Navier-Stokes Equations for Liquid Bridges

The fluid problem is governed by the Navier–Stokes equations for an incompressible flow, in which gravity is neglected, and by the energy equation. In dimensionless form, these can be written as
$$\begin{array}{rl}\mathbf{\nabla }\cdot \mathit{u}& \hspace{0.17em}=0,\end{array}$$
$$\begin{array}{rl}\frac{D\mathit{u}}{Dt}& \hspace{0.17em}=-\mathbf{\nabla }p+{\mathbf{\nabla }}^2\mathit{u},\end{array}$$
$$\begin{array}{rl}\text{and}\frac{DT}{Dt}& \hspace{0.17em}=\frac{1}{Pr}{\mathbf{\nabla }}^{2}T,\end{array}$$
where $\mathit{u}$ is the flow velocity vector, $p$ is the pressure, $T$ is the temperature and L, ν/ L, ρν²/ L², L²/ν and ΔT have been used as reference quantities for the geometrical coordinates, velocity, pressure, time (t) and temperature, respectively (L being a characteristic length). The non-dimensional temperature is defined as $(T-T_{\text{ref}})/\mathrm{\Delta }T$ , where $T_{\text{ref}}$ is a suitable reference temperature (T_cold shown in figure 1 in our case). The operator $D(\cdot )/Dt$ is the usual material derivative, while $Pr=\nu /\alpha$ is the Prandtl number, ratio between the fluid kinematic viscosity, $\nu$ , and the thermal diffusivity, $\alpha$ . This parameter is left unvaried throughout the whole study (Pr = 8, corresponding to NaNO₃). Moreover, in line with the majority of existing efforts on the study of PAS, the physical properties of the fluid are assumed to be constant. Closure of the problem, however, requires the addition of the tangential stress conditions at the interface:
$$[\mathbf{\nabla }\mathit{u}+{(\mathbf{\nabla }\mathit{u})}^{\text{T}}]\cdot \mathit{n}+\text{Re}{\mathbf{\nabla }}_{\text{s}}T=\mathbf{0},$$
where $\mathit{n}$ is the unit vector perpendicular to (pointing outward) to the interface, while ${\mathbf{\nabla }}_{\text{s}}T$ is the projection of the temperature gradient on the surface separating the liquid bridge and the surrounding environment. It should be noted that the latter is assumed to be a gas characterized by a viscosity much smaller than that of the liquid bridge.
Figure 1.

Schematic representation of the liquid bridge showing the temperature gradient and its projection along the interface (these two are coincident in this case as the interface is straight), and the unit vector perpendicular to the interface. (Online version in colour.)

Accordingly, its contribution is disregarded in the tangential stress conditions and in equations (2.1–2.3), thereby allowing the description of the multiphase interfacial flow in the framework of a single-fluid approach. Moreover, heat exchange with the gas is also neglected and the free interface is modelled as a perfect cylinder. The former assumption is generally considered valid when the two supporting discs are heated and cooled ‘symmetrically’ with respect to the ambient temperature, i.e. their temperatures are T_a+ ΔT/2 and T_a − ΔT/2, respectively, where T_a is the ambient temperature and ΔT is the overall temperature difference applied to the liquid bridge. In such circumstances, the temperature of the free surface (it is almost uniform with the exception of the changes that occur in proximity to the discs) is almost identical to that of the gas ambient, thereby minimizing the interfacial heat exchange. With regard to the latter hypothesis, the static curvature of the free interface can indeed be neglected as the liquid bridges are considered in microgravity conditions, their volume is identical to that of the corresponding cylinders having the same base and height and the wetting angle of the considered fluid is close to 90° (assuming the supporting discs to be coated with graphite, [8 (link),38 (link)]). Dynamic shape deformations are also ignored by considering that the so-called Capillary number, defined as $\text{Ca}={\sigma }_{\text{T}}\mathrm{\Delta }T/{\sigma }_{\text{o}}$ , is much smaller than 1 for the conditions considered in the present work ( ${\sigma }_{\text{o}}$ being the reference surface tension evaluated at T_cold, i.e. ${\sigma }_{\text{o}}\cong 1.15\times {10}^{-1}\text{N}{\text{m}}^{-1}$ ; ${\sigma }_{\text{T}}\cong 7\times {10}^{-5}\text{N}{\text{K}}^{-1}{\text{m}}^{-1}$ → $\text{Ca}\cong 6\times {10}^{-4}\hspace{0.17em}\mathrm{\Delta }T$ ). The dimensionless parameter $\text{Re}={\sigma }_T\mathrm{\Delta }TL/\rho {\nu }^2$ appearing in equation (2.4) is the Reynolds number based on the characteristic thermocapillary velocity $U_{\text{T}}={\sigma }_{\text{T}}\mathrm{\Delta }T/\rho \nu$ , where ${\sigma }_T=-\mathrm{\partial }\sigma /\mathrm{\partial }T$ is the derivative of the surface tension $\sigma$ with respect to the temperature, $\mathrm{\Delta }T$ is the aforementioned temperature difference between the two cylindrical rods supporting the liquid bridge, and L is the distance between them. It is usual practice to refer to the Marangoni number $\text{Ma}=\text{Re}Pr$ rather than to the Reynolds number to characterize the flow field, therefore, in the following this parameter is used instead of $\text{Re}$ to describe the results.
The additional thermal and kinematic boundary conditions for the fluid phase schematically shown in figure 1 can be turned into precise mathematical relationships as follows:
On the two supporting discs:
$$\begin{array}{rl}\hspace{0.17em}& \text{Cold disc }(y=0):\hspace{0.17em}T=0,\hspace{0.17em}\hspace{0.17em}\mathit{u}=\mathbf{0}.\end{array}$$
$$\begin{array}{rl}\hspace{0.17em}& \text{Hot disc }(y=1):T=1,\hspace{0.17em}\hspace{0.17em}\mathit{u}=\mathbf{0}.\end{array}$$
At the free surface:
$$\begin{array}{rl}\hspace{0.17em}& \mathrm{\partial }T/\mathrm{\partial }n=0\hspace{0.17em}\hspace{0.17em}(\text{adiabatic}\hspace{0.17em}\text{behaviour}).\end{array}$$
$$\begin{array}{rl}\hspace{0.17em}& \mathit{u}\cdot \mathit{n}=0\hspace{0.17em}\hspace{0.17em}\text{( no}\hspace{0.17em}\text{radial}\hspace{0.17em}\text{velocity) }.\end{array}$$