## 2 Governing Equations  
The bio-inspired motions often occur in the incompressible viscous flow according to the summarized characteristics above; the flow field is governed by the Navier-Stokes equations:  
\[\
abla\cdot\mathbf{u}=0, \tag{1}\]  
\[\frac{\partial\mathbf{u}}{\partial t}+\
abla\cdot(\mathbf{uu})=-\
abla\cdot(p\mathbf{I})+ \
abla\cdot\left[\
u\left(\
abla\mathbf{u}+(\
abla\mathbf{u})^{T}\right)\right]+\mathbf{f}, \tag{2}\]  
where \(\rho\), \(\mathbf{u}\), and \(p\) represent the density, velocity, and pressure in the flow field, respectively. \(\
u\) is the kinematic viscosity of the fluid. \(\mathbf{f}\) is the forcing term representing the mutual interaction between immersed objects and fluids. It will be handled in different forms in various IBMs. In the bionic simulations, there is a variety of governing equations to characterize the movement and deformation of bodies. Without loss of generality, we present some typical descriptions involving prescribed motion/deformation and passive ones that are affected by ambient fluids. We illustrate these four types of bio-locomotion in Figure 1, where the rigid and deformable properties are relative to the geometrical shapes in simulations, and active and passive characteristics determine the interaction between the structures and fluids. Active motion and deformation are prescribed without consideration of the hydrodynamical forces, while passive ones are affected by the ambient fluids. For rigid structures, the prescribed motion is often adopted in simulations of insect/bird flight relying on flapping wings. As shown in Figure 1, for a two-dimensional (2D) flapping foil with heaving and pitching motions, or a three-dimensional (3D) dragonfly flight with pitching or rolling motions, a series of harmonic functions are commonly used to describe these types of flapping motions [52; 53; 54]:  
\[\theta=\theta_{0}\sin(2\pi ft+\rho), \tag{3}\]  
where \(\theta\) is the attitude angle for pitching or rolling motions. \(\theta_{0}\) is the amplitude of harmonic motion with frequency \(f\) and phase difference \(\varphi\). In the 2D heaving and pitching motions, the translation displacement and pitching angle are also prescribed in the form  
Figure 1: Illustration of the typical descriptions of bio-locomotion. of Equation (3). Some other governing equations for prescribed motions can be found in the literature [55; 56]. In biological motion research, the passive motion of a rigid body is mainly reflected in self-propulsion by thrust, as shown in Figure 1. The propelling motion is governed by [PERSON]'s second law [54; 57; 58]:  
\[m\frac{d^{2}\mathbf{X}}{dt^{2}}=\mathbf{F}, \tag{4}\]  
where \(\mathbf{X}\) is the position vector, \(m\) is the mass of the swimming body, and \(\mathbf{F}\) is the hydrodynamic force. Note that this passive motion is usually driven by harmonic pitching in the form of Equation (3) around a pivot point. For deformable bodies, as shown in the lower part of Figure 1, the prescribed deformation of a swimming fish is often encountered in recent studies [6; 59; 60; 61; 62; 63; 64]. Some mathematical models are used to characterize the deformation of the swimming body periodically. Here we present one of them to elucidate the prescribed deformation [59], where the shape is determined by geometric parameters, and the backbone waveform is controlled by  
\[y(x,t)=\left(c_{1}x+c_{2}x^{2}\right)\sin(k_{w}x-\omega t+\varphi), \tag{5}\]  
where \(y(x,t)\) is the waveform along the backbone, \(k_{w}\) is the wavenumber, \(\omega\) is the circular frequency, the two coefficients \(c_{1}\) and \(c_{2}\) determine the amplitude envelope of straight-line swimming, \(\varphi\) is the phase difference in multibody problems. In Figure 1, we present some deformations governed by Equation (5), where the basic geometric shape is NACA0006 airfoil, and the parameters of the waveform are referred to [59]. This type of deformation is independent of the hydrodynamic forces exerted on the structures, called the active or prescribed deformation.