Nonlinear chemoconvection in the methylene-blue–glucose system: Two-dimensional shallow layers

Interfacial hydrodynamic instabilities arise in a range of chemical systems. One mechanism for instability is the occurrence of unstable density gradients due to the accumulation of reaction products. In this paper we conduct two-dimensional nonlinear numerical simulations for a member of this class of system: the methylene-blue–glucose reaction. The result of these reactions is the oxidation of glucose to a relatively, but marginally, dense product, gluconic acid, that accumulates at oxygen permeable interfaces, such as the surface open to the atmosphere. The reaction is catalyzed by methylene-blue. We show that simulations help to disassemble the mechanisms responsible for the onset of instability and evolution of patterns, and we demonstrate that some of the results are remarkably consistent with experiments. We probe the impact of the upper oxygen boundary condition, for fixed flux, fixed concentration, or mixed boundary conditions, and find significant qualitative differences in solution behavior; structures either attract or repel one another depending on the boundary condition imposed. We suggest that measurement of the form of the boundary condition is possible via observation of oxygen penetration, and improved product yields may be obtained via proper control of boundary conditions in an engineering setting. We also investigate the dependence on parameters such as the Rayleigh number and depth. Finally, we find that pseudo-steady linear and weakly nonlinear techniques described elsewhere are useful tools for predicting the behavior of instabilities beyond their formal range of validity, as good agreement is obtained with the simulations.

Figure 1

Experiments indicating a range of patterns, as seen from above. See [4, 15] for more information and other examples. The first two patterns are in Petri dishes which are open at the upper surface (the images are 13.5 cm wide) and the third is in a completely filled container (of diameter 5.1 cm) with an oxygen permeable lid.

Figure 2

(Color online) Five snapshots around the initial instability of simulations, for the standard parameter values with a fixed flux oxygen boundary condition at the upper surface (FFBC). Fluid velocity is superimposed upon each of the plots (if $\text{max}\left|\mathbf{u}\right|>{10}^{-3}$), which are arranged in groupings of (a) $A-⟨A⟩$, (b) $B$, and (c) $\Omega$. Snapshots are provided at times of 17, 18, 19, 20, and 21 nondimensional units from bottom to top. The maximum modulus of velocity is indicated on the right. Recall that there are ${10}^3$ numerical time steps between each recorded integer time $t_d\left(\approx 4\text{min}\right)$. (d) Plots of ${\text{Var}}_x\left(A\right)$ and ${\text{Var}}_x\left(B\right)$ (including data at time 1000 units) illustrating variations from a horizontally homogeneous solution (can be compared with perturbation solutions in linear and weakly nonlinear analyses).

Figure 3

Spatially averaged quantities (a) $⟨A⟩$, (b) $⟨B⟩$, (c) $⟨\Omega ⟩$, and (d) $⟨\left|\mathbf{v}\right|⟩$ from simulations with (1) fixed concentration, (2) fixed flux, (3) mixed boundary condition $\left({\beta }_c=1,{\beta }_f=0.2\right)$, (4) mixed boundary condition $\left({\beta }_c=1,{\beta }_f=3\right)$, and (5) mixed boundary condition $\left({\beta }_c=1,{\beta }_f=9\right)$. The insets depict early variation of the Var of the fields with time. Crosses, squares, etc., distinguish the line rather than indicate data points.

Figure 4

(Color online) Vertically averaged quantities (a) ${⟨A⟩}_z-⟨A⟩$, (b) ${⟨B⟩}_z$, (c) ${⟨\Omega ⟩}_z$, and (d) ${⟨\left|\mathbf{v}\right|⟩}_z$ for the standard parameter values with a fixed flux oxygen boundary condition at the upper surface (FFBC). The plots on the left are for 0–1000 nondimensional time units, and on the right for 0–100 nondimensional time units.

Figure 5

(Color online) Variation of the dominant wavenumbers with time. (a) the wavenumber with the greatest growth rate, $k_{\text{max}}$, (b) the maximum amplitude of the power spectrum, (c) power spectrum (left) and power spectrum normalized with respect to its maximum at each time (right). The normalized power spectrum is represented only for times at which its maximum is larger than ${10}^{-5}$. (a) and (b) are for standard parameter values with boundary conditions (1–5) as for Fig. 3, and (c) is for standard parameter values with a fixed flux oxygen boundary condition (FFBC).

Figure 6

(Color online) Plume interaction and annihilation for the standard parameter values with a fixed flux oxygen boundary condition at the upper surface (FFBC). Snapshots are presented at nondimensional times of 120, 130 and 140. The images portray a plume merging event.

Figure 7

(Color online) Results for standard parameter values with a fixed oxygen concentration upper boundary condition (FCBC). (a) and (b) are the vertically averaged quantities ${⟨A⟩}_z-⟨A⟩$, ${⟨B⟩}_z$, for early and long times, and (c) are snapshots of $A-⟨A⟩$ (top) and $B$ (bottom) at a time of 120 nondimensional units, with velocity superimposed. These images clearly indicate the horizontal asymmetry of cells.

Figure 8

(Color online) Snapshots of plume development for $⟨B⟩$ for the standard parameter values with mixed boundary conditions $\left({\beta }_c=1,{\beta }_f=3\right)$ at nondimensional times of 19 and 1000.

Figure 9

(Color online) ${⟨B⟩}_z$ from simulations varying Rayleigh number with mixed boundary conditions $\left({\beta }_c=1,{\beta }_f=3\right)$. Standard parameter values: $R=1.73$ (left), ${10}^{-1}$ (center), and ${10}^{-2}$ (right). $R\le 5\times {10}^{-4}=R_c$ provides no pattern.

Figure 10

(Color online) Plumes developing in a strong shear flow arising in a deep layer of fluid, $d=35$ (70 collocation points in the $z$ direction), for the standard parameter values with mixed boundary conditions $\left({\beta }_c=1,{\beta }_f=3\right)$. The snapshot of ${⟨B⟩}_z$ is presented at a nondimensional time of 1000.

Figure 11

$K_e$ vs $P_e$ for FFBC, FCBC, and a mixed boundary condition $\left({\beta }_c=1,{\beta }_f=3\right)$.