While the THW method is well established for homogeneous fluids, its applicability to two-phase systems such as fluid suspensions is still under development, and no reliable validity conditions for the latter exist so far (see Vadasz [30] for a discussion and initial study on the latter). As a result, one needs to refer to the two-equation model presented by Equations (1) and (3), instead of the one Fourier type equation that is applicable to homogeneous media.

Two methods of solution are in principle available to solve the system of Equations (

1) and (

3). The first is the elimination method while the second is the eigenvectors method. By means of the elimination method, one may eliminate

*T*_{f} from Equation (

1) in the form:

${T}_{\text{f}}=\frac{{\gamma}_{\text{s}}}{h}\frac{\partial {T}_{\text{s}}}{\partial {t}_{*}}+{T}_{\text{s}}$

(6)

and substitute it into Equation (

3) hence rendering the two Equations (

1) and (

3), each of which depends on both

*T*_{s} and

*T*_{f}, into separate equations for

*T*_{s} and

*T*_{f}, respectively, in the form:

${\tau}_{q}\frac{{\partial}^{2}{T}_{i}}{\partial {t}_{*}^{2}}+\frac{\partial {T}_{i}}{\partial {t}_{*}}={\alpha}_{\text{e}}\left[\frac{1}{{r}_{*}}\frac{\partial}{\partial {r}_{*}}\left({r}_{*}\frac{\partial {T}_{i}}{\partial {r}_{*}}\right)+\frac{{\tau}_{T}}{{r}_{*}}\frac{\partial}{\partial {r}_{*}}\left({r}_{*}\frac{{\partial}^{2}{T}_{i}}{\partial {r}_{*}\partial {t}_{*}}\right)\right]\text{for}i=\text{s},\text{f}$

(7)

where the index

*i* takes the values

*i* =

*s* for the solid phase and

*i* = f for the fluid phase, and the following notation was used:

$\begin{array}{ccc}{\tau}_{\text{q}}=\frac{{\gamma}_{\text{s}}{\gamma}_{\text{f}}}{h\left({\gamma}_{\text{s}}+{\gamma}_{\text{f}}\right)};& {\alpha}_{\text{e}}=\frac{{k}_{\text{f}}}{\left({\gamma}_{\text{s}}+{\gamma}_{\text{f}}\right)};& {\tau}_{\text{T}}=\frac{{\gamma}_{\text{s}}{k}_{\text{f}}}{h\left({\gamma}_{\text{s}}+{\gamma}_{\text{f}}\right){\alpha}_{\text{e}}}=\frac{{\gamma}_{\text{s}}}{h}\end{array}$

(8)

In Equation (

8), τ

_{q} and τ

_{T} are the heat flux and temperature-related time lags linked to Dual-Phase-Lagging [

22,

24–

27,

31], while

*α*_{e} is the effective thermal diffusivity of the suspension. The resulting Equation (

7) is identical for both fluid and solid phases. Vadasz [

22] used this equation in providing the solution. The initial conditions applicable to the problem at hand are identical for both phases, i.e. both phases' temperatures are set to be equal to the ambient temperature

*T*_{C}${t}_{*}=0:\phantom{\rule{0.5em}{0ex}}{T}_{i}={T}_{\text{C}}=\text{constant,}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}i=\text{s,f}$

(9)

The boundary conditions are

${r}_{*}={r}_{0*}:\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{T}_{\text{f}}={T}_{\text{C}}$

(10)

${r}_{*}={r}_{\text{w}*}:\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{\left(\frac{\partial {T}_{\text{f}}}{\partial {r}_{*}}\right)}_{{r}_{*}={r}_{\text{w}*}}=-\frac{{q}_{0}}{{k}_{\text{f}}}$

(11)

where

*r*_{0*} is the radius of the cylindrical container. Equation (

7) is second-order in time and second-order in space. The initial conditions (9) provide one such condition for each phase while the second-order Equation (

7) requires two such conditions. To obtain the additional initial conditions, one may use Equations (

1) and (

3) in combination with (9). From (9), it is evident that both phases' initial temperatures at

*t*_{*} = 0 are identical and constant. Therefore,

${\left({T}_{\text{f}}\right)}_{{t}_{*}=0}={\left({T}_{\text{s}}\right)}_{{t}_{*}=0}={T}_{\text{C}}=\text{constant}$, leading to

${\left({T}_{\text{f}}-{T}_{\text{s}}\right)}_{{t}_{*}=0}=0$ and

${\left[\partial /\partial {r}_{*}\left({r}_{*}\partial {T}_{\text{f}}/\partial {r}_{*}\right)\right]}_{{t}_{*}=0}=0$ to be substituted in (1) and (3), which in turn leads to the following additional initial conditions for each phase:

${t}_{*}=0:\phantom{\rule{0.5em}{0ex}}{\left(\frac{\partial {T}_{i}}{\partial {t}_{*}}\right)}_{{t}_{*}=0}=0\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}i=s,f$

(12)

The two boundary conditions (10) and (11) are sufficient to uniquely define the problem for the fluid phase; however, there are no boundary conditions set for the solid phase as the original Equation (

1) for the solid phase had no spatial derivatives and did not require boundary conditions. To obtain the corresponding boundary conditions for the solid phase, which are required for the solution of Equation (

7) corresponding to

*i* = s, one may use first the fact that at

*r*_{*} =

*r*_{0*} both phases are exposed to the ambient temperature and therefore one may set

${r}_{*}={r}_{0*}:\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{T}_{\text{s}}={T}_{\text{C}}$

(13)

Second, one may use Equation (

6) and taking its derivative with respect to

*r*_{*} yields

$\frac{{\gamma}_{\text{s}}}{h}\frac{\partial}{\partial {t}_{*}}\left(\frac{\partial {T}_{\text{s}}}{\partial {r}_{*}}\right)+\frac{\partial {T}_{\text{s}}}{\partial {r}_{*}}=\frac{\partial {T}_{\text{f}}}{\partial {r}_{*}}$

(14)

In Equation (

14), the spatial variable

*r*_{*} plays no active role; it may therefore be regarded as a parameter. As a result, one may present Equation (

14) for any specified value of

*r*_{*}. Choosing

*r*_{*} =

*r*_{w*} where the value of

${\left(\partial {T}_{\text{f}}/\partial {r}_{*}\right)}_{{r}_{\text{w}*}}$ is known from the boundary condition (11), yields from (14) the following ordinary differential equation:

$\frac{{\gamma}_{\text{s}}}{h}\frac{\text{d}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}}{\text{d}{t}_{*}}{\left(\frac{\partial {T}_{s}}{\partial {r}_{*}}\right)}_{{r}_{\text{w}*}}+{\left(\frac{\partial {T}_{\text{s}}}{\partial {r}_{*}}\right)}_{{r}_{\text{w}*}}=-\frac{{q}_{0}}{{k}_{\text{f}}}$

(15)

At steady state, Equation (

15) produces the solution

${\left(\frac{\partial {T}_{\text{s},\text{st}}}{\partial {r}_{*}}\right)}_{{r}_{\text{w}*}}=-\frac{{q}_{0}}{{k}_{\text{f}}}$

(16)

where

*T*_{s,st} is the steady-state solution. The transient solution

*T*_{s,tr} =

*T*_{s} -

*T*_{s,st} satisfies then the equation:

$\frac{{\gamma}_{\text{s}}}{h}\frac{\text{d}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}}{\text{d}{t}_{*}}{\left(\frac{\partial {T}_{\text{s,tr}}}{\partial {r}_{*}}\right)}_{{r}_{\text{w}*}}+{\left(\frac{\partial {T}_{\text{s,tr}}}{\partial {r}_{*}}\right)}_{{r}_{\text{w}*}}=0$

(17)

subject to the initial condition

${\left[{\left(\frac{\partial {T}_{\text{s,tr}}}{\partial {r}_{*}}\right)}_{{r}_{\text{w}*}}\right]}_{{t}_{*}=0}={\left[{\left(\frac{\partial {T}_{\text{s}}}{\partial {r}_{*}}\right)}_{{r}_{\text{w}*}}\right]}_{{t}_{*}=0}=0$

(18)

because

${\left[\partial {T}_{\text{s}}/\partial {r}_{*}\right]}_{{t}_{*}=0}=0$ for all values of

${r}_{*}\in \left[{r}_{\text{w}*},{r}_{\text{0}*}\right]$ given that according to (9) at

*t*_{*} = 0:

${\left(\phantom{\rule{0.5em}{0ex}}{T}_{\text{s}}\right)}_{{t}_{*}=0}={\left({T}_{\text{f}}\right)}_{{t}_{*}=0}={T}_{\text{C}}=\text{constant}$. Equation (

17) can be integrated to yield

${\left(\frac{\partial {T}_{\text{s,tr}}}{\partial {r}_{*}}\right)}_{{r}_{\text{w}*}}=A\mathrm{exp}\left(-\frac{h}{{\gamma}_{\text{s}}}t\right)$

(19)

which combined with the initial condition (18) produces the value of the integration constant

*A* = 0 and therefore the transient solution becomes

${\left(\frac{\partial {T}_{s,tr}}{\partial {r}_{*}}\right)}_{{r}_{w*}}=0$

(20)

The complete solution for the solid temperature gradient at the wire is therefore obtained by combining (20) with (16) leading to

${\left(\frac{\partial {T}_{\text{s}}}{\partial {r}_{*}}\right)}_{{r}_{\text{w}*}}=-\frac{{q}_{0}}{{k}_{\text{f}}}$

(21)

producing the second boundary condition for the solid phase, which is identical to the corresponding boundary condition for the fluid phase. One may therefore conclude that the solution to the problem formulated in terms of Equation (7) that is identical to both phases, subject to initial conditions (9) and (12) that are identical to both phases, and boundary conditions (10), (11), and (13), (21) that are also identical to both phases, should be also identical to both phases, i.e. *T*_{s} (*t*_{*},*r*_{*}) = *T*_{f} (*t*_{*},*r*_{*}). This, however, may not happen because then *T*_{f} - *T*_{s} = 0 leads to conflicting results when substituted into (1) and (3). The result obtained here is identical to Vadasz [32] who demonstrated that a paradox revealed by Vadasz [33] can be avoided only by refraining from using this method of solution. While the paradox is revealed in the corresponding problem of a porous medium subject to a combination of Dirichlet and insulation boundary conditions, the latter may be applicable to fluids suspensions by setting the effective thermal conductivity of the solid phase to be zero. The fact that in the present case the boundary conditions differ, i.e. a constant heat flux is applied on one of the boundaries (such a boundary condition would have eliminated the paradox in porous media), does not eliminate the paradox in fluid suspensions mainly because in the latter case the steady-state solution is identical for both phases. In the porous media problem, the constant heat flux boundary condition leads to different solutions at steady state, and therefore the solutions for each phase even during the transient conditions differ.

The elimination method yields the same identical equation with identical boundary and initial conditions for both phases apparently leading to the wrong conclusion that the temperature of both phases should therefore be the same. A closer inspection shows that the discontinuity occurring on the boundaries' temperatures at *t* = 0, when a "ramp-type" of boundary condition is used, is the reason behind the occurring problem and the apparent paradox. The question that still remains is which phase temperature corresponds to the solution presented by Vadasz [22]; the fluid or the solid phase temperature?

By applying the eigenvectors method as presented by Vadasz [

32], one may avoid the paradoxical solution and obtain both phases temperatures. The analytical solution to the problem using the eigenvectors method is obtained following the transformation of the equations into a dimensionless form by introducing the following dimensionless variables:

$\mathit{q}=\frac{{\mathit{q}}_{*}}{{q}_{0}},\phantom{\rule{0.5em}{0ex}}{\theta}_{i}=\frac{\left({T}_{i}-{T}_{\text{C}}\right){k}_{\text{f}}}{{q}_{0}{r}_{0*}},\phantom{\rule{0.5em}{0ex}}r=\frac{{r}_{*}}{{r}_{0*}},\phantom{\rule{0.5em}{0ex}}t=\frac{{\alpha}_{\text{e}}{t}_{*}}{{r}_{0*}^{2}}$

(22)

where the following two dimensionless groups emerged:

${\text{Fo}}_{q}=\frac{{\alpha}_{\text{e}}{\tau}_{q}}{{r}_{0*}^{2}};{\text{Fo}}_{T}=\frac{{\alpha}_{\text{e}}{\tau}_{T}}{{r}_{0*}^{2}}$

(23)

representing a heat flux Fourier number and a temperature Fourier number, respectively. The ratio between them is identical to the ratio between the time lags, i.e.

$\beta =\frac{{\text{Fo}}_{T}}{{\text{Fo}}_{q}}=\frac{{\tau}_{T}}{{\tau}_{q}}=\frac{{\gamma}_{\text{s}}+{\gamma}_{\text{f}}}{{\gamma}_{\text{f}}}$

(24)

Equations (

1) and (

3) expressed in a dimensionless form using the transformation listed above are

${\text{Fh}}_{\text{s}}\frac{\partial {\theta}_{\text{s}}}{\partial t}=\left({\theta}_{\text{f}}-{\theta}_{\text{s}}\right)$

(25)

${\text{Fh}}_{\text{f}}\frac{\partial {\theta}_{\text{f}}}{\partial t}=\frac{1}{{\text{Ni}}_{\text{f}}}\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial {\theta}_{\text{f}}}{\partial r}\right)-\left({\theta}_{\text{f}}-{\theta}_{\text{s}}\right)$

(26)

where the following additional dimensionless groups emerged:

${\text{Fh}}_{\text{s}}=\frac{{\alpha}_{\text{e}}{\gamma}_{\text{s}}}{h{r}_{0*}^{2}}={\text{Fo}}_{T}=\frac{{\gamma}_{\text{s}}+{\gamma}_{\text{f}}}{{\gamma}_{\text{f}}}{\text{Fo}}_{q}=\beta {\text{Fo}}_{q}$

(27)

${\text{Fh}}_{\text{f}}=\frac{{\gamma}_{\text{s}}+{\gamma}_{\text{f}}}{{\gamma}_{\text{s}}}{\text{Fo}}_{q}=\frac{{\alpha}_{\text{e}}{\gamma}_{\text{f}}}{h{r}_{0*}^{2}}=\frac{{\text{Fo}}_{T}}{\left(\beta -1\right)}=\frac{\beta}{\left(\beta -1\right)}{\text{Fo}}_{q}$

(28)

${\text{Ni}}_{\text{f}}=\frac{h{r}_{0*}^{2}}{{k}_{\text{f}}}=\frac{\left(\beta -1\right)}{{\beta}^{2}{\text{Fo}}_{q}}$

(29)

where Ni

_{f} is the fluid phase Nield number. The solutions to Equations (

25) and (

26) are subject to the following initial and boundary conditions obtained from (9), (10) and (11) transformed in a dimensionless form:

$t=0:\phantom{\rule{0.5em}{0ex}}{\theta}_{i}=0\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}i=\text{s,f}$

(30)

The boundary conditions are

$r=1:\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{\theta}_{\text{f}}=0$

(31)

$r={r}_{\text{w}}:\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{\left(\frac{\partial {\theta}_{\text{f}}}{\partial r}\right)}_{r={r}_{\text{w}}}=-1$

(32)

No boundary conditions are required for

*θ*_{s}. The solution to the system of Equations (

25)-(

26) is obtained by a superposition of steady and transient solutions

*θ*_{i,st}(

*r*) and

*θ*_{i,tr}(

*t*,

*r*), respectively, in the form:

${\theta}_{i}\left(t,r\right)={\theta}_{i,\text{st}}\left(r\right)+{\theta}_{i,\text{tr}}\left(t,r\right)\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}i=\text{s},\text{f}$

(33)

Substituting (33) into (25)-(26) yields to the following equations for the steady state:

$\left({\theta}_{\text{f,st}}-{\theta}_{\text{s,st}}\right)=0$

(34)

$\frac{1}{{\text{Ni}}_{\text{f}}}\frac{1}{r}\frac{d\phantom{\rule{0.5em}{0ex}}}{dr}\left(r\frac{d{\theta}_{\text{f,st}}}{dr}\right)-\left({\theta}_{\text{f,st}}-{\theta}_{\text{s,st}}\right)=0$

(35)

leading to the following steady solutions which satisfy the boundary conditions (31) and (32):

${\theta}_{\text{f,st}}\left(r\right)={\theta}_{\text{s,st}}\left(r\right)=-{r}_{\text{w}}\mathrm{ln}r$

(36)

The transient part of the solutions

*θ*_{i,tr}(

*t*,

*r*) can be obtained by using separation of variables leading to the following form of the complete solution:

${\theta}_{i}=-{r}_{\text{w}}\mathrm{ln}r+{\displaystyle \sum _{n=1}^{\infty}{S}_{\text{in}}\left(t\right){R}_{\text{on}}\left(r\right)}\phantom{\rule{0.5em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}i=\text{s},\text{f}$

(37)

Substituting (37) into (25)-(26) yields, due to the separation of variables, the following equation for the unknown functions

*R*_{on} (

*r*):

$\frac{1}{r}\frac{d}{dr}\left(r\frac{d{R}_{\text{on}}}{dr}\right)+{\kappa}_{n}^{2}{R}_{\text{on}}=0$

(38)

subject to the boundary conditions

$r=1:\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{R}_{\text{on}}=0$

(39)

$r={r}_{\text{w}}:\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{\left(\frac{d{R}_{\text{on}}}{dr}\right)}_{r={r}_{\text{w}}}=0$

(40)

and the following system of equations for the unknown functions

*S*_{in} (

*t*), (

*i =* s,f), i.e.

$\{\begin{array}{c}\frac{d{S}_{\text{s}n}}{dt}=a{S}_{\text{s}n}-a{S}_{\text{f}n}\\ \frac{d{S}_{\text{f}n}}{dt}=c{S}_{\text{s}n}+{d}_{n}{S}_{\text{f}n}\end{array}$

(41)

where

$a=-{\text{Fh}}_{\text{s}}^{-\text{1}}=-\frac{1}{\beta {\text{Fo}}_{q}}\phantom{\rule{0.5em}{0ex}};\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}c={\text{Fh}}_{\text{f}}^{-1}=\frac{\left(\beta -1\right)}{\beta {\text{Fo}}_{q}};\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{d}_{n}=-\frac{\left({\kappa}_{n}^{2}+{\text{Ni}}_{\text{f}}\right)}{{\text{Ni}}_{\text{f}}{\text{Fh}}_{\text{f}}}=-\beta {\kappa}_{n}^{2}-\frac{\left(\beta -1\right)}{\beta {\text{Fo}}_{q}}$

(42)

and where the separation constant ${\kappa}_{n}^{2}$ represents the eigenvalues in space.

Equation (

38) is the Bessel equation of order 0 producing solutions in the form of Bessel functions

${R}_{\text{on}}\left({\kappa}_{n},r\right)={Y}_{0}\left({\kappa}_{n}\right){J}_{0}\left({\kappa}_{n}r\right)-{J}_{0}\left({\kappa}_{n}\right){Y}_{0}\left({\kappa}_{n}r\right)$

(43)

Where

*J*_{0}(

*κ*_{
n
}*r*) and

*Y*_{0}(

*κ*_{
n
}*r*) are the order 0 Bessel functions of the first and second kind, respectively. The solution (43) satisfies the boundary condition (39) as can easily be observed by substituting

*r* = 1 in (43). Imposing the second boundary condition (40) yields a transcendental equation for the eigenvalues

*κ*_{
n
}in the form:

${J}_{0}\left({\kappa}_{n}\right){Y}_{1}\left({\kappa}_{n}{r}_{\text{w}}\right)-{Y}_{0}\left({\kappa}_{n}\right){J}_{1}\left({\kappa}_{n}{r}_{\text{w}}\right)=0$

(44)

where

*J*_{1}(

*κ*_{
n
}*r*_{w}) and

*Y*_{1}(

*κ*_{
n
}*r*_{w}) are the order 1 Bessel functions of the first and second kind, respectively, evaluated at

*r* =

*r*_{w}. The compete solution is obtained by substituting (43) into (37) and imposing the initial conditions (30) in the form

${\left({\theta}_{i}\right)}_{t=0}=-{r}_{\text{w}}\mathrm{ln}r+{\displaystyle \sum _{n=1}^{\infty}{S}_{\text{in}}\left(0\right){R}_{\text{on}}\left(r\right)}\phantom{\rule{0.5em}{0ex}}=0\text{for}\phantom{\rule{0.5em}{0ex}}i=\text{s},\text{f}$

(45)

At

*t* = 0, both phases' temperatures are the same leading to the conclusion that

${S}_{\text{s}n}\left(0\right)={S}_{\text{f}n}\left(0\right)={S}_{no}$

(46)

Multiplying (45) by the orthogonal eigenfunction

*R*_{
om
}(

*κ*_{
m
}*,r*) with respect to the weight function

*r* and integrating the result over the domain [

*r*_{w},1], i.e.

${\int}_{{r}_{\text{w}}}^{1}(\u2022){R}_{om}({\kappa}_{m},r)}\phantom{\rule{0.5em}{0ex}}r\phantom{\rule{0.5em}{0ex}}dr$ yield

${r}_{\text{w}}{\displaystyle \underset{{r}_{\text{w}}}{\overset{1}{\int}}r\mathrm{ln}r{R}_{om}\left({\kappa}_{m},r\right)dr}={\displaystyle \sum _{n=1}^{\infty}{S}_{no}{\displaystyle \underset{{r}_{\text{w}}}{\overset{1}{\int}}r{R}_{on}\left({\kappa}_{n},r\right){R}_{om}\left({\kappa}_{m},r\right)dr}}$

(47)

The integral on the right-hand side of (47) produces the following result due to the orthogonality conditions for Bessel functions:

$\underset{{r}_{\text{w}}}{\overset{1}{\int}}r\phantom{\rule{0.5em}{0ex}}{R}_{on}\left({\kappa}_{n},r\right){R}_{om}\left({\kappa}_{m},r\right)dr}=\{\begin{array}{cc}0\phantom{\rule{0.5em}{0ex}}& \text{for}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}n\ne m\\ N\left({\kappa}_{n}\right)& \text{for}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}n=m\end{array$

(48)

where the norm

*N*(

*κ*_{
n
}) is evaluated in the form:

$N\left({\kappa}_{n}\right)={\displaystyle \underset{{r}_{\text{w}}}{\overset{1}{\int}}r{R}_{on}^{2}\left({\kappa}_{n},r\right)dr}=\frac{2}{{\pi}^{2}}\frac{\left[{J}_{1}^{2}\left({\kappa}_{n}{r}_{w}\right)-{J}_{0}^{2}\left({\kappa}_{n}\right)\right]}{{\kappa}_{n}^{2}{J}_{1}^{2}\left({\kappa}_{n}{r}_{w}\right)}$

(49)

The integral on the left-hand side of (47) can be evaluated using integration by parts and the equation for the eigenvalues (44) to yield

$\underset{{r}_{\text{w}}}{\overset{1}{\int}}r\mathrm{ln}r{R}_{on}\left({\kappa}_{n},r\right)dr}=\frac{1}{{\kappa}_{n}^{2}}\left[{J}_{0}\left({\kappa}_{n}\right){Y}_{0}\left({\kappa}_{n}{r}_{w}\right)-{Y}_{0}\left({\kappa}_{n}\right){J}_{0}\left({\kappa}_{n}{r}_{w}\right)\right]$

(50)

Substituting (48) and (50) into (47) yields the values of

*S*_{in} at

*t* = 0, i.e.

*S*_{
no
}=

*S*_{
sn
}(0) =

*S*_{fn}(0)

$\frac{{r}_{\text{w}}}{{\kappa}_{n}^{2}}\left[{J}_{0}\left({\kappa}_{n}\right){Y}_{0}\left({\kappa}_{n}{r}_{\text{w}}\right)-{Y}_{0}\left({\kappa}_{n}\right){J}_{0}\left({\kappa}_{n}{r}_{\text{w}}\right)\right]={S}_{no}N\left({\kappa}_{n}\right)$

that need to be used as initial conditions for the solution of system (41)

${S}_{no}=\frac{{\pi}^{2}{r}_{\text{w}}{J}_{1}^{2}\left({\kappa}_{n}{r}_{\text{w}}\right)\left[{J}_{0}\left({\kappa}_{n}\right){Y}_{0}\left({\kappa}_{n}{r}_{\text{w}}\right)-{Y}_{0}\left({\kappa}_{n}\right){J}_{0}\left({\kappa}_{n}{r}_{\text{w}}\right)\right]}{2\left[{J}_{1}^{2}\left({\kappa}_{n}{r}_{\text{w}}\right)-{J}_{0}^{2}\left({\kappa}_{n}\right)\right]}$

(51)

to produce the explicit solutions in time. With the initial conditions for

*S*_{in} evaluated (

*i* = s,f), one may turn to solving system (41) that can be presented in the following vector form:

$\frac{d{\mathit{S}}_{\mathit{n}}}{dt}=A{\mathit{S}}_{\mathit{n}}$

(52)

where the matrix

*A* is explicitly defined by

$A=\left|\begin{array}{cc}a& -a\\ c& {d}_{n}\end{array}\right|$

(53)

with the values of

*a*,

*c* and

*d*_{
n
}given by Equation (

42), and the vector

S_{
n
}defined in the form

S_{
n
}= [

*S*_{
sn
},

*S*_{
fn
}]

^{T}. The eigenvalues

*λ*_{
n
}corresponding to (52) are obtained as the roots of the following quadratic algebraic equation:

${\lambda}_{n}^{2}-\left(a+{d}_{n}\right){\lambda}_{n}+a\left({d}_{n}+c\right)=0$

(54)

leading to

$\begin{array}{ccc}{\lambda}_{1n}=\frac{a+{d}_{n}}{2}+\frac{1}{2}\sqrt{{\left(a-{d}_{n}\right)}^{2}-4ac}& \text{and}& {\lambda}_{2n}=\frac{a+{d}_{n}}{2}-\frac{1}{2}\sqrt{{\left(a-{d}_{n}\right)}^{2}-4ac}\end{array}$

which upon substituting

*a*,

*c* and

*d*_{
n
}from Equation (

42) yields

${\lambda}_{1n}=-\frac{\left(1+\beta {\text{Fo}}_{q}{\kappa}_{n}^{2}\right)}{2{\text{Fo}}_{q}}\left[1+\sqrt{1-\frac{4{\text{Fo}}_{q}{\kappa}_{n}^{2}}{{\left(1+\beta {\text{Fo}}_{q}{\kappa}_{n}^{2}\right)}^{2}}}\right]$

(55)

${\lambda}_{2n}=-\frac{\left(1+\beta {\text{Fo}}_{q}{\kappa}_{n}^{2}\right)}{2{\text{Fo}}_{q}}\left[1-\sqrt{1-\frac{4{\text{Fo}}_{q}{\kappa}_{n}^{2}}{{\left(1+\beta {\text{Fo}}_{q}{\kappa}_{n}^{2}\right)}^{2}}}\right]$

(56)

The following useful relationship is obtained from (55) and (56):

${\lambda}_{1n}{\lambda}_{2n}=\frac{{\kappa}_{n}^{2}}{{\text{Fo}}_{q}}$

(57)

The corresponding eigenvectors

*υ*_{1}_{
n
}and

*υ*_{2}_{
n
}are evaluated in the form:

$\begin{array}{ccc}{v}_{1n}=\left[\frac{1}{\frac{\left(-{\lambda}_{1n}+a\right)}{a}}\right]& \text{and}& {v}_{1n}=\left[\frac{1}{\frac{\left(-{\lambda}_{2n}+a\right)}{a}}\right]\end{array}$

(58)

leading to the following solution:

${S}_{n}={\mathit{v}}_{1\mathit{n}}{C}_{1n}{e}^{{\lambda}_{1n}t}+{\mathit{v}}_{2\mathit{n}}{C}_{2n}{e}^{{\lambda}_{2n}t}$

(59)

and explicitly following the substitution of (58) and the initial conditions

*S*_{in} (

*i* = s,f), at

*t* = 0, i.e.

*S*_{
sn
}(0) =

*S*_{fn}(0) =

*S*_{
no
}with the values of

*S*_{
no
}given by Equation (

51)

${S}_{\text{s}n}=\frac{{S}_{no}}{\left({\lambda}_{2n}-{\lambda}_{1n}\right)}\left[{\lambda}_{2n}{e}^{{\lambda}_{1n}t}-{\lambda}_{1n}{e}^{{\lambda}_{2n}t}\right]$

(60)

${S}_{fn}=\frac{{S}_{no}}{\left({\lambda}_{2n}-{\lambda}_{1n}\right)}\left[{\lambda}_{2n}\left(1+\beta F{o}_{q}\phantom{\rule{0.5em}{0ex}}{\lambda}_{1n}\right){e}^{{\lambda}_{1n}t}-{\lambda}_{1n}\left(1+\beta F{o}_{q}\phantom{\rule{0.5em}{0ex}}{\lambda}_{2n}\right){e}^{{\lambda}_{2n}t}\right]$

(61)

Substituting (57) into (60) and (61) and the latter into the complete solution (37) yields

${\theta}_{\text{s}}=-{r}_{\text{w}}\mathrm{ln}r+{\displaystyle \sum _{n=1}^{\infty}{B}_{n}\left[{\lambda}_{2n}{e}^{{\lambda}_{1n}t}-{\lambda}_{1n}{e}^{{\lambda}_{2n}t}\right]{R}_{\text{on}}\left(r\right)}$

(62)

${\theta}_{\text{f}}=-{r}_{\text{w}}\mathrm{ln}r+{\displaystyle \sum _{n=1}^{\infty}{B}_{n}\left[\left({\lambda}_{2n}+\beta {\kappa}_{n}^{2}\right){e}^{{\lambda}_{1n}t}-\left({\lambda}_{1n}+\beta {\kappa}_{n}^{2}\right){e}^{{\lambda}_{2n}t}\right]{R}_{\text{on}}\left(r\right)}$

(63)

where

*B*_{
n
}is

${B}_{n}=\frac{{S}_{no}}{\left({\lambda}_{2n}-{\lambda}_{1n}\right)}=\frac{{\pi}^{2}{r}_{\text{w}}{J}_{1}^{2}\left({\kappa}_{n}{r}_{\text{w}}\right)\left[{J}_{0}\left({\kappa}_{n}\right){Y}_{0}\left({\kappa}_{n}{r}_{\text{w}}\right)-{Y}_{0}\left({\kappa}_{n}\right){J}_{0}\left({\kappa}_{n}{r}_{\text{w}}\right)\right]}{2\left({\lambda}_{2n}-{\lambda}_{1n}\right)\left[{J}_{1}^{2}\left({\kappa}_{n}{r}_{\text{w}}\right)-{J}_{0}^{2}\left({\kappa}_{n}\right)\right]}$

(64)

Comparing the solutions obtained above with the solution obtained by Vadasz [22] via the elimination method, one may conclude that the latter corresponds to the solid phase temperature *θ*_{s}.

The Fourier solution is presented now to compare the solution obtained from the Dual-Phase-Lagging model to the former. The Fourier solution is the result obtained by solving the thermal diffusion equation

$\frac{1}{\beta}\frac{\partial \theta}{\partial t}=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial \theta}{\partial r}\right)$

(65)

subject to the boundary and initial conditions

$t=0:\phantom{\rule{0.5em}{0ex}}\theta =0$

(66)

$r=1:\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\theta =0$

(67)

$r={r}_{\text{w}}:\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{\left(\frac{\partial \theta}{\partial r}\right)}_{r={r}_{\text{w}}}=-1$

(68)

where the same scaling as in Equation (

22) was applied in transforming the equation into its dimensionless form, hence the reason for the coefficient 1/

*β* in the equation. The Fourier solution for this problem has then the form [

34]

$\theta =-{r}_{\text{w}}\mathrm{ln}r+{\displaystyle \sum _{n=1}^{\infty}{C}_{n}{e}^{-\beta {\kappa}_{n}^{2}t}{R}_{\text{on}}\left(r\right)}$

(69)

where

${C}_{n}=\frac{{\pi}^{2}{r}_{\text{w}}{J}_{1}^{2}\left({\kappa}_{n}{r}_{\text{w}}\right)\left[{J}_{0}\left({\kappa}_{n}\right){Y}_{0}\left({\kappa}_{n}{r}_{\text{w}}\right)-{Y}_{0}\left({\kappa}_{n}\right){J}_{0}\left({\kappa}_{n}{r}_{\text{w}}\right)\right]}{2\left[{J}_{1}^{2}\left({\kappa}_{n}{r}_{\text{w}}\right)-{J}_{0}^{2}\left({\kappa}_{n}\right)\right]}={S}_{no}$

(70)

and the eigenvalues

*κ*_{
n
}are the solution of the same transcendental Equation (

44) and the eigenfunctions

*R*_{on}(

*r*) are also identical to the ones presented in Equation (

43). The relationship between the Fourier coefficient

*C*_{
n
}and the Dual-Phase-Lagging model's coefficient

*B*_{
n
}is

${C}_{n}=\left({\lambda}_{2n}-{\lambda}_{1n}\right){B}_{n}$

(71)