Surface tension per unit length Can shear stress be expressed as surface tension per unit length? How do I interpret it physically?
 A: Surface tension has the dimension of a force per unit length, stress has the dimension of a force per unit surface.
You can imagine the surface tension as the integral of stress over a surface with one infinitely small dimension,
$\Delta \mathbf{F} = \displaystyle \int_{\Delta S} \mathbf{t_n} dS = \displaystyle \int_{\Delta S} \underbrace{\mathbf{t_n} dh}_{\boldsymbol{\gamma}} \ d\ell = \displaystyle \int_{\Delta \ell} \boldsymbol{\gamma} d \ell$,
being:

*

*$\mathbf{t_n} = \mathbf{\hat{n}} \cdot \mathbb{T}$ the stress vector acting on a surface with unit normal vector $\mathbf{\hat{n}}$, where the stress tensor is $\mathbb{T}$

*$\boldsymbol{\gamma} = \mathbf{t_n} \ dh$ the force acting per unit length at the interface surface.

You can imagine the stress vector due to surface tension written using a delta in space $\mathbf{t_n} = \boldsymbol{\gamma} \delta(\mathbf{r} - \mathbf{r}_S)$, whose contribution is non-zero only at the points $\mathbf{r}_S$ of the interface, whose integral on the surface thickness gives
$\displaystyle \int_{\Delta S} \mathbf{t_n} dS  = \displaystyle \int_{\Delta S} \boldsymbol{\gamma} \delta(\mathbf{r} - \mathbf{r}_S) dS = \displaystyle \int_{\Delta \ell} \boldsymbol{\gamma} d \ell$,
to derive the results given above, a bit more rigorously.
