The second term \(\boxed{\nabla \cdot \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right)}\) in \(\boxed{\frac{\partial \rho(\mathbf{x}, t)}{\partial t} + \boxed{\nabla \cdot \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right)} = 0}\) is the divergence of the mass-density flux \( \rho \mathbf{u} \)
Such flux divergence terms frequently arise in conservation statements and
can be interpreted as the net loss at a point due to divergence of a flux. For example, the
local \( \rho \) will decrease with time if \( \nabla \cdot (\rho \mathbf{u}) \)
is positive.
Flux divergence terms are also called transport terms because
they transfer quantities from one region to another without making a net contribution over the entire
field. When integrated over the entire domain of interest, their contribution vanishes if there are no
sources at the boundaries
The material derivative \( D/Dt \) is the total time derivative in the Eulerian description of fluid motion. It is composed of unsteady part \( \partial F / \partial t \) and advective part \( \mathbf{u} \cdot \nabla F \): \[ \frac{DF}{Dt} \equiv \frac{\partial F}{\partial t} + \mathbf{u} \cdot \nabla F, \quad \text{or} \quad \frac{DF}{Dt} \equiv \frac{\partial F}{\partial t} + u_i \frac{\partial F}{\partial x_i} \]
Product rule for divergence:\[ \nabla \cdot (\rho \mathbf{u}) = \rho \nabla \cdot \mathbf{u} + (\mathbf{u} \cdot \nabla) \rho \]
The continuity equation may be written using the definition of \( D/Dt \) and \( \partial (\rho u_i)/\partial x_i = u_i \partial \rho / \partial x_i + \rho \partial u_i / \partial x_i \): \[ \frac{1}{\rho(\mathbf{x}, t)} \frac{D}{Dt} \rho(\mathbf{x}, t) + \nabla \cdot \mathbf{u}(\mathbf{x}, t) = 0 \]