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An elastic sublayer model for drag reduction by dilute solutions of linear macromolecules

Published online by Cambridge University Press:  29 March 2006

P. S. Virk
P. S. Virk
Affiliation:
Department of Chemical Engineering Massachusetts Institute of Technology
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Abstract

Further evidence of a universal maximum drag reduction asymptote is presented. In the elastic sublayer model, inferred therefrom, the mean velocity profile during drag reduction is approximated by three zones: the usual viscous sublayer, an elastic sublayer where the mixing-length constant is derived from the maximum drag reduction asymptote, and an outermost region with Newtonian mixinglength constant. Upon integration the model yields a friction factor relation, parametric in elastic sublayer thickness, which properly reproduces the known features of turbulent dilute polymer solution flow. The dependence of elastic sublayer thickness upon flow and polymeric parameters is inferred from experimental data revealing two hitherto unknown relationships: namely that on Prandtl co-ordinates, 1/f½vs. log Re f½ the difference in slope between a polymer solution and solvent is proportional to the square root of molar concentration and to the three-halves power of backbone chain links in the macromolecule. The proportionality constant in the preceding relationship is approximately the same for several different polymer species of carbon-carbon or similar skeletal structure in various thin solvents; there is an indication that this constant further depends upon the product of solvent viscosity times the cube of the effective bond length per chain link of the polymer species. Some recent results regarding the onset of drag reduction are also summarized.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 45 , Issue 3 , 15 February 1971 , pp. 417 - 440
Copyright
© 1971 Cambridge University Press

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References

Beech, D. R. & Booth, C. 1969 J. Polym. Sci. A 2, 7, 575.
Cerf, R. 1967 C. r. Acad. Sci. (Paris), C265, 791.
Cottrell, F. R. 1968 Sc.D. Thesis, M.I.T., Cambridge, Mass.
Cottrell, F. R., Merrill, E. W. & Smith, K. A. 1969 J. Polymer. Sci. A 2, 7, 1415.
Elata, C., Lehrer, J. & Kahanovitz, A. 1966 Israel J. Technol. 4, 87.
Elata, C. & Tirosh, J. 1965 Israel J. Technol. 3, 1.
Fabula, A. G., Lumley, J. L. & Taylor, D. W. 1966 In The Mechanics of Continua. (Ed. S. Eskinazi). Academic.
Flory, P. J. 1953 Principles of Polymer Chemistry. Cornell University Press.
Fruman, D. & Sulmont, P. 1968 Rep. to Lab. d'Aërodynamique, Faculté des Sciences, Orsay.
Gilbert, C. G. & Ripken, J. F. 1969 In Viscous Drag Reduction. (Ed. C. S. Wells). Plenum.
Goren, Y. & Norbury, J. F. 1967 A.S.M.E. J. Basic Engng, 89, 814.
Hershey, H. C. & Zakin, J. L. 1967 Chem. Engng Sci. 22, 1847.
Hoyt, J. W. & Fabula, A. G. 1964 Paper to the 5th Symposium on Naval Hydrodynamics.
Koleske, J. V. & Kurath, S. F. 1964 J. Polym. Sci. A, 2, 4123.
Kurata, M. & Stockmayer, W. H. 1963 Fortschr. Hochpolymer Forsh. 3, 192.
Laufer, J. 1954 NACA Rep. no. 1174.
Liaw, G. C. 1968 Ph.D. Thesis, Univ. of Mo. at Rolla.
Lumley, J. L. 1967 Appl. Mech. Rev. 20, 1139.
Lumley, J. L. 1969 Ann. Rev. Fluid Mech. 1, 367.
McNally, W. A. 1968 Ph.D. Thesis, Univ. of Rhode Island.
Metzner, A. B. & Park, M. G. 1964 J. Fluid Mech. 20, 291.
Meyer, W. A. 1966 A.I.Ch.E. J. 12, 522.
Oldroyd, J. G. 1948 Proc. 1st Int. Congress on Rheology, V 2, 130. North Holland.
Patterson, G. K. & Florez, G. L. 1969 In Viscous Drag Reduction. (Ed. S. C. Wells). Plenum.
Peterlin, A. 1963 J. Chem. Phys. 39, 224.
Pruitt, G. T., Rosen, B. & Crawford, H. R. 1966 Clearinghouse AD 642441.
Ram, A., Finklestein, E. & Elata, C. 1967 Ind. Engng Chem. Proc. Des. Dev. 6, 309.
Rodriguez, J. M., Zakin, J. L. & Patterson, G. K. 1967 Soc. Petrol. Engng J. 7, 325.
Rouse, P. E. 1953 J. Chem. Phys. 21, 1272.
Savins, J. G. 1964 Soc. Petrol. Engng J. 4, 203.
Seyer, F. A. & Metzner, A. B. 1969 A.I.Ch.E. J. 15, 425.
Shin, H. 1965 Sc.D. Thesis, M.I.T., Cambridge, Mass.
Spangler, J. G. 1969 In Viscous Drag Reduction. (Ed. C. S. Wells). Plenum.
Toms, B. A. 1948 Proc. 1st Int. Congress on Rheology, V 2, 135. North Holland.
Treolar, L. R. G. 1958 The Physics of Rubber Elasticity, 2nd ed. Clarendon.
Virk, P. S. 1971 J. Fluid Mech. 45, 225.
Virk, P. S. & Baher, H. 1970 Chem. Engng Sci. (to be published).
Virk, P. S. & Merrill, E. W. 1969 In Viscous Drag Reduction. (Ed. C. S. Wells). Plenum.
Virk, P. S., Merrill, E. W., Mickley, H. S., Smith, K. A. & Mollo-Christensen, E. L. 1967 J. Fluid Mech. 30, 305.
Virk, P. S., Mickley, H. S. & Smith, K. A. 1970 ASME J. Appl. Mech. 37, 488.
Walsh, M. 1967 Ph.D. Thesis, Calif. Inst. Tech., Pasadena, Calif.
Wells, C. S. 1965 A.I.A.A. J. 3, 1800.
Wells, C. S., Harkness, J. & Meyer, W. A. 1968 A.I.A.A. J. 6, 250.
Whittsitt, N. F., Harrington, L. J. & Crawford, H. R. 1968 Clearinghouse AD 677467.
White, A. 1967 Nature, Lond. 216, 994.
Zimm, B. H. 1956 J. Chem. Phys. 24, 264.