Difference between Cahn-Hilliard models I came across the "classical" Cahn-Hilliard model which can be formulated as follows:
Let $\phi$ be the order paramter, $\mu$ the chemical potential, $M$ the Cahn-Hilliard mobility, $\lambda$ the mixing energy parameter and $f$ the Helmholtz free energy. Then, we obtain the Cahn-Hilliard equations:
\begin{align*}
\partial_t \phi = \nabla \cdot M (\nabla \mu)\\
 \mu = -\lambda \nabla^2 \phi + f'(\phi)
\end{align*}
My confussion lays here: I also found a slightly different CH model with an additional term $\nabla \cdot (u \, \phi)$ in the setting of Cahn-Hilliard-Navier-Stokes model:
\begin{align}
\partial_t \phi + \nabla \cdot (u \, \phi) &= \nabla \cdot M (\nabla \mu)\\
 \mu &= -\lambda \nabla^2 \phi + f'(\phi)\\
\rho (\partial_t u + (u\cdot \nabla) u)&= - \nabla p + \nabla \cdot (\eta(\nabla u + (\nabla u)^T)) + \rho \cdot g\\
\nabla \cdot u &= 0
\end{align}
where $u$ denotes the velocity and the last two equations are the Navier-Stokes equations.
What does this additional term mean, in a physical and mathematical sense?
Edit: Is it some sort of "advection term"? I came across the "advected Cahn-Hilliard model" which also contains this additional term.
 A: If $\boldsymbol u$ is divergence-free, you can write $\nabla\cdot(\boldsymbol u \phi)=\boldsymbol u\cdot \nabla \phi$, and you can write the LHS as
$$\frac{\partial\phi}{\partial t}+\boldsymbol u\cdot\nabla\phi\equiv \frac{\mathrm D\phi}{\mathrm Dt}$$
Which is commonly referred to as the Material derivative. I think I have derived this before somewhere else on this site, but I can't find it, so I'll do it again.
Imagine you have a function $\boldsymbol u$ such that the velocity of a fluid particle that is located at the position $\boldsymbol x$ at the time $t$ has the velocity $\boldsymbol u(t,\boldsymbol x)$.
Now imagine that you track the path of a single particle, and it has position $\boldsymbol x(t)$ at time $t$. We now define a new function that returns the value of $\phi$ at the position of the particle, at the time $t$:
$$\varphi(t)=\phi(t,\boldsymbol x(t))$$
Using the chain rule, you can see
$$\dot{\varphi}(t)=(\partial_t\varphi)(t,\boldsymbol x(t))+\left(\frac{\mathrm dx^i}{\mathrm dt}\frac{\partial}{\partial x^i}\phi\right)(t,\boldsymbol x(t))$$
And of course since $\dot{\boldsymbol x}(t)=\boldsymbol u(t,\boldsymbol x(t))$, this is
$$\dot{\varphi}(t)=(\partial_t\phi)(t,\boldsymbol x(t))+(\boldsymbol u\cdot\nabla\phi)(t,\boldsymbol x(t))=\left(\frac{\mathrm D\phi}{\mathrm Dt}\right)(t,\boldsymbol x(t))$$
When deriving the Navier-Stokes equations, we track the forces acting on some arbitrary fluid parcel using the above construction, then apply that reasoning to the entire flow field. The $\nabla\cdot(\phi\boldsymbol u)$ term comes from this cosntruction.
