00:01
So in this video, we'll be looking at three questions.
00:05
The first question is showing that the reynolds number is a ratio of inertial force to viscous force, which is a dimensional analysis.
00:15
So let's look at the part a of this question.
00:20
So we need to prove that the reynolds number, re, is a function is a ratio of inertial force divided by the viscous force now the reynolds number is given by the definition rho v l by mu now rho is the the density of the fluid.
01:01
V is the velocity of the fluid.
01:07
L is the characteristic length.
01:16
The reason it is said as characteristic length is because it's not just pure length.
01:22
It depends on the objects.
01:24
For instance, if there is a flow that is flowing inside a pipe, you're looking at the characteristic length to be the diameter of the pipe.
01:32
If there is a flow that is flowing over a car you're looking at the characteristic length of the car to be either the length of the car itself or the height of the car now considering that we are looking at the reynolds number as a ratio of inertial force to viscous force let's let's take a let's take a quick diagram so assuming that you have a you have a plate okay and the flow is flowing through this plate yeah so what would happen here is i'll draw slightly below so that it's much easier so if you look at a flow that's flowing inside let's assume that there's a pipe or something so you've got a pipe here and this is the center line of the pipe assuming that there is a flow so the flow is going to behave like this so closer to the pipe the velocity of the flow is going to be zero and away from the pipe so if i draw a normal to it this is the normal point okay away from the pipe okay so you've got a flow like this away from the pipe which is towards the center of the pipe the velocity will be very high closer to the pipe because it's close to the pipe the fluid will try and stick to the pipe because it's sticking to the pipe it has a viscous effect closer to the pipe more strongly than away from the pipe so center line of the pipe will have a faster flow compared to the flow that is closer to the pipe.
03:26
Now with this information in place we can say that the velocity v, if the velocity is v as said before and there is some boundary layer which means that you've got some viscous effects closest to the pipe and this is the length of the pipe or the length that we are looking at in terms of the characteristic let us take length l, the inertial force, the numerator can be given as rho v multiplied by dv divided by dx, where this direction here is your x.
04:20
So how velocity changes with respect to x is given by the inertial force multiplied by the velocity and the density.
04:31
Now, the viscous force is given by mu, which is the viscosity, multiplied by the second order, d squared v by dx squared.
04:49
D squared v by dx squared, which is the change, the rate at which the velocity is changing.
04:58
Now, basically, what they're saying is, if you take the ratio of these two, the inertial force let's call this to be iv and discuss for which is vf i'm sorry uh if um then we call this to be vf so if you take so what they're saying is prove that uh re which is the reynolds number which is if by vf so if you take that ratio so you are getting rho v um um dv by dx divided by mu d squared v by dx squared now in terms of so so what they've said this is in the question they've clearly said that use the dimensional analysis okay so in terms of dimensional analysis you can see that um um rho v and the dimensions of dv by dx would be nothing but v by l and the other dimension is mu v by l squared because that's how the dimensional analysis plays between dv by dx for v by l and d squared v by dx squared for v by l squared.
06:21
Now, with this in place, you can see that the dimensions of rho are retained because there's nothing to cancel.
06:31
V will still be there.
06:34
But then you've got these two vs canceling off and one of these lengths canceling off here and giving you l.
06:47
So therefore, you would end up getting rho v l divided by, because this is in the denominator, that will go in the numerator.
07:00
Mu is already in the denominator.
07:02
So you would get re to be equal to rho v l by mu.
07:06
So this is the definition of reynolds number anyway.
07:08
So if you look at how we have defined rho vl by mu, this is the rho vl by mu.
07:13
Therefore, we have proven that the reynolds number is nothing but the ratio of inertial force divided by the viscous force.
07:20
So that basically solves problem a.
07:22
Now problem b talks about under what conditions flow is likely to be turbulent or laminar.
07:29
Now, and they've said that, give two examples.
07:32
Now we'll talk about two types of flows.
07:34
We'll talk about internal flows.
07:38
And external flows.
07:43
The condition for internal flows is that your re should be greater than 2600.
07:50
For external flows, flow over a car, flow over a bus, etc., your reynolds number should be much greater, greater than probably 500 ,000...