Derivation of the equations of change by integral theorems (Fig. 3D.1)

(a) A fluid is flowing through some region of 3-dimensional space. Select an arbitrary "blob" of this fluid that is, a region that is bounded by some surface S(t) enclosing a volume V(t), whose elements move with the local fluid velocity. Apply Newton's second law of motion to this system to get in which the terms on the right account for the surface and volume forces acting on the system. Apply the Leibniz formula for differentiating an integral (see SA.5), recognizing that at all points on the surface of the blob, the surface velocity is identical to the fluid velocity. Next apply the Gauss theorem for a tensor (see SA.5) so that each term in the equation is a volume integral. Since the choice of the "blob" is arbitrary, all the integral signs may be removed, and the equation of motion in Eq. 3.2-9 is obtained. 

(b) Derive the equation of motion by writing a momentum balance over an arbitrary region of volume V and surface S, fixed in space, through which a fluid is flowing. In doing this, just parallel the derivation given in S3.2 for a rectangular fluid element, the Gauss theorem for a tensor is needed to complete the derivation. This problem shows that applying Newton's second law of motion to an arbitrary moving "blob" of fluid is equivalent to setting up a momentum balance over an arbitrary fixed region of space through which the fluid is moving. Both (a) And (b) Give the same result as that obtained in S3.2. (c) Derive the equation of continuity using a volume element of arbitrary shape, both moving and fixed, by the methods outlined in (a) and (b). 

Transcribed Image Text:

--Sm pvdV (n T]dS + pgdV VI) V(t + At) V(1) ds