## 5 Mathematical Models
### Models at the Macro-Scale
#### 5.1.2 The Two-Phase Flow Problem  
Two-phase flow describes the combined behavior of two different fluids (or phases) that are immiscible at a molecular level and both occupy domain \(\Omega\in\Re^{n}\). For example, this method was used to describe a water droplet in the air or the description of cellular aggregates in the surrounding hydrogel. Within each phase, all physical quantities are taken to be continuous; however, at the interface, at least one physical quantity is discontinuous. Modeling and simulating multiphase fluid flows have been a big challenge over the years; we will describe some of the related models. We denote the domain occupied by the two liquids with \(\Omega_{1}\) and \(\Omega_{2}\), respectively, such that \(\Omega=\Omega_{1}\cup\Omega_{2}\) and such that they are separated by an interface \(\Gamma=\Omega_{1}\cap\Omega_{2}\). In general, the phase index \(i=1,2\) is used for all quantities of the liquid phase \(i\). If the fluids are in motion, the domains \(\Omega_{i}(t),i=1,2\), as well as the interface \(\Gamma(t)\), are unknown at the time instant \(t\) getting a so-called _free boundary problem_. When it comes to the numerical treatment of free boundary problems, we can often identify two different kinds of issues [98]. 1. The representation and evolution in time of the interface. 2. The way the imposed surface boundary conditions are treated. The first issue above will be treated at the following point \(1\), whereas point \(2\) deals with the second issue above. 1. _Representation and time evolution of the interface._ To represent the two-phase flows with shifting inter-phase boundaries, various methodologies have been developed in response to the problem's complexity, both numerically and physically. In principle, the methods can be divided into two classes: the so-called _front-tracking (FT) methods_ and _front-capturing (FC) methods_. In these methods, two-phase flow is treated as a single flow with the density and viscosity smoothly varying across the moving interface which is captured in a Eulerian framework (FC) or in a Lagrangian, framework (FT), with the terms Eulerian and Lagrangian frameworks, meaning, meaning the following [100]: _Lagrangian framework_: In classical field theories, the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. \begin{table}
\begin{tabular}{c c} \hline \hline _Outflow_ & In order to prescribe boundary conditions at the outlet, many technics exist. Practicable importance can be attributed to the so-called “do-nothing” method, which is described by \\  & the following equation \\  & with a prescribed normalized pressure \(p\)[98]. \\ \hline \multirow{4}{*}{_Periodic_} & In the case of a one-directional periodical problem with periodic length \(\pi\), the velocities and the pressure have to be equal on both sides of the periodical direction. If we assume that the problem is periodic in the \(x_{i}\) direction, it follows that [98] \\  & \(v(x_{1},\ldots,x_{i},\ldots,x_{n},t)\) & \(v(x_{1},\ldots,x_{i}+\pi,\ldots,x_{n},t)\) \\  & \(\partial_{x_{1}}v(x_{1},\ldots,x_{i},\ldots,x_{n},t)\) & \(\partial_{x_{1}}v(x_{1},\ldots,x_{i}+\pi,\ldots,x_{n},t)\) \\  & \(p(x_{1},\ldots,x_{i},\ldots,x_{n},t)\) & \(p(x_{1},\ldots,x_{i}+\pi,\ldots,x_{n},t)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._Eulerian framework: The Eulerian specification of the flow field is a way of looking at fluid motion that focuses on specific locations in the space through which the fluid flows as time passes. The front-tracking method is based on a formulation in which a separate unstructured grid (surface grid) with nodes converging at a local velocity is used to represent the interface. One benefit of the technique is that interfacial conditions can be easily included because the interface is precisely established by the surface grid's position. Moreover, a good approximation of the curvature allows for the consideration of surface tension force.