# A Review: Factors Affecting Internal Two-Phase Flow-Induced Vibrations
## 2 Factors Affecting FIV in Internal Two-Phase Flow
### Effect of Physical Properties  
The effect of physical properties on excitation force exerted at 90\({}^{\circ}\) bend due to the two-phase flow was investigated by [PERSON] and [PERSON] [73]. It was concluded that by reducing surface tension by 32%, there was no notable effect on forces exerted on bend due to the two-phase flow. Similarly, no notable effect was observed when liquid phase viscosity was increased by 2.62%. Moreover, it was reported that liquid holdup reduces by reducing liquid surface tension, while there was no effect of liquid viscosity on liquid holdup. Furthermore, these results are supported by the empirical correlation proposed by [PERSON] et al. [48]. An investigation [8, 73] on forces on bends was extended by [PERSON] et al. [48] to distinguish the effect of dimensionless parameters. The dependence of dimensionless parameters on RMS force was presented by [PERSON], as shown in Equation (4). This can be further simplified to Equation (12), as shown below:  
\[F_{RMS}=b\big{(}V_{m},\beta,D,\rho_{l},\rho_{g},\sigma,\mu,g\big{)} \tag{4}\]  
where \(b\) represents the unknown function of mentioned dimensional parameters, \(V_{m}\) is the mixture velocity, \(\beta\) is the void fraction, \(D\) is the pipe diameter, \(\sigma\) is the surface tension, \(\mu\) is the dynamic viscosity, \(g\) is the gravitational acceleration, \(\rho_{l}\) and \(\rho_{g}\) are liquid and gas phase densities, respectively. With the use of the standard dimensional analysis from [PERSON] and [PERSON] [90], Equation (4) can be written as follows:\[\frac{F_{RMS}}{F_{stat}}=B\left(\beta,\frac{\rho_{l}}{\rho_{g}},We,Re,Fr\right) \tag{5}\]  
\[F_{stat}=\rho_{l}(1-\beta){V_{m}}^{2}\left(\pi\frac{D^{2}}{4}\right) \tag{6}\]  
\[We=\frac{\rho_{l}{V_{m}}^{2}D}{\sigma} \tag{7}\]  
\[Re=\frac{\rho_{l}{V_{m}}D}{\mu} \tag{8}\]  
\[Fr=\frac{V_{m}}{\sqrt{\rho_{l}\beta}} \tag{9}\]  
where \(B\) only depends on dimensionless parameters and can be assumed to vary with \(\frac{1}{(1-\beta)}\). [PERSON] who used his experimental results [48] as well as conclusions of [PERSON] and [PERSON] [8] and of [PERSON] and [PERSON] [73] to study the effect of each parameter, concluded that the formulation can be extended as follows:  
\[\frac{F_{RMS}}{\rho_{l}(1-\beta){V_{m}}^{2}\left(\pi\frac{D^{2}}{4}\right)}= \frac{B}{(1-\beta)}We^{-0.4} \tag{10}\]  
This is further simplified into a dimensionless form using the Bukingham pi-theorem [91], as follows:  
\[\overline{F_{RMS}}=\frac{F_{RMS}}{\rho_{l}{V_{m}}^{2}\left(\pi\frac{D^{2}}{4 }\right)} \tag{11}\]  
\[\overline{F_{RMS}}=CWe^{-0.4} \tag{12}\]  
where \(C\) is a constant, \(F_{stat}\) is the stationary component of force, \(We\), \(Re\), \(Fr\) are dimensionless Weber, Reynolds, and Froude numbers, respectively. From the experimental data analysis [8; 63; 73], the suggested value of \(C\) is 10. Later, [PERSON] et al. [76] tried to correlate experimental data of excitation forces on bends to the Weber number alone. The effort was unsuccessful and a reasonable accuracy was not achieved. [PERSON] and [PERSON] [8] suggested the use of dimensionless parameters, such as We, Fr, and Re, while developing the correlation of excitation forces on bends. This stipulated that the incorporation of surface tension, viscosity, and gravitational effect into the \(F_{RMS}\) correlation may be important, although they appear to be unimportant. A very limited literature is available on the effects of physical properties on FIV. In addition, the available studies are focused on slug flow regime only and the findings are for limited diameter ranges. For a better explanation on the effects of physical properties in wider scale systems and various flow regimes, further research is needed.