  In this article, we study an inverse problem for the following convective
Brinkman-Forchheimer (CBF) equations:
  \begin{align*}
  \boldsymbol{u}_t-\mu
\Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla
p=\boldsymbol{F}:=f \boldsymbol{g}, \ \ \ \nabla\cdot\boldsymbol{u}=0,
  \end{align*}
  in a bounded domain $\Omega\subset\mathbb{R}^2$ with smooth boundary
$\partial\Omega$, where $\alpha,\beta,\mu>0$ and $r\in[1,3]$. The investigated
inverse problem consists of reconstructing the vector-valued velocity function
$\boldsymbol{u}$, the pressure field $p$ and the scalar function $f$. For the
divergence free initial data $\boldsymbol{u}_0 \in \mathbb{L}^2(\Omega)$, we
prove the existence of a solution to the inverse problem for two-dimensional
CBF equations with the integral overdetermination condition, by showing the
existence of a unique fixed point for an equivalent operator equation (using an
extension of the contraction mapping theorem). Moreover, we establish the
uniqueness and Lipschitz stability results of the solution to the inverse
problem for 2D CBF equations with $r \in[1,3]$.
