Boundary Conditions
 

I am not sure of boundary condition (BC) types, Dirichlet, Neumann, Cauchy and Robin in our transport phenomenon. How are they classified and how do they differ from each other? I know that when the conditions at the surface is defined it is Dirichlet type BC (for example no-slip) but the moving plate boundary condition (Couette flow, the gradient at wall = constant) is Neumann BC, isn’t this BC was defined at the wall shouldn’t it be Dirichlet? what about BC in the annular fluid flow case, where we define one BC that gradient is equal to shear stress for both fluids (dv/dr|_Ri = τ_1; dv/dr|_Ro = τ_2), so isn’t this should be Dirichlet boundary condition?

It's really simple. It is all about types and forms. It has nothing to do with where, but what is being specified.

If you specify the value of the variable then it is Dirichlet. If you specify the gradient then it is Neumann. Robin is a linear combination of Dirichlet and Neumann. Mixed type is not really a type but where you specify Dirichlet on sub-parts of a boundary and Neumann in sub-parts of a boundary. Cauchy is where you specify both the Value and the Gradient simultaneously. When you write down the BC's mathematically using symbols, it is readily apparent what type it is because it's really easy to recognize f=something versus df=something. Words on the the other hand, can easily confuse people.

No-slip means the velocity of the fluid at the wall is equal to the wall velocity. Since you specify the velocity, it is Dirichlet. The moving wall is actually still a no-slip condition and is still a Dirichlet type. The difference between moving wall is the moving wall as a non-zero wall velocity whereas the stationary wall is the null case (velocity = 0). Both are the same.

Your annular flow example is a bit special. The conditions at Ri and Ro are interface conditions and not actual boundary conditions if you consider the entire pipe volume as the control volume. However, if you redefine your control volumes such that one is around only the outer fluid and another around only the inner fluid... since you are fixing the gradients, it is actually a Neumann condition and not a Dirichlet condition. Again, mathematically it is very easy to figure out what type it is. dv/dr= something immediately means Neumann. If you wrote v=something then it would have been Dirichlet.