A Mathematical Study of Flow Dynamics and Stability of Bounded Cartesian Plumes
- 1Department of Basic and Applied Science, College of Applied and Health Sciences, A’Sharqiyah University, Ibra 400, Oman
Abstract
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 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, particularly in the high-Prandtl-number limit where thermal diffusion is drastically weaker than viscous diffusion. 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. This effect is heavily influenced by large Prandtl numbers, which narrow the thermal boundary layer and modify the spatial distribution of temperature perturbations. 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 both the high Prandtl number constraints and the plume’s position relative to the nearest sidewall.
Related
- Wavenumber-explicit stability of three-dimensional Cartesian UPMLs with piecewise-constant profiles for Helmholtz and Maxwell equations
- A new quantile regression model for bounded responses: mathematical derivation, estimation, and fitting bounded datasets
- Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below
- Stability Analysis of Bounded Solutions to 2-D Incompressible Navier-Stokes Equations in L^2_uloc Spaces
- Global Existence and Stability of 3D Stochastic NSEs in Bounded and Unbounded Domains Driven by a Special Multiplicative Wiener Process
BibTeX
@article{almashrafi2026mathematical,
title = {{A Mathematical Study of Flow Dynamics and Stability of Bounded Cartesian Plumes}},
author = {Khaled S. AlMashrafi and Ahmed A. Al Kasbi and Jamal Salah},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14193590},
url = {https://www.mdpi.com/2227-7390/14/19/3590},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A Mathematical Study of Flow Dynamics and Stability of Bounded Cartesian Plumes
AU - Khaled S. AlMashrafi
AU - Ahmed A. Al Kasbi
AU - Jamal Salah
PY - 2026
DA - 2026/10/02
JO - Mathematics
DO - 10.3390/math14193590
UR - https://www.mdpi.com/2227-7390/14/19/3590
AB - 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 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, particularly in the high-Prandtl-number limit where thermal diffusion is drastically weaker than viscous diffusion. 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. This effect is heavily influenced by large Prandtl numbers, which narrow the thermal boundary layer and modify the spatial distribution of temperature perturbations. 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 both the high Prandtl number constraints and the plume’s position relative to the nearest sidewall.
SN - 2227-7390
ER -