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A282581
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Decimal expansion of the limiting Nusselt Number for laminar flow in a cylindrical pipe with constant wall temperature
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0
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3, 6, 5, 6, 7, 9, 3, 4, 5, 7, 7, 6, 3, 2, 9, 2, 3, 6, 1, 9, 7, 9, 4, 3, 7, 5, 0, 6, 0, 8, 8, 4, 5, 2, 4, 3, 9, 5, 2, 2, 7, 4, 5, 2, 0, 4, 6, 4, 8, 8, 1, 4, 5, 4, 9, 8, 1, 6, 2, 0, 3, 5, 1, 8, 8, 3, 7, 1, 3, 9, 1, 6, 3, 7, 2, 1, 8, 0, 2, 1, 8, 4, 3, 0, 9, 1, 9, 9, 6, 9, 6, 8, 5, 9, 5, 3, 6, 0, 0, 3
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OFFSET
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1,1
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COMMENTS
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Study of the heat transfer in cylindrical pipes with fully-developed laminar flow lwith constant inlet temperature and constant wall temperature (the Graetz-Nusselt problem) leads to the dimensionless equation 2 * (1-r^2) * dT/dz = 1/r * dT/dr + d^2T/dr^2 subject to the boundary conditions T(z=0) = 1, T(r=1) = 0, (dT/dr)(r=0) = 0.
The solution to this equation, obtained using separation of variables, is (where M is Kummer's M function and beta an eigenvalue) T = M(1/2 - 1/2 * beta, 1, 2*beta*r^2) * exp(- beta*r^2) * exp( - beta^2*z).
The first eigenvalue is calculated from the condition that the function value is zero for r=1: M(1/2 - 1/2 * beta[1], 1, 2*beta[1]) = 0.
The Nusselt number then is Nu = 2*beta[1]^2.
The Nusselt number was named after the German engineer Wilhelm Nusselt (1882-1957). - Amiram Eldar, May 18 2021
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REFERENCES
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Theodore L. Bergman and Adrienne S. Lavine, Fundamentals of Heat and Mass Transfer, Wiley, 2017, section 8.4, p. 491.
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LINKS
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EXAMPLE
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Nu = 3.6567934577632923619...
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MAPLE
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fsolve(KummerM(1/2-1/2*beta, 1, 2*beta), beta=1..2)^2*2
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MATHEMATICA
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RealDigits[2 * (x /. FindRoot[Hypergeometric1F1[1/2 - x/2, 1, 2*x], {x, 1}, WorkingPrecision -> 120])^2, 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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