Find max flow in a network with all capcities of $\sqrt 2$ and one with 2
First, you can divide all capacities by $\sqrt{2}$ so that all capacities are now $1$ except for a single edge $e'$ which has capacity $\sqrt{2}$.

Call this instance $G$. Create $G_1$ which is a copy of $G$ where the capacity of $e'$ has been replaced with $1$, and find the max flow of this graph, with a value $F_1$.

Now, consider $G_2$ which is also a copy of $G$ where the capacity of $e'$ has been replaced with $2$ and has a max flow of $F_2$. 

If $F_2 &gt; F_1$, this means that starting from the flow assignment of maximum flow in $G_1$ there is an augmenting path in $G_2$ along which you can push $1$ additional unit of flow. This augmenting path necessarily corresponds to an augmenting path in $G$ where you can push $\sqrt{2}-1$ additional units of flow. Thus, the max flow for $G$ is $F=F_1 + \sqrt{2}-1$. 

If $F_2 = F_1$, this means that starting from the flow assignment of maximum flow in $G_1$ there is no augmenting path in $G_2$ along which you can push any amount of flow. This also means that this holds for $F$, as the capacity increase of $e'$ between $F_1$ and $F$ is lower than between $F_1$ and $F_2$. Thus, the max flow for $G$ is $F=F_1$.

(You just need a single search of augmenting path to determine if $F_2 &gt; F_1$ or $F_2 = F_1$ once you have computed the flow for $F_1$)

Then, to get the answer to your original problem, just scale the flow back by multiplying $F$ by $\sqrt{2}$.