Abstract
Asymptotic models are constructed for the solidification process in a highly viscous film flow on the surface of a cone with a given mass supply at the cone apex. In the thin-layer approximation, the problem is reduced to two parabolic equations for the temperatures of the liquid and the solid coupled with an ordinary differential equation for the solidification front. For large Péclet numbers, an analytical steady-state solution for the solidification front is found. A nondimensional parameter which makes it possible to distinguish flows (i) without a solid crust, (ii) with a steady-state solid crust, and (iii) with complete solidification is determined. For finite Péclet numbers and large Stefan numbers, an analytical transient solution is found and the time of complete flow solidification is determined. In the general case, when all the governing parameters are of the order of unity, the original system of equations is studied numerically. The solutions obtained are qualitatively compared with the data of field observations for lava flows produced by extrusive volcanic eruptions.
Similar content being viewed by others
References
A. Oron, S.H. Davis, and S.G. Bankoff, “Long-Scale Evolution of Thin Liquid Films,” Rev. Mod. Phys. 324, 261–286 (1996).
J.R. King, D.S. Riley, and A. Sansom, “Gravity Currents with Temperature-Dependent Viscosity,” Comp. Assist. Mech. Eng. Sci. 7, 251–277 (2000).
A. Sansom, J.R. King, and D.S. Riley, “Degenerate-Diffusion Models for the Spreading of Thin Non-Isothermal Gravity Currents,” J. Eng. Math. 48, 43–68 (2004).
D. Bercovici, “A Theoretical Model of Cooling Viscous Gravity Currents with Temperature Dependent Viscosity,” Geophys. Res. Lett. 21, 1177–1180 (1994).
N.J. Balmforth and R.V. Craster, “Dynamics of Cooling Domes of Viscoplastic Fluid,” J. Fluid Mech. 422, 225–248 (2000).
N.J. Balmforth, R.V. Craster, and R. Sassi, “Dynamics of Cooling Viscoplastic Domes,” J. Fluid Mech. 499, 149–182 (2004).
T.G. Myers, J.P.F. Charpin, and S.J. Chapman, “The Flow and Solidification of a Thin Film on an Arbitrary Three-Dimensional Surface,” Phys. Fluids 14, 2788–2803 (2002).
W.R. Smith, “The Propagation and Basal Solidification of Two-Dimensional and Axisymmetric Viscous Gravity Currents,” J. Eng. Math. 50, 359–378 (2004).
H.E. Huppert, “The Propagation of Two-Dimensional and Axisymmetric Viscous Gravity Currents over a Rigid Horizontal Surface,” J. Fluid Mech. 121, 43–58 (1982).
R.W. Griffiths, “The Dynamics of Lava Flows,” Annu. Rev. Fluid Mech. 32, 477–518 (2000).
A.A. Osiptsov, “Non-Isothermal Lava Flows over a Conical Surface,” Fluid Dynamics 40, 221–232 (2005).
J.R. Lister and P.J. Dellar, “Solidification of Pressure-Driven Flow in a Finite Rigid Channel with Application to Volcanic Eruptions,” J. Fluid Mech. 323, 267–283 (1996).
P.T. Delaney and D.D. Pollard, “Solidification of Basaltic Magma during Flow in a Dyke,” Amer. J. Sci. 282, 856–885 (1982).
P.M. Bruce and H.E. Huppert, “Thermal Control of Basaltic Fissure Eruptions,” Nature 342, 665–667 (1989).
J.R. Lister, “The Solidification of Buoyancy-Driven Flow in a Flexible-Walled Channel. Part 1. Constant-Volume Release,” J. Fluid Mech. 272, 21–44 (1994).
J.R. Lister, “The Solidification of Buoyancy-Driven Flow in a Flexible-Walled Channel. Part 1. Continual Release,” J. Fluid Mech. 272, 45–65 (1994).
L.G. Loitsyanskii, Mechanics of Liquids and Gases (Nauka, Moscow, 1978) [in Russian].
A.A. Osiptsov, “Steady Film Flow of a Highly Viscous Heavy Fluid with Mass Supply,” Fluid Dynamics 38, 846–853 (2003).
L. Graetz, “Über die Wärmeleitungsfähigkeit von Flüssigkeiten,” Ann. Phys. Chem. 25, 337–357 (1885).
W. Nusselt, “Die Abhängigkeit der Wärmeübergangszahl von der Rohrlänge,” Z. Ver. Deut. Ing. 54, 1154–1158 (1910).
M. Epstein and F.B. Cheung, “Complex Freezing-Melting Interfaces in Fluid Flow,” Annu. Rev. Fluid Mech. 15, 293–319 (1983).
R.D. Zerkle R.D. and J.E. Sunderland, “The Effect of Liquid Solidification in a Tube upon Laminar-Flow Heat Transfer and Pressure Drop,” J. Heat Transfer 90, 183–190 (1968).
M.A. Lévêque, “Les Lois de la Transmission de Chaleur par Convection,” Ann. Mines Mem. 13, 201–299, 305–362, 381–415 (1928).
F. Olver, Asymptotics and Special Functions (Acad. Press, New York, 1974).
Handbook of Mathematical Functions (Eds. M. Abramowitz and I. Stegun) (Nation. Bureau Stand., Appl. Math. Ser. 55, 1964).
C.A.J. Fletcher, Computational Techniques for Fluid Dynamics. V. 1. (Springer-Verlag, Berlin et al., 1988).
D.W. Peterson D.W., R.T. Hollcomb, R.I. Tilling, and R.L. Christiansen, “Development of Lava Tubes in the Light of Observations at Mauna Ulu, Kilauea Volcano, Hawaii,” Bull. Volcanol. 56, 343–360 (1994).
P.W. Lipman and N.G. Banks, “A Flow Dynamics, Mauna Loa 1984,” US Geol. Surv. Prof. Pap. 1350, 1527–1567 (1987).
K.V. Cashman, H. Pinkerton, and P.J. Stephenson, “Long Lava Flows,” J. Geophys. Res. 103, 27281–89 (1998).
M. Van-Dyke, Perturbation Methods in Fluid Mechanics (Acad. Press, New York, London, 1964).
R.W. Griffiths, R.S. Kerr, and K.V. Cashman, “Patterns of Solidification in Channel Flows with Surface Cooling,” J. Fluid Mech. 496, 33–62 (2003).
A.J.L. Harris and S.K. Rowland, “FLOWGO: a Kinematic Thermo-Rheological Model for Lava Flowing in a Channel,” Bull. Volcanol. 63, 20–44 (2001).
Additional information
Original Russian Text © A.A. Osiptsov, 2007, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2007, Vol. 42, No. 2, pp. 24–38.
Rights and permissions
About this article
Cite this article
Osiptsov, A.A. Asymptotic models of solidification in cooling thin-layer flows of a highly viscous fluid. Fluid Dyn 42, 170–183 (2007). https://doi.org/10.1134/S0015462807020032
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0015462807020032