## 4 Reconstruction of Flow
### Discussion  
Numerical approximation allows the calculation of the singular value decomposition (SVD) of the forward map, for axial flow and constant \(\mathbf{B}\) field. The resulting singular vectors in flow show the flow distributions that complete measurement of boundary potential is sensitive to, with relative sensitivity set by the relative size of the associated singular values [24]. This analysis provides a theoretical limit to flowmeters, and establishes the extent to which they have the capability to image the spatial flow distribution at a given SNR. We find that the geometry analyzed provides a dominant singular vector of flow that has appreciable sensitivity to bulk flow, while the remaining singular vectors have increasing sensitivity only near the boundary. The examples presented in the paper show that flowmeters can be used to reconstruct the spatial flow distribution using a TSVD. This method provides a simple kind of regularization that maps singular vectors from measurement-space to flow-space. Examples of flow tomography use a truncated singular value reconstruction, which enables reconstruction of the flow field, though with noticeable truncation error near the boundary due to Gibb's phenomenon. Applying the SVD to the 2-dimensional flowmeter has not been performed to the authors' knowledge and provides new insights to the sensitivities of the system. While these results show that there are theoretically many pieces of information detectable in the flowmeter system, the vast majority of that information is at the edge of the pipe. Figure 10: Absolute total flow error between the shear flow and the reconstructed flow for different \(k\). Figure 9: The truncated basis spectrum of the reconstructed flow (Figure 8 right). This means that, if one were, say, interested in detecting the change in the flow profile near the edge of the pipe, then a flowmeter with many electrodes would be very useful. This is evident in previous works using Shercliff's weight function in Figure 1, where the weight function increases in magnitude closer to the electrode positions. As most flowmeter arrangements have the electrodes around the edge, there must be increased sensitivity to flow changes near the edges. For the case computed in the examples, of 2-dimensional reconstruction of axial flow in a circular pipe with constant \(\mathbf{B}\) field, we found that around 150 singular vectors needed to be included in the TSVD to achieve an accurate (within 1%) estimate of bulk flow when no assumptions are made on the distribution of axial flow. Accordingly, more than 150 measurements of boundary potential would be required to make such a reconstruction, and estimate of bulk flow, which seems an excessively large number. It therefore seems desirable to increase the sensitivity of EM flowmeters to flow away from the boundary, or, equivalently reduce sensitivity to flow near the boundary. Since the Neumann Green's function in Equation (14) is fixed by geometry and the Ohmic properties of the fluid, this therefore requires reducing the term \(\mathbf{v}\times\mathbf{B}\) near the boundary. The non-shear boundary condition \(\mathbf{v}=0\) on \(\partial D\) assumes this term is 0 at boundaries, but requires assumptions on the flow that may not hold. Another solution is to ensure that \(\mathbf{B}=0\) at measurement electrodes, that is the imposed \(\mathbf{B}\) field is produced in a way that places electrodes at nulls of the magnetic field, perhaps by a suitable arrangement of electromagnets. This is an area for future research and analysis. When the primary aim is to detect the total flow rate of a fluid, perhaps in a range of geometries, Figure 6 provides useful insights. This shows that approximately all singular vectors that have a non-zero inner product with the uniform flow, or, in other words, contribute to a net positive or negative flow. The most significant one is the first singular vector with the next being the fifth, at a quarter the value of the first.