Three-dimensional numerical simulation of Marangoni instabilities in non-cylindrical liquid bridges in microgravity

Lappa, M. and Savino, R. and Monti, R. (2001) Three-dimensional numerical simulation of Marangoni instabilities in non-cylindrical liquid bridges in microgravity. International Journal of Heat and Mass Transfer, 44 (10). pp. 1983-2003. ISSN 0017-9310 (

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Instability of Marangoni convection in non cylindrical (convex or concave) liquid bridges of low Prandtl number fluids is investigated by direct three-dimensional and time-dependent simulation of the problem. Body-fitted curvilinear coordinates are adopted; the non cylindrical original physical domain in the (r,z,fi) space is transformed into a cylindrical computational domain in a (csi,eta,fi) space. The geometry of the domain is transformed using a coordinate transformation method by surface fitting technique. The field equations are numerically solved explicitly in time and with a finite difference technique in a staggered grid. The numerical results are analyzed and interpreted in the general context of the bifurcation's theory. The computations show that for semiconductor melts the first bifurcation is characterized by the loss of spatial symmetry rather than by the onset of oscillatory flow and that it is hydrodynamic in nature. The flow field azimuthal organization related to the critical wave number, depends on the geometrical aspect ratio A=L/D of the liquid bridge and on the shape factor S (convex S>1, concave S<1) of the free surface. The critical azimuthal wave number increases when the geometrical aspect ratio of the bridge is decreased and, for a fixed aspect ratio, can be shifted to higher values by increasing the volume (convex bridges) or to lower values by decreasing the volume (concave bridges). This behaviour is explained on the basis of the relation between the typology of the azimuthal disturbances and the structure of the fluid-dynamic field.A generalized law is found to correlate the critical azimuthal wave number of the instability to the geometrical aspect ratio and to the shape factor.A second oscillatory (Hopf) bifurcation occurs when further increasing the Marangoni number. Experimental results available in literature on this second bifurcation are considered for comparison. The experimental and numerical results show good a agreement.