Let v(t, x, y, z) be a continuously differentiable vector field over the region D in space

Question:

Let v(t, x, y, z) be a continuously differentiable vector field over the region D in space and let p(t, x, y, z) be a continuously differentiable scalar function. The variable t represents the time domain. The Law of Conservation of Mass asserts thatimage


where S is the surface enclosing D.


a. Give a physical interpretation of the conservation of mass law if v is a velocity flow field and p represents the density of the fluid at point (x, y, z) at time t.


b. Use the Divergence Theorem and Leibniz’s Rule,image


to show that the Law of Conservation of Mass is equivalent to the continuity equation,image


(In the first term ∇ · pv, the variable t is held fixed, and in the second term δp/δt, it is assumed that the point (x, y, z) in D is held fixed.)

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For  book-img-for-question

Thomas Calculus Early Transcendentals

ISBN: 9780321884077

13th Edition

Authors: Joel R Hass, Christopher E Heil, Maurice D Weir

Question Posted: