Tl;dr dust sticks to the bottom of ceiling fans because of static electricity and a condition in fluid dynamics called the no-slip condition, which says that air just around a moving object moves with the same speed as the object.
The dust sticks to fans for the same reason that paper clippings stick to your ruler after you’ve rubbed it really hard against your hair: electrostatics. In a similar manner, the frictional forces between the fan blades and air result in a build-up of static electricity. If you don’t know about static electricity, watch this.
The foremost edge of the blade in the direction its spinning in, usually grows the thickest layer of dust because this edge encounters the most friction as it is the first direct contact with air and develops the most charge.
Diverse materials have diverse attractions to electrons. By rubbing a range of materials against one another and challenging their resultant contact with substances of known charge, the tested things can be arranged according to their attraction to electrons. This is known as the triboelectric series.
However, the charge induced in the blade can only exert a weak attractive force – these are called Van Der Waals forces. These forces should soon discharge of their dipole and the dust would just fall off eventually. But they don’t.
Why the dust doesn’t get off
Once the dust has settled on the blades of the fan, there is no force to push or throw the dust particles off the blades. Gravity doesn’t pull them off for the same reason that we have suspended particulate matter in the air: the particles are very light. When a lot of dust accumulates, it falls down because gravity starts affecting the accumulated dust.
You might be wondering why the fan itself doesn’t push off the dust. The fan itself doesn’t push it off because the air around the blade isn’t moving. This is called the no slip boundary condition. This is a consequence of fluid dynamics that claims that at the border the velocity of the air is 0 with respect to the blade – essentially, air around the plate is standing still.
(But then where was the friction, if the air wasn’t moving right at the edges of the fan blade? It was between this steady layer of air and the other moving layers of air). Also, when particles of a fluid are at a solid-fluid interface they experience stronger solid-fluid attraction (adhesion) than fluid-fluid attraction (cohesion). The fluid in this case being the dust particles flowing with the air, experience strong adhesion to the fan surface, just like they do to your table or computer screen.
What is the No-Slip Boundary Condition
Particles close to a surface do not move along with a flow when adhesion is stronger than cohesion. At the fluid-solid interface, the force of attraction between the fluid particles and solid particles (Adhesive forces) is greater than that between the fluid particles (Cohesive forces). This force of adhesion brings down the fluid velocity to zero with respect to the object at the boundary. (Source)
The no slip condition is only defined for viscous flows and where continuum concept is valid.
Fluids stick to a surface due to the presence of microscopic obstacles on its surface. This is exactly what scientists try to overcome to achieve hydrophobicity – reduce the size of the irregularities to the extent that the liquid molecules do not get stuck in between them.
You can read more about the No-Slip condition in this article.
This theory is not so prevalent but it may still be a contributing factor which is that in most cities and housing facilities today, there are tons of oils and other micro-particles in the air that adhere to dust and many stationary surfaces. These generally assemble in a thin sheet as they get caught along the leading edge of the fan blade making it adhesive and then accumulates dust. Furthermore, humid air can enhance the stickiness. This is the same mechanism that leads to your face getting dusty despite your constant moving and facing air friction; oils and skin sebum traps dust on your skin leading to what I consider my greatest nemesis: acne.