While learning fluid-dynamics, I stumbled across this problem that requires me to calculate the volumetric flow rate Q of a thin layer of oil leaking from a small hole in a barge, as depicted in the following figure:
If I understand the problem correctly, this is a Poiseuille-flow problem, as the oil is going up thanks to the difference in pressure caused by the hydrostatic pressure of the water.
I tried then to obtain the Navier-Stokes equation in the x direction for this problem: (my reference frame is such that the x axis is aligned with the direction of flow of the oil)
$$0=-\frac{p_{water}gcos\theta}{\rho_{oil}}-gcos\theta +\nu\frac{d^2u}{dy^2}$$
Where I used the fact that $\frac{\partial p}{\partial x}=\rho_{water}gcos\theta$ and that $u=u(y)$ only depends on y
Integrating using the boundary condition that u(0)=0 (non-slip condition) I get:
$$u=\frac{gcos\theta}{2\nu}(1-\frac{\rho_{water}}{\rho_{oil}})y^2+c_1 y$$
But I still have one constant left, how can I get rid of it? I tried equating the shear stresses in the oil-water interface but could not get anything helpful at all. Any help will be appreciated.