## 2 Governing Equations  
A circular cylinder experiences time varying lift and drag, which are related to the vortex shedding, when placed in a flowing fluid. Thus, time varying vibration will occur when the cylinder is constrained to move on flexible supports, which is known as vortex-induced vibration (VIV). Moreover, the fluid around the cylinder is altered by this vibration, which affects the induced hydrodynamic forces in turn and the structure response is then changed. Therefore, this problem is associated with fully coupled fluid-structure interactions. For control of VIV in an electrically low-conducting fluid, the actuators on the cylinder surface consists of two half cylinders mounted with alternating electrodes and magnets as shown in Figure 1. The momentum of the fluid round the cylinder surface is increased with the application of Lorentz force, which is directed parallel to the cylinder surface. Figure 1: Lorentz force on the cylinder surface. In this way, the electro-magnetic force \(\mathbf{F}\) can be written in the dimensionless form:  
\[\mathbf{F}^{*}=\mathbf{J}^{*}\times\mathbf{B}^{*},\]  
where the superscript \"\({}^{**}\)\" refers to the dimensional form, no superscript refers to the dimensionless form. This can be written in dimensionless form [36; 38]  
\[\mathbf{F}^{*}=N\mathbf{F}, \tag{1}\]  
with  
\[F_{r}=0,\]  
\[F_{\theta}=e^{-\alpha(r-1)}g(\theta)\text{ with }g(\theta)=\left\{ \begin{array}{ ll}1&\text{covered with actuatoron upper surface}\\ -1&\text{covered with actuator on lower surface}\\ 0&\text{elsewhere}\end{array} \right.\]  
where \(r\) and \(\theta\) represent polar coordinates, \(r\) and \(\theta\) denote the components in \(r\) and \(\theta\) directions, respectively. \(\alpha\) is a constant, denoting the effective depth of Lorentz force in the fluid. The interaction parameter is termed as \(N=\frac{j_{0}\hbar a}{\rho\hbar a_{\infty}^{2}}\), with the current density \(j_{0}\), \(B_{0}\) the magnetic field, \(\rho\) the fluid density, and \(a\) the cylinder radius. The exponential-polar coordinates system \((\xi,\eta)\) is introduced here defined as \(r=e^{2\pi\xi}\), \(\theta=2\pi\eta\). Then, the stream function-vorticity equations in the dimensionless form, which express the flow considering an applied Lorentz force with the coordinate system attached on the moving cylinder, are termed as  
\[H\frac{\partial\Omega}{\partial t}+\frac{\partial(U_{r}\Omega)}{\partial\xi }+\frac{\partial(U_{\theta}\Omega)}{\partial\eta}=\frac{2}{\text{Re}}(\frac{ \partial^{2}\Omega}{\partial\xi^{2}}+\frac{\partial^{2}\Omega}{\partial\eta^{ 2}})+NH^{\frac{1}{2}}(\frac{\partial F_{\theta}}{\partial\xi}+2\pi F_{\theta} -\frac{\partial F_{r}}{\partial\eta}), \tag{2}\]  
\[\frac{\partial^{2}\psi}{\partial\xi^{2}}+\frac{\partial^{2}\psi}{\partial\eta^ {2}}=-H\Omega, \tag{3}\]  
We need to mention that the above equations (Equations (2) and (3)) have the same forms as the equations in the absolute coordinate system. The stream function \(\psi\) is defined as \(\frac{\partial\psi}{\partial\eta}=U_{r}=H^{\frac{1}{2}}u_{r}\), \(-\frac{\partial\psi}{\partial\xi}=U_{\theta}=H^{\frac{1}{2}}u_{\theta}\), while the vorticity \(\Omega\) is \(\Omega=\frac{1}{H}(\frac{\partial U_{\theta}}{\partial\xi}-\frac{\partial U_{ r}}{\partial\eta})\), with \(u_{r}\) and \(u_{\theta}\) indicating the velocity components in \(r\) and \(\theta\) directions, respectively. Moreover, \(H=4\pi^{2}e^{4\pi\xi}\), \(\text{Re}=\frac{2u_{\theta}a}{v}\), \(u_{\infty}\) is the free-stream velocity, \(v\) is the kinematic viscosity, and the dimensionless time is \(t=\frac{1+u_{\infty}}{a}\). The sketch of shear flow, which shows the change of the velocity in the cross-section plane of the cylinder, with a linear velocity profile \(u=u_{\infty}+Gy\)[43] over a cylinder in two-dimensional domain is shown in Figure 2, where \(u_{\infty}\) is the free-stream velocity at the center-line \(\theta=0\), \(y\) is the coordinate in the lateral direction with \(y=0\) at the center of the cylinder, and \(G\) is the lateral velocity gradient.