## Appendix A
A summary of input parameters to evaluate Equations (22) and (30) is shown in Table 2.  
Equation (1) was evaluated with magnetic flux density: \(B_{0}\) values from 1 to 0 T, i.e., \(B_{0}\) ranging from 10,000 to 0 G, which apply to permanent magnets. The algorithm to compute the z-component of the velocity \(\mathrm{W_{z}}=w_{l}(r,t)/W_{0}\) is shown in Figure A2, in which, \(B0\), \(r\_d\), and \(t\) stand for the reduced zero \(\beta_{n}=\gamma_{k}/R\), the flux density \(B_{0}\), the axial spatial coordinate \(r=r_{d}\), and time, respectively. The superscripts k, i, and p were used to index the reduced zeros, radius, and magnetic flux density values, respectively. The sum over the index k was stored in the variable X, which at the end of the iteration becomes \(\mathrm{W_{z}}\). Objects in bold correspond to a matrix or vector; otherwise they are scalars. The algorithm is subdivided into three parts. 1. Variable Initialization * Definition of variables to be used: \(r\_d\) for the axial spatial coordinate \(r=r\_d\), ranging from 0 to \(R\), \(\mathbf{B0}\) for the values of the magnetic flux density (\(B_{0}\)), \(\mathbf{B}\) for the zeros of the Bessel function of the first kind (\(J_{0}\)), and \(\mathbf{t}\) for the time values at which the velocity is evaluated. * Assignment of geometrical and medium parameters. 2. Velocity Calculation The algorithm returns a \(W\)z matrix of dimensions of length of \(r\_d\) times length of \(\mathbf{B0}\). Firstly, initial values are loaded, namely, \(r\_d\), \(\mathbf{B0}\), and \(\mathbf{t}\). Variables k, i, and p are initialized ([\(k\), \(i\), \(p\), \(d=0\)]). In the following, \(X^{k}\) is assigned as a function that depends on \(B^{k}\), \(r\_d^{i}\), \(B0^{p}\), and \(t^{d}\). The superscripts represent a counter in a nested iterating loop. The lower level loop allows for establishing at which point convergence is obtained, i.e., \(|X^{k}-X^{k-1}|\to 0\). If such a condition is fulfilled, the \(X^{k}\) at the \(k\) iteration is saved in \(Wz^{k,d}\); otherwise, the \(k\) is incremented and \(X\) is evaluated at the new \(B^{k+1}\) value. Next, at the second loop, \(i\) represents the pointer at the flask radius vector, \(r\_d\). This loop ends when the radius reaches its maximum value. The third loop is used to handle the columns of the final matrix. Different values of magnetic flux density are evaluated until \(\mathbf{B0}\) reaches its maximum. Finally, the fourth loop is related to the time, \(t\). It ends the iteration when its maximum is reached. A three-dimensional matrix is obtained, with dimensions of the length of \(r\_d\) times length of \(\mathbf{B0}\) times length of \(\mathbf{t}\). Power Calculation  
The portion of the flask geometry where the fluid is circulating and power is dissipated can be obtained by considering the volume of a sphere and subtracting the region above the water level (upper cap), the region below the bottom of the flask (lower cap), and the cylinder. Thus, the volume is divided into four domains, which are defined according to the \(r\) and \(z\) coordinates: the maximum radius (r_d) at \(z=0\), the upper radius corresponding to the water level (\(R_{1}\)) at \(z=Z_{1}\), the lower radius found at the bottom of the flask (\(R_{2}\)) at \(z=-h_{0}\), and the cylinder radius within the flask (\(R_{\rm c}\)) for \(-Z_{1}\leq z\leq Z_{2}\), representing the tube that introduces air. According to the defined regions, the dissipated power is computed based on the four terms at the right side of Equation (2). Thus, \(\left|P(t)\right|\) results from calculating the dissipation within the region of a sphere of radius r (int_1) and subtracting the last three terms, associated to the upper cap (int_2), lower cap (int_3), and the cylinder (int_c). Using the Algorithm in MATLAB  
As an example, we attached a simplified code in MATLAB language, which can compute the axial component of the liquid phase velocity and power dissipation. The variables and geometric parameters are loaded. In this example, k runs from 0 to 150.