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The Zeros of J21(ζ) - J0(ζ)J2(ζ) = 0 with an Application to Swirling Flow in a Tube

D. A. Macdonald
SIAM Journal on Applied Mathematics
Vol. 51, No. 1 (Feb., 1991), pp. 40-48
Stable URL: http://www.jstor.org/stable/2101844
Page Count: 9
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The Zeros of J21(ζ) - J0(ζ)J2(ζ) = 0 with an Application to Swirling Flow in a Tube
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Abstract

The roots of the equation J21(ζ) - J0(ζ)J2(ζ) = 0 are presented correctly to six decimal places, or better. The first 25 roots in the first quadrant of the zeta plane are tabulated, and an asymptotic formula that describes all subsequent roots correctly to at least six decimal places is derived. The roots are used to plot representative streamlines for the steady motion of a viscous fluid in a long tube, of constant radius, which rotates about its axis (the ẑ axis) with an angular velocity that changes discontinuously at ẑ = 0 from one constant value to another of the same sign. The results are discussed with reference to the vortex breakdown phenomenon.

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