I am struggling to set up the weak form of the following PDE (nondimensional form):
$$\frac{\partial c}{\partial \tau}=\epsilon ^{2}\frac{\partial^2 c}{\partial x^2}+\frac{\partial^2 c}{\partial y^2}-p\epsilon y(1-y)\frac{\partial c}{\partial x}$$
I would need the weak formulation to perform FEM for the simulation of a combination of diffusion and mass transfer in a microfluidic system.
2-dimensional scheme of microfluidic system
There are four boundary conditions (one for every side of the system): $$\frac{\partial c (\tau,x,y)}{\partial y}=0 \enspace at \enspace y=1$$ $$h \frac{C\_T}{R\_T}\frac{\partial c (\tau,x,y)}{\partial y}=\kappa\_a c(\tau,x,0)(1-b(\tau,x))-\kappa\_d b(\tau,x) \enspace at \enspace y=0$$ $$c(\tau,0,y)=1 \enspace at \enspace x=0$$ $$\frac{\partial c (\tau,x,y)}{\partial x}=0 \enspace at \enspace x=1$$
Is here someone who is able to help?
Best regards, Marvin