Insects were the first creatures to take to the air in powered flight - 350 million years or so ago, back when our ancestral line were all still fishes - and they still seem to have a few aerodynamic tricks to teach us. The number one problem of heavier than air flight is how to support oneself against gravity, and the universal answer for powered flight is the wing. Flight is accomplished by dragging a wing (fixed, rotating, or flapping) through the air at an angle and thereby causing air to be accelerated downward.
The details are in Newton’s second and third laws. Law three says that if a wing exerts a downward force is exerted on air, and equal opposite upward force will be exerted on the wing by the air. The second law says that keeping an object of mass m aloft against gravitational force mg requires transferring downward momentum to air at a rate of dP/dt = d(Mv)/dt = mg where M is the mass of the air and is its speed. For example, if dM/dt is the rate at which air is recruited for downward acceleration, it will need to be accelerated to a speed of v = mg/(dM/dt).
Of course I have left out a few details, like propulsion and drag. Propulsion is similarly accomplished by accelerating air in a backwards direction, and drag is that combination of friction and other forces exerted by the air that hold you back.
Flight poses some special challenges for the smallest flyers, the first of which is that viscous drag forces increase with decreasing Reynolds number Re = UL/υ, where U is speed, L a characteristic length, and υ, the kinematic viscosity of air, is roughly 10^(-5) (m^2/s). For the very tiniest flyer of all, a parasitic wasp only 10^(-4) m long, with a flight speed of a few 10s of cm/s (10^(-1) m/s), Reynolds numbers may be in the single digits, whereas for a 747 Re is in the tens of millions.
When you move your hand through water, or a wing through the air, there are a couple of kinds of forces you need to overcome. One is the inertia of the water (or air), that you have to accelerate to push out of the way. The other is the frictional drag that the water or air exerts on us as it slides past. Roughly speaking, the Reynolds number is a measure of the ratio of the inertial forces to the frictional forces. For people sized objects, or 747s, the inertial forces are nearly always huge compared to viscous (frictional) forces in air and water – though molasses or wet cement would be another case.
More familiar insects, say honeybees, operate in an intermediate aerodynamic zone , with U around a m/s and L about 10^(-2) m, so that Reynolds numbers are in the thousands – out of the wet cement zone but still small enough to result in much larger drag forces than experienced by larger flyers. That fact is responsible for the familiar claim that “aerodynamics proves that honeybees can’t fly.” Of course they do, and very well, but they need to depend on a few aerodynamic tricks that aren’t in the bag for larger flyers like albatrosses or 747s.
A second challenge stemming from their small size is that rather small forces can toss them about. Moment of inertia is a measure of the resistance of an object to being rotated, and moment of inertia scales like the fifth power of length. Thus a 747, which is about 10^4 (10,000) times the length of a honeybee is (10^4)^5 = 10^20 = 100 billion-billion times as hard to flip over as a honeybee. Rather small gusts of wind can and do flip honeybees about.
Evolution, however, has equipped them well to deal with such challenges. In particular, their flight instrumentation includes special purpose horizon sensors, onboard navigation, and highly integrated visual and airspeed sensors.
Take a look at the honeybee eye on the left. Notice all the fine hairs projecting out of those eyes. Why in the world would a creature evolve hair in its eyes? One plausible answer is as follows: both the hairs and the eyes serve as speed sensors – the eyes measuring ground speed through optical flow and the hairs measuring airspeed by means of the deflection by air flow. The honeybee’s distant and much smaller cousin, the fruit fly has physically separated air speed (the antenna) and ground speed sensors (eyes) – of course they, and other flies, also have inertial navigation systems - , resulting in a millisecond scale neural path delay of the signals that makes its flight slightly herky-jerky. By collocation, the honeybee perhaps can more quickly integrate these signals for optimal flight. On the other hand, the bee also has hairs almost everywhere else, so the real explanation may be less interesting.
Back to the fundamental problem of lift generation – sliding that wing through the air directs a stream of air downwards, but also generates drag. One type of drag, called lift-induced drag, can be thought of as deriving from the need to expend energy to accelerate that downward stream of air. If you think about the energy and momentum balance equations, you can see that it’s energetically ideal to accelerate the largest practical volume of air to the lowest speed compatible with the need to supply a momentum change at a rate mg. In practice, that means that faster flight (giving you a shot at more air to accelerate) is more efficient than slower. It also shows why hovering flight is so energetically expensive (for helicopters, hummingbirds, and dragonflies, for example).
The other type of drag, sometimes called parasitic drag, results ultimately from work against the friction of the air. Much of this energy goes into generating vortices in the air, though some of it, especially for very small flyers, is direct work against friction. This drag, by the way, increases roughly as the square of velocity.
I mentioned that bees and other insect flyers have other tricks for getting lift. Essentially these reduce to harnessing some of the energy is being pumped into vortices to gain additional lift (accelerate more air downwards). At least one of these tricks, so-called delayed stall, has been put to work for a high performance fighter jet.