2nd order derivative in boundary conditions
 

I am solving the basic flow of a 2-D sessile liquid droplet resting on a substrate, and neglecting the vaporization. The flow is governed by the non-dimensional Navier-Stokes, continuity and energy equations in the Boussinesq approximation.

I am using finite difference method and a streamfunction-vorticity formulation to exclude the pressure from the formulation; therefore, along the free surface, where the thermocapillary forces must be balanced by viscous tangential stresses, a 2nd order derivative is appeared in boundary conditions. Since I am using a very odd coordinate system (Toroidal coordinates), I faced problem to implement it in Matlab. Anyone has some experience in this type of boundary conditions?

Thank you in advance!

I am interested to take a look at it
 

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Mehul