| dM/dt = rate of increase of fluid inside the surface, = flux across the surface into the interior = − ∫∫ ρq cos θ dS, |
(1)
the integral equation of continuity.
In the Eulerian notation u, v, w denote the components of the velocity q parallel to the coordinate axes at any point (x, y, z) at the time t; u, v, w are functions of x, y, z, t, the independent variables; and d is used here to denote partial differentiation with respect to any one of these four independent variables, all capable of varying one at a time.
To transfer the integral equation into the differential equation of continuity, Green’s transformation is required again, namely,
| ∫∫∫ ( | dξ | + | dη | + | dζ | ) dx dy dz = ∫∫ (lξ + mη + nζ) dS, |
| dx | dy | dz |
(2)
or individually
| ∫∫∫ | dξ | dx dy dz = ∫∫ lξ dS, ..., |
| dx |
(3)