## 2 Neumann Green's Function and Dipole Kernel for EM Flowmeters  
In this section, we define and give the [PERSON]'s function for the Laplacian in the unit disk \(D=\{\mathbf{x}=(x,y):\|x\|<1\}\). This allows the solution of the flowmeter Equation (1) using Equation (2) for general flow in circular pipes. We go on to develop a dipole form of the forward map, and show its relationship to the weight functions given by [PERSON] [11] and Bevir [15], in the restricted setting of axial flow. The [PERSON]'s function, also known as [PERSON]'s function, is the [PERSON]'s function for the Laplacian on \(D\) with Neumann boundary conditions in \(\partial D\), i.e.,  
\[\
abla^{2}u=f\ \text{in}\ D,\quad\partial_{n}u=\psi\ \text{on}\ \partial D. \tag{7}\]  
It is well known that the solution of Neumann boundary value problem (BVP) (7) is not unique; however, it is unique up to an additive constant, or, equivalently, the statement of a reference potential. We will require \(\int_{\partial D}u\ \text{d}S=0\), i.e., the average potential on the boundary is zero. The fulfillment of the following condition, corresponding to conservation of current  
\[\iint_{D}f(\mathbf{x})\ \text{d}\ V=\int_{\partial D}\psi(\mathbf{x})\ \text{d}S,\]  
is a necessary and sufficient for the existence of a solution to this problem [14; 23]. Necessity follows by an application of the divergence theorem to the boundary value problem (BVP) (7), while sufficiency follows by construction of the solution, as follows. The Neumann Green's function for the unit disk \(D\) has the explicit form [23; 24]  
\[g_{\text{N}}(\mathbf{x}|\boldsymbol{\xi})=\frac{1}{2\pi}\Big{(}\log\|\mathbf{ x}-\boldsymbol{\xi}\|+\log\|\mathbf{x}-\boldsymbol{\xi}^{\prime}\|+\log\|\boldsymbol{ \xi}\|\Big{)} \tag{8}\]  
where \(\boldsymbol{\xi}^{\prime}=\boldsymbol{\xi}/\|\boldsymbol{\xi}\|^{2}\) is the inverse point of \(\boldsymbol{\xi}\) with respect to the unit disk, and satisfies the following properties, that may be established by direct calculation [12; 13; 24]. \(\
abla^{2}g_{\text{N}}(\mathbf{x}|\boldsymbol{\xi})=\delta(\mathbf{x}- \boldsymbol{\xi})\), \(\forall\boldsymbol{\xi}\in D\). \(g_{\text{N}}(\mathbf{x}|\boldsymbol{\xi})=g_{\text{N}}(\boldsymbol{\xi}| \mathbf{x})\), \(\forall\boldsymbol{\xi},\boldsymbol{\xi}\in D\). \(g_{\text{N}}(\mathbf{x}|\boldsymbol{\xi})\) is harmonic in \(D\setminus\{\boldsymbol{\xi}\}\ \forall\boldsymbol{\xi}\in D\). \(g_{\text{N}}(\mathbf{x}|\boldsymbol{\xi})-\frac{1}{2\pi}\log\|\mathbf{x}- \boldsymbol{\xi}\|\) is harmonic in \(D\), \(\forall\boldsymbol{\xi}\in D\). \(\partial_{n}g_{\text{N}}(\mathbf{x}|\boldsymbol{\xi})=1/(2\pi)\) on \(\partial D\), \(\forall\boldsymbol{\xi}\in D\). \(\int_{\partial D}g_{\text{N}}(\mathbf{x}|\boldsymbol{\xi})\ \text{d}\ S(\mathbf{x})=0\), \(\forall\boldsymbol{\xi}\in D\). The second property, i.e., self-adjointness of the (real) Neumann Green's function, is often called _reciprocity_ in physics. Property 4 shows that \(g_{\text{N}}\) is a fundamental solution of the form (3). The Neumann Green's function allows the solution of the Neumann BVP (7) via a simplified form of Equation (2) stated in the following Lemma [23]. **Lemma 2**. : _The Neumann problem (7) has solution_  
\[u(\mathbf{x}_{0})=\iint_{D}g_{N}(\mathbf{x}|\mathbf{x}_{0})f(\mathbf{x})\ \text{d}\,\mathbf{x}-\int_{\partial D}g_{N}(\mathbf{x}|\mathbf{x}_{0})\psi( \mathbf{x})\ \text{d}\,S \tag{9}\]  
_that satisfies \(\int_{\partial D}u(\mathbf{x})\ \text{d}\ S=0\)._  
**Remark 1**. : _When flow occurs in an electrically insulating pipe then \(\partial_{n}u=0\) on \(\partial D\) and the boundary integral in (9) is omitted._  
**Remark 2**. : _Unbounded regions may also be treated using the Neumann Green's function and the solution in (9), such as when implementing an EM flowmeter in the natural environment.