# A Mathematical and Numerical Framework for the Simulation of Oscillatory Buoyancy and Marangoni Convection in Rectangular Cavities with Variable Cross Section

Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 50)

## Abstract

It is often assumed that two-dimensional flow can be used to model with an acceptable degree of approximation the preferred mode of instability of thermogravitational flows and thermocapillary flows in laterally heated shallow cavities for a relatively wide range of substances and conditions (essentially pure or compound semiconductor materials in liquid state for the case of buoyancy convection and molten oxide materials or salts and a variety of organic liquids for the case of Marangoni convection). In line with the general spirit of this book, such assumption is challenged by comparing two-dimensional and three-dimensional results expressly produced for such a purpose. More precisely, we present a general mathematical and numerical framework specifically developed to (1) explore the sensitivity of such phenomena to geometrical “irregularities” affecting the liquid container and (2) take advantage of a reduced number of spatial degrees of freedom when this is possible. Sudden variations in the shape of the container are modelled as a single backward-facing or forward-facing step on the bottom wall or a combination of both features. The resulting framework is applied to a horizontally extended configuration with undeformable free top liquid-gas surface (representative of the Bridgman crystal growth technique) and for two specific fluids pertaining to the above-mentioned categories of materials, namely molten silicon (Pr  <  1) and silicone oil (Pr  >  1. The assumption of flat interface is justified on the basis of physical reasoning and a scaling analysis. The overall model proves successful in providing useful insights into the stability behaviour of these fluids and the departure from the approximation of two-dimensional flow. It is shown that the presence of a topography in the bottom wall can lead to a variety of situations with significant changes in the emerging waveforms.

## Keywords

Buoyancy convection Marangoni convection Flow instability Numerical framework

## References

1. 1.
Borcia, R., Bestehorn, M.: Phase-field model for Marangoni convection in liquid-gas systems with a deformable interface. Phys. Rev. E 67(6), 066307 (2003)
2. 2.
Bucchignani, E.: Numerical characterization of hydrothermal waves in a laterally heated shallow layer. Phys. Fluids 16(11), 3839–3849 (2004)
3. 3.
Burguete, J., Mukolobwiez, N., Daviaud, N., Garnier, N., Chiffaudel, A.: Buoyant-thermocapillary instabilities in extended liquid layers subjected to a horizontal temperature gradient. Phys. Fluids 13(10), 2773–2787 (2001)
4. 4.
Chorin, A.J.: Numerical solutions of the Navier-Stokes equations. Math. Comput. 22, 745–762 (1968)
5. 5.
Dupret, F., Van der Bogaert, N.: Modelling Bridgman and Czochralski growth. In: Hurle, D.T.J. (ed.) Handbook of Crystal Growth, vol. 2, pp. 877-1010. North-Holland, Amsterdam (1994)Google Scholar
6. 6.
Gelfgat, AYu., Bar-Yoseph, P.Z., Yarin, A.L.: On oscillatory instability of convective flows at low Prandtl number. J. Fluids Eng. 119, 823–830 (1997)
7. 7.
Gelfgat, Ayu, Bar-Yoseph, P.Z., Yarin, A.L.: Stability of multiple steady states of convection in laterally heated cavities. J. Fluid Mech. 388, 315–334 (1999)
8. 8.
Gelfgat, AYu.: Time-dependent modeling of oscillatory instability of three-dimensional natural convection of air in a laterally heated cubic box. Theor. Comput. Fluid Dyn. 31, 447–469 (2017)
9. 9.
Gill, A.E.: A theory of thermal oscillations in liquid metals. J. Fluid Mech. 64(3), 577–588 (1974)
10. 10.
Golovin, A.A., Nepomnyashchy, A.A., Pismen, L.M.: Nonlinear evolution and secondary instabilities of Marangoni convection in a liquid–gas system with deformable interface. J. Fluid Mech. 341, 317–341 (1997)
11. 11.
Gresho, P.M., Sani, R.T.: On pressure boundary conditions for the incompressible Navier-Stokes equations. Int. J. Num. Methods Fluids 7, 1111–1145 (1987)
12. 12.
Gresho, P.M.: Incompressible fluid dynamics: some fundamental formulation issues. Ann. Rev Fluid Mech. 23, 413–453 (1991)
13. 13.
Guermond, J.-L., Quartapelle, L.: On stability and convergence of projection methods based on pressure Poisson equation. Int. J. Numer. Meth. Fluids 26, 1039–1053 (1998)
14. 14.
Guermond, J.-L., Minev, P., Shen, J.: An Overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195, 6011–6045 (2006)
15. 15.
Harlow, F.H., Welch, J.E.: Numerical calculation of time-dependent viscous incompressible flow with free surface. Phys. Fluids 8, 2182–2189 (1965)
16. 16.
Hart, J.E.: Stability of thin non-rotating Hadley circulations. J. Atmos. Sci. 29, 687–697 (1972)
17. 17.
Hart, J.E.: A note on the stability of low-Prandtl-number Hadley circulations. J. Fluid Mech. 132, 271–281 (1983)
18. 18.
Kasperski, G., Labrosse, G.: On the numerical treatment of viscous singularities in wall-confined thermocapillary convection. Phys. Fluids 12(11), 2695–2697 (2000)
19. 19.
Kuo, H.P., Korpela, S.A.: Stability and finite amplitude natural convection in a shallow cavity with insulated top and bottom and heated from the side. Phys. Fluids 31, 33–42 (1988)
20. 20.
Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. Gordon and Breach, New York, London (1969)
21. 21.
Lappa, M., Savino, R.: Parallel solution of the three-dimensional Marangoni flow instabilities in liquid bridges. Int. J. Num. Meth. Fluids 31, 911–925 (1999)
22. 22.
Lappa, M.: Thermal convection and related instabilities in models of crystal growth from the melt on earth and in microgravity: past history and current status. Cryst. Res. Technol. 40(6), 531–549 (2005)
23. 23.
Lappa, M.: Assessment of the role of axial vorticity in the formation of Particle Accumulation Structures (PAS) in supercritical Marangoni and hybrid thermocapillary-rotation-driven flows. Phys. Fluids 25(1), 012101 (2013). 11 pages
24. 24.
Lappa, M.: Stationary solid particle attractors in standing waves. Phys. Fluids 26(1), 013305 (2014). 12 pages
25. 25.
Lappa, M.: On the onset of multi-wave patterns in laterally heated floating zones for slightly supercritical conditions. Phys. Fluids 28(12), 124105 (2016). 22 pages
26. 26.
Lappa, M.: Patterning behaviour of gravitationally modulated supercritical Marangoni flow in liquid layers. Phys. Rev. E 93(5), 053107 (2016). 13 pages
27. 27.
Lappa, M.: On the oscillatory hydrodynamic modes in liquid metal layers with an obstruction located on the bottom. Int. J. Thermal Science 118, 303–319 (2017)
28. 28.
Lappa, M.: Hydrothermal waves in two-dimensional liquid layers with sudden changes in the available cross-section. Int. J. Num. Meth. Heat Fluid Flow 27(11), 2629–2649 (2017)
29. 29.
Lappa, M., Savino, R., Monti, R.: Influence of buoyancy forces on Marangoni flow instabilities in liquid bridges. Int. J. Num. Meth. Heat Fluid Flow 10(7), 721–749 (2000)
30. 30.
Lappa, M.: Thermal Convection: Patterns, Evolution and Stability. Wiley, Chichester, England (2009)
31. 31.
Lappa, M., Yasushiro, S., Imaishi, N.: 3D numerical simulation of on ground Marangoni flow instabilities in liquid bridges of low Prandtl number fluid. Int. J. Num. Meth. Heat Fluid Flow 13(3), 309–340 (2003)
32. 32.
Laure, P., Roux, B.: Linear and non linear study of the Hadley circulation in the case of infinite cavity. J. Cryst. Growth 97(1), 226–234 (1989)
33. 33.
Li, Y.R., Peng, L., Akiyama, Y., Imaishi, N.: Three-dimensional numerical simulation of thermocapillary flow of moderate Prandtl number fluid in an annular pool. J. Cryst. Growth 259, 374–387 (2003)
34. 34.
Li, Y.R., Peng, L., Shi, W.Y., Imaishi, N.: Convective instabilities in annular pools. Fluid Dyn. Mater. Process. 2(3), 153–166 (2006)
35. 35.
Melnikov, D.E., Shevtsova, V.M., Legros, J.C.: Route to aperiodicity followed by high Prandtl-number liquid bridge. 1-g case. Acta Astronaut. 56(6), 601–611 (2005)
36. 36.
Melnikov, D.E., Shevtsova, V.M., Legros, J.C.: Onset of temporal aperiodicity in high Prandtl number liquid bridge under terrestrial conditions. Phys. Fluids 16(5), 1746–1757 (2004)
37. 37.
Monberg E.: Bridgman and related growth techniques. In: Hurle, D.T.J. (ed.) Handbook of Crystal Growth, vol. 2, pp. 53–97. North-Holland, Amsterdam (1994)Google Scholar
38. 38.
Okada, K., Ozoe, H.: The Effect of Aspect ratio on The critical Grashof number for oscillatory natural convection of zero Prandtl number fluid: numerical approach. J. Cryst. Growth 126, 330–334 (1993)
39. 39.
Peltier, L., Biringen, S.: Time-dependent thermocapillary convection in a rectangular cavity: numerical results for a moderate Prandtl number fluid. J. Fluid Mech. 257, 339–357 (1993)
40. 40.
Priede, J., Gerbeth, G.: Influence of thermal boundary conditions on the stability of thermocapillary-driven convection at low Prandtl numbers. Phys. Fluids 9, 1621–1634 (1997)
41. 41.
Quartapelle, L.: Numerical Solution of the Incompressible Navier-Stokes Equations, International Series of Numerical Mathematics, vol. 113. Birkäuser (1993)
42. 42.
Schwabe, D., Moller, U., Schneider, J., Scharmann, A.: Instabilities of shallow dynamic thermocapillary liquid layers. Phys. Fluids A 4, 2368–2381 (1992)
43. 43.
Schwabe, D., Zebib, A., Sim, B.C.: Oscillatory thermocapillary convection in open cylindrical annuli.Part 1.Experiments under microgravity. J. Fluid Mech. 491, 239–258 (2003)
44. 44.
Shevtsova, V.M., Nepomnyashchy, A.A., Legros, J.C.: Thermocapillary-buoyancy convection in a shallow cavity heated from the side. Phys. Rev. E 67, 066308 (2003). 14 pages
45. 45.
Shevtsova, V.M., et al.: Onset of hydrothermal instability in liquid bridge. Experimental benchmark. Fluid Dyn. Mater. Process 7(1), 1–28 (2011)
46. 46.
Shi, W.Y., Imaishi, N.: Hydrothermal waves in differentially heated shallow annular pools of silicone oil. J. Cryst. Growth 290, 280–291 (2006)
47. 47.
Smith, M.K.: Instability mechanism in dynamic thermocapillary liquid layers. Phys. Fluids 29(10), 3182–3186 (1986)
48. 48.
Smith, M.K., Davis, S.H.: Instabilities of dynamic thermocapillary liquid layers. Part 1: convective instabilities. J. Fluid Mech. 132, 119–144 (1983)
49. 49.
Squire, H.B.: On the stability of three-dimensional disturbances of viscous flow between parallel walls. Proc. R. Soc. Lond. Ser. A 142, 621–628 (1933)
50. 50.
Tang, Z.M., Hu, W.R.: Hydrothermal wave in a shallow liquid layer. Microgravity Sci. Tech. 16(1), 253–258 (2005)
51. 51.
Temam, R.: Uneméthoded’approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France 98, 115–152 (1968)
52. 52.
Tollmien, W.: General instability criterion of laminar velocity distributions. Tech. Memor. Nat. Adv. Comm. Aero., Wash. No. 792 (1936)Google Scholar
53. 53.
Ueno, I., Tanaka, S., Kawamura, H.: Oscillatory and chaotic thermocapillary convection in a half-zone liquid bridge. Phys. Fluids 15(2), 408–416 (2003)
54. 54.
Wang, T.M., Korpela, S.A.: Longitudinal rolls in a shallow cavity heated from a side. Phys. Fluids A 32, 947–953 (1989)
55. 55.
Xu, J., Zebib, A.: Oscillatory two- and three-dimensional thermocapillary convection. J. Fluid Mech. 364, 187–209 (1998)