# Investigation of Fluid-Structure Interaction Induced Bending for Elastic Flaps in a Cross Flow
## 2 Material and Method
### Numerical Model  
FSI modeling mainly deals with the information about how the fluid domain acts as a function of structure and how structural domain behaves as a function of fluid domain, at the interface, information is shared between the structural and fluid solver thus founding FSI [38]. In the present case, an elastic structure was surrounded by incompressible Newtonian fluid flow. \(\Omega_{\text{f}}\)(t) is defined as a time dependent fluid domain and \(\Omega_{\text{s}}\)(t) is defined as a time dependent structural domain. It is assumed that for all the time t, \(\Omega_{\text{f}}\)(t) is entirely occupied by the fluid and \(\Omega_{\text{s}}\)(t) is entirely occupied by the structure. Besides, \(\Gamma_{\text{o}}\)(t) = \(\Omega_{\text{f}}\)(t) \(\cap\)\(\Omega_{\text{s}}\)(t) is an interface where the elastic structure interacts with the fluid. A schematic of FSI in terms of structural domain, fluid domain and interface for a present case of an elastic flap immersed in a rectangular fluid domain is shown in Figure 1.  
For the established FSI solution, the kinematic, dynamic and geometric conditions at the common interface of fluid and structure should be satisfied, numerically at the interface \(\Gamma_{\text{o}}\)(t), the following conditions are assumed for all the time t:  
\[v_{i}^{s} = v_{i}^{f} \tag{1}\] \[\sigma_{ij}^{s}n_{i} = \sigma_{ij}^{f}\,n_{i}. \tag{2}\]  
Here, \(v^{s}\) and \(v^{f}\) represent the structure and fluid velocity constituting the kinematic stability. Dynamic stability is attained by the continuous normal stresses of solid \(\sigma^{s}\) and fluid \(\sigma^{f}\) at the interface. \(n\) being the unit vector normal at the interface and the subscript \(i\) indicates its direction. According to the geometric condition stability fluid-solid domain always match and do not overlap [38].  
Equations of motion for the structure and fluid are expressed as follows:  
**Structure equations**  
The flexible flap is modeled as an elastic continuum therefore the balance of linear momentum is given by:  
\[\rho^{s}\,\dot{v_{i}^{s}}=\sigma_{ij,j}^{s}\,\text{ in }\Omega_{\text{s}}( \text{t}). \tag{3}\]  
For linear elastic material, the linear Hook's law is followed by the structural stress as:  
\[\sigma_{ij}^{s}=2H\varepsilon_{ij}+\lambda\varepsilon_{kk}\delta_{ij}, \tag{4}\]  
Figure 1: A schematic of FSI in the present work.  
where \(\rho^{s}\) is the structural density and \(\sigma^{s}_{ij,j}\) is the structural stress. The subscript _ij_ of the stresses in Equations (3), (4), (6) and (7) represents the components of stress tensor whereas the subscript \(j\), indicates the differential term involved in the linear elastic model.  
The structural stress is a function of strain \(\varepsilon\) and the Lame parameters \(H\) and \(\lambda\), which are defined as:  
\[\begin{array}{c}\varepsilon_{ij}=\frac{1}{2}\Big{(}u_{i.j}+u_{j,i}\Big{)}\\ H=\frac{E}{2\left(1+\,\mathrm{V}\right)}\\ \lambda=\frac{E\,\mathrm{V}}{\left(1+\,\mathrm{V}\right)\left(1-2\,\mathrm{V} \right)},\end{array}\]  
where \(u\) represents the displacement, \(E\) and \(v\) are [PERSON]'s modulus and Poisson ratio, respectively [38; 39; 40].