# Effect of Low Rotation Rate on Steady Convection During the Solidification of a Ternary Alloy

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## Abstract

We consider the problem of steady convective flow during the directional solidification of a horizontal ternary alloy system rotating at a constant and low rate about a vertical axis. Under the limit of large far-field temperature, the flow region is modeled to be composed of two horizontal mushy layers, which are referred to here as a primary layer over a secondary layer. We first determine the basic state solution and then carry out linear stability analysis to calculate the neutral stability boundary and the critical conditions at the onset of motion. We find, in particular, that there are two flow solutions and each solution exhibits two neutral stability boundaries, and the flow can be multi-modal in the low rotating rate case with local minima on each neutral boundary. The critical Rayleigh number and the wave number as well as the vertical volume flux increase with the rotation rate, but the flow is found to be less stabilizing as compared to the binary alloy counterpart flow. The effects of low rotation rate increase the solid fraction and the liquid fraction at certain vertically oriented fluid lines, and the highest value of such increase is at a horizontal level close to the interface between the two mushy layers.

## Keywords

Convection Rotating flow Ternary solidification Convective flow## References

- Aitta, A., Huppert, H.E., Worster, M.G.: Diffusion-controlled solidification of a ternary melt from a cooled boundary. J. Fluid Mech.
**432**, 201–217 (2001a)Google Scholar - Aitta, A., Huppert, H.E., Worster, M.G.: Solidification in ternary systems. In: Ehrhard, P., Riley, D.S., Steen, P.H. (eds.) Interactive Dynamics of Convection and Solidification, pp. 113–122. Kluwer, Alphen aan den Rijn (2001b)CrossRefGoogle Scholar
- Amberg, G., Homsy, G.M.: Nonlinear analysis of buoyant convection in binary solidification with application to channel formation. J. Fluid Mech.
**252**, 79–98 (1993)CrossRefGoogle Scholar - Anderson, D.M.: A model for diffusion-controlled solidification of ternary alloys in mushy layers. J. Fluid Mech.
**483**, 165–197 (2003)CrossRefGoogle Scholar - Anderson, D.M., Worster, M.G.: Weakly nonlinear analysis of convection in mushy layers during the solidification of binary alloys. J. Fluid Mech.
**302**, 307–331 (1995)CrossRefGoogle Scholar - Anderson, D.M., Schulze, T.P.: Linear and nonlinear convection in solidifying ternary alloys. J. Fluid Mech.
**545**, 213–243 (2005)CrossRefGoogle Scholar - Anderson, D.M., Mcfadden, G.B., Coriell, S.R., Murray, B.T.: Convective instabilities during the solidification of an ideal ternary alloy in a mushy layer. J. Fluid Mech.
**647**, 309–333 (2010)CrossRefGoogle Scholar - Bloomfield, L.J., Huppert, H.E.: Solidification and convection of a ternary solution cooled from the side. J. Fluid Mech.
**489**, 269–299 (2003)CrossRefGoogle Scholar - Busse, F.H.: Nonlinear properties of thermal convection. Rep. Prog. Phys.
**41**, 1929–1967 (1978)CrossRefGoogle Scholar - Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Clarendon, Oxford (1961)Google Scholar
- Chung, C.A., Chen, F.: Onset of plume convection in mushy layers. J. Fluid Mech.
**408**, 53–82 (2000)CrossRefGoogle Scholar - Guba, P.: On the finite-amplitude steady convection in rotating mushy layers. J. Fluid Mech.
**437**, 337–365 (2001)CrossRefGoogle Scholar - Riahi, D.N.: Nonlinear steady convection in rotating mushy layers. J. Fluid Mech.
**485**, 279–306 (2003)CrossRefGoogle Scholar - Riahi, D.N.: On three-dimensional nonlinear buoyant convection in ternary solidification. Transp. Porous Media
**103**, 249–277 (2014)CrossRefGoogle Scholar - Roper, S.M., Davis, S.H., Voorhees, P.W.: An analysis of convection in a mushy layer with a deformable permeable interface. J. Fluid Mech.
**596**, 335–352 (2008)CrossRefGoogle Scholar - Sample, A.K., Hellawell, A.: The mechanisms of formation and prevention of channel segregation during alloy solidification. Metall. Trans. A
**15**, 2163–2173 (1984)CrossRefGoogle Scholar - Skarda, J.R.L., McCaughan, F.E.: Exact solution to stationary onset of convection due to surface tension variation in multi-component fluid flow with interfacial deformation. Int. J. Heat Mass Transf.
**42**, 2387–2398 (1999)CrossRefGoogle Scholar - Tait, S., Jahrling, K., Jaupart, C.: The planform of compositional convection and chimney formation in a mushy layer. Nature
**359**, 406–408 (1992)CrossRefGoogle Scholar - Thompson, A.F., Huppert, H.E., Worster, M.G.: A global conservation model for diffusion-controlled solidification of a ternary alloy. J. Fluid Mech.
**483**, 191–197 (2003a)Google Scholar - Thompson, A.F., Huppert, H.E., Worster, M.G.: Solidification and compositional convection of a ternary alloy. J. Fluid Mech.
**497**, 167–199 (2003b)CrossRefGoogle Scholar - Veronis, G.: Cellular convection with finite amplitude in a rotating fluid. J. Fluid Mech.
**5**, 401–435 (1959)CrossRefGoogle Scholar - Worster, M.G.: Instabilities of the liquid and mushy regions during solidification of alloys. J. Fluid Mech.
**237**, 649–669 (1992)CrossRefGoogle Scholar