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Lagrangian approach to time-dependent laminar dispersion in rectangular conduits. Part 1. Two-dimensional flows

Published online by Cambridge University Press:  21 April 2006

Shimon Haber  and
Roberto Mauri
Shimon Haber
Affiliation:
Department of Mechanical Engineering, Technion, Israel Institute of Technology, Israel
Roberto Mauri
Affiliation:
Department of Mechanical Engineering, Technion, Israel Institute of Technology, Israel
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Abstract

Time-dependent mean velocities and dispersion coefficients are evaluated for a general two-dimensional laminar flow. A Lagrangian method is adopted by which a Brownian particle is traced in an artificially restructured velocity field. Asymptotic expressions for short, medium and long periods of time are obtained for Couette flow, plane Poiseuille flow and open-channel flow over an inclined flat surface. A new formula is suggested by which the Taylor dispersion coefficient can be evaluated from purely kinematical considerations. Within an error of less than one percent, over the entire time domain and for various flow fields, a very simple analytical expression is derived for the time-dependent dispersion coefficient \[ \tilde{D}(\tau) = D + D^T\left(1-\frac{1-{\rm e}^{-\alpha\tau}}{a\tau}\right), \] where D is the molecular diffusion coefficient, DT denotes the Taylor dispersion coefficient, τ stands for the non-dimensional time π2Dt/Y/, Y is the distance between walls and a = (N + 1)2 is an integer which is determined by the number of symmetry planes N that the flow field possesses. For Couette and open-channel flow there are no planes of symmetry and a = 1; for Poiseuille flow there is one plane of symmetry and a = 4.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 190 , May 1988 , pp. 201 - 215
Copyright
© 1988 Cambridge University Press

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References

Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A 235, 67–77.Google Scholar
Brenner, H. 1980 A general theory of Taylor dispersion phenomena. Physico-Chem. Hydrodyn. 1, 91–123.Google Scholar
Brenner, H. 1982 A general theory of Taylor dispersion phenomena II. Physico-Chem. Hydrodyn 3, 139–157.Google Scholar
Budak, B. M., Samarski, A. A. & Tikhonov, A. N. 1964 A Collection of Problems of Mathematical Physics. Pergamon.
Chandrasekhar, S. 1943 Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89.Google Scholar
Dewey, R. & Sullivan, P. J. 1979 Longitudinal dispersion in flows that are homogeneous in streamwise direction. Z. Angew. Math. Phys. 30, 601–613.Google Scholar
Dewey, R. & Sullivan, P. J. 1982 Longitudinal dispersion calculations in laminar flows by statistical analysis of molecular motions. J. Fluid Mech. 125, 203–218.Google Scholar
Dill, L. H. & Brenner, H. 1983 A general theory of Taylor dispersion phenomena VI. J. Colloid Interface Sci. 93, 343–365.Google Scholar
Doshi, M. R., Daiya, P. M. & Gill, W. N. 1978 Three-dimensional laminar dispersion in open and closed rectangular conduits. Chem. Engng Sci. 33, 795–804.Google Scholar
Gradshteyn, J. S. & Ryzhik, J. M. 1965 Tables of Integrals, Series and Products. Academic.
Griffiths, A. 1911 On the movement of a coloured index along a capillary tube and its application to the measurement of the circulation of water in a closed circuit. Proc. Phys. Soc. Lond. 23, 190–197.Google Scholar
Lifschitz, E. M. & Pitaevski, L. P. 1981 Physical Kinetics. Pergamon.
Mauri, R. & Haber, S. 1983 Brownian motion and the condition of local equilibrium. Technion Internal Rep. TME 420.
Oppenheim, I. & Mazur, P. 1964 Brownian motion in systems of finite size. Physica 30, 1833–1845.Google Scholar
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186–203.Google Scholar
Van Den Broeck, C. 1982 A stochastic description of longitudinal dispersion in uniaxial flows. Physica A 112, 343–352.Google Scholar
Van Kampen, N. G. 1981 Stochastic Processes in Physics and Chemistry. North-Holland.