  In this paper we study an anisotropic expanding flow of smooth, closed,
uniformly convex hypersurfaces in $\mathbb{R}^{n+1}$ with speed
$\psi\sigma_k(\lambda)^{\alpha}$, where $\alpha$ is a positive constant,
$\sigma_k(\lambda)$ is the $k$-th elementary symmetric polynomial of the
principal radii of curvature and $\psi$ is a preassigned positive smooth
function defined on $\mathbb{S}^n$. We prove that under some assumptions of
$\psi$, the solution to the flow after normalisation exists for all time and
converges smoothly to a solution of the well-known $L^p$ Christoffel-Minkowski
problem $u^{1-p}( x ) \sigma_k \left( \nabla^2u+uI\right)=c\psi(x)$ for $p>1$.
