(4)
| Dw | = | dw | + u | dw | + v | dw | + w | dw | , |
| dt | dt | dx | dy | dz |
(5)
leading to the equations of motion above.
If F (x, y, z, t) = 0 represents the equation of a surface containing always the same particles of fluid,
| DF | = 0, or | dF | + u | dF | + v | dF | + w | dF | = 0, |
| dt | dt | dx | dy | dz |
(6)
which is called the differential equation of the bounding surface. A bounding surface is such that there is no flow of fluid across it, as expressed by equation (6). The surface always contains the same fluid inside it, and condition (6) is satisfied over the complete surface, as well as any part of it.
But turbulence in the motion will vitiate the principle that a bounding surface will always consist of the same fluid particles, as we see on the surface of turbulent water.
24. To integrate the equations of motion, suppose the impressed force is due to a potential V, such that the force in any direction is the rate of diminution of V, or its downward gradient; and then