## 2 Governing Equations
Figure 2: Sketch of shear flow over circular cylinder. Defining the shear rate \(K\) as \(K=2 Ga/u_{\infty}\), \(a\) refers to the cylinder radius. Only the circumstance of a positive shear rate, i.e., \(K>0\), needs to be discussed here, indicating that the flow velocity of the upper side prevails over that of the lower side. Initially, the flow is regarded as inviscid. When the cylinder can only move in a cross flow direction, the initial flow field in the moving frame of reference can be described by  
\[\psi=-2 sh(2\pi\xi)\Big{[}\sin(2\pi\eta)+\tfrac{k}{2}(2 ch(2\pi\xi)\cos(4\pi\eta )-e^{2\pi\xi})\Big{]} \tag{4}\]  
and \(\Omega=K\)  
As for \(t>0\), the cylinder is constrained to vibrate along the transverse direction with the effect of the vortex shedding. Based on the Galilean velocity decomposition and the stream function definition, we have  
\[\psi=\psi^{\prime}+\frac{dl(t)}{dt}e^{2\pi\xi}\cos(2\pi\eta)\]  
where the superscript \"\({}^{\prime}\)\" denotes the absolute coordinate (an inertial frame of reference), and no superscript denotes the coordinate (a non-inertial frame of reference) which is attached on the cylinder moving with the velocity \(\frac{dl(t)}{dt}\), where \(l\) is the cylinder displacement in the transverse direction. The relative angle of the incoming flow direction is defined as \(\theta_{0}=\tan^{-1}\Big{[}\frac{dl(t)}{dt}\Big{]}\), then  
\[\psi=\psi^{\prime}+(\tan\theta_{0})e^{2\pi\xi}\cos(2\pi\eta) \tag{5}\]  
when \(\xi\to\infty\), \(e^{2\pi\xi}\to 2 sh(2\pi\xi)\), from Equation (5), we have  
\[\psi=-2 sh(2\pi\xi)\Big{[}\frac{\sin(2\pi\eta-\theta_{0})}{\cos\theta_{0}}+ \frac{K}{2}(2 ch(2\pi\xi)\cos(4\pi\eta)-2 sh(2\pi\xi))\Big{]}\]  
which depends on the shear rate \(K\) and the vibration of cylinder, and  
\[\Omega=K \tag{6}\]  
On \(\xi=0\), \(-\frac{1}{H}\frac{\partial^{2}\psi}{\partial\xi^{2}}=-\frac{1}{H}\frac{ \partial^{2}\psi^{\prime}}{\partial\xi^{2}}-\tan\theta_{0}\cos(2\pi\eta)\)  
\[-\frac{1}{H}\frac{\partial^{2}\psi}{\partial\eta^{2}}=-\frac{1}{H}\frac{ \partial^{2}\psi^{\prime}}{\partial\eta^{2}}+\tan\theta_{0}\cos(2\pi\eta) \tag{7}\]  
and \(\Omega^{\prime}=-\frac{1}{H}(\frac{\partial^{2}\psi^{\prime}}{\partial\xi^{2 }}+\frac{\partial^{2}\psi^{\prime}}{\partial\eta^{2}})=-\frac{1}{H}(\frac{ \partial^{2}\psi}{\partial\xi^{2}}+\frac{\partial^{2}\psi}{\partial\eta^{2}})=\Omega\)  
Since \(\psi=0\), so that  
\[\psi^{\prime}=-(\tan\theta_{0})\cos(2\pi\eta)\text{ and }\frac{\partial^{2} \psi^{\prime}}{\partial\eta^{2}}=H\tan\theta_{0}\cos(2\pi\eta)\]  
where \(H=4\pi^{2}\). From Equation (7),  
\[\frac{\partial^{2}\psi}{\partial\xi^{2}}+\frac{\partial^{2}\psi}{\partial\eta^ {2}}=-H\Omega, \tag{3}\]0  
Thus, one finally obtains  
\[\frac{\partial^{2}\psi}{\partial\xi^{2}}+\frac{\partial^{2}\psi}{\partial\eta^ {2}}=-H\Omega, \tag{3}\]1  
In addition, the applied Lorentz force has no relationship with the flow field, and involves two types which can be defined as the field Lorentz force \(\left.F_{\theta}\right|_{\xi>0}\) and the wall Lorentz force \(\left.F_{\theta}\right|_{\xi=0}\). The field Lorentz force does not appear in any above equations, but acts as a source term affecting the boundary layer fluid and accordingly leads to the variations of hydrodynamic force. The wall Lorentzforce, however, is independent of the flow field due to the non-slip boundary and affects directly on the surface of cylinder.