  Given a $t$-$(v, k, \lambda)$ design, $\mathcal{D}=(X,\mathcal{B})$, a
zero-sum $n$-flow of $\mathcal{D}$ is a map $f : \mathcal{B}\longrightarrow
\{\pm1,\ldots, \pm(n-1)\}$ such that for any point $x\in X$, the sum of $f$
over all blocks incident with $x$ is zero. For a positive integer $k$, we find
a zero-sum $k$-flow for an STS$(u w)$ and for an STS$(2v+7)$ for $v\equiv
1~(\mathrm{mod}~4)$, if there are STS$(u)$, STS$(w)$ and STS$(v)$ such that the
STS$(u)$ and STS$(v)$ both have a zero-sum $k$-flow. In 2015, it was
conjectured that for $v>7$ every STS$(v)$ admits a zero-sum $3$-flow. Here, it
is shown that many cyclic STS$(v)$ have a zero-sum $3$-flow. Also, we
investigate the existence of zero-sum flows for some Steiner quadruple systems.
