Exact Solution to Navier-Stokes Equation for Developed Radial Flow between Parallel Disks
Publication: Journal of Engineering Mechanics
Volume 143, Issue 6
Abstract
Laminar radial flow between two parallel disks is a fundamental nonlinear fluid mechanics problem described by the Navier-Stokes (NS) equation, but is unsolved because (1) an exact solution is not found even with extensive references, and (2) it is unclear why radial flow remains laminar at high Reynolds numbers. This paper first presents exact velocity distribution solutions for developed radial inflows and outflows, proving that both flows are described by brief Jacobi elliptic sine-squared functions but with different characteristics. For inflow, a stable velocity distribution forms; for outflow, the velocity distribution may have an inflection point inducing flow instability or separation. Both velocity distributions become the classic parabolic law at low Reynolds numbers, but uniform (similar to turbulent velocity distributions) at high Reynolds numbers. Furthermore, both pressure and boundary shear stress follow an inverse-square law, but the friction factor is invariant. These results are instructive for studying nonuniform open-channel flow for which nonlinear inertia is of importance.
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Acknowledgments
This research was supported by the U.S. Federal Highway Administration Hydraulics Research and Development Program (Contract No. DTFH61-11-D-00010) through the Genex System to the University of Nebraska–Lincoln, the open fund research program (Contract No. HESS-1604) at the State Key Lab of Hydraulic Engineering Simulation and Safety, Tianjin University, China, and the open fund research program at the State Key Lab of Hydraulics and Mountain River Engineering (Contract No. SKHL1511), Sichuan University, China.
References
Boyack, B. E., and Rice, R. (1970). “An integral solution for the laminar radial outflow of a viscous fluid between parallel stationary discs.” J. Basic Eng., 92(3), 662–663.
Chatterjee, A., and White, D. (1989). “Radial entry flow of a Newtonian fluid.” J. Phys. D: Appl. Phys., 22(7), 915–924.
Clay Mathematics Institute. (2016). “Millennium problems.” ⟨http://www.claymath.org/millennium-problems⟩ (Dec. 14, 2016).
Elkouh, A. F. (1967). “Inertia effect in laminar radial flow between parallel plates.” Int. J. Mech. Sci., 9(5), 253–255.
Fang, D. Z. (1977). Mathematical manual, China Higher Education Press, Beijing (in Chinese).
Ghaly, W., and Vatistas, G. H. (1997). “Numerical solution of the flow between two discs.” ASME Proc., Computer and Information in Engineering, ASME, New York, 1–6.
Guo, J., and Zhang, J. (2016). “Velocity distributions in laminar and turbulent vegetated flows.” J. Hydraul. Res., 54(2), 117–130.
Hayes, W. F., and Tucker, H. G. (1973). “Theoretical radial pressure distribution for viscous fluid inflow within a thin disc chamber.”, National Research Council of Canada, Ottawa.
Jackson, J. D., and Symmons, G. R. (1965). “An investigation of laminar radial flow between two parallel discs.” Appl. Sci. Res. Section A, 15(1), 59–75.
Kreith, F. (1965). “Reverse transition in radial source flow between two parallel planes.” Phys. Fluids, 8(6), 1189–1190.
Launder, B. E. (1964). “Laminarization of the turbulent boundary layer in a severe acceleration.” J. Appl. Mech., 31(4), 707–708.
Launder, B. E., and Jones, W. P. (1969). “Sink flow turbulent boundary layers.” J. Fluid Mech., 38(04), 817–831.
Lee, P.-M., and Lin, S. (1985). “Pressure distribution for radial inflow between narrowly spaced disks.” J. Fluids Eng., 107(3), 338–342.
Li, P., Mirza, S., and Lin, S. (1989). “Pressure distribution in radial flow between disks.” J. Eng. Mech., 210–215.
Liu, Z.-B., and Wang, Z.-Q. (1984). “An analysis on entrance region effect of the laminar radial flow between two parallel disks.” Appl. Math. Mech., 5(1), 1057–1070.
Livesey, J. L. (1960). “Inertia effects in viscous flows.” Int. J. Mech. Sci., 1(1), 84–88.
McDonald, K. T. (2000). “Radial viscous flow between two parallel annular plates.” J. Princeton, 1–4.
McGinn, J. H. (1955). “Observations on the radial flow of water between fixed parallel plates.” Appl. Sci. Res., 5(4), 255–264.
Mohn, P. E. (1930). “Some phenomena of radial flow between discs.” M.S. thesis, Univ. of Illinois, Urbana, IL.
Moller, P. S. (1966). “A radial diffuser using incompressible flow between narrowly spaced discs.” J. Fluids Eng., 88(1), 155–162.
Morgan, P. G., and Saunders, A. (1960). “An experimental investigation of inertia effects in viscous flow.” Int. J. Mech. Sci., 2(1–2), 8–12.
Murphy, H. D., Chambers, F. W., and McEligot, D. M. (1983). “Laterally converging flow. Part 1: Mean flow.” J. Fluid Mech., 127(1), 379–401.
Murphy, H. D., Coxon, M., and McEligot, D. M. (1978). “Symmetric sink flow between parallel plates.” J. Fluids Eng., 100(4), 477–484.
Oliveira, J. C., and Amon, C. H. (1998). “Pressure-based semi-analytical solution of radial stokes flows.” J. Math. Anal. Appl., 217(1), 95–114.
Patel, V. C., and Head, M. R. (1968). “Reversion of turbulent to laminar flow.” J. Fluid Mech., 34(2), 371–392.
Savage, S. B. (1964). “Laminar radial flow between parallel plates.” J. Appl. Mech., 31(4), 594–596.
Singh, A., Vyas, B. D., and Powle, U. S. (1999). “Investigations on inward flow between two stationary parallel disks.” Int. J. Heat Fluid Flow, 20(4), 395–401.
Tsifourdaris, P. (2003). “On the flows developed within the gap of two parallel discs.” Ph.D. thesis, Concordia Univ., Montreal.
Vatistas, G. H. (1988). “Radial flow between two closely placed flat disks.” AIAA J., 26(7), 887–889.
Vatistas, G. H., Ghila, G. H., and Zitouni, G. (1995). “Radial inflow between two flat disks.” Acta Mech., 113(1–4), 109–118.
White, F. M. (1991). Viscous fluid flow, McGraw-Hill, New York.
Willis, R. R. (1828). “On the pressure produced on a flat plate when opposed to a stream of air issuing from an orifice in a plane surface.” Trans. Cambridge Philos. Soc., 3(1), 121–140.
Wilson, S. D. R. (1972). “A note on laminar radial flow between parallel plates.” Appl. Sci. Res., 25(1), 349–354.
Woolard, H. W. (1957). “A theoretical analysis of viscous flow in narrowly spaced radial diffuser.” J. Appl. Mech., 24(1), 9–15.
Yen, R. T. (1965). “Radial flow between two parallel discs.” M.S. thesis, Kansas State Univ., Manhattan, KS.
Zitouni, G., and Vatistas, G. H. (1997). “Purely accelerating and decelerating flows within two flat disks.” Acta Mech., 123(1), 151–161.
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Published In
Journal of Engineering Mechanics
Volume 143 • Issue 6 • June 2017
Copyright
©2017 American Society of Civil Engineers.
History
Received: Jul 17, 2016
Accepted: Nov 4, 2016
Published online: Feb 16, 2017
Published in print: Jun 1, 2017
Discussion open until: Jul 16, 2017
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