Developing laminar momentum boundary layer for channel flow at low Reynolds numbers

Theoretical developments for fully developed laminar flow between horizontal parallel plates and for a developing boundary layer over a horizontal flat plate have traditionally been treated separately. Yet, the two problems are related since flow over a horizontal flat plate is simply flow between horizontal parallel plates that are infinitely spaced. Similarly, at sufficient distances downstream, developing flow between horizontal parallel plates should converge to the fully developed horizontal parallel plate solution. However, traditional boundary layer approaches result in unbounded solutions for the boundary layer thickness downstream. To date, a unified approach for handling the general class of developing flow problems between horizontal parallel plates has not been established.; In this dissertation, I develop a general theory that is applicable to both the flat plate and parallel plate problems. For the case of developing flow between horizontal parallel plates, I use an order of magnitude analysis to simplify the Navier-Stokes equations for low Reynolds number flow and impose conservation of momentum across the channel to enforce a bounded boundary layer thickness downstream. I prescribe a new and more exact definition of the boundary layer thickness and solve for the x- and y-velocities, the boundary layer thickness, and the pressure in the developing region. These values are compared to theoretical limits, finite element solutions, and previously published experimental results.; The results show that my theory is consistent with the traditional formulations for the horizontal parallel plate and horizontal flat plate problems. The theory results in non-trivial expressions for both the x- and y-momentum equations, which gives rise to vorticity within the boundary layer. The imposed conservation of momentum across the channel results in an accelerating boundary velocity which disallows the traditional use of geometrically similar solutions for velocity profiles. This results in non-zero expressions for the pressure gradient across the channel and causes entry lengths to vary significantly with the Reynolds number.