i am new to cfd and have a fundamental question. the momentum equations along with the continuity equation are 4 equations with 4 unknowns (u, v, w, p) for incompressible fluids. So on each boundary we need to specify the values for all these 4 variables. but most often we specify u,v,w on the inlet and just the pressure on the exit. So what happens to the other variables needed on the boundary.

The Navier-Stokes equations are partial differential equations and not algebraic equations - the way you've added up the number of conditions is incorrect; e.g. if the flow is inviscid then you still have four equations and four unknowns but the number of boundary conditions is reduced because of the reduction in order of the equations (the equations also change type from parabolic to hyperbolic).

You need to read an elementary book on partial differential equations along with a book such as "Elementary fluid dynamics" by Acheson.

Inlet and outlet boundaries are artificial boundaries as compared to a solid wall. Because the computational domain cannot be infinite, we usually truncate it somewhere. And then we have to specify meaningful boundary conditions. These boundary conditions can also depend on the actual physical conditions that exist in a particular situation. Sometimes the pressure might be specified at an inlet also because it might be maintained at some pressure by the presence of a reservoir. Sometimes you want to specify the velocity profile at inlet because you already know what it is like. I have found that boundary conditions are given very little consideration in publications and most never mention what type of boundary conditions are used. Its probably something of an art because these artificial boundary conditions (ABC) cannot be rigorously derived from the governing equations.

All the variables must of course be updated at all grid points. Sometimes you have to guess what the author might have done since they dont mention the boundary conditions.

I can only comment on the Finite Volume Method where we have the simple fundamental transport equation (steady state)

Sum of fluxes over inside faces + Sum of Fluxes over boundary faces = source term

Specification of boundary conditions boils down to the calculation of the flux over the boundary faces, which is either convective flux or diffusive flux or a combination of the two. Convective flux require the value of your variable at the boundary face. Diffusive flux required the gradient of your variable accross the boundary face. Depending on your specific problem setup you will either specify a boundary value or gradient explicitly or calculate it from the cell values available inside the domain. The boundary flux is then moved to the source term and the link to coefficient of the cell "outside the domain" broken by setting it equal to zero.

Hope this helps a bit. Regards Tom