Previously we were interested in gross-average view point (control volume)

1. DIFFERENTIAL VIEW OF FLOW MECHANICS Previously we were interested in gross-average view point (control volume) Now we are interested in a point-by-point view point of an infinitesimal region of the flow. So, Lets start with a control volume but make it very small and apply mass conservation.

2. Control volume of size dx, dy, dz Z Y X

3. Note we used Taylor’s Series expansions in previous set up. Recall that a Taylor’s Series expansion is of the form… Upon substitution we get

4. ASIDE: The last term on the right comes from the Taylor’s Series expansion that consists of terms like Collecting all the terms we get Divide by dxdydz and let them go to zero

5. THIS IS KNOWN AS THE EQUATION OF CONTINUITY CAN ALSO WRITE IN COMPACT FORM AS

6. … AND FOR YOU VECTOR CALCULUS FREAKS OUT THERE (YOU KNOW WHO YOU ARE) We could have used Gauss’s Divergence Theorem which states And performed a direct substitution into To get

7. Since the equation holds regardless of control volume size is must also hold pointwise, leaving