## 1 Introduction  
Bluff structures such as offshore spar, marine risers, overhead transmission lines and heat exchangers are subjected to vortex-induced vibration (VIV) when exposed to a flowing fluid, which contributes to the fatigue life reduction of structures and may produce structure damage under certain unfavorable conditions. The interactions between the shedding among various cylinders are much more complicated and they are also relevant to a number of problems. One of the problems where this is critical is concentrated solar power (CSP) collectors, where the shedding from one row affects the next one, leading to mechanical problems due to vibration and fatigue [1; 2]. The vibrations arise from the time-periodic fluid force associated with the time-periodic shedding vortex, and subsequently alter the flow field, which will change the flow-induced force in turn. Such fluid-structure interactions increase the complexity of the fluid mechanisms. Extensive research on VIV has been conducted. The earliest studies have been the subject of interests with lock-in phenomenon of VIV (e.g., [3; 4; 5; 6; 7; 8; 9; 10]). The time varying parameters such as cylinder response, flow force, vortex shedding and the influences of cylinder motion on the vortex structure are further studied [11; 12; 13; 14], and the vortex-formation modes have been focused on later by [15; 16; 17; 18], which paid attention to the interactions with multiple cylinders, and showed that the arrangement or gap had a significant effect on the VIV system responses. However, a large fluctuation of flow forces, and the increase of drag, acoustic noise and even structure damage are usually generated by the undesirable flow separations and vibrations of the body in the previous cases. Therefore, modern flow control approaches and technologies are considered to suppress the above phenomena. Flow control usually involves two broad categories. For one, called passive control, the flow is modified without external energy input as helical strokes [19], fin [20], rope and bump [21], thick fairings [22] (p. 19377), splitter plates [23], guide foils [24] and so on. For the other, called active control, energy requires to be injected into the flow as momentum injections [25; 26; 27], moving-wall [28], synthetic jet actuators [29], suction and blowing [30; 31; 32; 33; 34] and so on. The Lorentz force, one of the active approaches, has been subjected to various studies since the 1960s. In the recent years, the Lorentz force has attracted more attention due to its promising applications in engineering fields. [PERSON] and [PERSON] [35] has numerically investigated that the Lorentz force can eliminate the flow separation when the flow past a stationary cylinder, and the suppressing effect of the Lorentz force has been confirmed by [PERSON] et al. [36] with both experiments and calculations. Later, in the cases of [PERSON] and [PERSON] [37] and [PERSON] [38], it is found that both of the continuous and pulsed Lorentz forces can suppress the force fluctuation and stabilize the flow. Recently, optimal and closed-loop control have been developed by [PERSON] et al. [39; 40; 41] aiming to improve the control efficiency of the cylinder wake, and the VIV of the shear incoming flow was also investigated preliminarily [42]. However, there has not been discussion in the literature on the control of VIV by Lorentz forces in shear flow. In this paper, control of VIV with Lorentz forces for the shear flow has been numerically investigated. The stream function-vorticity equations, the initial and boundary conditions, distribution of hydrodynamic force and the cylinder motion equation are deduced in the exponential-polar coordinate with the coordinate at the moving cylinder. The Lorentz force can be classified into the field Lorentz force and the wall Lorentz force. On the other hand, the evolution of VIV starting from rest to vibration, and suppression are all presented.