Viscosity Concept - Derivation and Discussion

Today, I'm going to talk about something called viscosity. It is important that you learn this within this course because typically, in the undergraduate curriculum in mechanical engineering, this is the only course where you will be exposed to viscosity, and it's a very important fluid parameter that will determine the way the fluid behaves. And solids do not have any viscosity, so this may be a new concept for us. Okay, so before I go ahead and make a mathematical derivation of the formula, I would like to talk about what it is, okay. So typically, what when people say viscosity, they talk about fluidity. The definition could be said as this: quantifies the resistance of a fluid being sheared. So what it says, it is quantify, so it puts a mathematical number, how resistant the fluid is being to sheared. Okay? So, one of the best examples that I can give is between water and honey. Let's say that I have two exact same volumes of cups, okay. One is full with water, the other one is it honey, or you can call it maple syrup as well. So if I'm interested in pouring them out, okay, so I tilt this 90 degrees and I'm pouring it out. You will all know that the water is going to be poured out much faster than honey. So this is what I'm talking about when I say viscosity. That's the difference between those two liquids, okay. But now, the goal here is to quantify it, because as we’re engineers, I need to put some number to it, okay. That's what viscosity is. One of the other common examples is from the oil industry. Okay? Let's say that you are changing your own oil. You go to an auto parts store to buy your oil. Let's say that you pick a particular brand. You're not actually quite done yet. The thing that you will see is that there will be some writings on the oil bottles. It will say 5W30, 0W30, etc. So actually, these numbers that I’m showing here refers to the viscosity numbers. So how do they accomplish is a huge project by itself? But basically, there are polymers that are coiled up over here, and then when the operating temperature is raised, it’s a polymers unwind. This is beyond the scope of my class, introductory fluid mechanics class, but I want you to realize. And some other thing that I hear, as engineers you maybe you should know this is these are just kind of like ranges. Okay? The viscosity is not five or thirty. Viscosity numbers is we will find out is fairly actually low, 10 to the minus 3, 10 to the minus 5, etc. Okay? Instead of saying that 1.12 times 10 to the minus 3W 1.6 times 10 to the minus 3, it's not going to go well, right? So we are just making up some numbers associated with it. Okay? And I see sometimes online people talk about: “ Oh, Zero? I'm not going to put zero into my engine, I might as well put water." Those are not quite fair comments. We talked about quantification, well. Let's quantify, okay. So, what I have here is two parallel plates. The bottom plate doesn't move. The top plate moves with a velocity, let's call this capital U. Let's determine this is the x direction, and this is the y direction, it doesn’t have to be. The distance between the plates are B. So now, I need to introduce you a condition. Okay? I'm going to call this no-slip condition. Let me write it down. At a solid surface, the fluid velocity component tangent to the surface is equal to that of the surface, okay. When you first read this, this may sound like a very complicated statement. At a solid surface, the fluid velocity component tangent, virtually tangential, to the surface is equal to that of the surface. Well, actually, this is not as complicated as it sounds. Okay, so basically what it's saying is, right I'm over here. Obviously, I have some fluid over here, right, that's what I'm assessing. And the very first fluid particle, and remember that everything is made of molecules, right, and atoms, the very first molecule right over here will stick to the surface. There'll be no slip between that very first molecule and the solid surface, so they will move with the same velocity. So, in this particular case, the velocity of the fluid over here will be zero. And from no slip condition, I will understand that the velocity of the fluid right up over here is capital U as well. Okay, remember, I need boundary conditions, but I need boundary conditions of the fluid. Basically, I established that this is, let's call this t is equal to zero. And let's see what happens. The velocity here is zero, and let's say that the velocity here is capital U, this much, right? Let's say capital U, this is this, right? So let me plot the velocity distribution. So there’s multiple options over here, okay? This can be parabolic, this can be linear, right, in real life. So the first thing I'm gonna do is I'm gonna take a linear velocity profile. I'll talk about parabolic after this. Okay, so what happens is, this is my velocity profile, and not every single fluid will follow this linear velocity profile. And I'll talk about that as well. Let's say that time has passed and t becomes dt. So now, question is, why do I call it dt? What is so significant about it? Basically, what dt signifies is that the time that passed is fairly small, okay? Let's say one microsecond later. So what will happen to the point over here? Is it gonna move to the right? The answer is no, because the velocity is zero there from the no slip condition. Bottom plate is stationary. How about the top plate? Is this gonna go to the right? The answer is yes. And what is the distance that it will travel? The top plate right over here, it will travel some distance. And what is this distance is my question to you? Well, if you remember from your physics, distance is equal to velocity times the time. So, if you think about it, if I'm traveling 60 miles per hour, in one hour I'm gonna travel 60 miles, right? From the exact same logic, except it looks a little bit more complicated, this is gonna have capital U times dt. That's how much amount of time has passed, very little amount. I'll also talk about why we need to say very little amount. And it’s right now. This angle that I make, I'm gonna call this d alpha. So why do I call it d alpha, not a regular alpha like 5 degrees, 10 degrees, 45 degrees? Well, here's the reason. I call that this is dt, and this is a very small number. So this distance that I'm looking from here to here will be very small. As a result, this angle will be very small as well. Okay, then let's go ahead and write my tangent of that angle. I'm interested in tangent of d alpha, see what happens. So what will it be? It will be, as you remember, the definition of tangent, it's this divided by this, right? So that will be U, capital U dt divided by b, that's the distance between those two plates. So I need to do one more step as well. So now I'm gonna take a much closer look. Instead of writing this U dt by this b, I want to just write it as the angle is constant, d alpha is constant. I want to write in this region, okay? So I want to take a little bit distance from the surface, and I'm gonna call this dy, very little amount of surface. Okay? What will be the corresponding velocity when the y is equal to dy? That would be du, the velocity will be small, du. So I can also write this way as well. Instead of the exact due to the tangent definition, I will get myself a small, you know, this distance divided by this distance, right, tangent. So as you can see, this will be the du dt this time because the velocity is du, the time is the same, it doesn't matter whether I'm taking a large scale or small scale, divided by the distance pointing up, and that would be dy. One of the oldest tricks in mathematics is this. When the angle is small, tangent of alpha is equal to itself, alpha. This is an approximation. This is not 100 percent accurate, but it works well. From here, I'm gonna get tangent, let’s write it in here. Tangent of d alpha will be equal to d alpha. And so, d alpha is equal to, let's write the second one, du dt. du times dt divided by dy. So let me arrange this a little bit better way. d alpha dt will be du dy. If I want to write it for the large scale, it's gonna be capital U divided by B. I'm still there, that part is fine from the mathematics derivation point of view. This is as far as I can go. The next step is to talk about a particular theorem. You may remember this person's name. He is everywhere. His name is Newton. He is here as well. Newton's law of viscosity. So what he says, I'm gonna just write it now, I'll talk about it. The shear stress on a surface tangent to the flow direction is proportional, not the same, proportional to the rate of change of shear strain. And let me just put something you will understand what I have been in this d alpha dt. The shear stress on a surface tangent to the flow direction is proportional to the rate of change of shear strain. So basically, what it's saying is the shear stress, I'm gonna call this at the wall, W, because it says at a surface. Or you can call S as well. It's saying that this is proportional, not equal to d alpha dt. So that's what this law says. So let me see and make a connection between those two. Look here. I said that d alpha dt is equal to du dy. So I want to actually, this is much more convenient for me, nothing more, nothing less. I'm gonna change that to du dy. So this is also equal to du dy. There has been experiments done to really assess whether this law is valid. Note that I didn't say linearly proportional, I said only proportional, right? So I'm gonna take a one additional step, and I'm gonna say that these are linearly proportional. And let me discuss. Is this applicable to every single fluid out there? The answer is no. The fluids that obeys this law, we call that Newtonian. So this is for Newtonian. And this mu is called absolute viscosity. Some people also call this dynamic viscosity. Now, so I said that this is only applicable for Newtonian fluids. So what are they? Are they something that we don't really encounter day today? I have some good news. The good news is, majority of the fluids that we encounter day today will follow this. Water, oil, air, all three will follow this. And note that when I say linear, the proportionality constant becomes the viscosity. Some fluids do not follow this. I'll talk about that briefly as well. Like one of the examples in my research area is human blood is not really Newtonian, okay? So that's a limitation from my end. There is also an interesting observation between the. Okay there's actually a two path over here, interestingly. Okay? So one is, if this is a liquid, what happens is this becomes negative value. What happens is, when I increase my temperature, the viscosity reduces for liquids. On the other hand, gases, this is opposite. When I increase the temperature, the viscosity of a gas increases. In the first segment about this module, I mentioned that what is the differences between liquids and gases. I said it's incompressible, one is compressible, right? This is another one that I would like to now note. There is another version of the viscosity, and that is also commonly used. Basically, this is the nu. It will be the mu divided by Rho, which is the density. This is referred to as kinematic viscosity. Question is this. You may very well ask me this. Hey, you're taking a constant because it is at a temperature and pressure, this is a constant number. And I look at the density, also it's a number. So why are you dividing two numbers and giving me another number? Well, it's by convenience. This occurs a lot. Again, we'll talk about these all in the future modules that you will go through. But this occurs a lot in fluid dynamics, so I don't want to repeat myself. Okay, there is something called inviscid flow. Basically, what it means is my viscosity is equal to zero, inviscid. I'd like you to note that there is no fluid in real life which has viscosity equal to zero. There are some fluids that has very low viscosity values, but you cannot simply say they are zero. So this is a good thing. If we want a first-order approximation of our analysis, and we can assume inviscid for basically analytical treatment of the fluid flow phenomena. This is the Newtonian fluids, right? This relationship is only for Newtonian fluid. So let's look at the cases for non-Newtonian fluids, okay? Or all the fluids? What will happen is there's something called a power law. Ostwald-de Waele, typically power law is how they refer to this. What I get is this: shear stress at the wall is gonna be K, this time around. Del u del Y, just like the one, to the power of n. n is a non-dimensional number. K is a flow consistency index is typically was referred as. These are basically material properties that you can obtain. Okay, I want to talk a little bit more about non-Newtonian fluids in general. So what we said is, I want to plot a graph like this here. So this will be the shear stress at the wall, and I want the x-axis to be du dy. And du dy is the rate of shearing strain. So if I have, so basically what I'm saying is, K is the viscosity, n is equal to 1, and I call this Newtonian. So now, there's a different type of fluids as well. The first one, let's look at this case when n is larger than 1. Okay, so it's gonna be like this. You can see the curve to it, right? So the n will be larger than 1 over here. And we call this shear thickening fluid, or we call this dilatant as well. And I'll give some examples from this. A typical example is water-starch mixture or quicksand. One thing that you may need to note is, when I move more, when I move more towards over here, I got deep moat inside, right? So that's the reason for dilatant. On the other hand, I have pseudoplastic like this. So you can see in here, n is now less than 1. And we call this pseudoplastic. You'll see how it's related to the plastic as well in a minute. We also call this shear-thinning fluid. Examples are from my research area. It's blood plasma, many of the polymer solutions, colloidal suspensions, paint — all follow the pseudoplastic. So it's fairly common. The other thing, interesting thing, there are some fluids that doesn't act like a fluid until I increase my shear stress to up to a level. Let's say that I go over here, and then it goes like this. This is called Bingham plastic. The typical example for this is toothpaste, mud, clay. Think about the clay. It acts like a solid until if you spin it. And if you want to make a vase off of that, you can, right? Because you're increasing the speed of it. So it acts like a more fluid. So you can give it a shape. Same thing with mud, or the toothpaste as well. When toothpaste is in the tube, it acts like a solid, right? But when squeeze it, comes out, like a fluid. That's what I'm talking about here.