Flow between parallel plates h Y Q water flow Perforated plate inflow inside u+ou You have a channel system in which the water leaky-wall BC flows through and is bounded with two perforated plates at the bottom and the top. Consider steady, incompressible, laminar flow between two infinite parallel horizontal plates as shown in the figure. h is the half of the hydraulic aperture (m). At the top and the bottom, perforated blocks are connected on both side plates a liquid inflows and mixes into the main water flow channel system and you would like to evaluate the total volumetric flow rate Q (m³ s¹). бл 0- 2 -by Perforated plate inflow inside leaky-wall BC P+5P The flow is in the x-direction, and there is no velocity component in either the y-or z-direction (i.e., v = 0 and w=0). In steady-state conditions applying the simple form of the Navier-Stokes equations (or similar to Poiseuille's law), along with the assumptions that v = 0, w = 0, and u = u(y) yields 88 др μ др др =-pg, =0 Oz. Ex by² by The dynamic viscosity is μ, The water density is p, and g denotes the gravitational acceleration. Integration of aPlaz=-pg leads to P=-pgy + g(x). Therefore, the water pressure is only a function of x, and the water flow velocity is the function of y. 1000|16 a) Please take integration twice of the equation below to determine the velocity field u(y). While taking the integration, you can assume that aPlax is a constant and simplify the equation (2 points). ди др by Ox b) After that assuming you have a leaky-wall boundary condition on both sides of the plates which demonstrates that the two opposing faces of the channel flow system are adjacent to some blocks from there a liquid inflows inside. By setting a leaky-wall boundary condition to represent crossflow between the blocks and the main channel flow system. So, after you take the integration twice, you will have two integral constants. You need to solve the problem with these two integral constants. Please apply the velocity conditions given below obtain these two integral constants and set them into the equation you solved above for 'u(y)'. Ahole is the total cross- section area of hole spaces (perforated area) that are connected to the main channel flow system (6 points). @y=h @y=-h u(h) - Ale = др μέχ u(-h)= OP μέχ c) After you evaluate the velocity field 'u(y)' that accounts for cross-flow between the channel and blocks on both opposing faces of the channel, please determine the volumetric flow rate with the following equation: Where you can use h=1x104 m, Apore = 1×10-7 m², p= 1×10-3 Pa.s, and aPlax = 5x10° Pa. (2 points)