 So, far we have discussed about various conservation equations and we have seen that how these equations can be derived and subsequently we will see that how these equations can be used to address various problems related to microfluidics. So, entering into the specific topics of microfluidics, we will first discuss about pressure driven microfluids. Why we are interested about pressure driven microfluids is because pressure driven microfluids are interesting and important from many points of view. Like if you think of classical flows that is fluid flows occurring over macroscopic scales, one of the most obvious ways by which you drive flow through a system, maybe a piping system is by pumping, so that is a pressure driven flow. So, in microfluidics of course we do not use the same type of pump that we use in macrophlos, but you can generate pressure gradients by various mechanisms in microflos to actuate the flow. While there are certain limitations, I should not say disadvantages, better to say limitations associated with this method, there are still some obvious advantages. So, we will start with this in terms of flow actuation mechanisms in micro scale flows. So, just like for any topic we are interested about first the motivation, why are we interested to study pressure driven flows in microfluidic systems. So several such applications are there, like for example you can have pressure driven microfluids in biophysical systems in general and human body mechanics in human body blood flow dynamics in particular. So, how is it possible? As I discussed earlier that in human body you have a hierarchical structure of the sizes of the blood vessels. So, you have large arteries, large veins, small arteries, small veins, arterioles, venioles and micro capillaries. Micro capillaries are nothing but like human body microfluidic channels, micro channels in human bodies. Of course there are several complexities associated with those microfluidic channels as compared to engineering made channels or engineered channels. So we will not get into those details right now, we will discuss about those issues when we talk about fabrication of microfluidic channels, that what are the complications that come into the picture as you go from the paradigm of microfluidic channel from an engineering point of view to a biomimetic microfluidic channel, a microfluidic channel that mimics the blood vessels of human bodies and how to fabricate those channels, these are still very open topics in research, in modern day research. In all these systems you have pulsatile flow of blood, so that flow is driven by a pressure gradient, then you can have various technologies in which you can achieve microfluidic mixing. So for example like one of the possibilities is that you introduce two fluids by a driving pressure gradient and allow the fluids to mix by stretching and folding of the fluid elements as the fluid passes through an undulated passage or a corrugated passage. So that is one way in which you can achieve good mixing in a microfluidic system where achieving mixing is otherwise a very difficult proposition because the Reynolds number is pretty low. We will see what is the consequence of Reynolds number in microfluidics and the corresponding hydrodynamics in many cases is classically known as lower Reynolds number hydrodynamics. Flow focusing is another application where you can see that you can basically have a fluid in the core and two fluids injected from the sides and this essentially creates a microfluidic stream, so that is called as flow focusing. So flow focusing is used in many applications like flow focusing is commonly used for generating droplets for example, so that you can see in one of these slides. So starting from this physiology to engineering applications, so you can see the wide gamut of applications of pressure driven microfluidics starting from the human body dynamics, understanding the mechanics of human bodies to like engineering devices for practical applications like mixing, droplet generation and all, I mean pressure driven flow is inevitable. Many times this pressure gradient is generated by a device which is like, which is a so called pump but in most of the cases a positive displacement pump rather than a rotodynamic pump and these types of pumps are called as syringe pump or peristaltic pump. So for example, this is a pictorial view of a syringe pump. So this syringe pump like the basic mechanism by which a pressure gradient, driving pressure gradient is generated in the same way in which an injection syringe operates. So that kind of technology is used. So we will not get into the technology of pressure generation, pressure gradient generation at this stage because we will be having a separate lab demonstration. So the way in which we have planned for the lab demonstration for this online video course is that we will have recording of the lab experiments which are going on in our lab and then we will demonstrate the videos with those recordings of the lab experiments. So as far as practicable, we will include some very basic experiments in microfluidics which we will supplement with the theoretical discussion that we are having so that it makes a sort of a complete package for the beginners. Now we will get into some of the fundamentals related to laminar pressure driven flow. Now these fundamentals are important to study microfluidics. Not necessarily these fundamentals are the only sort of theoretical basis on which micro flows are based but the very important fact is that these fundamentals are common to classical fluid mechanics with which most of you are familiar. So we will start with that. So let us say, let us go to the board maybe and discuss about this. Let us say that you have a channel like this. Fluid enters the channel and when it enters the channel, let us say that the fluid enters the channel with a uniform velocity profile just for conceptual understanding. Now when it enters the channel, the fluid is subjected to a resistance. The resistance is offered by the walls of the channel. So when the walls of the channel provide some resistance, what happens to the flow? If you consider, we will consider one section here and try to draw how the axial component of velocity varies and if we draw the vectors with the axial component of velocity that is called as the velocity profile. Remember many times it is an illusion that when we draw the velocity profile, it gives an impression that as if that is the only component not necessarily. There could be other components which are not as strong as the predominant axial component and that is why we are presenting the axial component as a representative of the velocity field but that is not the complete picture of the velocity field. Nevertheless we are presenting the axial velocity component in a vector diagram. So at the wall, what happens to the flow velocity? At the wall, there is 0 relative velocity between the fluid and the solid by virtue of no slip boundary condition. Now this statement sometimes is considered as a very serious offense when we are studying microfluidics because there have been many observations and many theoretical studies which have revealed that that may not be the scenario in microfluidic applications. So we will start with this assumption but we will study a chapter later on on the issues of slip or no slip boundary condition at the wall and there we will study it more carefully and more elaborately. For the time being assume that the no slip boundary condition may be one of the possibilities, one of the possibilities. Still in many microfluids no slip boundary condition works. So you cannot definitely say that the no slip boundary condition will work or will not work. Depends on several other things, what are those things we will discuss in details later on. Now the velocity here is 0 because the velocity at the wall is 0. Now if you come to a section like this, a station, a location like this, here the velocity is somewhat greater than 0 but not equal to this free stream velocity. Same thing if we assume that it is symmetrical, same thing is for the upper plate. In this way a section will come or rather it is better to say a location will come because it is in the same section, a location will come, a transverse location where the velocity comes to a maximum and it does not change beyond that maximum okay. So then it remains as a maximum like this. So if we draw a velocity profile, the velocity profile is something like this because of some issues with drawing by sketching by hand, it may appear to be asymmetric but it is I mean assume that it is symmetric in both sides of the channel. Now let us consider another section which is located here. So what we will do is we will draw the velocity profiles at the corresponding locations. Let us call it section 1 and let us call it section 2. We will draw the velocity profiles at the corresponding locations at which we have drawn the velocity profile for section 1. What is the velocity profile? If you join the tip of the axial velocity vectors, this red colored line that you see that is called the velocity profile. Now you come to this section. Now can you tell that at this location, whatever was the axial velocity at section 1, at section 2 will it be more or less? It has to be less because now fluid has come in more intense contact with the solid boundary. So it is more seriously decelerated right. So it will be less. That means it will take a greater distance from the wall to come to the maximum. Let us say that distance is here. Similar things we draw on the other side, draw some vectors and then draw the velocity profile. So the velocity profile shows that there is a section over which the velocity profile is uniform but beyond that there is a variation. So we can demarcate the flow field into 2 parts. One portion where there is a velocity gradient, why is there a velocity gradient? Because fluid in that region adhering to the wall fills the effect of the wall through viscosity and there is a outer region which does not directly fill the effect of the wall. So the inner region is called as the boundary layer and the outer region in this case is called as the core region. It is not free stream because it is still a bounded flow but it is called as a core region. So like if you now make a sketch of the demarcating boundary of the 2 regions. So let us say that just for clarity maybe I will draw it by a thick line instead of a dotted line. So this is the demarcating line between the 2 zones of fluid. This is the boundary layer region, this is the core region and this line that demarcates these 2 regions is called as the edge of the boundary layer. Now this region, in this region there is something which is quite interesting that takes place. First of all what happens to the core region? Let us say that uc1 is the velocity in the core region at section 1 and let us say uc2 is the velocity in the core region at section 2. Now the question is which one of these is more, uc1 or uc2? Now to understand that let us assume that the density is constant to simplify the situation. So if the density is constant that means to have the same mass flow rate through the 2 sections the volume flow rate must be constant. Volume flow rate is given by what? Integral of the velocity profile through the cross section. So integral of the velocity profile when we say integral of the velocity profile over section 1 is same as integral of the velocity profile over section 2. So in section 1 what happens? In section 1 if you see that there is a region in which the slowing down effect of the wall is failed that region is say of thickness delta 1. In section 2 that region is of thickness delta 2. Which one is more? Delta 2 is more. So what the integral is the same? So because delta 2 is more that span over which the fluid slowed down is slowed down is more that means in the core region the velocity should be higher at section 2 as compared to section 1 to compensate for that loss over this section where it is slowed down. So because I am repeating the logic again because the region over which the velocity is slowed down happens to be more for section 2 that needs to be compensated by higher velocity in the core region. So that means our first conclusion is uc2 is greater than uc1. When uc2 is greater than uc1 what can we conclude about the pressure gradient okay. First of all when uc2 is greater than uc1 that shows that in the region outside the boundary layer the fluid is accelerating right this is one of the most fundamental differences between an internal flow and an external flow. If you have flow over a single flat plate then the free stream u infinity is a constant. Here the so called u infinity u infinity is replaced by uc that uc is a function of x okay. So uc is a function of x if x is the axial coordinate. Now uc is a function of x that means what? The fluid is accelerating I mean uc is a function of x means it is either accelerating or decelerating but because uc is increasing with x that means it is accelerating assume steady flow when it is accelerating it cannot accelerate until and unless there is a net driving force. So what is that which is making it to accelerate that is nothing but a driving piezometric pressure gradient. So a driving piezometric pressure gradient or a better to say a driving head the head may be pressure related the head may be kinetic energy related the head may be potential energy related. If the kinetic energy head is same at the 2 sections then it is basically the sum total of so if you think of the head let us write so p by rho g okay. Of course you have to remember that when we are talking about this head all of you understand this right I am not going to that level of undergraduate fluid mechanics but if you feel that we need to discuss I can briefly summarize what we are talking about here when we write p by rho g we are essentially meaning that this is flow energy or flow work per unit mass. So that means it is the energy that is associated with the flow so that it can sustain the flow in presence of pressure per unit mass that is called as flow energy or flow work. When we write p actually we are talking about this entire section. So we are assuming that p does not vary with y we will see later on that when is the scenario when p does not vary with y but that is a very common practical engineering scenario. So that this p is basically p anywhere at a given section this v average square by 2g is called as the kinetic energy head of course from the expression it is clear and the meaning is obvious but many times we have to correct it with a factor which is called as kinetic energy correction factor why is it necessary it is necessary because like to get the kinetic energy you have to integrate the kinetic energy at a over the section instead of that what we have done is we have written the kinetic energy in terms of the average velocity over the section that gives an error that error is adjusted with the kinetic energy correction factor and this is the potential energy head. So some total of these 3 is the energy mechanical energy rather not total energy the mechanical energy transferred by the flow as the fluid is moving from section 1 to section 2 okay. Now you can define something which is called as p star is equal to p plus rho gz this is called as piezometric pressure. So if you have roughly the same value of this for sections 1 and 2 then fluid will flow from where see many times we make loose statements in signs and one of the looser statements in fluid mechanics is that fluid flows from high pressure to low pressure this is not necessarily true. So fluid will always flow from higher head or higher energy to lower head or lower energy. Now it happens to be from higher pressure to lower pressure if these 2 are same for sections 1 and 2 then this is the only driving difference but it has to be from a higher energy to lower energy. So when you have this value roughly the same for sections 1 and 2 let us assume that although because of the difference in velocity profiles the kinetic energy correction factors over the 2 sections are not the same. So there will be some difference but let us say that difference is very slight just for practical consideration. So if that difference is slight then it is the gradient in this p star that is driving the flow and it is important to keep the gravity effect into account because the channel may be vertical also and where the gravity effect is very important. So let us not forget about the gravity effect and then this is called as piezometric pressure. In all the derivations that we will be discussing in this chapter I will write the symbol p but mentally you take it as p star because every time it is difficult to write star and we may forget writing star. So instead of that I will just write p but assume that the rho gz is included in that p okay. So now you can clearly see that the driving force is the pressure gradient. So if you see piezometric pressure gradient in this case so if you see is increasing that means you have p star increasing with x. If p star is increasing with x or sorry decreasing with x sorry I am very sorry p star is decreasing with x because it is from higher p star to lower p star, p star is decreasing with x. So if p star is decreasing with x then what is d p star dx? This is negative. This is called as favorable pressure gradient. Why it is called as favorable? Because this kind of pressure gradient favours the flow okay and of course the opposite one if this becomes positive that is called as adverse pressure gradient because that decelerates the flow. So these names you have learnt earlier but I am just recapitulating whatever is possible within this short time so that we come to the right perspective. Now what happens to the pressure gradient? The pressure gradient is the pressure the p star is decreasing with x and then what happens to the wall shear stress? We will learn that subsequently. So 3 important quantities we are interested in are the core velocity d p star dx and tau wall. Why are we interested in d p star dx and tau wall? Because eventually these parameters dictate that what is the power required to drive the flow. As an engineer see we are very interested to know that what is the power required to maintain the flow because you have to make your investments accordingly. So that is what we are bothered about. Now this come back to the edges of the boundary layer. When the edges of the boundary layer have met each other and where they will meet? If it is symmetrical in the upper half and the lower half they have to meet on the centre line of the channel. Let us say it meets here. When it meets here what is the significance? The significance is that now the entire section of the channel feels the effect of the wall. Prior to this only the boundary layer region is explicitly feeling the effect of the wall. The outer region was not feeling the effect of the wall basically it is as if it would not know that there is a wall. But the entire fluid here has now understood that what is the yes there is a wall. I understand that I am being slowed down because of the effect of the wall. So then we say that the flow has become hydrodynamically fully developed or simply fully developed. So from here we say that this is fully developed. What is the parameter on which this it depends on how long it will take to get fully developed. So to do that I will typically refer to one order of magnitude estimate of how delta varies with x. So if you consider a single plate delta by x scales with Reynolds number to the power minus half okay. There is not enough opportunity to derive this because like you know these things we commonly discuss in the boundary layer theory now here our scope or objective of discussion is different. So that means you can clearly see that if the Reynolds number is small if the Reynolds number is small then actually this so typically when at what x will the flow become fully developed. The flow will become fully developed when x is of the order of the let us say this is h half height of the channel when x is equal to h right when sorry when delta is equal to h not x equal to h we have to find out x when delta is equal to h right. So when delta equal to h then we can figure out that what is the x at which the order of magnitude of x at which the flow has become fully developed. Now very qualitatively we can understand that as the boundary layer is growing because the micro channels are very thin it is possible that very soon from the entrance the boundary layer reaches the center line the edge of the boundary layer reaches the center because the channel itself is very thin. That means in many analysis actually fully developed flow type of analysis in microfluidic channel works well but we will discuss about the general analysis later on where we will not presume fully developed flow but fully developed flow is very common in microfluidic applications and we will start with that. So when the flow has become fully developed if you draw the velocity profile let us extend this channel a little bit. So if you draw the velocity profile the velocity profile when the flow has become fully developed from here onwards the hallmark is that u is not equal to a function of x why now why should the velocity profile change because entire fluid has failed already the effect of the wall. So it has sort of come to a state where there is no further change that is called as fully developed flow and the region prior to that is called as the flow in this region is called as developing flow and the length of this region is called as entrance length these are some terminologies. Let us say that we are interested to find out how the center line velocity dp star dx and tau wall vary along this regions. So basically we were interested to make a plot of center line velocity versus x dp star dx or let us say p star p star versus x and let us say tau wall versus x maybe I will draw it a little bit elevated so that it comes in the frame tau wall versus x. So u center line versus x how will it look so initially u center line is increasing as we move along x till the flow becomes fully developed and then it becomes a constant. What happens to p star to understand that let us assume that this is a section of a pipe and let us consider a small section dx one important hallmark of the fully developed flow is that the flow is not accelerating because the flow is not accelerating that means resultant force acting on a control volume must be 0 recall the navier stokes equation the left hand side is mass into acceleration per unit volume right hand side force per unit volume on the control volume. So if the fluid is not accelerating that means what the resultant force is 0 so if the resultant force is 0 that means let us say that this is let us say here let us say the radius of the pipe is r. See many times for discussion we will use a circular geometry many times we will use a parallel plate geometry but the physical concept is the same only the mathematics involved is different. So if you say that if r is the radius of this pipe or tube so in microfluidics of course we will not commonly use the term pipe because typically these are narrow so these are often called as microtubes like an example of a microtubes commonly used for medical applications is a microneve. So microtube let us say r is the radius of the tube then what are the forces acting on this element so let us say this is force due to pressure p star into what? Pi r square right here p star plus dp star into pi r square then there is a wall shear stress this is basically distributed over the circumference so tau wall times 2 pi r dx this element is fluid element is non-accelerating under the action of these forces so these forces must be balanced that means we can write p star minus p star plus dp star into pi r square minus tau wall into 2 pi r dx equal to 0 right. So you have tau wall is equal to minus r by 2 dp star dx this is only for fully developed flow fully developed flow because we have assumed this is 0 non-accelerating you can see that this equation nicely tells you what is the physics basically there is a driving piezometric pressure gradient that drives the flow and there is a opposing viscous resistance that opposes the flow and these 2 are exactly balanced when the flow is fully developed. So when this is taking place this kind of balance takes place now from here can you tell that what happens to p star versus x when the flow is fully developed when the flow is fully developed the question is how does tau wall vary with x how does tau wall vary with x it does not vary with x because tau wall is given by the gradient of the velocity profile at the wall and because the velocity profile is not changing with x it is gradient at the wall is also not changing with x therefore tau wall is constant. So let us come back to this come to this third plot we have tau wall is equal to constant in the fully developed region that we draw if tau wall is constant dp star dx is constant right that means p star is linearly varying with x so again as I tell in all my classes I must confess that whenever I am trying to draw a straight line it becomes curved in the board. So please assume that this is a straight line eventually whenever I want to draw some curve it looks like a straight line. So you have to be careful about that whatever I draw and whatever it appears. So this is a linear variation now the interesting point what happens in the developing region. So in the developing region what about tau wall. So tau wall see tau wall scales with what mu into uc by delta right mu del u del y like that. So this is change in u from 0 to uc over the distance from 0 to delta. So the velocity gradient scale is uc by delta. Now the interesting thing is that as you go along x uc increases with x as you go along x delta also increases with x. So how will you know that this ratio whether it increases with x or decreases with x right you get my question that here is an expression where the numerator also increases with x denominator also increases with x. So how do you know that the resultant fraction or whatever expression need not be fraction resultant expression increases with x or decreases with x. So for that we have to use a very critical judgment see what is delta as x tends to 0 at x equal to 0 actually delta is not defined that is a point of singularity but x tends to 0 plus little bit away from the entrance infinitesimal distance away from the entrance at that location what is delta delta will be tending to 0 but uc is not unmounted uc is still finite. So some finite number divided by something tending to 0 will be very large and then delta becomes suddenly finite. So that means from a large it should come down to a small value. So it should be something like this and analogously here dp star dx because you have an acceleration term dp star dx has to be non-linear. So it will be something like this okay. So these pictures are summarizations of what goes on in an internal flow and what do we understand about a fully developed flow. One important point we have to keep in mind we have to carefully keep in mind that we have discussed this by assuming that the flow is laminar flow. We are not considering turbulent flow as a part or as a scope of this particular course which essentially handles low Reynolds number flows. I mean there are possibilities by which you can sort of discuss like turbulent flows over relatively lower Reynolds numbers and that is a very important area of research in the microfluidics domain right now but for a basic coursework perspective we are not coming into that analysis. So now whatever we have learnt to summarize the part of that let us go to the slide. So if you recall so there is a channel. So these dotted lines are the edges of the boundary layer and this is the inviscid core region and the boundary layers merge beyond that the flow becomes fully developed. So the physics of the developing and the developed region can be understood by this sort of simple statements which I have jotted down only for the beginners. These are not very serious rigorous mathematical statements but just to give you a qualitative understanding that as the fluid enters the channel wall resistances come into play. The driving pressure gradient is aimed to overcome the viscous resistances. When we say the driving pressure gradient we essentially mean the driving piezometric pressure gradient. Beyond a wall adjacent viscous layer the fluid does not feel the effect of the wall right which is the that viscous layer is called as the boundary layer. To ensure that fluid flows at the same rate over each section fluid in the core region must accelerate and edges of the boundary layers from the opposite faces meet at the center line at a section beyond which there is no further change in the velocity profile and that is the fully developed velocity. So of course like you can also have fully developed turbulent flow but there you have to talk in terms of the statistical description of turbulent flow when you talk about fully developed otherwise like a turbulent flow will always have all the velocity components and the velocity components function of time. When you only statistically average that information or statistically process that information you can discuss about something which is called as fully developed turbulent flow which is a sort of a misnomer I would say because turbulent flow I mean in terms of the instantaneous velocities it will never be fully developed. So next we will start with a very simple derivation of fully developed velocity profile in a parallel plate channel. We will work this out in the board. So the question is why is the parallel plate channel interesting? Why should we study parallel plate channel? What is a parallel plate channel? So parallel plate channel is a channel made of 2 parallel plates where the dimensions are let us say this is 2h this is 2w the axial direction if the axial dimension if the flow is fully developed will not be important because velocity will not vary along the axial direction if the flow has become fully developed. Now if this w is much much greater than h then interesting velocity gradients take place only along the direction of h because along the direction of w that is say the z direction the dimension is so large the gradient is what? Change by the dimension the scale of the gradient that gradient is very small because the dimension is large. So primary gradient takes place along the smallest or the shortest dimension and that kind of situation is very common in micro channel. See micro channel does not mean that all the dimensions are microns like you can have one of the important features which gives the characteristic length scale of the order of microns but others may be millimeters. So the width may be millimeter but this h may be of the order of microns. So then you can abstract the problem like a flow between 2 parallel plates separated by a distance of 2h the problem is translationally invariant along x because the flow is fully developed but your interesting variation is the variation along y. So it is important for us to figure out that how the velocity varies with x and what component of velocity the axial component of sorry how the velocity varies with y what varies with y how u varies with y. So in our next lecture we will begin with this issue and see that how u varies with y. We will not stop there but we will see that if the flow is not fully developed but still of Lorentz number then how do the consequences of that relate to this concept of fully developed flow. So we will take a short break now and we will continue with the next lecture after the break. Thank you.