EFFECT OF NON-UNIFORM VELOCITY DISTRIBUTION 47 



The change per second of the horizontal component of the momentum 

 of the water passing through the area dA is 



UWndA I I w I I 



\Vn\ = -Vn\Vn\dA 



g g 



The absolute value sign is necessary to insure that those filaments for 

 which the flow is upstream will be subtracted in getting the total for 

 the entire cross section, which is 



gJ 



and not 



wQV 



g 



as is usually assumed. Coriolis introduced a coefficient by which the 

 latter value should be multiplied to equal the correct result. 



- I Vn\Vn\dA = a — V a = -2-p [404] 



g J g V^A 



For a non-uniform distribution, a will always be greater than unity. 

 The kinetic energy carried through dA, per unit of time, will be 



VndA ' v^ 



2g 



which expression should be integrated over the whole cross section 

 to determine the total kinetic energy of the water passing downstream 

 through the section per unit of time. Since this is customarily written 



2g 

 a coefficient is again needed, defined by 



-Jv.vdA=a--V -=—i^s^ [405] 



Some writers advocate use of this coefficient in the application of 

 Bernoulli's theorem 



Zi + ^ + a/ ^ = Z2 + ^ + as' -^ + /^/ [406] 



w 2g w 2g 



For normal velocity distributions the value of a is nearly unity, so 

 that to introduce this coefficient has very little effect. If one of the 



