## 5 Mathematical Models
### Models at the Macro-Scale
#### 5.1.2 The Two-Phase Flow Problem
The method's drawback is that it requires the use of extra algorithms to compute flows with substantial interface deformations such bubble break-up and coalescence since the surface grid will be severely distorted. In contrast, the front-capturing approach uses a scalar function on a stationary grid to capture the interface as part of its implementation of the interface evolution. The ability to compute phenomena like bubble break-up and coalescence is made possible by the implicit interface capture provided by front-capturing techniques. The two main difficulties in using this approach are keeping a clear boundary between the different fluids and accurately calculating the surface tension forces [98]. The _Level Set_ approach was developed as a result of the front-capturing approach [59; 60]. A different method, known as the Immersed Boundary (IB) method [58; 101; 102; 103; 104], was developed for flows across challenging geometries. It can be thought of as a combination of front-capturing and front-tracking procedures. **Level Set Method** The idea on which the level set method is based is quite simple. Given an interface \(\Gamma\) in \(\Re^{n}\) of codimension one, bounding an open region \(\Omega\), the LS method intends to analyze and compute its subsequent motion under a velocity field \(\mathbf{v}\). This velocity depends on the position, time, the geometry of the interface (e.g., its normal or its mean curvature) and the external physics. The idea is to define a smooth function \(\phi(x,t)\) representing the interface as the set where \(\phi(x,t)=0\). The level set function \(\phi(x,t)\) has the following properties:  
\[\phi(x,t)>0 \text{for }x\in\Omega_{1}\] \[\phi(x,t)<0 \text{for }x\in\Omega_{2}\] \[\phi(x,t)=0 \text{for }x\in\Gamma(t)\]  
Thus, the interface is to be captured for all times \(t\), by defining the set of points \(\Gamma(t)\) for which the \(\phi\) values (levels) obey the following equation  
\[\frac{\partial\phi}{\partial t}+\mathbf{v}\cdot\
abla\phi=0. \tag{19}\]  
Coupling the level set method with problems related to two-phase Navier-Stokes incompressible flow needs the reformulation of Problem 1 as follows  
**Problem 2**. : _Equations of two-phase Navier-Stokes incompressible flow defined in \(\Omega\times\Theta\) using the level set method_  
\[\frac{\partial}{\partial t}(\rho\mathbf{v})+(\rho\mathbf{v})\odot\
abla\bm {v} = -\
abla p+\text{div }\mathbf{T}+\rho\mathbf{g}+\delta(\phi)\sigma\kappa\mathbf{N} \tag{20}\] \[\text{div }\mathbf{v} = 0\] (21) \[\frac{\partial\phi}{\partial t}+\mathbf{v}\cdot\
abla\phi = 0\] (22) \[\mathbf{v}_{|t=0} = \mathbf{v}_{0}\] (23) \[p_{|t=0} = p_{0} \tag{24}\]\[\phi(x,t)>0 \text{for }x\in\Omega_{1}\]0  
_where \(v\) is the velocity field, \(p\) is the pressure field, \(g\) is the gravitational acceleration, \(\rho(\phi)\) and \(\mathbf{T}(\phi)\) are the piecewise constant fluid densities and stress tensors described by the Heaviside function, \(\sigma\) is the surface tension coefficient, \(\kappa\) is the curvature of the interface, \(\mathbf{N}\) is the unit normal vector outward to \(\Gamma(t)\), and \(\delta(\phi)\) is a delta function. The following equations hold [59]: With the symbol \(|u|\), we denote the modulus of the vector \(u\in\Re^{n}\), that is,_  
\[\phi(x,t)>0 \text{for }x\in\Omega_{1}\]1 \[\phi(x,t)>0 \text{for }x\in\Omega_{1}\]2 \[\phi(x,t)>0 \text{for }x\in\Omega_{1}\]3 (27) \[H_{1}(x,t) = \left\{ \begin{array}{ll}1&\text{for }x\in\Omega_{1}(t)\\ 0&\text{for }x\
otin\Omega_{1}(t)\end{array} \right.\] (28) \[\delta(\phi(x,t)) = \left\{ \begin{array}{ll}1&\text{for }x\in\Gamma(t)\\ 0&\text{for }x\
otin\Gamma(t)\end{array}