# Experimental and Numerical Study of Vacuum Resin Infusion for Thin-Walled Composite Parts
## 3 Model of the Resin Infusion: Problem Statement  
The fundamental difference between the proposed approach and those considered above is the interdependence of the equations of the coupled problem, the solution of which describes the dynamics of liquid resin in a porous preform. The approaches discussed above used the [PERSON]' equation as the leading (ruling) equation:  
\[\mathbf{u}=-[K]/\mu\cdot\
abla p \tag{1}\]  
where \(\mathbf{u}\) is the superficial resin velocity, \([K]\) is the permeability tensor of the porous medium, \(\mu\) is the resin viscosity and \(\
abla p\) is the pressure gradient in the liquid phase. The proposed approach is based on the use as the leading equation, describing the motion of the interface between two contacting phases--the front of the moved resin. In this case, the calculation of the velocity field, depending on the pressure gradient in the liquid, will be made based on the result obtained when determining the shape and position of the front. To some extent, this approach repeats the methodology used in [47], where a new numerical algorithm is proposed for modeling the dynamics of the interface between two immiscible phases. This interface is a thin layer, across which its physical properties change sharply, but continuously, and its evolution is determined by a phase field variable, which obeys the Cahn-Hilliard convection-diffusion-type equation. Its solution allows directly monitoring the interface and provides the correct interfacial tension due to the free energy accumulated in the separation layer. The phase field method does not require a regular FE mesh, a computational domain of simple form and spatial periodicity, which is required by other methods (e.g., the FE/CV method). This is important for modeling large-scale stream systems with complex and multiple interfaces, especially in three-dimensional (3D) cases. Let the system includes two immiscible phases A and B. The phase field variable \(\phi\) is introduced so that the concentrations of components A (resin) and B (void, vacuum) are \((1+\phi)\)/2 and \((1-\phi)\)/2, respectively. Note that hereinafter everywhere, in order to avoid unacceptable confusion of quantities, the phase field variable will be denoted as \(\phi\) and preform porosity as \(V_{\phi}\). For the mixing energy, we take a form analogous to the form of the Lagrangian of the Ginzburg-Landau equation for the complex field of Cooper's pairs:  
\[f_{mix}(\phi,\
abla\phi)=\frac{1}{2}\lambda\big{|}\
abla\phi\big{|}^{2}+\frac {\lambda}{4\epsilon^{2}}\big{(}\phi^{2}-1\big{)}^{2} \tag{2}\]  
Figure 8: The non-impregnated edges of the cured tail panel in its corners (**right**) and on the periphery (**left**). The full view of the ready defective part is shown in the center photo. where \(\lambda\) is the mixing energy density with force dimension and \(\varepsilon\) is the capillary width, which scales with the thickness of the diffuse interface. As \(\varepsilon\to 0\), the ratio \(\lambda/\varepsilon\) gives the interfacial tension \(\sigma\) in the classical sense:  
\[\sigma=2\sqrt{2}\lambda/3\varepsilon \tag{3}\]  
The equation describing the evolution of a phase field variable is the Cahn-Hilliard equation [48]:  
\[\partial\phi/\partial t+\mathbf{u}\cdot\
abla\phi=\
abla\cdot\gamma\
abla G \tag{4}\]  
where \(G\) (Pa) is a chemical potential:  
\[G=\lambda\left[-\
abla^{2}\phi+\phi\left(\phi^{2}-1\right)/\varepsilon^{2}\right] \tag{5}\]  
and \(\gamma\) (m\({}^{3}\)s/kg) is the phases mobility. Mobility \(\gamma\) determines the time scale of the Cahn-Hilliard diffusion and should be large enough to maintain a constant interfacial thickness, but small enough so that the convective terms are not excessively damped. Mobility \(\gamma\) is a function of the thickness of the interface according to \(\gamma\) = \(\chi\varepsilon^{2}\), where \(\chi\) is the mobility setting parameter. The Cahn-Hilliard equation forces a value for \(\phi\) of 1 or \(-1\), with the exception of a very thin region at the interface between two media.