Accurate study of normal pressure distribution in entrance region of channel

There are few implicit solutions available for the pressure distribution in the y-direction. Thus, for the flow between parallel plates, the pressure distribution in the entrance region was studied, focusing on the pressure gradient in the y-direction at Reynolds numbers (Re) between 100 and 5000. In the numerical method, the vorticity transport equation is first solved and then Poisson's equation for pressure distribution is solved without any assumptions taken for pressure distribution. Consequently, the difference in pressure between the wall and the centerline existed near the inlet and decreased as Re increased. The pressure at the wall is lower than that in the central core for Re <= 5000. This result shows that (i) the boundary-layer assumptions do not hold for Re <= 5000 and (ii) the pressure distribution is contrary to Bernoulli's law across parallel plates, although the law does not apply to viscous flow.