Chapter 6.10, Problem 15P

Suppose the density $\rho$ of a fluid varies from point to point as well as with time, that is, $\rho =\rho (x,y,z,t).$ If we follow the fluid along a streamline, then x, y, z are functions of $t$ such that the fluid velocity is $v=\text{i}\frac{dx}{dt}+\text{j}\frac{dy}{dt}+\text{k}\frac{dz}{dt}$

Show that then $d\rho /dt=\partial \rho /\partial t+v\cdot \nabla \rho .$ Combine this equation with (10.9) to get $\rho \nabla \cdot v+\frac{d\rho }{dt}=0$ $(Physically,$ d \rho / d t is the rate of change of density with time as we follow the fluid along a streamline; $\partial \rho /\partial t$ is the corresponding rate at a fixed point.) For a steady state (that is, time-independent), $\partial \rho /\partial t=0,$ but $d\rho /dt$ is not necessarily zero. For an incompressible fluid, $d\rho /dt=0;$ show that then $\nabla \cdot v=0.$ (Note that incompressible does not necessarily mean constant density since $d\rho /dt=0$ does not imply either time or space independence of $\rho ;$ consider, for example, a flow of water mixed with blobs of oil.)