A Mathematical Study of Flow Dynamics and Stability of Bounded Cartesian Plumes

This paper examines the mathematical model governing the behavior of a buoyant fluid column rising through a less buoyant ambient fluid within a confined region bounded by two vertical walls. The study represents a bounded-domain extension of the compositional plume model previously developed by Eltayeb and Loper [9 – 11] for an unconfined environment. The system is characterized by five dimensionless parameters: (i) the Grashoff number, representing the ratio of buoyancy forces generated by concentration differences between the plume and the surrounding fluid to viscous forces; (ii) the Prandtl number, defined as the ratio of viscosity to thermal diffusivity; (iii) the plume thickness; (iv) the separation distance between the two vertical walls; and (v) the dimensionless distance between the plume and the nearest wall, with all length scales normalized using the salt-finger length scale. The principal aim of the study is to assess how the presence of boundaries modifies the solutions obtained for an unbounded ambient fluid. The analysis shows that the basic-state solution does not depend on either the Grashoff or Prandtl numbers. Unlike the unbounded case, where symmetry is preserved, the presence of sidewalls generally breaks this symmetry unless the plume is positioned exactly midway between the two boundaries. In an unbounded domain, the plume is always unstable and exhibits two independent modes of instability: the varicose (V) mode and the sinuous (S) mode. The introduction of sidewalls significantly changes the stability characteristics of the flow. A strongly unstable region emerges when a thin plume is located near a boundary. In other areas of the parameter space, the instability growth rate remains comparable to that of the unbounded configuration but decreases as the distance between the walls becomes smaller. Instability continues to appear in two distinct classes of solutions related to the classical varicose and sinuous modes, although these modes are modified by the plume’s position relative to the nearest sidewall.