 So, now that I have the Navier Stokes equation let me now precisely define what I mean by this Reynolds number. So, let me write it nicely del v del t plus v dot del times v is equal to minus 1 by rho gradient of p plus eta by rho del square. Now, let us say I have some body of size L let us say I have some rigid body, rigid body of size L which is moving through this fluid at a speed u. The strength of the inertial terms will be given by this right. So, the inertial terms inertial terms is this v dot del of v which to an order of magnitude. So, this is two velocity terms right v into v which means if it is moving with a speed u with like u square it has a spatial derivative which means it is something like an u square over L. The viscous terms are given over here. So, the strength of the viscous terms strength of the viscous terms is over there which is eta by rho this is two derivatives with respect to space and one velocity. So, one velocity and two derivatives L square ok. So, this characterizes the strength of the inertial terms in this equation this characterizes the strength of the viscous terms in this equation. If you take the ratio of these that gives me remember my Reynolds number which characterizes how important the inertial terms are compared to the viscous terms right. So, that is like inertial is u square by L viscous is eta by rho u by L square right which means it is rho L u by eta. This is what is called my Reynolds number. So, Reynolds number for a body of length L or dimensions L moving with a velocity u in a fluid of density rho and viscosity eta is this rho times L times u divided by eta. So, this Reynolds number which determines the physics like we saw in these Taylor's experiments G i Taylor's experiments is not simply a function of the fluid properties only. It is not simply a property of the viscosity of the fluid, but it also depends on the body that is moving through this fluid. It depends on the dimensions of this body, it depends on the velocity of this body ok. So, the dimensions of that die nozzle or the velocity of that die for example, which is why even if you take the same fluid for example, water and you move through it and you are like 6 feet tall or whatever nobody is 6 feet tall. You are 5 feet tall versus let us say an E coli which is a micron long that L term over here will change from 5 feet to 1 micron and that will bring down your Reynolds number even though the fluid is exactly the same. The viscosity of the fluid is the same let us say the density of the fluid is the same ok. So, the same fluid will have completely different Reynolds number by orders of magnitude right 5 orders of magnitude 6 orders of magnitude depending on what body is moving through that fluid right. So, depending on depending on the velocity of course, so if you move very fast if the E coli were to speed through then it would not then to might increase its Reynolds number for a typical, but for a typical swimming speed which is again tens of microns, typical length which is again microns the Reynolds number for these objects are going to be very small. So, typically in biological fluids I think will if you put in the numbers you will get Reynolds numbers. So, of the order of maybe 10 to the power of minus 5 minus 4 something like that and for this macroscopic world that we are used to you will get Reynolds numbers in the regimes of 1000s or tens of 1000s ok. So, these are many many orders of magnitude apart and so depending on whether this term is small or this term is large depends on whether this term will play an important role or this term will play an important role. If the Reynolds number is very small ok which means that this inertial forces are very small you can maybe neglect this inertial forces ok neglect these inertial forces then the equation that you will have will comprise only of the special gradient force and this viscous force ok and that is what is called actually the Stokes limit or the Stokes equation. So, we will do that we will do the Stokes flow at least in a simple case next class, but depending on whether you can neglect this inertial terms or not the solutions of this equation will be very different ok. So, that is the whole idea that when you are looking at these biological systems they live in a world which has very very low Reynolds number which means we can neglect these inertial terms and once we neglect these inertial terms you get the solutions of this Navier Stokes equation that you get or the Stokes equation in that case are very different from the solutions of this full Navier Stokes equation. And in fact, the Navier Stokes equation the difficulty in the Navier Stokes equation arises mainly from this term. This term as you can see is a non-linear term right it has two velocities v square which is why it makes general solutions of this Navier Stokes very difficult and in fact, if you can solve it you get 1 million dollars I think. It is one of the clay millennial problems if you can show that a solution exists you get 1 million dollars. So, it is an open on the general solution is an open unsolved problem. But as long as you are talking of Stokes flow of these low Reynolds number flows we do not need to worry about these inertial terms you only need to worry about these terms plus any external forces. All right. So, we will continue on this next class let me see. So, since I have 5 minutes let me just define for you although we will not talk about it let me just define. So, these remember I derived all of these for Newtonian fluids let me just define a few types of non-Newtonian fluids and how the viscosity comes in that case. So, for a Newtonian fluid remember So, for a Newtonian fluid you have force upon area which is the stress let me call that tau that is equal to eta the viscosity times del v the gradient of the velocity which is like my strain rate right. So, this is my stress this is my stress that is my strain rate and I have a linear relation between them for a Newtonian fluid and with the constant being the viscosity. If you have a non-Newtonian fluid you can have different types. So, one for example, is called the shear thickening fluid shear thickening fluid and in that case you get this tau is equal to eta del v del y to the power of some n some exponent where this n is larger than 1 ok. So, the effective if you talk of some effective viscosity that sort of increases the more stress you put on it ok. Those are called shear thickening fluids for example, this corn starch in water is a shear thickening fluid. On the other hand if you you can have a shear thinning fluid you can have shear thinning fluids the same expression, but now with n less than 1. So, here the effective the viscosity decreases with increasing stress and things like paint and blood and so on are shear thinning fluids right. Paint for example, if you paint it as you apply the shear force on with your brush it flows, the moment you stop it it stops flowing. So, as you increase the shear force the viscosity decreases and it starts flowing if there is you decrease the force it stops flowing. So, it is a shear thinning fluid same for blood. If you were a mass murderer you could try an experimental with blood as a non-Newtonian property. You could sort of try to apply a shear force and see whether it flows when it flows and when it does not flow. There is another for example, you could there is another class which are called Bingham fluids which are called Bingham fluids where it does not flow until a certain critical stress and then starts flowing. So, del v del y is 0 if your stress is less than some critical stress and then it is tau minus tau c by eta if your stress is greater than its critical stress. So, this for example, I think is toothpaste is an example of this needs certain amount of critical force for it to sort of start flowing below that it does not flow. So, there are different kinds of non-Newtonian fluids that you can get and for this this Navier-Stokes equation will not be valid you need to make corrections in the derivation to account for this non-Newtonian character, but we limit ourselves for the time being to this Newtonian fluids we will assume that all fluids are Newtonian and we will simply try to look at what are the effects of this low Reynolds number. So, what happens when these Reynolds numbers are very low if I just write down this Stokes equation then what sort of solutions can I come up with and what does that basically tell me for these biological microorganisms that are swimming in these in these sort of low Reynolds number environments ok. So, that is what we will do for the next couple of classes. All right. So, I will see you.