A reciprocal integral identity of coupled Poisson and Laplace equations in two arbitrary domains sharing a common boundary

In solving the coupled vapor and liquid unidirectional flows in micro heat pipes, we discovered numerically an integral identity. After asymptotic and polynomial expansions, the coupled flows yield two reciprocal systems of equations. In system A, a vapor velocity $U_A$ obeys the Poisson equation and drives, through an interfacial boundary condition, a liquid velocity $W_A$ that satisfies the Laplace equation. In reciprocal system B, a liquid velocity $W_B$ obeys the Poisson equation and drives, through another interfacial boundary condition, a vapor velocity $U_B$ that satisfies the Laplace equation. We found that the vapor volume flow rate of $U_B$ is numerically equal to the liquid volume flow rate of $W_A$ for seven different pipe shapes. Here, a general proof is presented for the integral identity, and some interesting implications of this identity are discussed.