Answer Happy

A control volume is an arbitrary volume of space through which fluid can flow. Let the control volume of a spherical shaped blob in a flowing fluid is V. Suppose the fluid has density ρ(r,t) and velocity field u​. Then applying mass conservation to the blob, Rate of change of mass in =− Mass out flow. The total mass of fluid entering to the blob in a control volume V is m=∭V​ρdV The mass out flow is given by, ∬S​(ρu​)dS where, S is the surface bounded by the control volume V. Apply the Gauss Divergence Theorem to show that, ∂t∂ρ​+∇​⋅(ρu​)=0.