My question comes from the fact that when studying laminar flow between parallel plates, we generally see studies of fully developed flows in textbooks. I have seen that as per the formula for Reynolds number given as $Re = \frac{u_{avg}D}{\nu}$ where $D$ is the characteristic length or diameter. In pipe flows or channel flows its usually called hydraulic diameter and is given a formula as $D = \frac{4ab}{2(a+b)}$ for channels. My question is does this formula extend to flow between parallel plates? If so in a 2-dimensional case, how do we get the hydraulic diameter?
Say there is a height $h$ between the plates (along $y$) and the plates have a finite length $'L'$ (along x) and no dimension along the $z$-axis since its a 2D case, if so does the magnitude of length affect the Reynolds number? Or will it only have an effect on development of the flow (entrance length)?
Also is there an empirical relationship between Re and the entrance length? If how do we establish it in this case? I have seen equations like $L/D = [(0.635)^{1.6}+ (0.0442Re)^{1.6}]^{1/1.6}$ in journals. Are these valid for 2D parallel cases like this?