70 CHAPTER 4. SYSTEMS OF EQUATIONS
θ
V
BA
The angle is called the angle of incidence, B is the span of the wing and A is called thechord. Denote by l the lift. Then this should depend on various quantities like θ ,V,B,Aand so forth. Here is a table which indicates various quantities on which it is reasonable toexpect l to depend.
Variable Symbol Unitschord A mspan B mangle incidence θ m0kg0 sec0
speed of wind V msec−1
speed of sound V0 msec−1
density of air ρ kgm−3
viscosity µ kgsec−1 m−1
lift l kgsec−2 m
Here m denotes meters, sec refers to seconds and kg refers to kilograms. All of theseare likely familiar except for µ . One can simply decree that these are the dimensions ofsomething called viscosity but it might be better to consider this a little more.
Viscosity is a measure of how much internal friction is experienced when the fluidmoves. It is roughly a measure of how “sticky” the fluid is. Consider a piece of areaparallel to the direction of motion of the fluid. To say that the viscosity is large is to saythat the tangential force applied to this area must be large in order to achieve a given changein speed of the fluid in a direction normal to the tangential force. Thus
µ (area)(velocity gradient) = tangential force.
Hence(units on µ)m2
( msecm
)= kgsec−2 m
Thus the units on µ are kgsec−1 m−1 as claimed above.Then one would think that you would want
l = f (A,B,θ ,V,V0,ρ,µ)
However, this is very cumbersome because it depends on seven variables. Also, it doesn’tmake very good sense. It is likely that without much care, a change in the units such asgoing from meters to feet would result in an incorrect value for l. The way to get aroundthis problem is to look for l as a function of dimensionless variables multiplied by some-thing which has units of force. It is helpful because first of all, you will likely have fewerindependent variables and secondly, you could expect the formula to hold independent ofthe way of specifying length, mass and so forth. One looks for
l = f (g1, · · · ,gk)ρV 2AB