The linear stability of a compositional plume in the presence of material diffusion

One of the problems the iron casting industry faces is the appearance of freckles in iron bars leading to their weakness. When iron ore is poured into ingots the trapped air escapes in the form of thin air channels which when the iron bar solidifies appear as very thin black strips along the iron bar. These air pockets lead to a weakness in the iron bar. Experiments of directional solidification using fluid alloys chilled from below showed that the formation of these channels is a result of instability of a mushy layer that forms at the bottom of the fluid alloy. These channels are now termed compositional plumes and are found to occur in a variety of situations in geophysics and the environment in addition to industry. It is then of importance to study the dynamics of these plumes. Recent work has shown that a single plume is unstable indicating that it can break up into blobs or widen and become bigger.

The studies on the dynamics of compositional plumes use the equations of conservation of mass, and conservation of momentum in the presence of Boussinesq approximation together with equations of heat, concentration of light material, and state. In particular, we examine the stability of a fully developed plume rising in a fluid of larger dimensions of different material composition. The dimensionless width of the plume, , the Grashoff (Reynold) number ( = , where and are units of velocity and length , respectively , and is the kinematic viscosity ), the Prandtl number, ( , where is the thermal diffusivity) and the Schmidt number, (= , where is the material diffusion) govern the stability of the plume. The stability is investigated in the parameter space ( ) for small values of the Grashoff (Reynold) number. The work extends previous stability analyses to include the material diffusion. It is shown that the presence of material diffusion has a stabilizing influence except in a small range of the Schmidt number. However, overall effect of material diffusion is tends to stabilise the plume with the consequences that the growth rate is reduced but the plume is remains unstable. Moreover, it was found that the influence of material diffusion is independent of the type of the mode considered. The results are presented in graphical form in the parameter space and their significance is discussed.