Abstract
Two-fluid flow is examined analytically and numerically for increased flow rates through a channel with surface roughness or branching or both. The viscosity and density ratios of the fluids are of order unity. There is much concern in terms of applications as well as fluid dynamical phenomena in configurations where one fluid is present only as a thin layer near an outer wall, leaving the other fluid occupying the channel core and part of a viscous wall layer. The interactive dynamics in both regions is studied and numerical and asymptotic analyses are performed. The major situations examined are: the flow to two symmetrically bifurcating daughters and the flow in a single channel over a rough wall, as well as a combination of the two situations. The principal phenomena of interest are conditions for flow reversal, the presence of upstream influence and the trajectories of the injected fluid as the density or viscosity ratio varies. Special relatively thin or thinning wall layers are produced when the core fluid viscosity increases or when the fluid travels downstream into a daughter vessel.
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References
Balásházy I.: Simulation of particle trajectories in bifurcating tubes. J. Comput. Phys. 110, 11–22 (1994)
Baroud C.N., Tsikata S., Heil M.: The propagation of low-viscosity fingers into fluid-filled branching networks. J. Fluid Mech 546, 285–294 (2006)
Blyth M.G., Mestel A.J.: Steady flow in a dividing pipe. J. Fluid Mech. 401, 339–364 (1999)
Brotherton Ratcliffe, R.V.: PhD thesis University of London (1986)
Carlson A., Do Quang M., Amberg G.: Droplet dynamics in a bifurcating channel. Int. J. Multiphase Flow 36, 397–405 (2010)
Chisholm D.: Influence of pipe surface roughness on friction pressure gradient during two-phase flow. J. Mech. Eng. Sci. 20(6), 353–354 (1978)
Drew D.: Mathematical modeling of two-phase flow. Ann. Rev. Fluid Mech. 15, 261–291 (1983)
Gao E.E., Young W.L., Pile-Spellman J., Joshi S., Duong H., Stieg P.E., Ma Q.: Cerebral arteriovenous malformation feeding artery aneurysms; a theoretical model of intravascular pressure changes after treatment. Neurosurgery 41, 1345–1358 (1997)
Green J.E.F., Ovenden N.C., Smith F.T.: Models of motion of a single fluid through a branched network with moving walls. J. Eng. Math. 64, 353–365 (2009)
Hademenos G.J., Massoud T.F., Viñuela F.: A biomathematical model of intracranial arteriovenous malformations based on electrical network analysis: theory and hemodynamics. Neurosurgery 38, 1005–1015 (1996)
Issa R.I., Kempf M.H.W.: Simulation of slug flow in horizontal and nearly horizontal pipes with the two-fluid model. Int. J. Multiphase Flow 29, 69–95 (2003)
Joseph D.D., Bai R., Chen K.P., Renardy Y.Y.: Core annular flows. Ann. Rev. Fluid Mech. 29, 65–90 (1997)
Liu Y., So R.M.C., Zhang C.M.: Modeling the bifurcating flow in an asymmetric human pulmonary artery. J. Biomech. 36, 951–959 (2003)
Lussenhop A., Spence W.: Artificial embolization of cerebral arteries: report of use in a case of arteriovenous malformation. J. Am. Med. Assoc. 11, 1153–1155 (1960)
Mahlmann S., Papageorgiou D.T.: Buoyancy-driven motion of a two-dimensional bubble or drop through a viscous liquid in the presence of a vertical electric field. Theoret. Comput. Fluid Dyn. 23(5), 375–399 (2009)
Manga M.: Dynamics of drops in branched tubes. J. Fluid Mech. 315, 105–117 (1996)
Mazzeo M., Coveney P.: HemelB: a high performance parallel Lattice-Boltzmann code for large scale fluid flow in complex geometries. Comput. Phys. Commun. 178(12), 894–914 (2008)
Pedley T.J., Schroter R.C., Sudlow M.F.: Flow and pressure drop in systems of repeatedly branching tubes. J. Fluid Mech 46, 365–383 (1971)
Rosenhead, L. (ed.): Laminar Boundary layers. Dover Pubs., New York
Rothmayer, A.P., Smith, F.T.: Incompressible triple deck theory. In: Handbook of Fluid Dynamics, Chapters 23–25. CRC Press, Boca Raton (1998)
Seruga T.T.: Endovascular treatment of intercranial arteriovenous malformations. Radiol. Oncol. 36(3), 201–208 (2002)
Shi H., Kleinstreuer C., Zhang Z.: Nanoparticle transport and deposition in bifurcating tubes with different inlet conditions. Phys. Fluids 16(7), 2199–2213 (2004)
Shockling M.A., Allen J.J., Smits A.J.: Roughness effects in turbulent pipe flow. J. Fluid Mech. 564, 267–285 (2006)
Smith F.T.: Laminar flow over a small hump on a flat plate. J. Fluid Mech. 57, 803–824 (1973)
Smith F.T.: Pipeflows distorted by non-symmetric indentation or branching. Mathematika 23, 62–83 (1976)
Smith F.T.: Flow through symmetrically constricted tubes. J. Inst. Applies 21, 145–156 (1978)
Smith F.T.: Non-parallel flow stability of the Blasius boundary layer. Proc. Roy. Soc. A. 366(1724), 91–109 (1979)
Smith F.T., Brighton P.W.M., Jackson P.S., Hunt J.C.R.: On boundary-layer flow past two-dimensional obstacles. J. Fluid Mech. 113, 123–152 (1980)
Smith F.T., Jones M.A.: One to few and one to many branching tube flows. J. Fluid Mech. 423, 1–31 (2000)
Smith F.T., Jones M.A.: AVM modelling by multi-branching tube flow: large flow rates and dual solutions. IMA J. Math. Med. Biol. 20(2), 183–204 (2003)
Tadjfar M., Smith F.T.: Direct simulations and modelling of basic three-dimensional bifurcating tube flows. J. Fluid Mech. 519, 1–32 (2004)
Taylor G.I.: Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161–165 (1960)
Tillett J.P.K.: On the laminar flow in a free jet of liquid at high Reynolds numbers. J. Fluid Mech. 96, 1–26 (1967)
Timoshin S.N., Smith F.T.: Singularities encountered in three dimensional boundary layers under an adverse or favourable pressure gradient. Phil. Trans. A. 352, 45–87 (1995)
Werle M.J., Thomas Davis R.: His contributions to numerical simulation of viscous flows—Part I-historical perspective. Meccanica 22(3), 133–158 (1991)
White, A.H.: PhD Thesis, University of London (2008)
White, A.H., Smith F.T.: Mathematical modelling of the embolization process for the treatment of arteriovenous malformations, Proc. IMechE Part H J. Eng. Med. (in preparation)
Zhang Z., Kleinstruer C., Kim C.S.: Gas-solid two-phase flow in a triple bifurcation lung airway model. Int. J. Multiphase flow 28, 1021–1046 (2002)
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Communicated by: Hussaini.
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White, A.H., Smith, F.T. Wall shape effects on multiphase flow in channels. Theor. Comput. Fluid Dyn. 26, 339–360 (2012). https://doi.org/10.1007/s00162-011-0237-7
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DOI: https://doi.org/10.1007/s00162-011-0237-7