## 3 Hydrodynamic Forces  
Defining the net hydrodynamic force \(\mathbb{F}^{\theta}\) acting on a cylinder as  
\[C_{\mathrm{F}}^{\theta}=\frac{\mathbb{F}^{\theta}}{\frac{1}{2}\rho u_{\infty}^{ 2}}=\sqrt{\left(C_{\mathrm{\tau}}^{\theta}\right)^{2}+\left(C_{p}^{\theta} \right)^{2}}\]  
If the prime coordinate system indicates the absolute coordinate (the inertial frame of reference) in the previous reference, the shear stress \(C_{\mathrm{\tau}}^{\theta}\) is written as  
\[C_{\mathrm{\tau}}^{\theta}=\frac{\tau_{r\theta}}{\frac{1}{2}\rho u_{\infty}^{ 2}}=-\frac{4}{\mathrm{Re}H}\frac{\partial^{2}\psi^{\prime}}{\partial\xi^{ \prime 2}}\]  
As \(\frac{\partial^{2}\psi^{\prime}}{\partial\xi^{\prime 2}}+\frac{\partial^{2} \psi^{\prime}}{\partial\psi^{\prime 2}}=-H\Omega^{\prime}\), then  
\[C_{\mathrm{\tau}}^{\theta}=\frac{4}{\mathrm{Re}}(\Omega^{\prime}+\frac{1}{H} \frac{\partial^{2}\psi^{\prime}}{\partial\eta^{\prime 2}})\]  
On the cylinder surface,  
\[\psi^{\prime}=-\frac{dl}{dt}\cos(2\pi\eta)\text{ and }\Omega^{\prime}=\Omega\]  
Thus,  
\[C_{\mathrm{\tau}}^{\theta}=C_{\mathrm{\tau}F}^{\theta}+C_{\mathrm{\tau}V}^{ \theta} \tag{9}\]  
where \(C_{\mathrm{\tau}F}^{\theta}=\frac{4}{\mathrm{Re}}\Omega\)  
\[C_{\mathrm{\tau}V}^{\theta}=\frac{4}{\mathrm{Re}}\frac{dl}{dt}\cos(2\pi\eta)\]  
Therefore, the shear stress involves two components denoted as \(C_{\mathrm{\tau}F}^{\theta}\) and \(C_{\mathrm{\tau}V}^{\theta}\), where \(C_{\mathrm{\tau}F}^{\theta}\) is proportional to vorticity at the wall, whereas \(C_{\mathrm{\tau}V}^{\theta}\) induced by the cylinder motion in viscous flow, has no relationship with the vorticity field. The pressure \(C_{p}^{\theta}\) is termed as based on the definition [41]  
\[C_{p}^{\theta}=\frac{\mathbb{F}_{p}^{\theta}}{\frac{1}{2}\rho u_{\infty}^{2}}= \frac{p_{\theta}-p_{\infty}}{\frac{1}{2}\rho u_{\infty}^{2}}=P_{\theta}-P_{\infty}\]  
where the pressure is \(P=\frac{p}{\rho u_{\infty}^{2}/2}\) with \(p\) the pressure of the flow field. Then, \(C_{p}^{\theta}\) can be formulated further by following mathematical derivation. From the momentum equations in the moving coordinate of reference, one obtains  
\[P_{\theta}-P_{0}=\frac{4}{\mathrm{Re}}\int_{0}^{\eta}\frac{\partial\Omega}{ \partial\xi}d\eta+4\pi N\int_{0}^{\eta}\left.F_{\theta}\right|_{\xi=0}d\eta-4 \frac{d^{2}l(t)}{dt^{2}}\sin(2\pi\eta) \tag{10}\]  
and  
\[P_{\infty}-P_{0}=-4\pi\int_{0}^{\infty}\frac{\partial u_{r}}{\partial t}e^{2 \pi\xi}d\xi-1-2\int_{0}^{\infty}u_{\theta}\frac{\partial u_{r}}{\partial\eta} d\xi+4\pi\int_{0}^{\infty}u_{\theta}^{2}d\xi-\frac{4}{\mathrm{Re}}\int_{0}^{ \infty}\frac{\partial\Omega}{\partial\eta}d\xi \tag{11}\]  
Then,  
\[C_{p}^{\theta}=P_{\theta}-P_{\infty}=C_{pF}^{\theta}+C_{pW}^{\theta}+C_{pV}^{ \theta} \tag{12}\]  
where  
\[C_{\mathrm{\tau}}^{\theta}=\frac{\tau_{r\theta}}{\frac{1}{2}\rho u_{\infty}^{ 2}}=-\frac{4}{\mathrm{Re}H}\frac{\partial^{2}\psi^{\prime}}{\partial\xi^{ \prime 2}}\]0\[C_{\mathrm{\tau}}^{\theta}=\frac{\tau_{r\theta}}{\frac{1}{2}\rho u_{\infty}^{ 2}}=-\frac{4}{\mathrm{Re}H}\frac{\partial^{2}\psi^{\prime}}{\partial\xi^{ \prime 2}}\]1  
\[C_{\mathrm{\tau}}^{\theta}=\frac{\tau_{r\theta}}{\frac{1}{2}\rho u_{\infty}^{ 2}}=-\frac{4}{\mathrm{Re}H}\frac{\partial^{2}\psi^{\prime}}{\partial\xi^{ \prime 2}}\]2  
\[C_{\mathrm{\tau}}^{\theta}=\frac{\tau_{r\theta}}{\frac{1}{2}\rho u_{\infty}^{ 2}}=-\frac{4}{\mathrm{Re}H}\frac{\partial^{2}\psi^{\prime}}{\partial\xi^{ \prime 2}}\]3  
Here, pressure \(C_{p}^{\theta}\) consists of \(C_{pF}^{\theta}\) induced by the field Lorentz force, \(C_{pW}^{\theta}\) induced by the wall Lorentz force and \(C_{pV}^{\theta}\) induced by the inertial force. The hydrodynamic force is also regarded as a force consisting of drag force and lift force denoting the force components in the streamwise and the transverse directions, respectively.