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**A mathematical model of angular two-phase Jeffery Hamel flow in a geothermal pipe.**
*(English)*
Zbl 1468.76070

Summary: A model of an annular two-phase Jeffrey Hamel flow in a geothermal pipe is designed with silica particles present in the effluent. Thermophoresis is singled out as the source of movement for the particles. The phenomena is also used to determine the growth of silica particles. The power law model has been used to express vicosity in both phases whereby viscosity is a non-linear function of wedge angle in the gaseous phase while a function of wedge angle and temperature in the liquid phase. Equations of mass, momentum, heat transfer and species transfer are the governing equations for this fluid flow. These equations are transformed into nonlinear ordinary differential equations by introducing a similarity transformation. The resulting equations are then solved using the bvp4c MATLAB solver. This method saves on computational time but still provides accurate and convergent results. Skin friction, Nusselt number, Sherwood number, non-dimensional thermophoresis velocity and non-dimensional thermophoresis deposition velocity are determined. The effect of the flow parameters on the flow variables is examined and it is established that changes in velocity and temperature does affect the concentration of silica. An increase in the Reynolds number implies an increase in the fluid velocity which increases the non-dimensional thermophoresis velocity VTW of both the liquid and the gaseous phases. Increase in the VTW implies increased concentration of colloidal silica particles which deposit on the walls of the geothermal pipe.

### MSC:

76T10 | Liquid-gas two-phase flows, bubbly flows |

76T15 | Dusty-gas two-phase flows |

76M99 | Basic methods in fluid mechanics |

80A19 | Diffusive and convective heat and mass transfer, heat flow |

86A60 | Geological problems |

### Keywords:

gas-liquid flow; silica particle; non-linear viscosity; thermophoresis; mass deposition rate; similarity transform; collocation method; skin friction
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\textit{V. Ojiambo} et al., Int. J. Adv. Appl. Math. Mech. 6, No. 2, 1--13 (2018; Zbl 1468.76070)

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### References:

[1] | J. Polii and H. Abdurrachim,Model Development of Silica Scaling Prediction on Brine Flow Pipe, Proceedings, |

[2] | A. Hasan and C. Kabir, Modelling two-phase fluid and heat flows in geothermal wells, J. of Petroleum Science and Engineering 71(2003),(2010) 77-86. |

[3] | H. Palsson, E. Berghorsson and O. Palsson, Estimation and validation of models two phase flow geothermal wells, Proceedings of 10th international symposium of heating and cooling, (2006) |

[4] | H. Mazumder and A. Siddique, CFD analysis of two-phase flow characteristic in a 90 degree elbow, J. of mechanical engineering. University of Michigan. 3(3) (2011). |

[5] | K. Umar, A. Naveed, Z. Zaidi , S. Jan and T. Syed, On Jeffrey-Hamel flows, Int. J. of Modern Mathematical Sciences. 7(3)(2013) 236-267. |

[6] | Gerdroodbary, M. Barzegar, M. Rahimi. and D. Ganji, Investigation of thermal radiation Jeffrey Hamel flow to stretchable convergent/ divergent channels. Science direct, Elsevier case studies in thermal engineering. 6 (2015) 28-39. |

[7] | K. Umar, A. Naveed, S. Waseen , T. Syed and D. Mohyud, Jeffrey-Hamel flow for a non-Newtonian fluid, J. of Applied and Computational Mechanics. 2(1) (2016) 21-28. |

[8] | K. Nizami and Sutopo, Mathematical modeling of silica deposition in geothermal wells, 5th International Geothermal Workshop (IIGW2016), IOP Conference series: Earth and Environmental Science (2016)2016 |

[9] | R. Bosworth, A. Ventura, A. Ketsdever and S. Gimelchein, Measurement of negative thermophoretic force, Journal of Fluid Mechanics. 805(2016) 207-221. |

[10] | A. Rahman, M. Alam and M. Uddin, Influence of magnetic field and thermophoresis on transient forced convective heat and mass transfer flow along a porous wedge with variable thermal conductive and variable thermal conductivity and variable Prandtl number, Int. J. of Advances in Applied Mathematics and Mechanics. 3(4) (2016) 49-64. · Zbl 1367.76067 |

[11] | J. Nagler J., Jeffrey-Hamel flow on non-Newtonian fluid with nonlinear viscosity and wall friction, J. of Applied Mathematics and Mechanics. 38(6)(2017) 815-830. · Zbl 1367.76004 |

[12] | G. Batchelor and C. Shen, Thermophoretic deposition of particles in gas flowing over cold surface, J. of colloid interphace science, 107 (1985) 21-37. |

[13] | G.B. Jeffrey, Two dimensionalsteady motion of a viscous fluid, Philos Mag, 6 (1915) 455-465. |

[14] | G. Hamel, Spiralformige Bewgugen Zaher Flussigkeiten, Jahresbericht der Deutschen, 25 (1916) 34-60. · JFM 46.1255.01 |

[15] | L.Talbot, R.Cheng , A. Schefer, D. Wills, Thermophoresis of particles in a heated boundary layer, J. of fluid mechanics. 101 (1980) 737-758. |

[16] | M. A. Sattar, A local simialrity transformation for the unsteady two-dimensional hydrodynamic boundary layer equations of a flow past a wedge, Int. J. appl.Math. and Mech. 7 (2011) 15-28. · Zbl 1427.76061 |

[17] | M. S. Alam, M. N. Huda, A new approach for local similarity solutions of an unsteady hydromagnetic free convective heat transfer flow along a permeable flat surface, Int. J. Adv. Appl.Math.Mech. 1(2) (2013) 39-52. · Zbl 1360.76350 |

[18] | M. S. Alam, M. M. Haque, M. J. Uddin, Unsteady MHD free convective heat transfer flow along a vertical porous flat plate with internal heat generation, Int. J. Adv. Appl.Math.Mech. 2(2) (2014) 52-61. · Zbl 1359.76328 |

[19] | M. A. Sattar, Derivation of the similarity equation of the 2-D Unsteady Boundary Layer equations and the Corresponding similarity equations, American J. of Fluid Mech. 3(5) (2013) 135-142. |

[20] | M. H. Mkwizu, A. X. Matofali and N. Ainea, Entropy generation in a variable viscosity transient genralized Coutte flow on nanofluids with Navier Slip and Convective cooling, Int. J. of Adv. in Appied Math. and Mech., 5(4) (2018) 20-29. · Zbl 1469.76133 |

[21] | M. S. Alam, M. M. Haque and M. J. Uddin, Convective flow of nano fluid along a permeable stretching/shrinking wedge with second order slip using Buongiorno’s mathematical model, Int. J. of Adv. in Appied Math. and Mech., 3(3) (2016) 79-91. · Zbl 1367.76058 |

[22] | V. Ojiambo, M. Kimathi and M. Kinyanjui, A study of two phase Jeffrey Hamel flow in a geothermal pipe, Int. J. of Adv. in Applied Math. and Mech., 5(4) (2018) 21-32. · Zbl 1469.76134 |

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