(9)
with the two others
| dv | + u | dv | + v | dv | + w | dv | = Y − | 1 | dp | , | |
| dt | dx | dy | dz | ρ | dy |
(10)
| dw | + u | dw | + v | dw | + w | dw | = Z − | 1 | dp | . | |
| dt | dx | dy | dz | ρ | dz |
(11)
23. As a rule these equations are established immediately by determining the component acceleration of the fluid particle which is passing through (x, y, z) at the instant t of time considered, and saying that the reversed acceleration or kinetic reaction, combined with the impressed force per unit of mass and pressure-gradient, will according to d’Alembert’s principle form a system in equilibrium.
To determine the component acceleration of a particle, suppose F to denote any function of x, y, z, t, and investigate the time rate of F for a moving particle; denoting the change by DF/dt,
| DF | = lt· | F(x + uδt, y + vδt, z + wδt, t + δt) − F(x, y, z, t) |
| dt | δt |
| = | dF | + u | dF | + v | dF | + w | dF | ; |
| dt | dx | dy | dz |