Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T02:29:22.589Z Has data issue: false hasContentIssue false
  • English
  • Français

Streaming potential generated by a drop moving along the centreline of a capillary

Published online by Cambridge University Press:  12 November 2009

ETIENNE LAC  and
J. D. SHERWOOD
ETIENNE LAC*
Affiliation:
Schlumberger Doll Research, One Hampshire Street, Cambridge MA 02139-1578, USA
J. D. SHERWOOD
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK.
*
†Email address for correspondence: elac@slb.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The electrical streaming potential generated by a two-phase pressure-driven Stokes flow in a cylindrical capillary is computed numerically. The potential difference ΔΦ between the two ends of the capillary, proportional to the pressure difference Δp for single-phase flow, is modified by the presence of a suspended drop on the centreline of the capillary. We determine the change in ΔΦ caused by the presence of an uncharged insulating neutrally buoyant drop at a small electric Hartmann number, i.e. when the perturbation to the flow field caused by electric stresses is negligible.

The drop velocity and deformation, and the consequent changes in the pressure difference Δp and streaming potential ΔΦ, depend upon three independent parameters: the size a of the undeformed drop relative to the radius R of the capillary; the viscosity ratio λ between the drop phase and the continuous phase; and the capillary number Ca which measures the ratio of viscous to capillary forces. We investigate how the streaming potential depends on these parameters: purely hydrodynamic aspects of the problem are discussed by Lac & Sherwood (J. Fluid Mech., doi:10.1017/S0022112009991212).

The potential on the capillary wall is assumed sufficiently small so that the electrical double layer is described by the linearized Poisson–Boltzmann equation. The Debye length characterizing the thickness of the charge cloud is taken to be small compared with all other length scales, including the width of the gap between the drop and the capillary wall. The electric potential satisfies Laplace's equation, which we solve by means of a boundary integral method. The presence of the drop increases |ΔΦ| when the drop is more viscous than the surrounding fluid (λ > 1), though the change in |ΔΦ| can take either sign for λ < 1. However, the difference between ΔΦ and Δp (suitably non-dimensionalized) is always positive. Asymptotic predictions for the streaming potential in the case of a vanishingly small spherical droplet, and for large drops at high capillary numbers, agree well with computations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 640 , 10 December 2009 , pp. 55 - 77
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Antraygues, P. & Aubert, M. 1993 Self potential generated by two-phase flow in a porous medium: experimental study and volcanological applications. J. Geophys. Res. 98 (B12), 22273–22281.CrossRefGoogle Scholar
Baygents, J. C. & Saville, D. A. 1991 Electrophoresis of drops and bubbles. J. Chem. Soc., Faraday Trans. 87, 1883–1898.CrossRefGoogle Scholar
Brenner, H. 1971 Pressure drop due to the motion of neutrally buoyant particles in duct flows. II. Spherical droplets and bubbles. Indus. Engng Chem. Fundam. 10 (4), 537–543.CrossRefGoogle Scholar
Brunet, E. & Ajdari, A. 2006 Thin double layer approximation to describe streaming current fields in complex geometries: analytical framework and applications to microfluidics. Phys. Rev. E 73 (5), 056306.CrossRefGoogle Scholar
Bungay, P. M. & Brenner, H. 1973 The motion of a closely-fitting sphere in a fluid-filled tube. Intl J. Multiphase Flow 1, 25–56.CrossRefGoogle Scholar
Guichet, X., Jouniaux, L. & Pozzi, J.-P. 2003 Streaming potential of a sand column in partial saturation conditions. J. Geophys. Res. 108 (B3), 2141.Google Scholar
Hochmuth, R. M. & Sutera, S. P. 1970 Spherical caps in low Reynolds number tube flow. Chem. Engng Sci. 25, 593–604.CrossRefGoogle Scholar
Hunter, R. J. 1981 Zeta Potential in Colloid Science. Academic.Google Scholar
Jackson, M. D. 2008 Characterization of multiphase electrokinetic coupling using a bundle of capillary tubes model. J. Geophys. Res. 113, B04201.Google Scholar
Jackson, M. D., Saunders, J. H. & Addiego-Guevara, E. A. 2005 Development and application of new downhole technology to detect water encroachment towards intelligent wells, Paper 97063. In Society of Petroleum Engineers Annual Technical Conference, 9–12 October 2005, Dallas, TX.Google Scholar
Lac, E. & Sherwood, J. D. 2009 Motion of a drop along the centreline of a capillary in a pressure-driven flow. J. Fluid Mech. (in press) doi:10.1017/S0022112009991212.CrossRefGoogle Scholar
Liron, N. & Shahar, R. 1978 Stokes flow due to a Stokeslet in a pipe. J. Fluid Mech. 86 (4), 727–744.CrossRefGoogle Scholar
Martinez, M. J. & Udell, K. S. 1990 Axisymmetric creeping motion of drops through circular tubes. J. Fluid Mech. 210, 565–591.CrossRefGoogle Scholar
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1, 111–146.CrossRefGoogle Scholar
Morgan, F. D., Williams, E. R. & Madden, T. R. 1989 Streaming potential properties of westerly granite with applications. J. Geophys. Res. B 94, 12449–12461.CrossRefGoogle Scholar
Olbricht, W. L. 1996 Pore-scale prototypes of multiphase flow in porous media. Annu. Rev. Fluid Mech. 28, 187–213.CrossRefGoogle Scholar
Revil, A. & Cerepi, A. 2004 Streaming potentials in two-phase flow conditions. Geophys. Res. Lett. 31, L11605.CrossRefGoogle Scholar
Revil, A., Schwaeger, H., Cathles, L. M. III, & Manhardt, P. D. 1999 Streaming potential in porous media. Part 2. Theory and application to geothermal systems. J. Geophys. Res. 104 (B9), 20033–20048.CrossRefGoogle Scholar
Saunders, J. H., Jackson, M. D. & Pain, C. C. 2006 A new numerical model of electrokinetic potential response during hydrocarbon recovery. Geophys. Res. Lett. 33, L15316.CrossRefGoogle Scholar
Saunders, J. H., Jackson, M. D. & Pain, C. C. 2008 Fluid flow monitoring in oilfields using downhole measurements of electrokinetic potential. Geophysics 73 (5), E165–E180.CrossRefGoogle Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29, 27–64.CrossRefGoogle Scholar
Sherwood, J. D. 2007 Streaming potential generated by two-phase flow in a capillary. Phys. Fluids 19, 053101.CrossRefGoogle Scholar
Sherwood, J. D. 2008 Streaming potential generated by a long viscous drop in a capillary. Langmuir 24 (18), 10011–10018.CrossRefGoogle Scholar
Sherwood, J. D. 2009 Streaming potential generated by a small charged drop in Poiseuille flow. Phys. Fluids 21, 013101.CrossRefGoogle Scholar
Sprunt, E. S., Mercer, T. B. & Djabbarah, N. F. 1994 Streaming potential from multiphase flow. Geophysics 59 (5), 707–711.CrossRefGoogle Scholar
Tsai, T. M. & Miksis, M. J. 1994 Dynamics of a drop in a constricted capillary tube. J. Fluid Mech. 274, 197–217.CrossRefGoogle Scholar
Wurmstich, B. & Morgan, F. 1994 Modeling of streaming potential responses caused by oil well pumping. Geophysics 59 (1), 46–56.CrossRefGoogle Scholar