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Date: 13-7-2018
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Date: 13-7-2018
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Date: 23-7-2018
3079
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To solve the heat conduction equation on a two-dimensional disk of radius , try to separate the equation using
(1) |
Writing the and terms of the Laplacian in cylindrical coordinates gives
(2) |
so the heat conduction equation becomes
(3) |
Multiplying through by gives
(4) |
The term can be separated.
(5) |
which has a solution
(6) |
The remaining portion becomes
(7) |
Dividing by gives
(8) |
where a negative separation constant has been chosen so that the portion remains finite
(9) |
The radial portion then becomes
(10) |
(11) |
which is the spherical Bessel differential equation.
Consider disk or radius with initial temperature and the boundary condition . Then the solution is
(12) |
where is the th positive zero of the Bessel function of the first kind (Bowman 1958, pp. 37-39).
REFERENCES:
Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958.
Carslaw, H. S. and Jaeger, J. C. "Some Two-Dimensional Problems in Conduction of Heat with Circular Symmetry." Proc. London Math. Soc. 46, 361-388, 1940.
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مخاطر خفية لمكون شائع في مشروبات الطاقة والمكملات الغذائية
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"آبل" تشغّل نظامها الجديد للذكاء الاصطناعي على أجهزتها
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تستخدم لأول مرة... مستشفى الإمام زين العابدين (ع) التابع للعتبة الحسينية يعتمد تقنيات حديثة في تثبيت الكسور المعقدة
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