Shock-wave-like structures induced by an exothermic neutralization reaction in miscible fluids

We report shock-wave-like structures that are strikingly different from previously observed fingering instabilities, which occur in a two-layer system of miscible fluids reacting by a second-order reaction $A+B\to S$ in a vertical Hele-Shaw cell. While the traditional analysis expects the occurrence of a diffusion-controlled convection, we show both experimentally and theoretically that the exothermic neutralization reaction can also trigger a wave with a perfectly planar front and nearly discontinuous change in density across the front. This wave propagates fast compared with the characteristic diffusion times and separates the motionless fluid and the area with anomalously intense convective mixing. We explain its mechanism and introduce a new dimensionless parameter, which allows to predict the appearance of such a pattern in other systems. Moreover, we show that our governing equations, taken in the inviscid limit, are formally analogous to well-known shallow-water equations and adiabatic gas flow equations. Based on this analogy, we define the critical velocity for the onset of the shock wave which is found to be in the perfect agreement with the experiments.

Figure 1

Time evolution of the reaction front position obtained experimentally for different values of $K_{\rho }$. Solid line corresponds to the theoretical estimate for the critical value of the front speed (15) to trigger a shock wave.

Figure 2

Shock-wave structure observed in 150 s after the aqueous solutions of ${\mathrm{HNO}}_3$ (upper layer) and NaOH were brought into contact. The frames from left to right correspond to interferogram showing a refractive index distribution, velocity field revealed by the traces of light-scattering particles and pH distribution obtained due to universal indicator, respectively. The initial concentrations of acid and base are equal to 1.5 mol/l and 1.4 mol/l, respectively. The initial contact line is indicated by the horizontal band. $K_{\rho }=0.997$.

Figure 3

Map of chemoconvection modes at the plane of the initial concentrations $A_0$, $B_0$, and expansion coefficients ${\beta }_a$, ${\beta }_b$ of species. The shaded areas correspond to the shock-wave appearance at $K_{\rho }<1$. The open and filled circles indicate the CC and DC regimes experimentally observed for NaOH/${\mathrm{HNO}}_3$, respectively. The numbered circles correspond to the experimental results shown in Fig. 1.

Figure 4

Time evolution of a transverse averaged total density $\rho (z,t)$ for ${\mathrm{HNO}}_3$/NaOH: (a) the case when the density of the reaction zone is equal to those of upper layer: $K_{\rho }=1$; (b) the CC regime at $K_{\rho }=0.997$ (the circle $\mathit{2}$ in Fig. 3). In the last case, nonlinear simulations clearly shows the appearance of the shock wave for $t>0.5$ after the collapse of the depleted zone low in density.

Figure 5

(a) Time evolution of the spatial reaction rate computed as the number of points where salt $S(x,z,t)$ is larger than a given small threshold $S^{*}=0.001$ normalized by the area of the domain behind the traveling shock wave; (b) the frames from left to right show the salt concentration at successive times. $K_{\rho }=0.997$.

Figure 6

Schematic presentation of a Hele-Shaw cell.