Zt+At= water elevation at time t+At 



p = density of water 

 1,2 = subscripts denoting Sections 1 and 2, respectively 

 These quantities will be used in the derivations given next and in subsequent 

 discussion. The relationship between the channel top width B and area A is 

 B = dA/dz. 

 Mass conservation equation 



21. The mass of water entering the channel during a time interval At is 

 pQAt + pqAtAy, and the mass of water leaving the channel reach during the same 

 time interval is p(Q + dQ/dyti.y)^t. Assuming dQ/dy and q to be positive, there 

 is a net mass outflow from the reach. The law for conservation of mass 

 requires that the mass inside the reach be reduced because of the net mass 

 outflow during the time At, and, consequently, the water surface should fall. 

 The mass of water inside the reach of length A/ is pAAy, and the rate of 

 decrease of the mass can be expressed as -d/dt{pAAy) = -pAy dA/dt. The 

 reduction in mass during the interval At is -pLyi,dA/dt)t\t . To satisfy 

 conservation of mass, the net mass outflow must be equal to the reduction in 

 mass inside the reach, so that 



p\p + |2 Ay) At - (pOAt + pgAtAy) = - p|| AtAy (1) 



Simplifying, dQ/dy = -dA/dC + q, or 



22. Equation 2 is the equation of continuity and is the mathematical 

 expression for the law of conservation of mass in open channels. 

 Momentum equation 



23. Conservation of momentum is given by Newton's second law of motion, 

 which states that the rate of change of momentum is equal to the applied 

 force. In Figure 1, the net applied force on the element of volume in the 

 reach Ay is the resultant of the pressure, gravity, shear forces, and form 

 drag on the element. The water depth is h at Section 1 and h + dh/dyAy at 

 Section 2. The cross -sectional area is A at Section 1 and A + dA/dyAy at 

 Section 2 . 



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