  Let $I\subset A$ and $J\subset B$ be two monomial ideals, where $A$ and $B$
are two polynomial rings with disjoint variables. Considering a general set-up
of monomial filtrations, we study the behaviour of the $\mathrm{v}$-function
under binomial expansion. As an application, we get an explicit formula of
$\mathrm{v}((I+J)^{(k)})$ in terms of $\mathrm{v}(I^{(i)})$ and
$\mathrm{v}(J^{(j)})$, where $L^{(k)}$ denote the symbolic power of an ideal
$L$. Furthermore, an analogous formula is extended for the
$\mathrm{v}$-function of integral closure of $(I+J)^k$.
