Given a velocity field $\vec F(x,y)$, the flow along a curve $C$ is given by $$\int_C \vec F\cdot \vec T ds= \int_C \vec F\cdot d\vec r,$$ where $\vec r(t)$ is a parametrization of $C$.
What the units of flow? It seems the obvious answer is along the lines of "$m^2/s$", but I have no intuitive understanding of what that means. I understand that flow is measuring "how much" of something is moving "along the curve $C$." The units, though, befuddle me.
$\vec{F}$ would have the unit $\frac{\text{length}}{\text{time}}$, $\vec{T}$ is a unit vector with no units, and $ds$ would have units of $\text{length}$. So flow would have units of $\frac{\text{length}^2}{time}$ because of the product in the integral.