Linear stability analysis of bubble-induced convection in a horizontal liquid layer

We investigate with a linear analysis the stability of a horizontal liquid layer subjected to injection of gas bubbles through a bottom wall. The injection is assumed uniform in space and constant in time. Injected bubbles ascend in the liquid layer due to the Archimedean buoyancy force and are ejected from the top free surface of the liquid layer. Modeling this two-phase flow system as two interpenetrating liquid and gas continua, we show that homogeneous upward gas flows become unstable at large gas fluxes. We determine the critical conditions of this homogeneous-heterogeneous regime transition and show that the critical modes are made of stationary convection rolls, either multi- or whole-layered depending on liquid viscosity, the radius of bubbles, and the thickness of liquid layer. By examining the energy transfer from base to perturbation flows, we indicate that liquid convective motion is driven by the buoyancy on heterogeneously distributed bubbles. We also reveal that the lift forces on bubbles have significant stabilizing effects by homogenizing bubble distribution close to the bottom wall.

Figure 1

(a) Schematic illustration of the problem, and (b) (i) profiles of bubble ascending velocity ${\overline{w}}_G\left(z\right)$ and (ii) gas volume fraction ${\overline{\varepsilon }}_G\left(z\right)$ in the base state with different $\mathrm{Pr}$ defined as Eq. (19): the gas injection velocities and gas injection volume fractions at $z=-1$ of the profiles equal to $w_{G,0}=1000$ and ${\varepsilon }_{G,0}=0.02$, respectively.

Figure 2

(a) Marginal stability curves, above which the quiescent state is unstable, for different values of the Prandtl number $\mathrm{Pr}$ in the $k\text{−}\mathrm{Ra}$ space. The gas injection velocity and the bubble radius are fixed at $w_{G,0}=1000$ and $r_b=0.01$. The temporal natures of marginally stable mode are indicated on curves by $s$ (stationary) and $o$ (oscillatory). (b) Eigenvectors at the local minimum points (i)–(ix). The liquid velocity ${\mathbf{u}}^{\prime }=(u^{\prime },w^{\prime })$ and the gas fraction ${\varepsilon }_G^{\prime }$ are shown by vectors and colors, respectively. Bubble-rich and bubble-poor cells correspond to red and white zones. Some streamlines are also shown in (b) to visualize the structures of liquid flows.

Figure 3

Variation of critical parameters $(k_c,{\mathrm{Ra}}_c)$ as function of $\mathrm{Pr}$. The whole- and multilayer modes are critical on the left and right of a codimension-two point $\mathrm{Pr}={\mathrm{Pr}}^{*}$, respectively. The best-fitting power laws ${\mathrm{Ra}}_c\propto {\mathrm{Pr}}^n$ are shown in dashed lines. The exponent $n$ is 2.68 and 1.93 for the left and right lines.

Figure 4

Powers providing energy to perturbation gas flows at marginally stable conditions: $W_I$ (solid line), $W_D$ (dotted line), and $W_L$ (dashed line) represent the powers of liquid inertia and drag and lift forces, respectively. The powers have been normalized by twice the kinetic energy $K_G$ of perturbation flows. Different panels show results for different values of the Prandtl number $\mathrm{Pr}$. The values of critical wave number $k_c$ are marked on the $k$ axis. The temporal natures of marginally stable mode are indicated by $s$ (stationary) and $o$ (oscillatory) at the top of each panel.

Figure 5

Marginal stability curves, above which the quiescent state is unstable, for different values of Prandtl number $\mathrm{Pr}$ in the $k\text{−}\mathrm{Ra}$ space. The bubble radius and the bubble injection velocity are fixed at $r_b=0.01$ and $w_{G,0}=1000$. The drag coefficient is modeled by $C_D=16/{\mathrm{Re}}_b$ [38, 39].

Figure 6

Marginal stability curves for different values of bubble radius $r_b$ in the $k\text{−}\mathrm{Ra}$ space. Prandtl number $\mathrm{Pr}$ and the injection velocity are fixed at $\mathrm{Pr}=2.65\times {10}^{-5}$ and $w_{G,0}=1000$. The temporal natures of marginally stable mode are indicated on curves by $s$ (stationary) and $o$ (oscillatory).