Radial Viscous Flow between Two Parallel Annular Plates Mya San May Abstract We seek the approximate solution to the Navier-Stokes equation for the radial flow between two parallel annular plates since there is no exact solution which satisfies the boundary conditions for the problem. 1. Introduction In this paper, we shall deduce the velocity distribution of steady flow of an incompressible fluid of densit y ρ and viscosity μ between two parallel, coaxial annular plates of inner radius r1, outer radius r2 and separation h when pressure difference ∆p is applied between the inner and outer radii. As an exact solution of the Navier-Stokes equation appears to be difficult, it suffic es to gi ve an app roximat e soluti on assu ming that the v elocity is purely radial , rz) u(r, q , in a cylindrical coordinate system (r, φ , z) whose z-axis coincides with that of the two annuli. We will deduce a condition for validity of the approximation. This problem arises, for example, in considerations of a rotary joint between two sections of a pipe. Here, we ignore the extra complication of the effect of the rotation of one of the annuli on the fluid flow. 2. Two-Dimensional Flow between Parallel PlatesFigure (1)The velocity distribution obeys the continuity equation for an incompressible fluid, Dr, Tutor, Department of Mathematics, University of Magway z=0 z=h
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8/13/2019 Radial Viscous Flow between Two Parallel Annular Plates
Radial Viscous Flow between Two Parallel Annular Plates
Mya San May
Abstract
We seek the approximate solution to the Navier-Stokes equation for the
radial flow between two parallel annular plates since there is no exact
solution which satisfies the boundary conditions for the problem.
1. Introduction
In this paper, we shall deduce the velocity distribution of steady flow of an
incompressible fluid of densit y ρ and viscosity μ between two parallel, coaxialannular plates of inner radius r 1, outer radius r 2 and separation h when pressure
difference ∆p is applied between the inner and outer radii.
As an exact solution of the Navier-Stokes equation appears to be difficult, it
suffices to give an approximate solution assuming that the velocity is purely radial,
r ˆ
z)u(r,q , in a cylindrical coordinate system (r, φ , z) whose z-axis coincides with
that of the two annuli. We will deduce a condition for validity of the approximation.
This problem arises, for example, in considerations of a rotary joint betweentwo sections of a pipe. Here, we ignore the extra complication of the effect of the
rotation of one of the annuli on the fluid flow.
2. Two-Dimensional Flow between Parallel Plates
Figure (1)
The velocity distribution obeys the continuity equation for an incompressible
fluid,
Dr, Tutor, Department of Mathematics, University of Magway
z=0
z=h
8/13/2019 Radial Viscous Flow between Two Parallel Annular Plates
82 Magway University Research Journal 2012, Vol. IV, No. 1
There are only three examples in which analytic solutions to the equation have
been obtained when the non-linear term (q )q is non-vanishing [ 5 ].These three
cases are
( i ) the flow due to an infinite plane disk rotating uniformly about its axis,( ii ) the steady flow between two plane walls meeting at an angle α and
( iii ) the flow in a jet emerging from the end of a narrow tube into an infinite
space filled with the fluid.
References
Batchelor, G.K., 1997, “An Introduction to Fluid Dynamic”, Cambridge University Press, New York,
Burgess, D and Van Elst, H., 2002 “MAS 209: Fluid Dynamics”, University of
London, London.
Chorlton, F., 1967, “Fluid Dynamics”, D. Van Nostrand Co.Ltd, Londan.
Kirk McDonald, T., 2008, “ Radial Viscous Flow between Two Parallel Annular Plates” , Princeton
University, Princeton, June 25, 2000; updated July 30.
Landau, L. D and Lifshitz, E. M., 1987, “Fluid Mechanic”, 2 nd ed, chap (2) Pergamon Press, Oxford,
Wilson, D. H., 1964, “Hydrodynamics”, Edward Arnold (Publishers) Ltd, London.