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Aug 2005

Volume 17, Issue 8, Articles (08xxxx)

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back to top Viscous and Non-Newtonian Flows

Stalactite growth as a free-boundary problem

Martin B. Short, James C. Baygents, and Raymond E. Goldstein

Phys. Fluids 17, 083101 (2005); http://dx.doi.org/10.1063/1.2006027 (12 pages) | Cited 10 times

Online Publication Date: 11 August 2005

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Stalactites, the most familiar structures found hanging from the ceilings of limestone caves, grow by the precipitation of calcium carbonate from within a thin film of fluid flowing down their surfaces. We have recently shown [ M. B. Short, J. C. Baygents, J. W. Beck, D. A. Stone, R. S. Toomey III, and R. E. Goldstein, “Stalactite growth as a free-boundary problem: A geometric law and its Platonic ideal,” Phys. Rev. Lett. 94, 018501 (2005) ] that the combination of thin-film fluid dynamics, calcium carbonate chemistry, and carbon dioxide diffusion and outgassing leads to a local geometric growth law for the surface evolution which quantitatively explains the shapes of natural stalactites. Here we provide details of this free-boundary calculation, exploiting a strong separation of time scales among that for diffusion within the layer, contact of a fluid parcel with the growing surface, and growth. When the flow rate, the scale of the stalactite, and the chemistry are in the ranges typically found in nature, the local growth rate is proportional to the local thickness of the fluid layer, itself determined by Stokes flow over the surface. Numerical studies of this law establish that a broad class of initial conditions is attracted to an ideal universal shape, whose mathematical form is found analytically. Statistical analysis of stalactite shapes from Kartchner Caverns (Benson, AZ) shows excellent agreement between the average shape of natural stalactites and the ideal shape. Generalizations of these results to nonaxisymmetric speleothems are discussed.
Show PACS
91.55.-y Structural geology
47.54.-r Pattern selection; pattern formation
82.40.Ck Pattern formation in reactions with diffusion, flow and heat transfer
47.70.Fw Chemically reactive flows
68.15.+e Liquid thin films
68.43.Mn Adsorption kinetics
82.65.+r Surface and interface chemistry; heterogeneous catalysis at surfaces

Optimal probes for withdrawal of uncontaminated fluid samples

J. D. Sherwood

Phys. Fluids 17, 083102 (2005); http://dx.doi.org/10.1063/1.2006128 (10 pages) | Cited 1 time

Online Publication Date: 11 August 2005

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Withdrawal of fluid by a composite probe pushed against the face z = 0 of a porous half-space z>0 is modeled assuming incompressible Darcy flow. The probe is circular, of radius a, with an inner sampling section of radius αa and a concentric outer guard probe αa<r<a. The porous rock in 0 ⩽ z ⩽ βa is saturated with fluid 1, and the region z>βa is saturated with fluid 2; the two fluids have the same viscosity. It is assumed that the interface between the two fluids is sharp and remains so as it moves through the rock. The pressure in the probe is lower than that of the pore fluid in the rock, so that the fluid interface is convected with the fluids towards the probe. This idealized axisymmetric problem is solved numerically, and it is shown that an analysis based on far-field spherical flow towards a point sink is a good approximation when the nondimensional depth of fluid 1 is large, i.e., β⪢1. The inner sampling probe eventually produces pure fluid 2, and this technique has been proposed for sampling pore fluids in rock surrounding an oil well [A. Hrametz, C. Gardner, M. Wais, and M. Proett, U.S. Patent No. 6,301,959 B1 (16 October 2001)]. Fluid 1 is drilling fluid filtrate, which has displaced the original pore fluid (fluid 2), a pure sample of which is required. The time required to collect an uncontaminated sample of original pore fluid can be minimized by a suitable choice of the probe geometry α [J. Sherwood, J. Fitzgerald and B. Hill, U.S. Patent No. 6,719,049 B2 (13 April 2004)]. It is shown that the optimal choice of α depends on the depth of filtrate invasion β and the volume of sample required.
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47.56.+r Flows through porous media
47.27.T- Turbulent transport processes
47.55.Kf Particle-laden flows

Theory of shear-induced migration in dilute polymer solutions near solid boundaries

Hongbo Ma and Michael D. Graham

Phys. Fluids 17, 083103 (2005); http://dx.doi.org/10.1063/1.2011367 (13 pages) | Cited 45 times

Online Publication Date: 18 August 2005

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In this work, a continuum theory is developed for the behavior of flowing dilute polymer solutions near solid surfaces, using a bead-spring dumbbell model of the dissolved polymer chains. Hydrodynamic interactions between the chains and the wall lead to migration away from the wall in shear flow. At steady state, this hydrodynamic effect is balanced by molecular diffusion; an analytical expression for the resulting concentration profile is derived. It is shown that the depletion layer thickness is determined by the normal stresses that develop in flow and can be much larger than the size of the polymer molecule. The transient development of this depletion layer is also studied, as well as the spatial development downstream from an entrance. Numerical and similarity solutions in these cases show that the developing concentration profile generally displays a maximum at an intermediate distance from the wall.
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47.50.-d Non-Newtonian fluid flows
47.20.-k Flow instabilities
66.10.C- Diffusion and thermal diffusion
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