Simulations have been carried out adopting the ‘buoyantBoussinesqPimpleFoam’ transient solver available in OpenFOAM, properly complemented by the thermocapillary stress condition equation (2.4) (see e.g. [33 (link),37 (link)]). In the present case, obviously, the acceleration of gravity has been set to zero (microgravity conditions), which means that only the fluid-volume preserving abilities of the solver have been exploited (incompressible flow).
In OpenfFOAM, the governing equations for the fluid phase (i.e. equations 2.1–2.3 in our case) are discretized using the Finite Volume Method (FVM). Moreover, the solver relies on the PISO algorithm of Issa [41 (link)], where a collocated grid arrangement of the variables is used to integrate the momentum equations and enforce mass conservation. The energy equation (2.3) is subsequently solved in a segregated manner. Integration in time of both the thermo-flow field and the particles governing equation is based on the backward Euler scheme. Convective terms have been discretized using the second-order accurate, linear central-differences scheme.
All these kernels have already been used in the previous studies by Capobianchi and Lappa [32 (link),33 (link),37 (link)] to which the interested reader is referred for additional details about the numerical implementation. Here, we wish simply to recall that Capobianchi & Lappa [32 (link)] provided evidence for the reliability and accuracy of these kernels through focused comparison with the simulations by Melnikov et al. [13 (link)], and the independent numerical study by Lappa [22 (link)]. Excellent agreement was obtained in terms of fundamental properties of the supercritical Marangoni flow (the frequency of the hydrothermal wave for a liquid bridge with aspect ratio (height/diameter) A = 0.34 and Ma = 20 600) and the morphology of the emerging particle structures (for and St = 10−4).
As a concluding remark for this section, we wish to point out that all the numerical results presented in §3 have been obtained using a mesh having the M1 resolution defined in the earlier study by Capobianchi & Lappa [32 (link)] for the same value of the Prandtl number considered here (i.e. Pr = 8). Such a resolution was found to provide a good compromise between computational times and accuracy over an extended range of values of the Marangoni number (see Table III in [32 (link)]). The corresponding values taken by the ratio of the maximum particle diameter over the minimum computational-cell size for the conditions examined in §3 (particle Stokes number between and for A = 0.34 and A = 0.5, respectively) are less than 1 for all the considered circumstances, the only exception being the particle with diameter 80 µm in the A = 0.34 case (for which the particle-to-cell ratio slightly exceeds the unit value, which however we still consider acceptable given the one-way nature of the particle-fluid coupling implemented here).
In OpenfFOAM, the governing equations for the fluid phase (i.e. equations 2.1–2.3 in our case) are discretized using the Finite Volume Method (FVM). Moreover, the solver relies on the PISO algorithm of Issa [41 (link)], where a collocated grid arrangement of the variables is used to integrate the momentum equations and enforce mass conservation. The energy equation (2.3) is subsequently solved in a segregated manner. Integration in time of both the thermo-flow field and the particles governing equation is based on the backward Euler scheme. Convective terms have been discretized using the second-order accurate, linear central-differences scheme.
All these kernels have already been used in the previous studies by Capobianchi and Lappa [32 (link),33 (link),37 (link)] to which the interested reader is referred for additional details about the numerical implementation. Here, we wish simply to recall that Capobianchi & Lappa [32 (link)] provided evidence for the reliability and accuracy of these kernels through focused comparison with the simulations by Melnikov et al. [13 (link)], and the independent numerical study by Lappa [22 (link)]. Excellent agreement was obtained in terms of fundamental properties of the supercritical Marangoni flow (the frequency of the hydrothermal wave for a liquid bridge with aspect ratio (height/diameter) A = 0.34 and Ma = 20 600) and the morphology of the emerging particle structures (for and St = 10−4).
As a concluding remark for this section, we wish to point out that all the numerical results presented in §3 have been obtained using a mesh having the M1 resolution defined in the earlier study by Capobianchi & Lappa [32 (link)] for the same value of the Prandtl number considered here (i.e. Pr = 8). Such a resolution was found to provide a good compromise between computational times and accuracy over an extended range of values of the Marangoni number (see Table III in [32 (link)]). The corresponding values taken by the ratio of the maximum particle diameter over the minimum computational-cell size for the conditions examined in §3 (particle Stokes number between and for A = 0.34 and A = 0.5, respectively) are less than 1 for all the considered circumstances, the only exception being the particle with diameter 80 µm in the A = 0.34 case (for which the particle-to-cell ratio slightly exceeds the unit value, which however we still consider acceptable given the one-way nature of the particle-fluid coupling implemented here).
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