What is viscous induced drag?

In this video he talk at 17:04 about viscous inudced drag, induced drag comes exclusively from inviscid theory,by the definition this drag exist only in inviscid world, so what is viscous induced drag?

There's no such a thing as a "viscous" induced drag and that terminology should be avoided.

Drag coefficient for the entire airplane can be approximated with the well known expression:

$$C_d=C_{d_0}+kC²_l$$.

The first term is called "parasite drag" while the second term is called "induced drag".

The main source of induced drag is the "drag due to lift" due to pressure forces. The main source of parasite drag is the "skin friction" due to viscosity.

Anyway, pressure is also responsible for a (normally negligible) fraction of the parasite drag. And viscosity is also responsible for a (normally negligible) fraction of the induced drag. In short:

viscosity pressure
parasite $$C_{d_0}$$ skin friction drag due to stall, wave drag
induced $$kC²_l$$ profile drag drag due to lift

This latter should be what in the video is called "viscous induced drag" i.e. the part of the drag which is proportional to the lift and caused by the viscosity.

This drag is simply what is normally seen in a plot of the $$C_d$$ vs. $$C_l$$ for a generic 2D airfoil (plot source):

Unfortunately enough this drag is more or less proportional to the square of $$C_l$$ and is therefore grouped together with the "drag due to lift" to form the "induced drag" term $$kC²_l$$.

I said unfortunately because this "viscous induced drag" is actually not "induced" (by the lift) and has absolutely nothing to do with the production of lift: it is just a drag due to the growth of the boundary layer with the speed and the angle of attack. It is more correctly termed "profile drag" because the growth of the boundary layer virtually modifies the profile of the airfoil seen by the airflow. I have no idea why a new fancy terminology has been used in that video.

In a nutshell, when he talks about induced drag in the viscosity-free inviscid world, he's talking about the energy consumed in overcoming only the inertia of the air molecules to move them down from point A to point B in generating lift.

It's treating each molecule as an independent object in a vacuum, completely unaffected by adjacent molecules. You could say the inviscid world is only concerned with the energy required to move a single molecule from A to B in a vacuum, then applying that energy value to all the molecules, as if the molecules were coated with a super-lubricant that eliminated friction between them.

But since the molecules are actually packed together as a gas of a certain density, the molecules have to get past each other, inhibited by their own intermolecular friction, or resistance to shearing forces. To make lift in the real viscous world, energy is consumed overcoming the shearing forces in addition to energy consumed overcoming inertia.

So, what he's doing is calculating the energy to make the air move down solely to overcome inertial forces, without accounting for shearing or viscosity effects, the inviscid induced drag, then adding the energy required to overcome viscosity effects, the viscous induced drag, arriving at the total induced drag.

Remember how in 2D airfoil data -- like in Theory of Wing Sections -- the 2D drag looks like a parabola? Here he is bookkeeping that drag in with the induced drag.

It is a lift dependent drag -- that takes a parabolic form -- so this is reasonable to do. Furthermore, if you were measuring drag in a wind tunnel, it would be very difficult to separate that drag out from the true induced drag.

However, it is not a true induced drag (in terms of its physical origin).