×

Dispersion of solute in wall-bounded parallel shear flows. (English) Zbl 0754.76045

(Authors’ abstract.) For laminar dispersion in wall-bounded straight channels, a rigorous method of solution to the basic convective diffusion equation is developed by using the Green’s function. The method leads to a self-consistent and computationally useful procedure for the determination of the concentration distribution at arbitrary times. The dispersion approximation of W. N. Gill and R. Sankarasubramanian [Proc. R. Soc. London, Ser. A 322, 101-117 (1971; Zbl 0228.76117); 327, 191-208 (1972; Zbl 0236.76066), 329, 479-492 (1972; Zbl 0247.76081)] and the alternative approach of R. Smith [e.g.: J. Fluid Mech. 129, 347-364 (1983; Zbl 0516.76089)] for the transverse mean concentration are critically examined. It is shown that these latter methods are valid only under restricted conditions. For initial-value problems where the dispersion approximation applies, a purely algebraic method, distinctively different from the work of A. E. DeGance and L. E. Johns [e.g.: Appl. Sci. Res. 43, 239-274 (1987; Zbl 0661.76088); 42, 55-88 (1985; Zbl 0669.76115)] for the determination of the dispersion coefficients is developed.
The truncated dispersion approximation for the area-weighted transverse mean concentration of an inert solute due to prescribed initial conditions in flows without internal sources is analyzed in detail. It is shown that, by using the central moments of the mean concentration distribution, suitable criteria for the applicability of the second-order approximation can be reasonably established. Results pertaining to a concentrated initial distribution in Poiseuille pipe flow are presented.

MSC:

76F10 Shear flows and turbulence
76R50 Diffusion
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G.I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. Roy. Soc. Lond. A219 (1953) 186–203. · doi:10.1098/rspa.1953.0139
[2] R. Aris. On the dispersion of a solute in fluid flowing through a tube. Proc. Roy. Soc. Lond. A 235 (1956) 67–77. · doi:10.1098/rspa.1956.0065
[3] P.C. Chatwin, The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe. J Fluid Mech. 43 (1970) 321–352. · Zbl 0216.53401 · doi:10.1017/S0022112070002409
[4] M.J. Lighthill, Initial development of diffusion in Poiseuille flow, J. Inst. Maths Applics 2 (1966) 97–108. · doi:10.1093/imamat/2.1.97
[5] W.N. Gill and R. Sankarasubramanian, Exact analysis of unsteady convective diffusion. Proc. Roy. Soc. Lond. A 316 (1970) 341–350. · Zbl 0197.52704 · doi:10.1098/rspa.1970.0083
[6] W.N. Gill and R. Sankarasubramanian, Dispersion of a non-uniform slug in time dependent flow, Proc. Roy. Soc. Lond. A 322 (1971) 101–117. · Zbl 0228.76117 · doi:10.1098/rspa.1971.0057
[7] R. Sankarasubramanian and W.N. Gill, Dispersion from a prescribed concentration distribution in time-variable flow. Proc. Roy. Soc. Lond. A 329 (1972) 479–492. · Zbl 0247.76081 · doi:10.1098/rspa.1972.0125
[8] R. Sankarasubramanian and W.N. Gill, Unsteady convective diffusion with interphase mass transfer, Proc. Roy. Soc. Lond. A 333 (1973) 115–132. · Zbl 0259.76043 · doi:10.1098/rspa.1973.0051
[9] M.R. Doshi, P.M. Daiya and W.N. Gill, Three dimensional laminar dispersion in open and closed rectangular conduits. Chem. Eng. Sci. 33 (1978) 795–804. · doi:10.1016/0009-2509(78)85168-9
[10] A.E. DeGance and L.E. Johns, The theory of dispersion of chemically active solute in a rectilinear flow field, Appl. Sci. Res. 34 (1978) 189–225. · Zbl 0406.76075 · doi:10.1007/BF00418868
[11] A.E. DeGance and L.E. Johns, On the construction of dispersion approximations to the solution of the convective diffusion equation, AIChE J. 26 (1980) 411–419. · Zbl 0661.76088 · doi:10.1002/aic.690260313
[12] I. Frankel and H. Brenner, On the foundations of generalized Taylor dispersion theory, J. Fluid Mech 204 (1989) 97–119. · Zbl 0687.76086 · doi:10.1017/S0022112089001679
[13] J.S. Yu Laminar dispersion for flow through round tubes, ASME J. Appl. Mech. 46 (1979) 750–756. · Zbl 0418.76054 · doi:10.1115/1.3424648
[14] J.S. Yu. Dispersion in laminar flow through tubes by simultaneous diffusion and convection, ASME J. Appl. Mech. 48 (1981) 217–223. · doi:10.1115/1.3157600
[15] R. Smith, A delay-diffusion description for contaminant dispersion, J. Fluid Mech. 105 (1981) 469–486. · Zbl 0463.76086 · doi:10.1017/S0022112081003297
[16] R. Smith, Non-uniform discharge of contaminants in shear flows, J. Fluid Mech. 120 (1982) 79–89. · Zbl 0512.76103 · doi:10.1017/S0022112082002675
[17] P.H. Morse and H. Feshbach, Methods of Theoretical Physics (Chapters 2 and 7), McGraw-Hill (1953). · Zbl 0051.40603
[18] H.S. Wilf, Mathematics for the Physical Sciences, Wiley (1962). · Zbl 0105.26803
[19] R.S. Subramanian, Unsteady convective diffusion in capillary chromatography, J. Chromatogr. 101 (1974) 253–270. · doi:10.1016/S0021-9673(00)82843-5
[20] W.N. Gill, A note on the solution of translent dispersion problems, Proc. Roy. Soc. Lond. A 298 (1967) 335–339. · Zbl 0147.45305 · doi:10.1098/rspa.1967.0107
[21] G.S. Booras and W.B. Krantz, Disporsion in laminar flow of power-law fluids through straight tubes, Ind. Eng. Chem. Fundam. 15 (1976) 249–254. · doi:10.1021/i160060a004
[22] E.O. Brigham, The Fast Fourier Transform, Prentice-Hall (1974). · Zbl 0375.65052
[23] W.N. Gill and V. Ananthakrishnan, Laminar dispersion in capillaries: part 4, the slug stimulus, AIChE J. 13 (1967) 801–807. · doi:10.1002/aic.690130439
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.