Dissolution-driven transport in a rotating horizontal cylinder

TL;DR

The paper analyzes dissolution-driven transport in a rotating horizontal cylinder by coupling a moving-boundary Stefan problem to 2D incompressible flow with buoyancy under the Oberbeck–Boussinesq approximation. It develops a boundary-fitted numerical framework with an ADI scheme and Gibbs–Thomson interface regularization to simulate the evolving solute–fluid interface and quantify dissolution time, mixing, and interface morphology across a wide range of $Ra$ and $Ω$. Key findings show that rotation and buoyancy nonlinearly affect dissolution and mixing: rotation can slow dissolution by reducing the interfacial concentration gradient, but may enhance mixing and lead to near-axisymmetric interfaces when rotation dominates; a modified Rayleigh number $Ra_Ω=Ra/Ω^2$ governs the shape regime with circular interfaces for $\sqrt{Ra_Ω}\lesssim 250$. The results reveal an optimal combination of $Ra$ and $Ω$ for efficient mixing and illustrate how symmetry breaking and interface geometry evolve with the balance of buoyancy and rotation, offering insights for applications in drug release, chemical processing, and environmental dispersion.

Abstract

We study the combined effects of natural convection and rotation on the dissolution of a solute in a solvent-filled circular cylinder. The density of the fluid increases with the increasing concentration of the dissolved solute, and we model this using the Oberbeck-Boussinesq approximation. The underlying moving-boundary problem has been modelled by combining the Navier-Stokes equations with the advection-diffusion equation and a Stefan condition for the evolving solute-fluid interface. We use highly resolved numerical simulations to investigate the flow regimes, dissolution rates, and mixing of the dissolved solute for $Sc = 1$, $Ra \in [10^5, 10^8]$ and $Ω\in [0, 2.5]$. In the absence of rotation and buoyancy, the distance of the interface from its initial position follows a square root relationship with time ($r_d \propto \sqrt{t}$), which ceases to exist at a later time due to the finite-size effect of the liquid domain. We then explore the rotation parameter, considering a range of rotation frequency -- from smaller to larger, relative to the inverse of the buoyancy-induced timescale -- and Rayleigh number. We show that the area of the dissolved solute varies nonlinearly with time depending on $Ra$ and $Ω$. The symmetry breaking of the interface is best described in terms of $Ra/Ω^2$.

Figures (17)

• Figure 1: Schematic representation of the problem under consideration. (a) An infinitely long rotating horizontal cylinder filled with an incompressible solvent (blue) in which a solute (yellow) in the shape of a concentric circular cylinder is placed initially (at time $t = 0$), and (b) its $xy-$cross-section showing the two-dimensional nature of the problem under investigation in this study. (c) Location and geometry of the solute-solvent interface $\Gamma(t)$ at a later time ($t \gg 1$) along with its initial position, $\Gamma(0)$ (dashed line), and the cylinder wall, $\Gamma_w$.
• Figure 2: Layout of a typical $10\times 24$ grid in the (a) physical $(x,y)$ domain and (b) computational $(\xi,\eta)$ domain.
• Figure 3: Streamlines (white contours) overlaying the concentration contours of the dissolved solute at $A_d(t) = 10\%$, $30\%$, $60\%$, and $90\%$ (left to right) for $Ra = 10^5$ and $\Omega =$ 0 to 2 with an increment of 0.5 (top to bottom). The corresponding colour map illustrates the concentration variations in the domain at these instances.
• Figure 4: Streamlines (white contours) overlaying the concentration contours of the dissolved solute at $A_d(t) = 10\%$, $30\%$, $60\%$, and $90\%$ (left to right) for $\Omega =$1 and $Ra = 10^5$, $10^6$, and $10^7$ (top to bottom). The corresponding color map illustrates the concentration variations in the domain at these instances.
• Figure 5: Pathlines of two passive tracers, released initially at $(0.0745, -1.0115)$ and $(-0.0204, -1.9441)$, for rotation speeds ranging from $0$ to $1.5$ (shown left to right with an increment of $0.5$). The respective trajectories of these particles are shown in green and blue. The initial positions of the tracers are marked with empty markers, while their final positions are marked with the corresponding filled markers. Arrow markers along the pathlines denote the direction of particle motion. The red contour denotes the final shape of the undissolved solute, while $+$ represents the centre of the cylinder.