μd2vxdy2=−ΔPFL1.a)Integrating the governing equation twice yieldsvx(y)=−1μΔPFL1y22+C1y+C2.Since the velocity is U at y=0,C2=U,and since the velocity is 0 at y=h,0=−1μΔPFL1h22+C1h+UorC1=1μΔPFL1h2−Uh.The velocity profile is thereforevx(y)=1μΔPFL1h22yh−y2h2⎡⎣⎢⎤⎦⎥+U 1−yh⎛⎝⎜⎞⎠⎟.This expression gives the velocity profile in both regions (downstream and upstream of the verticalslot).The only difference is the negative pressure gradient in the two regions.In the forward flowregion,ΔPFL1=P1−P0L1,whereas in the backflow regionΔPBL2=P0−P1L2.b)In the backflow region, the velocity profile is againvx(y)=1μΔPBL2h22yh−y2h2⎡⎣⎢⎤⎦⎥+U 1−yh⎛⎝⎜⎞⎠⎟.