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A Burgers concentration dispersion equation

Published online by Cambridge University Press:  26 April 2006

Ronald Smith  and
Ian Walton
Ronald Smith
Affiliation:
Mathematical Sciences, University of Technology, Loughborough, LE11 3TU, UK
Ian Walton
Affiliation:
Tulsa R&D Center, Dowell Schlumberger Inc, 5051 South 129th East Avenue, Tulsa, OK 74134, USA
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Abstract

In a non-vertical borehole light particles tend to rise towards the upper side of the borehole. The resulting non-uniform density distribution tends to induce an upwards contribution to the longitudinal flow along that upper side of the flow, with a compensating downflow elsewhere. On average the particles experience an extra upflow proportional to the cross-sectionally averaged concentration of particles. Mathematically this concentration-related change of speed corresponds to the nonlinearity of the Burgers equation. Such is the strength of the buoyancy effect that in realistic flow conditions the Burgers nonlinearity can be significant for particle volume fractions of only one part per thousand.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 239 , June 1992 , pp. 65 - 80
Copyright
© 1992 Cambridge University Press

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References

Bugliarello, G. & Jackson, E. D., 1964 Random walk study of convective diffusion. J. Engng Mech. Div. ASCE 90, 49–77.Google Scholar
Burgers, J. M.: 1948 A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171–199.Google Scholar
Cole, J. D.: 1951 On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Maths 9, 225–236.Google Scholar
Elder, J. W.: 1959 The dispersion of marked fluid in turbulent shear flow. J. Fluid Mech. 5, 544–560.Google Scholar
Erdogan, M. E. & Chatwin, P. C., 1967 The effects of curvature and buoyancy on the laminar dispersion of solute in a horizontal tube. J. Fluid Mech. 29, 465–484.Google Scholar
Giddings, J. C.: 1968 Non-equilibrium theory of field-flow fractionation. J. Chem. Phys. 49, 81–85.Google Scholar
Herbolzheimer, E. & Acrivos, A., 1981 Enhanced sedimentation in narrow tilted channels. J. Fluid Mech. 108, 485–499.Google Scholar
Hopf, E.: 1950 The partial differential equation ut + uux = μuxx. Commun. Pure Appl. Maths 3, 201–230.Google Scholar
Leighton, D. & Acrivos, A., 1987 The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415–439.Google Scholar
Lightfoot, E. N., Chiang, A. S. & Noble, P. T., 1981 Field-flow fractionation (polarisation chromatography). Ann. Rev. Fluid Mech. 13, 351–378.Google Scholar
Muller, S. J. & Bittleston, S. H., 1992 Taylor & Aris dispersion of generalised Newtonian fluid for axial flow in an eccentric annulus (in preparation).
Sankarasubramanian, R. & Gill, W. N., 1971 Taylor diffusion in laminar flow in an eccentric annulus. Intl J. Heat Mass Transfer 14, 905–919.Google Scholar
Schlichting, H.: 1955 Boundary Layer Theory. Pergamon.
Smith, R.: 1990a Two-dimensional shear dispersion for skewed flows in narrow gaps between moving surfaces. J. Fluid Mech. 214, 211–228.Google Scholar
Smith, R.: 1990b Shear dispersion along an axle in a closely fitting shaft. J. Fluid Mech. 219, 647–658.Google Scholar
Snyder, W. T. & Goldstein, G. A., 1965 An analysis of fully developed laminar flow in an eccentric annulus. AIChE J. 11, 462–467.Google Scholar
Taylor, G. I.: 1910 The conditions necessary for discontinuous motion in gases. Proc. R. Soc. Lond. A 84, 371–377.Google Scholar
Taylor, G. I.: 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 216, 186–203.Google Scholar