Water flows through a duct of height 2h and width W as shown

Water flows through a duct of height 2h and width W, as shown in the figure below. The velocity varies across the duct according to U/U_m = 1 - y/h^2 where U_m and h are constants and U varies with y. Find the volume, mass, and momentum fluxes over the cross-sectional area of the duct.

Solution

given duct height = 2h and width = W take a small elemental area at distance of P from centerline and of thickness of dP .

volume flow through this elemental crossection = dP*W*Um*(1-(P/h)² )

volume flow through the entire crossection =2* ₀^h dP*W*Um*(1-(P/h)² ) = (4*W*Um*h)/3

volume flux = voulme flow/area of crossection = (4*W*Um*h)/(3*2h*W) = (2*Um)/3

mass flow through this elemental crossection = dP*W*Um*(1-(P/h)² )*row of water

since row of water is constant,mass flow rate through the entire crossection = row*(4*W*Um*h)/3

mass flux = mass flow/area of crosection = (row*(4*W*Um*h)/3) / 2hW = (2*Um*row)/3

momentum of flow through elemental crossection = row*dP*W*U² = row*W*Um² *(1-(p/h)²)² * dP

momentum of flow through entire crossection = 2* ₀^h row*W*Um² *(1-(p/h)²)² * dP = 2*row*W*Um² * 8h/15

= (16*row*W*Um² * h)/15

momentum flux = momentum of flow/area of crossection =( (16*row*W*Um² * h)/15)/2h*W = (8*row*Um² )/15