Flux boundary conditions on U

The mass flux sums are used to calculate Dirichlet velocity boundary conditions for air on the cathode and for fuel on the anode. Note that the calculations take place with fields defined on the anode interface, so patchToPatchInterpolation to the air side is required. Then, we have for U on the boundary

\[{{\left. \mathbf{U} \right|}_{\text{electrode}~\text{ }}}=-\frac{\sum{{{\dot{m}}''}}}{\rho }\frac{\mathbf{A}}{\left\| \mathbf{A} \right\|}~\]
where \(\mathop{\sum }^{}{{\dot{m}}^{''}}\) is the accumulated mass flux, ρ is the fluid density, and \({{\mathbf{A}}/{\left\| \mathbf{A} \right\|}\;}\) is the unit outward normal.