where the integrations extend throughout the volume and over the surface of a closed space S; l, m, n denoting the direction cosines of the outward-drawn normal at the surface element dS, and ξ, η, ζ any continuous functions of x, y, z.
The integral equation of continuity (1) may now be written
| ∫∫∫ | dρ | dx dy dz = ∫∫ (lρu + mρv + nρw) dS = 0, |
| dt |
(4)
which becomes by Green’s transformation
| ∫∫∫ ( | dρ | + | d(ρu) | + | d(ρv) | + | d(ρw) | ) dx dy dz = 0, |
| dt | dx | dy | dz |
(5)
leading to the differential equation of continuity when the integration is removed.
22. The equations of motion can be established in a similar way by considering the rate of increase of momentum in a fixed direction of the fluid inside the surface, and equating it to the momentum generated by the force acting throughout the space S, and by the pressure acting over the surface S.
Taking the fixed direction parallel to the axis of x, the time-rate of increase of momentum, due to the fluid which crosses the surface, is