation, with the phase three. And all these models are included in ComSol Multi-Physics. That's why I was putting that snapshot, I was doing it quickly. But you can see here, in the ComSol Multi-Physics itself, we have Navier-Stokes equations, and this one in the cannular equation. Well, the snapshot is not well-detended, but you can see that the surface tension force is related to the incompletion.