326 Forces and their Relationship to the Structure of the Ocean 



(3) The continuity equation for an incompressible medium will take the form 



S^cp c^cp 3^9 



Neglecting the Coriolis force and the frictional forces, the three Eulerian equations 

 of motion equation (X. 16), on multiplication by dx, dy and dz, respectively, and 

 taking further into account the identity 



du du I d ^ ^ 



rf^ = «7 + 2 8Tx("^ + '-^ + "'^) ('^■33) 



and by subsequent addition, can be compressed into the single equation 



where F(t) is an arbitrary function of t alone and Q is the potential of the external 

 forces. For a steady current 



(8u 8v 8w \ 

 di^ 8i ^ 8t ^^) 



in which the stream lines coincide with the trajectories of the fluid elements 



U^ + V^ + H'^ p 



^ +~+^=C, (X.35) 



where the quantity C is constant along each stream line but changes on passing from 

 one stream line to another. The equation (X. 35) is called the Bernoulli theorem 

 (equation). It shows that for steady motions the pressure at points along a stream line 

 is greatest where the velocity is smallest and vice versa. Considering that a fluid particle 

 on transfer from higher to lower pressure is subject to an acceleration (increase in 

 velocity) the above statement is readily understood. This is another way of expressing 

 the conservation of energy, since for unit mass the first term is the kinetic energy of 

 motion, the second is the work done against pressure and the third is the potential 

 energy; in a steady flow the sum of these energies along a stream line must be constant. 

 If the water movement is solely influenced by the gravity force, then since Q = gz, 

 the Bernoulli pressure equation will have the form 



^ + - + ?z = const., with m2 m f2 ^ ^^,2 - c\ (X.36) 



2 p 



For a two-dimensional potential flow it is convenient to introduce a stream function ifj 

 defined by the relations 



«=-^, v=+^^ (X.37) 



and therefore from (X. 32) 



Sep Si/» ^9 8ijj 



8x^ 8)'' 8y^ ~ 8x' 



