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Heat Conduction Equation--Disk

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To solve the heat conduction equation on a two-dimensional disk of radius a=1, try to separate the equation using

U(r,theta,t)=R(r)Theta(theta)T(t).
(1)

Writing the theta and r terms of the Laplacian in cylindrical coordinates gives

del ^2=(d^2R)/(dr^2)+1/r(dR)/(dr)+1/(r^2)(d^2Theta)/(dtheta^2),
(2)

so the heat conduction equation becomes

(RTheta)/kappa(dT)/(dt)=(d^2R)/(dr^2)ThetaT+1/r(dR)/(dr)ThetaT+1/(r^2)(d^2Theta)/(dtheta^2)RT.
(3)

Multiplying through by r^2/RThetaT gives

(r^2)/(kappaT)(dT)/(dt)=(r^2)/R(d^2R)/(dr^2)+r/R(dR)/(dr)+(d^2Theta)/(dtheta^2)1/Theta.
(4)

The theta term can be separated.

(d^2Theta)/(dtheta^2)1/Theta=-n(n+1),
(5)

which has a solution

Theta(theta)=Acos[sqrt(n(n+1))theta]+Bsin[sqrt(n(n+1))theta].
(6)

The remaining portion becomes

(r^2)/(kappaT)(dT)/(dt)=(r^2)/R(d^2R)/(dr^2)+r/R(dR)/(dr)-n(n+1).
(7)

Dividing by r^2 gives

1/(kappaT)(dT)/(dt)=1/R(d^2R)/(dr^2)+1/(rR)(dR)/(dr)-(n(n+1))/(r^2)=-1/(lambda^2),
(8)

where a negative separation constant has been chosen so that the t portion remains finite

T(t)=Ce^(-kappat/lambda^2).
(9)

The radial portion then becomes

1/R(d^2R)/(dr^2)+1/(rR)(dR)/(dr)-(n(n+1))/(r^2)+1/(lambda^2)=0
(10)
r^2(d^2R)/(dr^2)+r(dR)/(dr)+[(r^2)/(lambda^2)-n(n+1)]R=0,
(11)

which is the spherical Bessel differential equation.

Consider disk or radius a with initial temperature U(r,0)=0 and the boundary condition U(a,t)=1. Then the solution is

U(r,t)=1-2sum_(n=1)^infty(J_0((alpha_nr)/a))/(alpha_nJ_1(alpha_n))e^(-alpha_n^2kappat/a^2),
(12)

where alpha_n is the nth positive zero of the Bessel function of the first kind J_0(x) (Bowman 1958, pp. 37-39).

See also

Heat Conduction Equation

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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.

Cite this as:

Weisstein, Eric W. "Heat Conduction Equation--Disk." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HeatConductionEquationDisk.html

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