By putting

*θ* =

*c* –

*c* _{ s }, Eqs.

2.61,

2.62,

2.63, and

2.64 are written as follows:

$$ \partial \theta /\partial t = D\,\,\left\{ {{\partial^2}\theta /\partial {r^2} + (1/r) \cdot (\partial \theta /\partial r)} \right\} $$

(2.155)

$$ \partial \theta /\partial r = 0,\,\,\,\,\,\,{\text{at}}\,\,\,r = 0,\,\,\,\,\,t \geqslant 0 $$

(2.156)

$$ \theta = 0,\quad \quad {\text{at}}\quad r = R,\,\,\,\,\,\,t \geqslant 0 $$

(2.157)

$$ \theta = {c_b} - {c_s}\,\,\,\,\,\,\,{\text{at}}\,\,\,\,\,t = 0,\,\,\,\,0 \leqslant r \leqslant R $$

(2.158)

We may attempt to find a solution of Eq.

2.155 by putting

*θ*(

*r*,

*t*) =

*X*(

*r*) ·

*T*(

*t*), where the

*X* and

*T* are functions of

*r* and

*t*, respectively. Equations

2.159 and

2.160 are obtained as follows:

$$ X\prime\prime + (1/r)X\prime - kX = 0 $$

(2.159)

$$ T\prime - kDT = 0, $$

(2.160)

where

*k* is constant.

In the case of

*k* = 0, the solution in the ordinary differential equation of Eq.

2.159 is

*X* =

*C* _{1} *In*(

*r*) +

*C* _{2}. In this case, from the above boundary conditions,

*C* _{1} =

*C* _{2} = 0 is obtained. Then, the solution is

*X* = 0. That is, it is understood that

*k* = 0 does not provide a suitable solution. In the case of

*k* > 0, the solution of Eq.

2.159 is

\( X = {C_3}{I_0}(\sqrt {k} r) + {C_4}{K_0}(\sqrt {k} r) \), where

*I* _{0} and

*K* _{0} are the modified Bessel functions of the first and second kinds of order zero, respectively. From the boundary condition of Eq.

2.161 with a finite value and the conditions of

\( I_0^{\prime}(0) = {I_1}(0) = 0 \) and

\( K_0^{\prime}(0) = - {K_1} = - \infty \), so that

*C* _{4} = 0 is obtained, and from the boundary conditions of Eq.

2.162 and the condition of

*I* (0)|

_{ r=0} ≠ 0

*, C* _{3} = 0 is. Thus, the condition of

*k* > 0 is also an unsuitable choice since

*X* = 0 is. Finally, in the case of

*k* < 0, the solution of Eq.

2.159 is

\( X = {C_5}{J_0}(\sqrt {{ - k}} r) + {C_6}{Y_0}(\sqrt {{ - k}} r) \), where

*J* _{0} and

*Y* _{0} are Bessel functions of the first and second kinds of order zero, respectively. From the boundary condition of Eq.

2.161 and the conditions of

\( J_0^{\prime}(0) = \mathit{finite} \) and

\( Y_0^{\prime}(0) = \infty \), so that

*C* _{6} = 0 is obtained, and from the boundary condition of Eq.

2.162,

\( {C_5}{J_0}(\sqrt {{ - {k_n}}} R) = 0 \) must be held, where

*k* _{ n } should be the n-th positive root of

\( {J_0}(\sqrt {{ - {k_n}}} R) = 0 \). Then,

*X* is given as follows:

$$ X = \sum_{{n = 1}}^{\infty } {{J_0}} ({\lambda_n}r),\,\,\,\,\,\,{\lambda_n} \equiv \sqrt {{ - {k_n}}} $$

(2.163)

Then,

*θ*(

*r*,

*t*) is obtained as Eq.

2.165.

$$ \theta \,\,(r,\,t) = \sum_{{n = 1}}^{\infty } {{C_n}} {J_0}({\lambda_n}r)\,\,\,\mathit{exp}\,\,\,\left( { - \lambda_n^2Dt} \right) $$

(2.165)

From the initial condition of (

2.158), the following condition is obtained.

$$ \theta \,\,(r,\,\,0) = \sum_{{n = 1}}^{\infty } {{C_n}} {J_0}({\lambda_n}r) = {c_b} - {c_s} $$

(2.166)

In order to determine the values of

\( {C_n},\smallint_0^Rr{J_0}({\lambda_{{n\,}}}r)dr \) is multiplied on both sides of Eq.

2.166 and integrated from 0 to

*R*. By using the following characteristics of the orthogonal functions Eqs.

2.167,

2.168, and

2.169, the values of

*C* _{ n } are obtained as Eq.

2.170.

$$ \int_0^R {r{J_0}} ({\lambda_n}r)\,{J_0}\,({\lambda_m}r)\,dr = 0,\,\,\,{\text{n}} \ne {\text{m}} $$

(2.167)

$$ \int_0^R {r{{\left\{ {{J_0}({\lambda_n}r)} \right\}}^2}} dr = (1/2){R^2}J_1^2({\lambda_n}R) $$

(2.168)

$$ \int_0^R {r{J_0}({\lambda_n}r)\,} dr = R{J_1}({\lambda_n}R)/{\lambda_n}, $$

(2.169)

where

*J* _{1}(

*x*) is the Bessel function of the first order,

$$ {C_n} = ({c_b} - {c_s})\int_0^R {r{J_0}({\lambda_n}r)\,} dr\bigg/\!\int_0^R {r{{\left\{ {{J_0}({\lambda_n}r)} \right\}}^2}dr = 2({c_b} - {c_s})} /R{\lambda_n}{J_1}({\lambda_n}R) $$

(2.170)

Substituting Eq.

2.170 into Eq.

2.165 and using the relation of

*θ* =

*c* –

*c* _{ s } yields Eq.

2.65.

$$ c = \left\{ {2\,\,({c_b} - {c_s})/R} \right\}\,\,\sum_{{n = 1}}^{\infty } {\left\{ {{J_0}({\lambda_n}r)/{\lambda_n}{J_1}({\lambda_n}R)} \right\}\,\, \cdot \,\,\mathit{exp}\left( { - D\lambda_n^2t} \right)} + {c_s} $$

(2.65)