Solid Cylinder, Steady 2D, R0JZKL

Consider two-dimensional heat conduction in the finite cylinder. The temperature is given by

$$\frac{\partial ^{2}T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T}{\partial r} +\frac{\partial ^{2}T}{\partial z^{2}} = -\frac{g(r,z)}{k}$$ (12)

$$0<z<L;\;0<r<a$$

$$k_{i}\frac{\partial T}{\partial n_{i}}+h_{i}T = f_{i}\mbox{ \ for boundaries }i=1,\; 2,\mbox{ or }3$$

The temperature may be stated in the form of integrals with the method of Green's functions. If the Green's function, $G$, is known, the temperature is given by

$$T(r,z)=\int_{z^{\prime }=0}^{L} \int_{r^{\prime }=0}^{a} \fr... ...prime }\right.) \,2\pi r^{\prime } dr^{\prime } dz^{\prime }$$

(for volume energy generation)

$$+\sum_{j=1}^{3}\int_{s_{j}}\frac{f_{j}}{k} G(r,z\,\left\vert \,r_{j}^{\prime },z_{j}^{\prime }\right.) ds_{j}^{\prime }$$

(for boundary conditions of type 2 and 3)

$$-\sum_{i=1}^{3}\int_{s_{i}}f_{i}\frac{\partial G(r,z\,\left... ...^{\prime }\right.) }{\partial n_{i}^{\prime }}ds_{i}^{\prime }$$ (13)

(for boundary conditions of type 1 only).


The same Green's function appears in each integral but is evaluated at locations appropriate for each integral. Here position $(r_{i}^{\prime },z_{i}^{\prime })$ is located on surface $s_{i}$. Surface differential $ds_i^{\prime}$ is associated with appropriate surfaces of the cylinder: on surface $r=a$, $ds_i^{\prime}= 2 \pi a dz'$; and, at $z'=0$ or $z'=L$, $ds_i^{\prime}= 2 \pi r' dr'$.

GF, 2D Cylinder

The steady Green's function represents the response at point $(r,z)$ caused by a point source of heat located at $(r^{\prime },z^{\prime})$. The GF for the finite cylinder satisfies the following equations:

$$\frac{\partial ^{2}G}{\partial r^{2}}+\frac{1}{r}\frac{\partial G}{\partial r} +\frac{\partial ^{2}G}{\partial z^{2}} = -\frac{1}{r}\delta (r-r^{\prime }) \delta (z-z^{\prime })$$ (14)

$$0<r<a; \; 0<z<L$$

$$k_{i}\frac{\partial G}{\partial n_{i}}+h_{i}G = 0\mbox{ \ for faces }% i=1, \; 2, \;3$$

Note that the boundary conditions are homogeneous and of the same type as the temperature problem.

The Green's function for the cylinder with axisymmetric heat conduction is given by

$$G(r,z\,\left\vert \,r^{\prime },z^{\prime }\right. )= \frac{P_{00... ...eta_{0m}r) J_n(\beta_{0m}r^{\prime}) } {N_{r}(\beta_{0m})} P_{0m}(z,z^{\prime})$$ (15)

where $J_n$ are Bessel functions of order $n$. The norms, eigenconditions, and kernel functions are identical to those used for the three-dimensional GF. The above single-sum GF may be derived either by a direct solution of the defining equation for $G$, or, by integrating the 3D GF over $0 < \phi < 2\pi$. A physical interpretation of this approach, called the method of descent, is to distribute 3D point sources to form a ring-shaped source appropriate for axisymmetric 2D heating. A double-sum form of the GF may also be found from the transient GF by the limit method; see Beck et al. (1992, p. 249) for a discussion of case R01Z11.