1. Thermal energy
In [Reference Stojanović, Romanò and Kuhlmann3] we have claimed to take into account the temperature dependence of all thermophysical parameters. However, in the temperature equation (3.1c) of [Reference Stojanović, Romanò and Kuhlmann3] we have neglected the term describing the advection of
$c_p$
. Even though the advection of
$c_p$
has little effect (see below), we present the temperature equation which includes the advection of
$c_p$
, but still neglects the pressure variations. The correspondingly revised temperature equation reads
\begin{align} \partial _t \left (\rho c_p \hat T\right ) + \nabla \cdot \left (\rho c_p \hat {{{{{\ U}} }}} \hat T \right ) &= \nabla \cdot (\lambda \nabla \hat T) + \rho \hat T \cfrac {\,\text {D} c_p}{\, {D} t} \end{align}
or, equivalently [Reference Bird, Stewart and Lightfoot1],
\begin{align} \rho c_p \left ( \partial _t + \hat {{{{{\ U}} }}} \cdot \nabla \right ) \hat T &= \nabla \cdot (\lambda \nabla \hat T). \end{align}
As a result the equations (4.1)–(4.6) must be replaced by the following expressions, where we use the same equation numbering as in the original publication.
\begin{align} \rho c_p \left [ \partial _t + \left ({{{{\ u}}}}_0 + {{{{\ u}}}} \right )\cdot \nabla \right ] \left (T_0 + T\right ) &= \nabla \cdot (\lambda \nabla T_0)+\nabla \cdot (\lambda \nabla T). \end{align}
\begin{align} \rho c_p \left ( \partial _t T_0 + \partial _t T + {{{{\ u}}}}_0 \cdot \nabla T_0 + {{{{\ u}}}}_0 \cdot \nabla T + {{{{\ u}}}} \cdot \nabla T_0\right ) = \nabla \cdot (\lambda \nabla T_0)+\nabla \cdot (\lambda \nabla T). \end{align}
\begin{align} \rho _0c_{p0} \partial _t T_0 & + \left (\rho _0 c^{\prime}_p + \rho^{\prime}_0 c_{p0}\right )T\partial _t T_0 + \rho _0 c_{p0} \partial _t T + \left [\rho _0 c_{p0} + (\rho _0 c^{\prime}_{p0} + \rho^{\prime}_0 c_{p0})T \right ]{{{{\ u}}}}_0 \cdot \nabla T_0\nonumber \\& + \rho _0 c_{p0} \left ({{{{\ u}}}}_0 \cdot \nabla T + {{{{\ u}}}} \cdot \nabla T_0\right ) = \nabla \cdot (\lambda _0 \nabla T_0)+\nabla \cdot (\lambda^{\prime}_0 T \nabla T_0)+\nabla \cdot (\lambda _0 \nabla T). \end{align}
\begin{align} \rho _0 c_{p0} {{{{\ u}}}}_0 \cdot \nabla T_0 &= \nabla \cdot (\lambda _0 \nabla T_0). \end{align}
\begin{align} \rho _0 c_{p0} \partial _t T &+ (\rho _0 c^{\prime}_{p0} + \rho^{\prime}_0 c_{p0})T {{{{\ u}}}}_0 \cdot \nabla T_0 + \rho _0 c_{p0} \left ( {{{{\ u}}}}_0 \cdot \nabla T + {{{{\ u}}}} \cdot \nabla T_0 \right )\nonumber \\ &= \nabla \cdot (\lambda^{\prime}_0 T \nabla T_0)+\nabla \cdot (\lambda _0 \nabla T). \end{align}
\begin{align} {\underbrace {\vphantom {\bigl [} \rho _0 c_{p0} T\partial _t T}_{\texttt { T1}}}= &-{\underbrace {\vphantom {\bigl [} 0}_{\texttt { T2}}} -{\underbrace {\vphantom {\bigl [} \rho^{\prime}_0 c_{p0} T^2 {{{{\ u}}}}_0 \cdot \nabla T_0}_{\texttt { T3}}} -{\underbrace {\vphantom {\bigl [} \rho _0 c^{\prime}_{p0} T^2 {{{{\ u}}}}_0 \cdot \nabla T_0}_{\texttt { T4}}}\nonumber \\ &-{\underbrace {\vphantom {\bigl [} \rho _0 c_{p0} T {{{{\ u}}}}_0 \cdot \nabla T}_{\texttt { T5}}} -{\underbrace {\vphantom {\bigl [} \rho _0 c_{p0} T{{{{\ u}}}} \cdot \nabla T_0}_{\texttt { T6}}} +{\underbrace {\vphantom {\bigl [} T\nabla \cdot (\lambda^{\prime}_0 T \nabla T_0)}_{\texttt { T7}}} +{\underbrace {\vphantom {\bigl [} T\nabla \cdot (\lambda _0 \nabla T)}_{\texttt { T8}}}. \end{align}
As a result of the inclusions of the
$c_p$
-advection term, the term T2 in (4.6) of [Reference Stojanović, Romanò and Kuhlmann3] vanishes and the terms T3 to T6 are modified. The rate of change of thermal energy (4.8) formally remains the same, but with the following meaning.
\begin{align} D_{\text {th}} &= -\int _{V_i} \left (\text {part of}\ \texttt{T8} \right ) \,\text {d} V = \int _{V_i} \lambda _0 (\nabla T)^2\,\text {d} V, \end{align}
\begin{align} J &= - \int _{V_i} \texttt { T6} \,\text {d} V = -\int _{V_i} \rho _0 c_{p0} T (u\partial _r T_0 + w\partial _z T_0) \,\text {d} V, \end{align}
\begin{align} H_{\text {fs}} &= -\int _{V_i} \left (\text {part of}\ \texttt{T8} \right ) \,\text {d} V = \alpha _i\int _{A_{\text {fs}}} \lambda _0 T \nabla T \cdot {{{{\ n}}}}\,\text {d} S, \end{align}
\begin{align} K_{\text {G,th}} &= -\int _{V_i} \left (\text {part of}\ \texttt{T5} \right ) \,\text {d} V = -\frac {1-\alpha _i}{4}\int _{A_{\text {out}}}\rho _0c_{p0} T^2w_0 \,\text {d} S, \end{align}
\begin{align} \Pi _\rho &= -\int _{V_i} \texttt{T3}\,\text {d} V = -\int _{V_i} \rho^{\prime}_0 c_{p0} T^2 {{{{\ u}}}}_0 \cdot \nabla T_0 \,\text {d} V, \end{align}
\begin{align} \Pi _{c_p} &= -\int _{V_i} \left (\texttt{T4} + \text {part of}\ \texttt{T5} \right ) \,\text {d} V = -\frac {1}{2} \int _{V_i} \rho _0 c^{\prime}_{p0} T^2 {{{{\ u}}}}_0\cdot \nabla T_0 \,\text {d} V, \end{align}
\begin{align} \Pi _\lambda &= -\int _{V_i} \texttt { T7} \,\text {d} V = \alpha _i\int _{A_{\text {fs}}} \lambda^{\prime}_0 T^2 \nabla T_0\cdot {{{{\ n}}}} \,\text {d} S - \frac {1}{2}\int _{V_i} \lambda^{\prime}_0 \nabla T_0\cdot \nabla T^2 \,\text {d} V, \end{align}
\begin{align} \partial _t E^{\prime}_T &= -\int _{V_i} \texttt { T2} \,\text {d} V = 0. \end{align}
For the sake of completeness we have specified all subequations of (4.9) of [Reference Stojanović, Romanò and Kuhlmann3]. Note that equations (4.9a), (4.9c), (4.9d), (4.9g) and (4.9h) remain unchanged, while the subequations (4.9b), (4.9e), (4.9f) and (4.9h) are updated. The notation’part of T#’ in (4.9),
$\#\in [5,8]$
, should indicate that only part of the respective term
$ {T\#}$
from (4.6) enters the integral in (4.9).
We note that the integral over T5 yields
\begin{align*} \int _{V_i} \texttt { T5} \,\text {d} V &= \int _{V_i} \rho _0 c_{p0} T {{{{\ u}}}}_0 \cdot \nabla T \,\text {d} V = \overbrace {\int _{A_{\text {out}}} \rho _0 c_{p0} T^2 {{{{\ u}}}}_0 \cdot {{{{\ n}}}} \,\text {d} S}^{:= -2K_{\text {G,th}}} - \int _{V_i} T \nabla \cdot \left ( \rho _0 c_{p0} {{{{\ u}}}}_0 T \right )\,\text {d} V \\ &= -2K_{\text {G,th}} - \int _{V_i} T \rho _0 {{{{\ u}}}}_0 \cdot \left ( c_{p0} \nabla T + T \nabla c_{p0}\right )\,\text {d} V\\ &= -2K_{\text {G,th}} - \int _{V_i} \rho _0 c_{p0} T {{{{\ u}}}}_0 \cdot \nabla T \,\text {d} V - \int _{V_i} \rho _0 c^{\prime}_{p0} T^2 {{{{\ u}}}}_0 \cdot \nabla T_0 \,\text {d} V. \end{align*}
From which one concludes
\begin{align*} \int _{V_i} \rho _0 c_{p0} T {{{{\ u}}}}_0 \cdot \nabla T \,\text {d} V &= -K_{\text {G,th}} - \frac {1}{2}\int _{V_i} \rho _0 c^{\prime}_{p0} T^2 {{{{\ u}}}}_0 \cdot \nabla T_0 \,\text {d} V. \end{align*}
For a derivation of the remaining expressions in the revised equation (4.9) the Appendix A of [Reference Stojanović, Romanò and Kuhlmann3] is not required anymore and should be dropped.
2. Effect of
$c_p$
advection on the linear stability boundary
To demonstrate the effect of
$c_p$
advection on the linear stability boundary, we supplement the linear stability boundaries from table 3 of [Reference Bird, Stewart and Lightfoot1] by critical Reynolds numbers (and temperature differences) obtained including the term
$\rho \hat T \,\text {D} c_p/\,\text {D} t$
, i.e. the effect of
$c_p$
advection (superscript ’adv’). Table 1 reveals that the advection of
$c_p$
reduces the critical Reynolds number in the full-temperature-dependence model (FTD) by 1.6%. Considering the deviations from the critical Reynolds number in the Oberbeck–Boussinesq (OB) model we find that the temperature dependence of
$c_p$
without
$c_p$
advection reduces the critical Reynolds number by 2.2%. From table 3 of [Reference Bird, Stewart and Lightfoot1] we note that the temperature dependence of
$\rho$
and
$\lambda$
(including advection of these quantities) individually reduce
${\ {Re}}_c^{\text {OB}}$
by about 2%, while individually taking into account
$c_p(\hat T)$
(including
$c_p$
advection) reduces
${\ {Re}}_c^{\text {OB}}$
by to
$4.3$
percent, where the advection of
$c_p$
contributes about as much as the mere temperature dependence (without
$c_p$
advection) does. Therefore, among the three quantities
$\rho$
,
$\lambda$
and
$c_p$
,
$c_p$
has the largest share in reducing the critical Reynolds number relative to
${\ {Re}}_c^{\text {OB}}$
. Of course, the temperature dependence of the viscosity
$\mu$
alone dominates the reduction of the critical Reynolds number from
${\ {Re}}_c^{\text {OB}}$
. The change of the critical Reynolds number
$\Delta ^{(i)} {\ {Re}}_c := {\ {Re}}_c^{(i)} - {\ {Re}}_c^{\text {OB}}$
due to the temperature dependence of each individual thermophysical parameter, where
$i$
symbolizes
$i \in [\text {OB}+\rho, \, \text {OB}+\mu, \, \text {OB}+c_p^{\text {adv}},\, \text {OB}+\mu ]$
(including advection in all cases), is almost additive: For the case considered, the sum of the Reynolds number reductions amounts to
$\sum _i \Delta ^{(i)} {\ {Re}}_c = - 473$
with the deviation
${\ {Re}}_c^{\text {FTD, adv}} - {\ {Re}}_c^{\text {OB}} = 1448 - 1835 = - 387$
. An updated version of the code MaranStable which includes the effect of
$c_p$
advection is available under https://github.com/fromano88/MaranStable. The driver file for the old version is ‘main_v3d1.m’, while the one for the new version is ‘main_v3d2.m’.
Critical temperature difference
$\Delta T_c$
and critical Reynolds number
${\ {Re}}_c = \gamma \bar {\rho }_{\text {L}} \Delta T_c d/\bar {\mu }_{\text {L}}^2$
for a slender liquid bridge with
$\Gamma =0.66$
and
$\mathcal {V}=0.9$
made of 2-cSt silicone oil as in table 3 of [Reference Bird, Stewart and Lightfoot1]. The superscript ’adv’ indicates results when the advection of
$c_p$
is included in the governing equations. For all models, the critical wave number is
$m_c=3$
. The relative deviations
$\epsilon _c^{\text {FTD}} := ({\ {Re}}_c - {\ {Re}}_c^{\text {FTD}}) / {\ {Re}}_c^{\text {FTD}}$
and
$\epsilon _c^{\text {OB}} := ({\ {Re}}_c - {\ {Re}}_c^{\text {OB}}) / {\ {Re}}_c^{\text {OB}}$
are given in percent. For the definition of
$\Delta ^{(i)}{\ {Re}}_c$
, please see the text
3. Kinetic energy
Equation (B25) in [Reference Bird, Stewart and Lightfoot1] contains a sign error. The corrected equation (B25) reads
\begin{align*} \int _{V_i}\texttt { K7a} \,\text {d} V &= \int _{V_i} \mu _0 u_l\partial _m \partial _m u_l\,\text {d} V\\ &=\underbrace {\alpha _i\int _{A_{\text {fs}}} \mu _0 u_l n_m\partial _m u_l\,\text {d} S}_{:=M} - \int _{V_i} \mu _0 (\partial _m u_l)^2\,\text {d} V - \int _{V_i} u_l (\partial _m\mu _0) (\partial _m u_l)\,\text {d} V\\ &= - \int _{V_i} \mu _0 (\partial _m u_l)^2\,\text {d} V + M - \int _{V_i} u_l (\partial _m u_l)(\partial _m\mu _0)\,\text {d} V \\ &= - \int _{V_i} \mu _0 (\partial _m u_l)^2\,\text {d} V+ M_r+M_\varphi +M_z \\ &\quad -\alpha _i\int _{A_{\text {fs}}} \mu _0(h_0 w^2h_{0zz} -v^2)\,\text {d} \varphi \,\text {d} z - \int _{V_i} \mu^{\prime}_0{{{{\ u}}}}\cdot (\nabla {{{{\ u}}}})^{\text {T}}\cdot \nabla T_0\,\text {d} V. \end{align*}
As a consequence (B26) must read
\begin{equation*} \int _{V_i} \texttt { K7a}\,\text {d} V = -D_{\text {kin}}+M_r+M_\varphi +M_z - \frac {1}{2}\int _{V_i} \mu^{\prime}_0\cdot (\nabla {{{{\ u}}}}^2)\cdot \nabla T_0\,\text {d} V, \end{equation*}
and equation (5.8i) needs to be updated to
\begin{align*} \Lambda _\mu &=\int _{V_i} \mu^{\prime}_0{{{{\ u}}}}\cdot \nabla {{{{\ u}}}}\cdot \nabla T_0\,\text {d} V+\int _{V_i} (\mu^{\prime}_0 + \mu^{\prime\prime} _0 T_0 ) {{{{\ u}}}}\cdot [\mathcal {S}_0+(\nabla {{{{\ u}}}}_0)^{\text {T}}]\cdot \nabla T\,\text {d} V\\ &- \int _{V_i} \mu^{\prime}_0 T(\nabla {{{{\ u}}}}_0):(\nabla {{{{\ u}}}})\,\text {d} V+\alpha _i\int _{A_{\text {fs}}} \mu^{\prime}_0 w T\left (N^2 \partial _r w_0 - N^2 h_{0z} \partial _z w_0 - h_{0z}^2 h_{0zz} w_0\right )\,\text {d} \varphi \,\text {d} z. \end{align*}
We apologize for the errors made.
4. Related publications
Publications [Reference Stojanović, Romanò and Kuhlmann2] and [Reference Stojanović, Romanò and Kuhlmann4] also considered the stability of the thermocapillary flow in liquid bridges when the specific heat
$c_p(T)$
is temperature dependent. Similar as in table 1 above, we provide in table 2 the equivalent of table 5 of [Reference Stojanović, Romanò and Kuhlmann4], here supplemented by the result when the advection of
$c_p$
is taken into account (superscript ’adv’). Note that the deviations specified in the last column are taken relative to the full-temperature-dependence model (FTD) without
$c_p$
advection, as considered in [Reference Stojanović, Romanò and Kuhlmann4].
Table 2 shows that the inclusion of
$c_p$
advection reduces the critical Reynolds number of the FTD model by 1.2%. In contrast, the linearization of the temperature dependence of all material properties yields a 6.4% reduction, mainly caused by the insufficient representation of
$\mu (\hat T)$
by a linear function of
$\hat T$
. The percentage change of the critical Reynolds number due to
$c_p$
advection is further corroborated by considering the critical data for the same liquid near the global maximum of
${\ {Re}}_c$
from figures 5 (green square) and 9 (black square) of [Reference Stojanović, Romanò and Kuhlmann4]. Table 3 shows the corresponding comparisons. In both cases considered the advection of
$c_p$
slightly decreases the critical Reynolds number by 1.9 % (case A) and 1.1 % (case B) which is compatible with the trend seen in table 2. In contrast, the critical frequency is slightly increased.
Critical Reynolds numbers
${\ {Re}}_c$
and critical temperature differences
$\Delta T_c$
for a liquid bridge volume ratio
$\mathcal {V}=0.88$
and different approximations of the transport equations. Shown are the results of [Reference Stojanović, Romanò and Kuhlmann4] for the FTD, LTD and OB models (all exclusive of
$c_p$
advection) in comparison with the present results for the FDT
$^{\text {adv}}$
and LTD
$^{\text {adv}}$
models (both including the effect of
$c_p$
advection). The relative deviation
$\epsilon _c = ({\ {Re}}_c - {\ {Re}}_c^{\text {FTD}}) / {\ {Re}}_c^{\text {FTD}}$
is given in percent. All other parameters are identical to those for table 5 of [Reference Stojanović, Romanò and Kuhlmann4]: Shin-Etsu silicone oil with
$\nu (\hat T = 25^\circ \text {C})=2$
cSt,
$\Gamma =0.66$
,
${\mathcal{V}} = 1$
,
$\text {Bd}=0.363$
Critical Reynolds numbers
${\ {Re}}_c$
near their extrema for a liquid bridge from 2 cSt silicone oil and selected cases of [Reference Stojanović, Romanò and Kuhlmann4]. Case A:
$\Gamma =0.93$
,
${\mathcal{V}}=1$
,
$\text {Bd}=0.721$
,
${\ {Re}}_g=0$
. Case B:
$\Gamma =0.66$
,
${\mathcal{V}}=1$
,
$\text {Bd}=0.363$
,
${\ {Re}}_g=-500$
. Values are given for the full-temperature-dependence model without (FTD) and with inclusion of
$c_p$
advection (FTD,adv). The relative deviation
$\epsilon _c = ({\ {Re}}_c^{\text {FTD,adv}} - {\ {Re}}_c^{\text {FTD}}) / {\ {Re}}_c^{\text {FTD}}$
is given in percent; correspondingly for the critical frequency
$\omega _c$
We have demonstrated that the advection of
$c_p$
affects the critical data presented in [Reference Stojanović, Romanò and Kuhlmann2, Reference Stojanović, Romanò and Kuhlmann4] by reducing
${\ {Re}}_c$
by 1 to 2 percent. This is much less than the effect a temperature dependent viscosity of 2 cSt silicone oil has on the critical Reynolds number. Nevertheless, the
$c_p$
advection influences the critical data about as much as the inclusion or neglect of the temperature dependence of the liquid’s density, its thermal conductivity or of
$c_p$
itself has.
In particular, the stability curves obtained by including
$\rho \hat {T} \,\text {D}_t c_p$
in the governing equations are within a 2% tolerance level from the critical Reynolds numbers reported in [Reference Stojanović, Romanò and Kuhlmann2, Reference Stojanović, Romanò and Kuhlmann4] for
$\Delta T \leqslant 56$
K (see table 3). All conclusions drawn and discussions reported in [Reference Stojanović, Romanò and Kuhlmann3, Reference Stojanović, Romanò and Kuhlmann4] still hold true when the
$c_p$
advection term
$\rho \hat {T} \,\text {D}_t c_p$
is consistently included in the energy equation, except for small quantitative deviations discussed herein.
Open access
