# A New Analytical Procedure to Solve Two Phase Flow in Tubes

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

#### 1.1. Level Sets in Cylindrical Co-Ordinates

#### 1.2. A New Composite Velocity Formulation

#### 1.3. Special Case Solution

#### Consideration of Curvature Alone

## 2. Results and Discussion

## 3. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Adanhounme, V.; Adomoub, A.; Codo, F.P. Analytical solutions for Navier-Stokes equations in the cylindrical coordinates. Bull. Soc. Math. Serv. Stand.
**2012**, 2, 16–23. [Google Scholar] [CrossRef] - Pereira, L.M.; Guerrero, J.S.P.; Cotta, R.M. Integral transformation of the Navier-Stokes equations in cylindrical geometry. Comput. Mech.
**1998**, 21, 60–70. [Google Scholar] [CrossRef] - Taylor, C.; Hood, P. A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids
**1973**, 1, 73–100. [Google Scholar] [CrossRef] - Chorin, A.J. The numerical solution of the Navier-Stokes equations for an incompressible fluid. Bull. Am. Math. Soc.
**1967**, 73, 928–931. [Google Scholar] [CrossRef] - Osher, S.; Sethian, J. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comp. Phys.
**1988**, 79, 12–49. [Google Scholar] [CrossRef] - Sethian, J. Level Set Methods and Fast Marching Methods; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Duflot, M. A study of the representation of cracks with level sets. Int. J. Numer. Methods Eng.
**2007**, 70, 1261–1302. [Google Scholar] [CrossRef] - Caselles, V.; Catte, F.; Coll, T.; Dibos, F. A geometric model for active contours in image processing. Numer. Math.
**1993**, 66, 1–31. [Google Scholar] [CrossRef] - Li, C.; Huang, R.; Ding, Z.; Gatenby, J.C.; Metaxas, D.N. A Level Set Method for Image Segmentation in the Presence of Intensity Inhomogeneities with Application to MRI. IEEE Trans. Image Process.
**2011**, 20, 2007–2016. [Google Scholar] [PubMed] - Mitchell, I.; Tomlin, C.J. Level Set Methods for Computation in Hybrid Systems, International Workshop on Hybrid Systems: Computation and Control. In Proceedings of the International Workshop on Hybrid Systems: Computation and Control, Pittsburgh, PA, USA, 23–25 March 2000; pp. 310–323. [Google Scholar]
- Maitre, E.; Milcent, T.; Cottet, G.-H.; Raoult, A.; Usson, Y. Applications of level set methods in computational biophysics. Math. Comput. Model.
**2009**, 49, 2161–2169. [Google Scholar] [CrossRef][Green Version] - Sussman, M.; Smereka, P.; Osher, S. A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys.
**1994**, 114, 146–159. [Google Scholar] [CrossRef] - Olsson, E.; Kreiss, G. A conservative level set method for two phase flow. J. Comput. Phys.
**2005**, 210, 225–246. [Google Scholar] [CrossRef] - Ayati, A.A.; Farias, P.S.C.; Azevedo, L.F.A.; de Paula, I.B. Characterization of linear interfacial waves in a turbulent gas-liquid pipe flow. Phys. Fluids
**2017**, 29, 062106. [Google Scholar] [CrossRef] - Osher, S.; Fedkiw, R. Level Set Methods and Dynamic Implicit Surfaces; Springer: Berlin, Germany, 2003. [Google Scholar]
- Yaji, K.; Otomori, M.; Yamada, T.; Izui, K.; Nishiwaki, S.; Pironneau, O. Shape and topology optimization based on the convected level set method. Struct. Multidiscip. Optim.
**2016**, 54, 659–672. [Google Scholar] [CrossRef] - Xie, F.; Zheng, X.; Triantafyllou, M.S.; Constantinides, Y.; Zheng, Y.; Karniadakis, G.E. Direct numerical simulations of two-phase flow in an inclined pipe. J. Fluid Mech.
**2017**, 825, 189–207. [Google Scholar] [CrossRef]

**Figure 2.**f versus r for $\alpha =-$1000 in level set function $\varphi $, ${R}_{i}=0.85$ cm is the radial value at the gas/liquid interface.

**Figure 3.**f versus r for $\alpha =-$5000 in level set function $\varphi $, ${R}_{i}=0.85$ cm is the radial value at the gas/liquid interface.

**Figure 4.**f versus r for $\alpha =-$10,000 in level set function $\varphi $, ${R}_{i}=0.85$ cm is the radial value at the gas/liquid interface.

**Figure 5.**f versus r for $\alpha =-$100,000 in level set function $\varphi $, ${R}_{i}=0.85$ cm is the radial value at the gas/liquid interface.

**Figure 6.**f versus r for $\alpha =-$1,000,000 in level set function $\varphi $, ${R}_{i}=0.85$ cm is the radial value at the gas/liquid interface.

**Figure 7.**$L(r)$ versus r, for $\alpha =-$10,000 corresponding to Figure 4 at different values of $Fz$.

**Figure 8.**$L(r)$ versus r, for $\alpha =-$1,000,000 corresponding to Figure 6 at different values of $Fz$.

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Moschandreou, T.E. A New Analytical Procedure to Solve Two Phase Flow in Tubes. *Math. Comput. Appl.* **2018**, *23*, 26.
https://doi.org/10.3390/mca23020026

**AMA Style**

Moschandreou TE. A New Analytical Procedure to Solve Two Phase Flow in Tubes. *Mathematical and Computational Applications*. 2018; 23(2):26.
https://doi.org/10.3390/mca23020026

**Chicago/Turabian Style**

Moschandreou, Terry E. 2018. "A New Analytical Procedure to Solve Two Phase Flow in Tubes" *Mathematical and Computational Applications* 23, no. 2: 26.
https://doi.org/10.3390/mca23020026