76 HYDRAULICS AND ITS APPLICATIONS 



^ 



But v = g , and S s cos Q = 6 2, z being measured vertically upwards 



and S z being the difference of level of the two ends of the element. 



W 

 .'. Equation (1) becomes . v B v = W B z B p 



or - v Sv +WSz + Sp = 0. (2) 



t/ 



When & s, 5 p, are indefinitely small we get, in the limit, on integrating 



W tf 

 - . ->- + JF# + j> = constant 



9 * 



or ^7 + w + z ~ constant. (3) 



ART. 30. 



Since v is the velocity along a stream line, any attempt to apply the 

 equation to the motion of a mass of fluid by taking v as the mean velocity 

 of the mass will obviously lead to error unless the square of this mean 

 velocity, multiplied by the mass itself, is equal to the sum of the squares 

 of the various stream tube velocities, each multiplied by the mass 

 contained in its own stream tube ; and this is only true when all the 

 stream lines are parallel and have the same velocity, and when, in con- 

 sequence, no internal work is being performed against viscosity. 



While this state of affairs is never accurately realised in the case of the 

 motion of water in pipes or open channels, yet the equation may be made 

 to apply to such cases by the introduction of a term involving losses of 

 energy due to viscous resistance. 



E.g., if the suffixes (1) and (2) refer to two successive positions of a 

 particle of viscous fluid, we may say that 



P 1 j v * J_ 7 - P* j v ^ 7 TT 

 W + 2~g +Zl - W+2~g + Z * + lJ/2 ' 



where iH 2 represents this loss of energy between the positions (1) 

 and (2). 



Where we have turbulent motion between (1) and (2), this equation may 

 still be usefully applied, i7T 2 now including the loss of energy in eddy 

 formation. Applying this to the case of flow through a pipe or channel 

 of varying area, it is usual to assume that the equation still holds when 

 vi and t>2 are the mean velocities of flow in the direction of the axis at 

 the two sections (1) and (2). 1 



1 In the case of flow in a straight pipe with the transverse curve of velocities in the form of 

 a semi-ellipse, the actual kinetic energy of flow parallel to the axis is about 2 per cent, greater 

 than the value computed from the mean velocity, when the latter has a value of 5 feet 

 per second. 



