I am deriving a specialised form of the fluid momentum conservation equation for two-phase incompressible flow. One of the terms in the equation is of the following form: $$ \mathbf{C}(t) = \iiint_{V(t)}\! (\nabla\alpha\!\cdot\!\mathbf{x})\mathbf{u}\, dV, $$ where $V(t)$ is a time-varying fluid control volume enclosed by a smooth boundary $S(t)$, $\mathbf{x}$ is a position vector, $\mathbf{u} \equiv \mathbf{u}(\mathbf{x}, t)$ is the fluid velocity, and $\alpha \equiv \alpha(\mathbf{x}, t)$ is a linearly varying (within $V(t)$) scalar field. Thus, $\nabla\alpha$ is constant within $V(t)$. Is there a way that I can simplify this triple integral further?
Edit: I am essential trying to apply the finite volume method to all integral terms in the equation. This particular term is problematic as I cannot see how I can apply the Divergence theorem to it to cast it as an area integral. I am so far able to split it up into two terms - one problematic, the other not: $$ \mathbf{C}(t) = \iiint_{V(t)}\! [\nabla(\alpha\mathbf{x}) - \alpha\nabla\!\cdot\!\mathbf{x}]\mathbf{u}\, dV, $$