Suppose, for contradiction, that $\Gamma \subset D$ is a periodic orbit enclosing a region $A$ with outward unit normal $n$. Since $f$ is tangent to $\Gamma$ at every point, we have $f(x) \cdot n(x) = 0$ for all $x \in \Gamma$. Since $\phi$ is continuous and $\phi\, f$ has the same tangential direction as $f$ along $\Gamma$, it follows that $\phi\, f \cdot n = 0$ on $\Gamma$. Applying the divergence theorem: \begin{align*} 0 = \oint_\Gamma \phi\, f \cdot n\, d\ell = \int_A \nabla \cdot (\phi\, f)\, dA. \end{align*} But if $\nabla \cdot (\phi\, f)$ is strictly positive (or strictly negative) throughout $D$, then the integral over $A \subset D$ cannot be zero, a contradiction.