‫Welcome back.
‫So in order to rewrite this differential thrust in terms of differential lift and differential drank,
‫we first have to look at this fine angle.
‫You have this fine angle over here as well.
‫It's this one here in the same triangle is also here like this.
‫So for the differential thrust force, you only need to extract the vertical components of both of these
‫vectors.
‫And you can see that this deal has a positive vertical component like this and this deal has a negative
‫vertical component like this, so therefore the T equals the L times.
‫And in order to get this vertical component of DL, you simply have to multiply this DL by cosine fi.
‫And to get the negative vertical component of this director, you simply have to put here minus the
‫the Times and sine fi and that's it.
‫That's how you write this equation.
‫Now of course you might ask now why we did that.
‫The reason why we did that is because from aerodynamics we can take formulas for DL and so for DL.
‫The aerodynamics gives us this equation.
‫DL equals C sub L times one half times the air density row times you squared, which is this velocity
‫vector here, the resultant air velocity vector, then times C times D hour and then the differential
‫drag equals C D times one 1/2 times the air density times you squared times C times D.R.
‫So let's very quickly look at these terms and see what they are.
‫SEECP L is called lift coefficient.
‫This lift coefficient is a dimensionless quantity that essentially tells you how good the shape of your
‫airfoil is for lift generation.
‫A bigger lift coefficient means that your shape is better for lift generation.
‫And it makes sense, right?
‫The bigger the s.L, the bigger your deal.
‫However, S.L is not constant.
‫It depends on three things.
‫It depends on the angle of attack, which is this one here.
‫It also depends on the Mach number and on the Reynolds number, first of all, let's look at the angle
‫of attack.
‫I'm going to draw here a typical graph for s.L versus Alpha.
‫So this is S.L here and this is Alpha and a typical graph looks like this meaning as you increase your
‫angle of attack, which is this angle here, you also increase your lift coefficient.
‫However, when the angle of attack becomes too large, then your lift coefficient starts dropping,
‫and that's when, for example, an airplane enters into a stall.
‫So essentially, what you want for live generation, you want a smooth flow around the airfoil.
‫That's what we call laminar flow.
‫But when you're angle of attack is too large, then your flow will become turbulent.
‫Something like this, and as a result, you lose lift and gain a lot of drag.
‫And you will start falling down then.
‫And in order to avoid falling down, then you have to put your nose down again, meaning you have to
‫decrease your angle of attack.
‫And then you have something called the magic number and the magic number is the ratio of your velocity
‫over the speed of sound.
‫So your magic number equals your you velocity divided by the sound.
‫And so if you're below one, if your magic number is below one, then you travel under the speed of
‫sound.
‫And if it's above one, then you go supersonic.
‫And your lift coefficient also depends on the magic number.
‫And finally, you have something called a renals number, which also comes from aerodynamics.
‫The Reynold's number is a ratio that is used in aerodynamics to determine if the airflow over the wing
‫will be laminar or turbulent.
‫The difference between the laminar and turbulent flow is the following.
‫This would be a laminar flow.
‫It goes over the wing in a very smooth way and this is a turbulent flow.
‫It detaches itself from the wing and it becomes very chaotic and it's very hard to model this kind of
‫flow.
‫And so this Reynolds number, it helps us to predict what kind of flow we will have when the air goes
‫over some kind of wing.
‫So I mentioned that Reynolds number is a ratio.
‫And one way to write it is like this rule times, Soviet times all over new roadways, the air density.
‫Then why is the air velocity that would be Revy and then El is essentially the cross-sectional length
‫of the object.
‫It can also be called cord.
‫And then this mbewe here.
‫It measures the viscosity of your fluid.
‫So what is viscosity?
‫If you think about water versus honey, then you know that honey is more gluey, right?
‫Honey doesn't change its shape as easily as water.
‫And so honey is more viscous than water.
‫And so if you look at this ratio, then essentially the denominator, it contributes more to the fluid
‫being laminar.
‫So the bigger this new value, the more it contributes to the flow being laminar.
‫Why?
‫Because the viscosity is greater and viscosity is like gluing this and this glue in it will help the
‫fluid to be more laminar and not become chaotic, like in the case of a turbulent flow.
‫On the contrary, the numerator, the bigger it is, the more it contributes to the fluid becoming turbulent.
‫So, for example, the bigger the velocity, the easier it is for the flow to detach itself from the
‫wing and become turbulent.
‫And that's why when you have lower Reynold's numbers, then the flow tends to be laminar.
‫And that is very good for generating lift.
‫By the way, you want your flow to be laminar.
‫You don't want it to be turbulent.
‫You want it to be laminar because then you will have high lift forces and it is a lot easier to model
‫the airflow when it's laminar.
‫And so if your value for viscosity is greater, then this ratio becomes smaller.
‫And so a smaller ratio means a smaller Reynolds number.
‫And that's why laminar flows happen for smaller, rainless, no values.
‫But when your numerator becomes bigger relative to your denominator, then your renals number increases
‫and that contributes to the fluid being turbulent.
‫And so your lift coefficient also depends on your renals number.
‫In conclusion, your S.L depends on your angle of attack, your Mach number and your Reynold's number.
‫So that was the lift coefficient and now that's the air density that you have is just air density in
‫kilograms per cubic meter.
‫It's how dense your air is.
‫Then this you vector, that's your resultant air velocity vector, but not in this formula.
‫You have to square it.
‫Then see is the core length, which is this one in green.
‫And then finally, your D-R, that's your differential with the differential width of your airfoil,
‫so you can imagine that this Sea Times D.R, it's really your differential area, but not the differential
‫area of your upper surface.
‫Rather, if you think about it, then this is your court length, then if you multiply them together.
‫Then you will have this differential rectangle inside this airfoil.
‫That looks something like this, and that would be your differential area here.
‫All right, and finally, let's talk a little bit about the drag coefficient, see some somebody.
‫And so drag coefficient, just like lift coefficient, it's a dimensionless quantity that tells you
‫if you're airfoil shape contributes to drag force more or it contributes to it less.
‫So if you're see Savides bigger than your differential drag force would be bigger and vice versa.
‫So I've looked up to Shape's on Wikipedia.
‫One is a sphere, and so if you let air go through it like this, then the drag coefficient is zero
‫point forty seven.
‫And if you have a streamlined body like this.
‫Then your drag coefficient is zero point zero for.
‫And in aerospace, you want a very low drag force and therefore airfoils, they look like a streamlined
‫body like this one.
‫And also your drag coefficient depends on the angle of attack, the magic number and the Reynold's number.
‫So it's not constant.
‫Now, what I want you to understand is that s.L and said they are not about size, they are about shape
‫to objects having the same shape but different size, assuming that they're Alpha and M and the rainless
‫number, assuming that they're all the same.
‫Then the two objects with the same shape, but different sizes have still the same s.L and the same
‫CD.
‫So that's the point of the lift and drag coefficient.
‫They help us analyze the shape of an object without considering their sizes.
‫You want to know how the shape of an object contributes to the lift and drag force without caring about
‫the size of that object?
‫Now, you know that this new vector can be rewritten like this, you equals square root.
‫Or mega times are squared, plus the sub, the squared, and we got that thanks to Pythagoras Theorem.
‫That means that we can rewrite our differential lift and drag forces like that.
‫Diesel equals C sub L times one 1/2 times the air density, and now instead of writing you squared,
‫you can take this thing and you can square it.
‫And if you do that, then the square root will disappear.
‫So you will simply have.
‫Omega are squared, plus Visa V squared, and you put it in the brackets here and then times C times,
‫these are and you can rewrite your differential drag force like that as well.