# How do we define Reynolds number for incompressible laminar flow between 2D parallel plates?

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?

The definition of the Reynolds number in fluid dynamics is somehow arbitrary, you only need a value of density $$\rho$$, length $$L$$, velocity $$U$$ and viscosity $$\mu$$ representative of the fluid and the flow,

$$Re = \dfrac{\rho U L}{\mu}$$

If you're studying a laminar flow, where you know no instability and turbulence occurs, you are usually interested in a Reynolds number with the dimension $$L$$ perpendicular to the channel, as an example the distance of between the surfaces of the channel (the height of the channel). As a velocity you can use the average velocity, if you know it, or the maximum velocity. Usually, if you know (i.e. you measure) the volume flow (or the mass flow for incompressible constant density flows), you'd better use the height $$L$$ of the channel and the average velocity $$U$$, whose product gives you the volume flow,

$$Q = U L\qquad \rightarrow \qquad Re = \dfrac{\rho Q}{\mu} \qquad \qquad$$ or $$\qquad \qquad \overline{Q} = \rho U L\qquad \rightarrow \qquad Re = \dfrac{\overline{Q}}{\mu}$$.

If you're interested in flow experience instability and then turbulence originating from the boundary layer at the wall, you probably need a Reynolds number build with a streamwise length $$x$$,

$$Re_x = \dfrac{\rho U x }{\mu }$$,

to estimate, as an example, the critical length where instability and turbulence starts

$$Re_{cr} = \dfrac{\rho U x_{cr} }{\mu } \quad \rightarrow \quad x_{cr} = \dfrac{\mu Re_{cr}}{U \rho}$$