## 5 Mathematical Models
### Models at the Macro-Scale
#### 4.2.2 Immersed Boundary Method
: _Equations of the Navier-Stokes incompressible flow defined in \(\Omega_{f}\times\Theta\) surrounding an Immersed Body \(\Omega_{b}\):_  
\[\frac{\partial}{\partial t}(\rho\mathbf{v})+(\rho\mathbf{v})\odot\
abla\mathbf{v} = -\
abla p+\text{div}\ \mathbf{T}+\rho\mathbf{g}+\mathbf{f}^{IBM} \tag{40}\] \[\text{div}\ \mathbf{v} = 0\] (41) \[\mathbf{v}_{|t=0} = \mathbf{v}_{0}\] (42) \[p_{|t=0} = p_{0}\] (43) \[\frac{\delta X}{\delta t} = \int_{\Omega_{b}}\mathbf{v}(x,t)d(|x-X|)dx\] (44) \[\mathbf{F}(X,t)=-\frac{\wp E}{\wp X}, \tag{36}\]0 (45) \[\mathbf{F}(X,t)=-\frac{\wp E}{\wp X}, \tag{36}\]1  
_where \(\rho\) is the density of the fluid, \(\mathbf{v}\) is the velocity field, \(\mathbf{p}\) is the pressure field, \(\mathbf{g}\) is the gravitational acceleration, \(\mathbf{T}\) is the stress response to the deformation of the fluid, \(X\) is the Lagrangian coordinate of the generic point of the body, and \(E\) is the elastic potential energy of the body material._  
Equations (38) and (39) are interaction equations. They convert Lagrangian variables to Eulerian variables (Equation (38)) and vice versa (Equation (39)). 2. _Surface boundary conditions treatment_ Since the second derivative of a discontinuous function determines the curvature \(\kappa\), using front-capturing methods to compute the surface tension force requires particular considerations. Various methods were devised to precisely calculate the surface tension force. The most well-known method is presumably the _continuum surface force (CSF)_ model [61]. The model considers surface tension force as a continuous, 3D effect across an interface, rather than as a boundary value condition on the interface. The continuum method eliminates the need for interface reconstruction. A model of a  
\[\mathbf{F}(X,t)=-\frac{\wp E}{\wp X}, \tag{36}\]2 A generic body around which flow is to be simulated. The body occupies the volume \(\Omega_{b}\) with boundary \(\Gamma_{b}\) (**a**). Transfer of forcing \(\mathbf{F}\) from Lagrangian boundary point \(X\) to surrounding fluid nodes (**b**). diffuse interface is examined, in which the surface tension force is converted into a volume force that distributes across several cell layers. We note (see Equation (20)) that surface tension \(\mathbf{F}_{sv}\) per interfacial area at a point \(\mathbf{x}_{\Gamma}\) on the interface \(\Gamma\) is given by  
\[\mathbf{F}(X,t)=-\frac{\wp E}{\wp X}, \tag{36}\]3  
where \(\rho\) is the surface tension coefficient, \(\kappa\) is the curvature of the interface, and \(\mathbf{N}\) is the unit normal vector outward to \(\Gamma(t)\). Suppose that the interface \(\Gamma\) where the fluid changes from fluid 1 to fluid 2 discontinuously is replaced by a continuous transition. Applying a pressure jump at an interface brought on by surface tension is no longer appropriate. Instead, as can be shown in Figure 5, surface tension should be considered active everywhere in the transition area. As in [61], we choose a volume force \(\mathbf{F}_{sv}(\mathbf{x})\) at any point \(x\) of transition region as  
\[\mathbf{F}_{sv}(\mathbf{x})=\left\{ \begin{array}{ll}\rho\kappa(\mathbf{x})\
abla\tilde{c }(\mathbf{x})&\text{ for }|\mathbf{N}(\mathbf{x}_{\Gamma})(\mathbf{x}-\mathbf{x}_{\Gamma})|<h\\ 0&\text{ for }|\mathbf{N}(\mathbf{x}_{\Gamma})(\mathbf{x}-\mathbf{x}_{\Gamma})|\geq h \end{array} \right. \tag{48}\]  
where \(\tilde{c}(\mathbf{x})\) is a smooth _\"color function\"_ that approaches the _\"characteristic function\"_, \(c(\mathbf{x})\), as the width \(h\) of the region approaches to zero,  
\[\lim_{h\to 0}\tilde{c}(\mathbf{x})=c(\mathbf{x}), \tag{49}\]  
and where the _\"characteristic function\"_ can be defined as  
\[c(\mathbf{x})=\left\{ \begin{array}{ll}1&\text{ for }x\in\Omega_{1}\\ 0&\text{ for }x\in\Omega_{2}\\ \frac{1}{2}&\text{ for }x\in\Gamma\end{array} \right., \tag{50}\]  
where \(\Omega_{1}\) and \(\Omega_{2}\) represent the domains occupied by fluids 1 and 2, respectively.