## Appendix A  
The mathematical solution of a bubbly flow [29] is associated with an experimental setup, which is described in the following (Figure 11). A flat bottom glass flask filled with a biological medium, consisting of water with nutrients, was surrounded by an array of permanent magnets. Air was introduced near the bottom inside the flask through a thin tube. Consequently, bubbles were created, which, due to buoyancy, led to water circulation. A step-by-step solution was provided, where the main equations used were Equations (22) and (30). The axial component of the liquid phase velocity is denoted as \(w_{l}(r,t)\) showing the dependence of the radial component and time. This velocity is expressed in terms of the Bessel function of the first kind of order 0 (\(J_{0}\)) and order 1 (\(J_{1}\)), where \(\beta_{k}=\gamma_{k}/R\) and the values of \(\gamma_{k}\) are the zeros of \(J_{0}\) for \(k=1,2,3\ldots\infty\)[37]. The zeros were extracted from [65]. According to Equation (22), \(w_{l}\) is affected by the radius of the flask \(R\), water conductivity \(\sigma\), density \(\rho_{l}\), dynamic and kinematic viscosity \(\mu_{l}\) and \(\
u_{l}\), respectively, and the liquid phase volume fraction \(\Phi_{l}\), as well as the magnitude of the external magnetic flux density \(B_{0}\). The term in the exponential term is \({\eta^{2}}_{k}=\
u_{l}\big{[}{\gamma_{k}}^{2}/R^{2}+{\sigma B_{0}}^{2}/(\phi_{ l}\mu_{l})\big{]}\) and \(W_{0}\) is the initial value \(w_{l}\). The power dissipation experimented via the Joule effect is denoted as \(P\) with the general form \(P=-\int\mathbf{J}^{2}dV/(\sigma\Phi_{l})\), where \(\mathbf{J}\) is the vector form of the induced current in the medium. It was found that \(\mathbf{J}=-\sigma B_{0}w_{l}(r)\hat{\mathbf{e}}_{\theta}\), where \(\hat{\mathbf{e}}_{\theta}\) is the unit vector of the circumferential component. Replacing the induced current density in the power equation leads to \(P(t)=-{\sigma B_{0}}^{2}/\Phi_{l}\int_{V(r)}w_{l}(r,t)^{2}dV\). By expressing the volume \(V\) in terms of the generic \(r\)-coordinate, i.e., \(V=V(r)\), the volume element \(dV\) was obtained as a function of \(r\); thus, \(P\) can be evaluated numerically, using geometrical and input parameters. The missing geometrical parameters in Equation (30) to describe are \(h_{1}=R-Z_{1}\) and \(h_{2}=R-h_{0}\), where \(Z_{1}\) is the distance measured from the origin of the reference system to the top water level, and \(h_{0}\) is the depth measured from the origin of the reference system to the bottom of the flask (Figure A1a). The length of the cylinder inside the flask has a length \(l=Z_{1}+Z_{2}\) and a radius \(R_{c}\). Code files are attached. The following step after modeling the experiment was to collect information regarding the bubbly flow. Four flat bottom glass flasks were filled with fresh water, and air was injected through a tube. Bubbles were obtained as depicted in Figure A1. Based on the images, it was possible to perform measurements of the bubble diameter \(d_{i}\) (with \(i\), the \(i\)-th measured diameter) and the gas phase volume fraction as \(\phi_{g}=\sum_{i=1}^{n}{d_{i}}^{3}/\big{(}6R^{2}L\big{)}\)[42]. Photographs were taken for each individual flask as bubbles were rising, as shown in Figure A1b-d. To obtain the bubble diameter, two points were defined, considering that the flask diameter \(d\) is known. Thus, the equivalence for \(d\) in pixels (\(d=131\) mm = 1960 px) was found, leading to the calculation of the scale (14.9623 px/mm) and the measurement of the bubble diameter. In addition, bubbles were filmed in slow mode. The frames were processed with PhysMo Video Motion Analysis software to determine the whole profile of the vertical bubble velocity as a function of time. Two points were defined for which the distance between them was known. The origin was placed at a location where bubbles arise. The time base was 29.97 fps (frames per second), which corresponds to the slow camera configuration. Finally, the position for each bubble must be allocated at each frame.