Evolution of Bamboo Wire by Bulk Diffusion

If the grain boundary remains planar, the flux into the boundary must be uniform, 

Assuming that the solution to Eq. 16.19 is the product of functions of z and r and using the separation-of-variables method (Section 5.2.4), 

The bamboo segment can be approximated as a cylinder of average radius Rc, where 

The knRc quantities are the roots of the zeroth-order Bessel function of the first kind, 

The planar grain-boundary condition given by Eq. 16.18 is satisfied if 

---