# Numerical investigation of interface motion between two immiscible fluids in a channel

### Abstract

The main difficulty of the modeling of two immiscible viscous fluids flow in the channel (pipe, etc.), is the choice of the boundary condition on the line (contact line) formed by intersection of the interface between fluids with the solid surface. If the no-slip condition is used on the solid boundary to determine the flow produced when a fluid interface moves along a solid boundary, the viscous stress is approached to infinity at the vicinity of the contact line. It seems, unreasonable to continue to apply a continuum model at the vicinity of the contact line. Thus an inner region, close to the contact line, could be examined, where the molecular interactions between the two fluids and the solid must be studied, and this region matched to an outer region, where the Navier- Stokes equations would apply. Such an analysis would be very difficult, but it has been suggested that the likely outcome would be equivalent to replacing the no-slip boundary condition by a slip condition, and continuing to employ the Navier-Stokes equations. The effect of the slip on the interface motion is numerically investigated in this work. Also relation between steady-state contact angle and capillary number is investigated in this paper and compared with work [8].### References

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[2] de Gennes P.G. Wetting: statics and dynamics // Rev. Mod. Phys. - 1985. - Vol. 57. - P. 827-863.

[3] Huh C. and Scriven L. Hydrodynamic model of steady movement of a solid/liquid/fluid contact line // J. Coll. Interf. Sci. - 1971. - Vol. 35. - P. 85-101.

[4] Lauga E., Brenner M.P., Stone H.A. Microfluidics: The no-slip boundary condition, in Handbook of Experimental Fluid Dynamics (Chapter 19) // Springer, 2007.

[5] Vinogradova O.I. Slippage of water over hydrophobic surfaces // Int. J. Mineral Processing. - 1999. - Vol. 56. - P. 31-60.

[6] Teo C.J., Khoo B.C. Analysis of Stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves // Microfluid Nanofluid. - 2009. - Vol. 7. - P. 353-382.

[7] Greenspan H.P. On the motion of a small viscous droplet that wets a surface // J. Fluid Mech. - 1978. - Vol. 84. - P. 125-143.

[8] Bonn D., Eggers J., Indekeu J., Meunier J. and Rolley E. Wetting and spreading // Rev. Mod. Phys. - 2009. - Vol. 81. - P. 739-805.

[9] Pilliod J.E., Jr. and Puckett E.G. Second-order accurate volume-of-fluid algorithms for tracking material interfaces // J. Comput. Phys. - 2004. - Vol. 199. - P. 465-502.

[10] Scardovelli R. and Zaleski S. Analytical relations connection linear interfaces and volume fractions in rectangular grids // J. Comput. Phys. - 2000. - Vol. 164. - P. 228-237.

[11] Brown D.L., Cortez R. and Minion M.L. Accurate projection methods for the incompressible Navier-Stokes equations // J. Comput. Phys. - 2001. - Vol. 168. - P. 464-499.

[12] Christer B. and Johansson V. Boundary Conditions for Open Boundaries for the Incompressible Navier-Stokes Equation // J. Comput. Phys. - 1993. - Vol. 105. - P. 233-251.

[13] Tryggvason G., Scardovelli R. and Zaleski S. Direct Numerical Simulations Of Gas–Liquid Multiphase Flows // Cambridge University Press, 2011.

[14] Popinet S. The Gerris Flow Solver: http://gfs.sourceforge.net

[2] de Gennes P.G. Wetting: statics and dynamics // Rev. Mod. Phys. - 1985. - Vol. 57. - P. 827-863.

[3] Huh C. and Scriven L. Hydrodynamic model of steady movement of a solid/liquid/fluid contact line // J. Coll. Interf. Sci. - 1971. - Vol. 35. - P. 85-101.

[4] Lauga E., Brenner M.P., Stone H.A. Microfluidics: The no-slip boundary condition, in Handbook of Experimental Fluid Dynamics (Chapter 19) // Springer, 2007.

[5] Vinogradova O.I. Slippage of water over hydrophobic surfaces // Int. J. Mineral Processing. - 1999. - Vol. 56. - P. 31-60.

[6] Teo C.J., Khoo B.C. Analysis of Stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves // Microfluid Nanofluid. - 2009. - Vol. 7. - P. 353-382.

[7] Greenspan H.P. On the motion of a small viscous droplet that wets a surface // J. Fluid Mech. - 1978. - Vol. 84. - P. 125-143.

[8] Bonn D., Eggers J., Indekeu J., Meunier J. and Rolley E. Wetting and spreading // Rev. Mod. Phys. - 2009. - Vol. 81. - P. 739-805.

[9] Pilliod J.E., Jr. and Puckett E.G. Second-order accurate volume-of-fluid algorithms for tracking material interfaces // J. Comput. Phys. - 2004. - Vol. 199. - P. 465-502.

[10] Scardovelli R. and Zaleski S. Analytical relations connection linear interfaces and volume fractions in rectangular grids // J. Comput. Phys. - 2000. - Vol. 164. - P. 228-237.

[11] Brown D.L., Cortez R. and Minion M.L. Accurate projection methods for the incompressible Navier-Stokes equations // J. Comput. Phys. - 2001. - Vol. 168. - P. 464-499.

[12] Christer B. and Johansson V. Boundary Conditions for Open Boundaries for the Incompressible Navier-Stokes Equation // J. Comput. Phys. - 1993. - Vol. 105. - P. 233-251.

[13] Tryggvason G., Scardovelli R. and Zaleski S. Direct Numerical Simulations Of Gas–Liquid Multiphase Flows // Cambridge University Press, 2011.

[14] Popinet S. The Gerris Flow Solver: http://gfs.sourceforge.net

Published

2017-11-24

How to Cite

КУДАЙКУЛОВ, A. A.; ЖОЗЕРАНД, К.; КАЛТАЕВ, А..
Numerical investigation of interface motion between two immiscible fluids in a channel.

**KazNU Bulletin. Mathematics, Mechanics, Computer Science Series**, [S.l.], v. 89, n. 2, p. 75-86, nov. 2017. ISSN 1563-0277. Available at: <http://bm.kaznu.kz/index.php/kaznu/article/view/355>. Date accessed: 17 aug. 2018.
Section

Mechanics

**Keywords:**Navier-Stokes equations, flow of two immiscible fluids, Gerris program, slip boundary condition, volume-of-fluid (VOF) method, contact line, contact angle, capillary number