## 7 Conclusions  
With exponential-polar coordinates attached on a moving cylinder, we have deduced the related equations considering the electro-magnetic force for the shear flow, i.e., the stream function-vorticity equations, the initial and boundary conditions, the hydrodynamic forces on the cylinder surface, and the cylinder motion equation together with the mathematical expressions of the lift force \(C_{l}\). The applied Lorentz force was not dependent on the flow field and it involves two categories: the field Lorentz force \(F_{\theta}|_{g>0}\) and the wall Lorentz force \(F_{\theta}|_{g=0}\). As a source term, the field Lorentz force affected the flow field in the boundary layer, which leads to the variations of hydrodynamic force. The wall Lorentz force, however, is independent of the flow field due to the non-slip boundary and affects directly on the surface of cylinder.  
The momentum of the fluid round the cylinder surface is increased by Lorentz force, which overcomes the effect of the \"adverse\" pressure gradient. Therefore, the separation points move downstream, and even disappear on the cylinder surface. With the application of symmetrical Lorentz forces, the symmetric field Lorentz force can increase the drag, suppress the flow separation, decay the lift fluctuation, and, in turn, suppress the VIV, whereas the wall Lorentz force decreases the drag only, i.e. no effect on the lift. With the application of asymmetrical Lorentz forces besides above-mentioned effects, the field Lorentz force can strengthen additional lift (negative value) induced by shear flow, whereas the wall Lorentz force can counteract the additional lift, which dominates the total effect. Therefore, it is notable that the asymmetrical Lorentz force, which suppresses the lift vibration and enhances the lift value, can be applied to overcome the lift loss that is caused by the shear flow.  
From the above discussion, the aims for the vibration suppression, drag reduction and lift amplification have been obtained with application of Lorentz force. More complicated geometries, such as wing sections, can be controlled with application of this approach. The optimum performance can be obtained with changing the position of actuators in the applications based on the separation points, vortexes, the suction and pressure sides and so on. In addition, the value of Lorentz force varies with time according to the instantaneous flow field, which can be applied to control varying flow conditions, such as dynamically-changing angles of attack.  
The work was supported by the National Natural Science Foundation of China (Grant Nos. 11672135 and 11202102), the Fundamental Research Funds for the Central Universities (Grant No. 30916011347) and A Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant No. 201461). [PERSON] conceived the study, wrote the paper and interpreted the results. [PERSON] participated in data analysis and editing the manuscript. [PERSON] and [PERSON] reviewed the study plan and corrected the grammar mistakes. [PERSON] and [PERSON] helped in the simulation.  
Figure 20: The distribution of pressure induced by vortex \(C_{pF}^{\theta}\) on the surface of cylinder with the asymmetric Lorentz force (upper surface \(N=3\), lower surface \(N=2\)) for shear rate \(K=0.2\).