How does Drain current flow?

The inversion layer is already formed, apply positive voltage between the Drain and Source terminals. This will create an electric field that will drive the electrons from the Source to the Drain terminal through the inversion channel layer.

The movement of electrons represents Drain current from the Drain to the Source.

The figure 15 illustrates the flow of Drain current

Figure 15: Flow of Drain Current 

Initially it can be said that the current flow is ohmic and the current here is due to the flow of electrons. The flow of electrons is due to the application of the applied voltage. 

This is the phenomenon of a Resistor.

Let the total resistance between the Source and Drain be represented by RDS.

There are three regions through which the current flows - Source, Channel and Drain. All these three regions can be represented by three resistances.

Therefore the total resistance RDS can be represented by:

RDS = RSource+ RInversion +RDrain

It is known that the Source and Drain regions are heavily doped - huge number of electrons in the case of N Channel MOSFET and huge number of Holes in the case of P Channel MOSFET.  Due to this the resistance in these two regions is small.

Therefore 

RDS ≃ RInversion

The basic definition of resistance is repeated below:

R=⍴L/A

Extending this to our N Channel MOSFET, 

⍴ is replaced by the ⍴ of the inversion layer i.e. ⍴inv,  and the area A is product of W and t_inv.

The Drain current ids is given by:

The product ninv.tinv can be expressed as Sheet Concentration  of electrons in the inversion layer, let this product be termed as nS.

Also we know that 

Now insert sheet concentration, and rewrite the Drain current equation,

In the N Channel MOSFET the resistance is dependent on the Gate voltage.