  Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics
$g_{ij}(t)$ evolving by geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where
$S_{ij}(t)$ is a family of smooth symmetric two-tensors on $M$. In this paper
we derive differential Harnack estimates for positive solutions to the
nonlinear heat equation with potential: \begin{eqnarray*} \frac{\partial
f}{\partial t} = {\Delta}f + \gamma (t) f\log f +aSf, \end{eqnarray*} where
$\gamma (t)$ is a continuous function on $t$, $a$ is a constant and
$S=g^{ij}S_{ij}$ is the trace of $S_{ij}$. Our Harnack estimates include many
known results as special cases, and moreover lead to new Harnack inequalities
for a variety geometric flows.
