58 HYDRAULICS AND ITS APPLICATIONS 



The equations in full are deduced in the following pages, and the terms 

 involving viscosity are afterwards eliminated, so as to give the form as 

 usually stated for a non-viscous incompressible fluid. 



Afterwards, in considering the motion of fluids, it will be assumed 

 that with uniform steady motion the distribution of pressure is 

 unaffected, so that the pressure at any point is the equivalent of the 

 hydrostatic pressure due to its depth. This appears from the general 

 equations of motion. Also the further assumption is made, that if the 

 moving particles have the acceleration which they would have if acted 

 upon by their external forces alone i.e., if independent of the surround- 

 ing particles the pressure throughout is uniform. Thus, in a jet falling 

 freely through the atmosphere under the action of gravity, the pressure 

 throughout is sensibly uniform and equal to that of the atmosphere. 



The principles on which the following demonstrations are based may 

 be briefly indicated, and are as follow : 



The Principle of Linear Momentum. The rate of change of the com- 

 ponent of the linear momentum of any system in any direction is equal 

 to the parallel component of the applied forces. 



The Principle of Angular Momentum. The rate of change of the 

 component of angular momentum of a system about any axis is equal 

 to the moment of the applied forces about that axis. 



The Principle of the Conservation of Energy. The sum of the kinetic and 

 potential energies of any system is constant, except for the effect of such 

 dissipative forces as friction which convert mechanical energy into 

 heat. 



ART. 20. EQUATIONS OF MOTION FOR A Viscous FLUID. 



Taking a fixed point in the fluid as the origin of co-ordinates, let OX, 

 OY, OZ be three co-ordinate axes, and let u, v, iv be the components of 

 the velocity of a particle parallel to these three axes, u, v, and w, will be 

 supposed finite and continuous, and, since they vary with the position of 

 the particle and the time, are functions of x, y, z, and t. 



The velocity of a particle may be considered from two points of view. 

 Considering any fixed point, the velocity of successive particles as they 

 pass that point may vary, and since x, y, z are now constant, the rates of 

 variation parallel to the three axes are represented by the partial 



differentials ^ f etc. For steady motion these are separately zero. 

 If, however, we consider the variation of velocity of any one particle as 



