Here, \(\mathbf{v}\) is the velocity of the fluid at each point, \(\mathbf{B}\) is the imposed magnetic flux density, and \(u\) is the resulting electric potential throughout the moving fluid. The right-hand side is an effective current source distribution within the fluid resulting from the Lorentz force, or equivalently Faraday's Law, while the left-hand side combines Ohm's Law and conservation of current. This equation assumes that the conductivity of the fluid is finite, homogeneous, and isotropic, though, interestingly, the value of the conductivity cancels and does not appear in the flowmeter Equation (1). [PERSON] [10; 11] developed theory and solutions for a cylindrical geometry, with circular cross-section, assuming axial flow, i.e., that flow \(\mathbf{v}\) in a pipe occurs only parallel to the axis of the pipe taken to be the \(z\)-direction, assumed uniform \(\mathbf{B}\) field perpendicular to the axis of the pipe along the \(x\) direction, and electrodes sensing the voltage at the boundary located on the diameter along the \(y\) axis. These simplifying assumptions reduce the problem to a 2-dimensional problem in the circular domain \(D\) that is the cross-section of the pipe in \(\mathbf{x}=(x,y)\). Then, the flowmeter equation reduces to \(\
abla^{2}u=B\partial v/\partial y\) where \(B\) is the magnitude of \(\mathbf{B}\) and \(v\) is the \(z\)-component of \(\mathbf{v}\). This simplified theory, as summarized by [PERSON] in [10], breaks down as soon as the flow is no longer a simple distribution. This was widely known by leading proponents in the field at the time of his writing. For this reason, and the prevalence of liquid metal cooled nuclear power at the time, EM flowmeter theory development was confined mostly to cylindrically symmetric systems. Solving the flowmeter equation for the potential \(u\) may be performed by an application of [PERSON]'s theorem for [PERSON]'s equation, as in the following Lemma; see ([12], p. 229). Lemma 1: _If \(u\) is continuous and has continuous second-order derivatives within a 2-dimensional domain \(D\), with boundary \(\partial D\), containing the point \(\mathbf{x}_{0}\), then_  
\[u(\mathbf{x}_{0})=\iint_{D}h(\mathbf{x}|\mathbf{x}_{0})\
abla^{2}u(\mathbf{x })\ \mathrm{d}\ V+\int_{\partial D}\left(u(\mathbf{x})\partial_{n}h(\mathbf{x}| \mathbf{x}_{0})-h(\mathbf{x}|\mathbf{x}_{0})\partial_{n}u(\mathbf{x})\right) \ \mathrm{d}\ S \tag{2}\]  
_where \(h(\mathbf{x}|\mathbf{x}_{0})\) is any fundamental solution for \(\
abla^{2}\), that is_  
\[h(\mathbf{x}|\mathbf{x}_{0})=\frac{1}{2\pi}\log\|\mathbf{x}-\mathbf{x}_{0}\|+ v(\mathbf{x}|\mathbf{x}_{0}) \tag{3}\]  
_where \(v(\mathbf{x}|\mathbf{x}_{0})\) is harmonic in \(D\), and \(\partial_{n}\) denotes the outward normal derivative on the boundary \(\partial D\)._  
Strictly, the adjoint fundamental solution, and adjoint Green's function, is required in Lemma 1[13], though, typically the boundary value problem for EM flowmeters is self-adjoint so this detail can be overlooked. When each of of the terms \(\
abla^{2}u(\mathbf{x})\) for \(\mathbf{x}\in D\), \(u(\mathbf{x})\) for \(\mathbf{x}\in\partial D\), and \(\partial_{n}u(\mathbf{x})\) for \(\mathbf{x}\in\partial D\) are known, the solution can be determined by evaluating Equation (2). When the pipe is insulating, it follows that \(\partial_{n}u(\mathbf{x})=0\ \forall\mathbf{x}\in\partial D\) though the term \(u(\mathbf{x})\) for \(\mathbf{x}\in\partial D\) is unknown. For a bounded region \(D\), it is not possible to choose the harmonic function \(v(\mathbf{x}|\mathbf{x}_{0})\) in Equation (3) so that \(\partial_{n}h(\mathbf{x}|\mathbf{x}_{0})=0\ \forall\mathbf{x}\in\partial D\), since, by conservation of current, \(\int_{\partial D}\partial_{n}h(\mathbf{x}|\mathbf{x}_{0})\ \mathrm{d}\ S=1\); see, (e.g., [14], p. 286). Thus, the basic definition of the (adjoint) Green's function, as a fundamental solution that satisfies the homogeneous form of the (adjoint) boundary conditions, needs to be modified for the [PERSON] problem (7).