cube is 



(pX -jSx Sy Sz 



This force is now equal to the mass of the particle in the cube multiplied by its acceleration 

 in the a;-direction, which will be denoted by du/dt. Hence 



{'^-D 



^ I s^ »„, s^ _ . »^ s», ff-, du 



Sx Sy 82 = p 8x 8y 8z 



du ^x-l~ 

 dt P dx 



dt 



[3a] 



In this equation, u has reference to a certain particle of the fluid, which at a given in- 

 stant occupies a certain cube but whose position in space varies. Usually, however, it is of 

 no interest actually to follow a particle in its motion; it is more convenient to regard w as a 

 function of position and time or 



u{x, y, s, t) 



without regard to the identity of the particle whose coordinates at a particular time t are 

 represented by x, y, z. Viewed from this mathematical standpoint, the change in u during the 

 time dt at the particle just considered may be written 



du=^ dx+^dy + ^dz+^dt 

 dx dy dz dt 



Here dx represents the change in the a;-coordinate of the particle during the time dt\ hence 



dx - u dt. Similarly, dy = v dt, dz = w dt. Hence, after dividing through by dt, 



du ^du^^du_^^dv_^y^dw_ [3b] 



dt dt dx ^y c?2 



Here du/dt represents the rate of change of m at a given particle, whereas du/dt represents the 



rate of change in m at a fixed point in space. The last three terms represent an effect due to 



motion of the particle and are sometimes called convection terms. 



Thus, Equation [3a] may be written, together with the analogous equations for the y- 



and 3-directions, in the form known as Euler's equations: 



[3c] 

 [3d] 

 [3e] 



where X, Y , Z are the components of the external force per unit mass. These equations hold 

 whether the density of the fluid is constant or not; in general, p is a function ol x, y, z, and t. 



For a fluid of constant density. Equations [2b], [3c], [3d], and [3e] constitute four 

 differential equations in four unknowns: u, v, w, and p. As arbitrary constants and functions 

 enter into the solutions of differential equations, boundary conditions are required in order to 



^ + u^+v^+w^=X ■ 

 dt dx dy dz 



_ 1 ^ 

 ~ P dx 



d^ + u^ + v^+w-^^ Y- 

 dt dx dy dz 



1 dp 



P dy 



^+u^+v-^+w^=Z 

 dt dx dy ds 



1 dp 

 P dz 



