4.1. Overview of the reaction–diffusion–convection regimes

Analysis of the results of the experiments made with different acid–base pairs in a wide concentration range has revealed the development of reaction–diffusion–convection processes per two different scenarios, depending on the value of the reaction-induced buoyancy number ${K}_{\rho }$. These scenarios are general for all pairs studied and do not depend on the reagents used. Further, we overview both scenarios in detail and give some quantitative characteristics.

In the case of ${K}_{\rho }>1$ (figure 3), the stably stratified transition zone begins to form on either side of the reaction front, just after the layers were brought into contact. The concentration changes of both reagents and the reaction product occur within this transition zone, as can be seen from the interferograms presented in figure 3(a,d,g). The liquid inside the zone is immovable and the reagents and the product of reaction are transferred between the layer and the reaction front only by means of the diffusion mechanism. The reaction front moves slowly with characteristic velocity $O( {{10}^{-5}} \text {--} {{10}^{-4}} )$ cm s$^{-1}$. The zone broadens with time, remaining stably stratified. A weak convective flow with characteristic velocity of the order of $O({{10}^{-2}})$ cm s$^{-1}$ in the form of fingers always develops in the upper layer above the transition zone (figure 3b,e,h). A weak convective motion may also arise in the lower layer in a certain concentration range. The reasons for the formation of these instabilities and their structure will be considered in greater detail below. Since the convective motion is weak and localized far from the reaction front, being separated by the motionless transition zone, the reaction rate is defined mainly by diffusion processes. Within this scenario, the reaction lasts from a few hours to several days depending on the initial reagent concentration. The reaction rate, defined as the conversion rate of reagents, is on average $O( {{10}^{-8}} )$ mol s$^{-1}$. The temperature excess due to reaction exothermicity, as measured at the reaction front, is a few tenths of a degree. It is interesting that the characteristic diffusion time, estimated by the vertical size of either layer, is of the order of $O( {{10}^{6}} )$ s which almost coincides with the full reaction time within this scenario.

Figure 3.

Evolution of two-layer system observed within the framework of the DC regime (${K}_{\rho }=1.042$) for the pair of $\textrm {HNO}_3$–$\textrm {NaOH}$ reactants. From left to right: interference patterns, flow velocity fields, pH distributions. The initial concentrations of the acid and base are equal to 0.3 mol l$^{-1}$ and 2 mol l$^{-1}$, respectively. The initial contact line between the reactant solutions coincides with the horizontal black line (slot for the slider) in the middle of the cell. Time after the start of the experiment: top row – 110 s; middle row – 1500 s; lower row – 4630 s.

A completely different regime develops in the case of ${K}_{\rho }\le 1$ (figure 4). Right after the layers are brought into contact, the vigorous plumes are formed above the reaction front. Very soon, the intensive convective motion occupies the whole upper layer, homogenizing it. The characteristic velocity of the liquid motion is of the order of $O( {{10}^{-2}} \text {--}{{10}^{-1}} )$ cm s$^{-1}$, which is much faster than it is observed in the DC regime. At the same time, the lower layer remains always motionless and homogeneous. The reaction front moves downward with relatively high velocity, of the order of $O( {{10}^{-2}} )$ cm s$^{-1}$. When the reaction front reaches the bottom of the cuvette, the reaction stops. The reaction lasts from 7 to 15 min, depending on the initial concentration of the reagents. The reaction rate is, on average, $O( {{10}^{-5}} )$ mol s$^{-1}$. Due to the high conversion rate of the reagents, the temperature excess at the reaction front is a few degrees. Some important characteristics of the described CC and DC regimes are summarized in table 2 for comparison.

Figure 4.

Evolution of the two-layer system observed within the framework of the CC regime (${K}_{\rho }=0.997$) for the pair of $\textrm {HNO}_3$–$\textrm {NaOH}$ reactants. From left to right: interference patterns, flow velocity fields, pH distributions. The initial concentrations of the acid and base are equal to 1.5 mol l$^{-1}$ and 1.4 mol l$^{-1}$, respectively. The initial contact line between reactant solutions coincides with the horizontal black line (slot for the slider) in the middle of the cell. Time after the start of the experiment: top row – 3 s; middle row – 150 s; lower row – 700 s.

4.2. Stability maps

The reaction-induced buoyancy number $K_\rho$ can be precalculated for each concentration ratio and each pair of reagents to predict the development of one or other regime prior to the experiment. An example of such calculation for the pair $\textrm {HNO}_3$–$\textrm {NaOH}$ is presented in figure 5(a). The dimensionless quantities ${{\beta }_{a}}{{A}_{0}}$ and ${{\beta }_{b}}{{B}_{0}}$, being plotted along the $x$-axis and $y$-axis, are the relative density increments of the acid and base solutions, respectively. The curve, obtaining as a result of intersection of the three-dimensional surface ${{K}_{\rho }}=f( {{\beta }_{a}}{{A}_{0}},{{\beta }_{b}}{{B}_{0}} )$ with the plane ${{K}_{\rho }}=1$, gives us the boundary between the areas of parameters, in which the CC or DC regime takes place.

Figure 5.

Three-dimensional surface characterizing the dependence of $K_\rho$ on the dimensionless quantities $\beta _a A_0$ and $\beta _b B_0$ (a) and the map of the regimes (b) calculated for the pair of $\textrm {HNO}_3$–$\textrm {NaOH}$ reactants. The black solid line is the isopycnic line. The shaded region, adjoining the isopycnic line, is the area of parameters where the CC regime develops. The region, marked with white colour, corresponds to the DC regime. The open and filled symbols indicate the parameters at which the CC and DC modes have been observed in the experiments with the $\textrm {HNO}_3$–$\textrm {NaOH}$ system, respectively.

The map of regimes obtained in such a way for the $\textrm {HNO}_3$–$\textrm {NaOH}$ pair is shown in figure 5(b). The bisectrix on the map represents the isopycnic line, i.e. the line along which the initial densities of the upper and lower reagent solutions are equal. The equation of the isopycnic line is ${{\beta }_{a}}{{A}_{0}}={{\beta }_{b}}{{B}_{0}}$. Here, we neglect a weak nonlinearity of the dependence being of the opinion that, for all substances, the solutal expansion coefficient is constant in the concentration range used. This assumption looks justified, since the maximal error in the definition of the solution density connected with this simplification is within 0.2 %. In the region which lies above the isopycnic line, the upper layer is formed by the acid solution and the lower layer by the base solution. In the lower half of the map, the configuration of the two-layer system is the opposite. The closed area of parameters (shaded area in figure 5(b)), adjacent to the isopycnic line, is the area where $K_\rho <1$ and, therefore, the CC mode develops. For all other parameter values, $K_\rho >1$ and the DC mode is realized. The symbols in the map indicate the parameters at which the CC mode (open circles) or the DC mode (filled circles) have been observed in the experiments. It is seen that these results are in perfect agreement with the calculations based on (2.5).

The experimental results presented above, as well as their excellent agreement with the results of calculations, confirm the correctness of the physical interpretation of the observed phenomena and the choice of dimensionless governing parameter. Moreover, plotting the regime maps at the stage of planning experiments allows us to predict the regions of existence of different regimes in the problem parameter space, thereby significantly reducing the search stage of the study. In the next sections, we consider each regime in more detail.

It is worth noting here that the maps of regimes, calculated on the base of initial reagent concentrations $A_0$ and $B_0$, give us the possibility to predict one of the two global scenarios realized from the very beginning of the system's evolution. Later on, reduction of the reagent concentrations in the course of reaction leads to shifting of the point describing the position of the system in the regime map. If initial parameters are chosen very near the boundary between the zones, the change of the parameters can result even in a change of regime during later evolution of the system. An example of such transition between global scenarios will be considered in detail in § 4.5. If the initial position of the system in the map is located far from the boundary between the zones the reaction–diffusion–convection processes occur within the initial scenario.

4.3. CC mode

Figure 6 presents the maps of regimes for all reagent pairs. One can see that each map is individual and the position of the curve ${{K}_{\rho }}=1$, bounding the parameter area within which the CC regime exists changes under varying both an acid and a base in a pair of reagents. Each map was checked in experiments and the results demonstrated a perfect agreement with the calculation.

Figure 6.

Maps of regimes for $\textrm {HCl}$–$\textrm {MeOH}$ (a) and $\textrm {HNO}_3$–$\textrm {MeOH}$ (b) systems. The zone of CC regime existence is denoted with the coloured areas. The region, marked with white colour corresponds to the DC regime.

It is worth noting here that, not only the form of the area where the CC regime exists changes under replacement of one of the reagents, but its depth as well. The three-dimensional surface, shown in figure 5(a), has a pocket-like form with a minimum, always located on the isopycnic line. The depth of the pocket, i.e. the minimal value of ${{K}_{\rho }}$, depends on the reagents used. The curves, obtained as a result of the intersection of the ${{K}_{\rho }}=f( {{\beta }_{a}}{{A}_{0}},{{\beta }_{b}}{{B}_{0}} )$ surface by the vertical plane, containing the isopycnic line, are shown in figure 7. Since the values of ${{\beta }_{a}}{{A}_{0}}$ and ${{\beta }_{b}}{{B}_{0}}$ are the same along the isopycnic line, any of these quantities can be plotted along the $x$-axis. It is seen that the dependencies are non-monotonic, demonstrating the presence of a minimum. The minimal value of ${{K}_{\rho }}$ which can be achieved within the region and the location of the minimum significantly depend on the reagent pair.

Figure 7.

The dependence of the parameter $K_\rho$ on the relative density increment of the solution ${{\beta }_{b}}{{B}_{0}}$ calculated along the isopycnic line for different pairs of reactants.

Analysing figures 6 and 7 indicates that the size and shape of the parameter area, where the CC regime exists, change significantly when replacing both the acid and the base. Equation (2.5) shows that the value of the parameter ${{K}_{\rho }}$ depends on the ratio of the diffusion coefficients of the reagents. An increase in ${{\delta }_{D}}$ reduces the concentration of salt and residual reagent in the reaction zone after the reaction, which, in turn, results in reduction of ${{K}_{\rho }}$ and, consequently, in widening of the zone. Other important parameters are the ratios of the expansion coefficients of each of the reagents to that of salt ${{{\beta }_{a}}}/{{{\beta }_{s}}}$ and ${{{\beta }_{b}}}/{{{\beta }_{s}}}$. Indeed, an increase in the contribution of the salt into the reaction zone density as compared to that of either of the reagents leads to a faster growth of the reaction zone density and, consequently, a reduction in the zone of the CC regime.

The map of regimes and the dependence ${{K}_{\rho }}( {{\beta }_{b}}{{B}_{0}})$ were calculated to elucidate the contribution of each of the above quantities. The results of these calculations are presented in table 3. Each row of the table contains the results obtained at two different values of one of the above parameters that differ by 10 %, while the other parameters remained unchanged. It is seen that a decrease in each of the parameters results in a reduction of the zone of the CC mode existence. Moreover, the most significant effect is observed when the ratio of the diffusion coefficients of the reagents changes.

Table 3.

Influence of the parameters $\beta _{b}/\beta _s$, $\beta _{a}/\beta _s$ and $\delta _D$ on the size and the depth of the zone of CC regime existence.

These findings allow us to better understand the reasons the maps of regimes change on the variation of alkali ions in the pairs of reagents. Analysing the data for $\textrm {MeOH}\text {--}\textrm {HNO}_{3}$ pairs presented in table 1 shows that the growth of ${{{\beta }_{b}}}/{{{\beta }_{s}}}$ is compensated by a reduction of ${{{\beta }_{a}}}/{{{\beta }_{s}}}$ when switching from lithium hydroxide to potassium hydroxide. In this situation, the increase in the zone of CC regime existence (see figure 6) and its depth (see figure 7) are mainly due to an increase in ${{\delta }_{D}}$. Changing the base to caesium hydroxide leads to a significant decrease in ${{{\beta }_{a}}}/{{{\beta }_{s}}}$, which cannot be compensated by a slight increase in ${{{\beta }_{a}}}/{{{\beta }_{s}}}$ and ${{\delta }_{D}}$. As a result, a significant reduction of the size and depth of the zone of CC regime existence is observed for the $\text {CsOH--HNO}_{3}$ pair (see figures 6 and 7).

Another important feature of the regime maps is their asymmetry with respect to the isopycnic line. Analysing the data presented in table 1 shows that the zone of CC regime existence is always wider from the side where the ratio of the solutal expansion coefficient of the upper reagent to that of the salt is smaller. Indeed, the smaller this ratio, the slower the parameter ${{K}_{\rho }}$ grows as the concentration of the upper reagent increases. For all reagent pairs, except $\textrm {LiOH}\text {--}\textrm {HNO}_{3}, ({{{\beta }_{a}}}/{{{\beta }_{s}}})<({{{\beta }_{b}}}/{{{\beta }_{s}}})$, which leads to a wider zone in the upper half of the map, where the upper reagent is acid. The case of $\textrm {LiOH}\text {--}\textrm {HNO}_{3}$ is the only reagent pair where $({{{\beta }_{a}}}/{{{\beta }_{s}}})>({{{\beta }_{b}}}/{{{\beta }_{s}}})$ and the zone is wider in the lower half of the map. As a quantitative criterion responsible for the degree of asymmetry, one can choose the ratio of the zone width above ${{\varDelta }_{1}}$ and below ${{\varDelta }_{2}}$, the isopycnic line at a fixed value of ${{\beta }_{a}}{{A}_{0}}$ (see the explanations in figure 5(b)). Expressing the concentration values at the zone boundary from the condition ${{K}_{\rho }}=1$, we get

(4.1)\begin{equation} \frac{{{\varDelta}_{1}}}{{{\varDelta }_{2}}}=\frac{{{\beta }_{b}}( {{\delta }_{D1}}-1 )+{{\beta }_{a}}-{{\beta }_{s}}}{{{\beta }_{b}}( {{\delta }_{D2}}-1 )+{{\beta }_{a}}-{{\beta }_{s}}}\cdot \frac{{{\delta }_{D2}}{{\beta }_{b}}}{{{\beta }_{s}}-{{\beta }_{a}}}. \end{equation}

This expression is valid for those pairs of reagents for which ${{\beta }_{a}}<{{\beta }_{b}}$, i.e. the equimolar line lies above the isopycnic line. In the opposite case, when ${{\beta }_{a}}>{{\beta }_{b}}$, which is typical only for the pair $\textrm {LiOH}\text {--}\textrm {HNO}_{3}$

(4.2)\begin{equation} \frac{{{\varDelta }_{1}}}{{{\varDelta }_{2}}}=\frac{{{\beta }_{a}}( {{\delta }_{D1}}-1 )+{{\beta }_{b}}-{{\beta }_{s}}}{{{\beta }_{a}}( {{\delta }_{D2}}-1 )+{{\beta }_{b}}-{{\beta }_{s}}}\cdot ( {{\delta }_{D2}}-1 ). \end{equation}

Here, ${{\delta }_{D1}}$ and ${{\delta }_{D2}}$ are the values of this parameter at the zone boundary located above and below the isopycnic line, respectively. Neglecting the concentration dependence of the diffusion coefficient within the zone, i.e. assuming ${{\delta }_{D1}}\approx {{\delta }_{D2}}$, we obtain the following approximate expressions:

(4.3)\begin{gather} \frac{{{\varDelta }_{1}}}{{{\varDelta }_{2}}}={{\delta }_{D}}\frac{{{\beta }_{b}}}{{{\beta }_{s}}-{{\beta }_{a}}},\quad \text{if } {{\beta }_{a}}<{{\beta }_{b}}, \end{gather}

(4.4)\begin{gather}\frac{{{\varDelta }_{1}}}{{{\varDelta }_{2}}}=( {{\delta }_{D}}-1 ),\quad \text{if } {{\beta }_{a}}>{{\beta }_{b}}, \end{gather}

where ${{\delta }_{D}}$ is the average value within the zone. The value of this parameter, calculated by formula (4.3) or (4.4), which reflects the degree of asymmetry of the zone of CC mode existence, is presented in table 4 for all the pairs of reagents studied.

Table 4.

Values of the parameter that determines the degree of asymmetry of the zone of CC regime existence for all reagent pair used.

The parameter ${{K}_{\rho }}$ defines not only the development of one or another reaction–diffusion–convection regime but also the convective motion intensity. As a measure of the motion intensity, we used the average velocity in a group of plumes that appear above the reaction zone after the solutions were brought into contact. The velocity was measured at the initial stage when it was constant. Later, the transverse diffusion and interaction between the plumes reduced the plume velocity, and therefore we did not take these results into account. The dependence of the plume velocity, obtained in such a way, on the parameter ${{K}_{\rho }}$ is presented in figure 8(a). The results measured for different pairs of the reagents form a unified dependence, the best fit of which is $v\sim (K_{\rho }-0.97)^{-4}$. This unification of the experimental data, obtained for different pairs, indicates the universal character of ${{K}_{\rho }}$, as a quantity being responsible for the convective motion intensity in the system under study. It is seen that the convective motion rapidly weakens as it approaches the boundary between the CC and DC regimes.

Figure 8.

Variation of the plume average velocity measured at the initial stage of an experiment just after the contact of the regents ($a$) and the reaction front velocity ($b$) as a function of $K_{\rho }$. The solid lines represent approximating functions that best describe the experimental data.

The convective motion intensity affects the mass transfer rate near the reaction front, which causes the reaction front velocity to change. The variations of the reaction front velocity with change of ${{K}_{\rho }}$ are presented in figure 8(b). The position of the reaction front was determined from the spatial distributions of the pH value. Note that the value of ${{K}_{\rho }}$ uniquely defines the rate of reaction front motion. All experimental points obtained for different reagent pairs and different concentration ranges form a unified dependence, the best fit of which is $v_{f}\sim (K_{\rho }-0.98)^{-3}$. It is seen that the reaction front velocity decreases by more than two orders of magnitude when switching from the CC regime to the DC one, which is associated with changes in the mass transfer mechanism, from convective to primarily diffusive.

4.4. DC mode

In this section, we describe some important features of the development of reaction–diffusion–convection processes within the DC regime. Despite of the DC regime's name, mechanical equilibrium is not the only state of the system. Diffusion is only the prevailing mass transfer mechanism near the reaction front, which defines the characteristic time of the reaction–diffusion–convection processes. However, hydrodynamic instabilities can develop at a certain distance from the reaction front due to differential-diffusion effects.

Let us consider these processes in more detail. The fragment of the two-layer system near the reaction front is schematically shown in figure 9. At the reaction front, the reagents are consumed and the reaction product is formed at certain rates. Since the density distribution across the layers is stable at the initial stage in the DC regime, both reagents transfer towards the front only by diffusion. The reaction product, arising at the front during the reaction, spreads upward and downward due to diffusion as well. Taking into account the extremely high rate of the neutralization reaction, one can exclude the penetration of any reagent into the adjacent layer. Then we can assume that each layer contains only two components, one of the reagents and the reaction product, which diffuse towards each other. In this case, the reaction does not affect, to a first approximation, the diffusion processes, and the development of differential-diffusion effects can be examined in each layer independently.

Figure 9.

Schematic image of the diffusion processes near the reaction front.

The situation, when two solutes diffuse at different rates towards each other in a vertical direction, was considered in numerous works (Stern & Turner Reference Stern and Turner1969; Taylor & Veronis Reference Taylor and Veronis1996; Pringle, Glass & Cooper Reference Pringle, Glass and Cooper2002; Sorkin, Sorkin & Leizerson Reference Sorkin, Sorkin and Leizerson2002; Trevelyan et al. Reference Trevelyan, Almarcha and De Wit2011). When the upper species diffuse faster, a DLC instability develops. In the opposite case, when the lower species diffuse faster, a DD instability develops in the parameter region ${{\delta }_{b}}>{{( \varphi {{R}_{b}} )}^{{2}/{3}}}$, where ${{\delta }_{b}}={{{D}_{l}}}/{{{D}_{u}}}$ is the ratio of the diffusion coefficients of the lower and upper species, $\varphi ={{{C}_{l}}}/{{{C}_{u}}}$ is the ratio of the initial concentrations of the lower and upper species and ${{R}_{b}}={{{\beta }_{l}}}/{{{\beta }_{u}}}$ is the ratio of the solutal expansion coefficients of the lower and upper species (Trevelyan et al. Reference Trevelyan, Almarcha and De Wit2015). When $1\le {{\delta }_{b}}\le {{( \varphi {{R}_{b}} )}^{{2}/{3}}}$, the system remains stable for all time.

Using the above inequalities, the parameter regions where the DD or DLC instability occurs can be calculated in advance. Figure 10 presents the regime maps calculated for the $\textrm {HNO}_3$–$\textrm {MeOH}$ systems, where the areas of parameters for which different differential-diffusion instabilities occur within the DC regime are indicated. The maps in figure 10(a,c,e,g) were plotted for the upper layer and in figure 10(b,d,f,h) for the lower layer. The light-grey and dark-grey areas indicate zones where the DLC or DD instability develops, respectively. The mechanical equilibrium zone is denoted by the white colour.

Figure 10.

Parameter areas for different differential-diffusion instabilities in the DC regime, calculated for the $\textrm {HNO}_3$–$\textrm {MeOH}$ systems: (a) $\textrm {HNO}_3$–$\textrm {NaOH}$, (b) $\textrm {HNO}_3$–$\textrm {LiOH}$, (c) $\textrm {HNO}_3$–$\textrm {KOH}$, (d) $\textrm {HNO}_3$–$\textrm {CsOH}$. Panels (a,c,e,g) are plotted for the upper layer and panels (b,d,f,h) for the lower layer. The light-grey and dark-grey areas indicate the zones where the DLC or DD instability develops, respectively. The mechanical equilibrium zone is denoted by the white colour.

In all cases, except for the $\textrm {HNO}_3$–$\textrm {LiOH}$ pair, which will be discussed separately, the reagent diffuses faster than salt. That is the reason why the DLC instability always develops in the upper layer at any concentration ratio, regardless of whether the acid (upper half in the maps, presented in figure 10(a,c,e,g)) or the base (lower half in the maps, presented in figure 10(a,c,e,g)) is contained in the upper layer. In the lower layer (figure 10b,d,f,h), where the salt diffusing downwards from the reaction front always has a diffusion coefficient lower than that of the reagent, a DD instability may develop (dark-grey areas in figure 10). At the same time, there are wide parameter areas (white areas in figure 10), where the lower layer remains stationary all the times. The maps were thoroughly tested in a series of experiments, the results of which demonstrated good agreement with calculations. The interferograms, illustrating the development of differential-diffusion instabilities in the areas indicated in the map, are presented in figure 10.

A more interesting situation is observed for the $\textrm {HNO}_3$–$\textrm {LiOH}$ pair. As in the above cases, the salt $\textrm {LiNO}_3$ diffuses slower than the acid. However, the diffusion coefficient ratio for the salt and base is not so unambiguous. The salt diffuses slower than the base in the concentration range below 1 mol l$^{-1}$, and it diffuses faster in the higher concentration range. This causes both DLC and DD instabilities to occur in the layer, where the salt and base diffuse towards each other, depending on a base concentration. Figure 10($b$) presents the regime maps found in the $\textrm {HNO}_3$–$\textrm {LiOH}$ system. It is seen that, in the configuration when the nitric acid is placed in the upper layer, the DLC instability develops both in the upper (upper half in the left map in figure 10(b)) layer and in the lower (upper half in the right map in figure 10(b)) layer. The interferograms illustrating this situation are inserted in the corresponding halves of the maps. In the reverse configuration, i.e. when the base is located in the upper layer, other situations may occur. If the base concentration is low enough ${{\beta }_{b}}{{B}_{0}}<0.025$, then the DLC instability develops in the upper layer (lower half in the left map in figure 10(b)), whereas the lower layer (upper half in the right map in figure 10(b) remains stationary in most of the parameter area. At high base concentration, the DD instability develops in both layers. The interferograms inserted in the suitable parts of the maps illustrate the experimental observations.

In the framework of DC regime, we have found a new type of instability, which results in the formation of convective patterns, which has not ever been observed in the previous studies on this subject. This type of instability was briefly described in our previous work (Bratsun et al. Reference Bratsun, Kostarev, Mizev and Mosheva2015), and here we discuss it in more detail. The evolution of the two-layer system, observed experimentally within this mode, is as follows. At the initial stage, the process develops, as usual, within the DC regime. Immediately after the layers are brought into contact, the stable transition zone forms above and below the reaction front. In the upper layer above the transition zone, the DLC instability results in weak convective motion formation (see figure 11a–d). A few minutes after the experiment starts, the zone with rarefaction of interference fringes appears just above the reaction front, where the horizontal row of convective cells forms later (figure 11e–h). Above and below the cells, the liquid is stationary, which indicates the formation of a local potential well in the vertical density profile, within which the density stratification is unstable. The cellular structure does not interact with the DLC structure in the upper layer, being separated from it by the immobile liquid layer. The study of spatial pH allocation (figure 11d,h,l) shows that the areas located above and below the cells remain almost homogeneous (acidic or alkaline), whereas a more complex pH distribution is observed within the cellular structure. The descending flows in the convective cells are enriched with the acid, which comes from the upper layer through the stationary part of the liquid due to diffusion. The ascending flows in the convective cells are saturated with salt, which points out that the reaction occurs at the lower edge of the cellular structure. The location of the reaction front is also evidenced by the temperature distribution measured along the vertical direction, which shows the maximum heat release at the lower boundary of the cells.

Figure 11.

Evolution of the two-layer system within the framework of CDD instability for the pair of $\textrm {HCl}$–$\textrm {NaOH}$ reactants. From left to right: flow structure, flow velocity field, interference pattern, pH distribution. The initial concentrations of the acid and base are equal to 1.4 mol l$^{-1}$ and 1 mol l$^{-1}$, respectively. The initial contact line between the reactant solutions coincides with the horizontal black line (slot for the slider) in the middle of the cell. Time after the start of the experiment: top row - 38 s; middle row - 722 s; lower row - 1965 s.

This regime lasts a few hours. The zone within which the cellular structure develops widens with time, which, in turn, results in the change of both the vertical and horizontal size of the convective cells. Figure 12 presents the temporal evolution of non-dimensional wavelength $\varLambda ={\lambda }/{d}$, where $\lambda$ is the dimensional wavelength of the cellular structure, and $d$ is the vertical size of the zone within which the cells exist. The time is counted from the moment the cells starts to form. It is seen that, at the initial stage, the wavelength and zone height vary proportionally one to another so that the non-dimensional wavelength remains constant. It is interesting to note that the value is close to the wavelength of convective cells in the Rayleigh–Bénard problem for a layer with free boundaries. This underlines the fact that both instabilities have a similar origin. Later on, as seen from figure 12, the non-dimensional wavelength starts to vary, gradually decreasing over time according to the root law. This is attributed to the decay of descending and ascending streams, forming the cells, and their separation into separate plumes, which can be seen well in the interferograms of figure 11.

Figure 12.

Temporal evolution of non-dimensional wavelength of the cellular structure caused by the CDD instability. The time is counted from the moment the cells starts to form. The numbers correspond to the experiments with different reactants pairs and initial concentrations: 1–1 mol l$^{-1}\ \textrm {HNO}_3\text {--}1$ mol l$^{-1}\ \textrm {NaOH}$; 2–1.5 mol l$^{-1}\ \textrm {HNO}_3\text {--}1.5$ mol l$^{-1}\ \textrm {NaOH}$; 3–2 mol l$^{-1}\ \textrm {HNO}_3\text {--}2$ mol l$^{-1}\ \textrm {NaOH}$; 4–1.5 mol l$^{-1}\ \textrm {HNO}_3\text {--}2$ mol l$^{-1} \textrm {KOH}$; 5–3 mol l$^{-1}\ \textrm {HNO}_3$–4 mol l$^{-1}\ \textrm {KOH}$; 6–1 mol l$^{-1}\ \textrm {HCl}$–0.6 mol l$^{-1} \textrm {NaOH}$.

The above regime develops in a confined region of parameters within the DC regime. Figure 13 presents the maps for some reagent pairs, where the parameters’ area, within which this regime was found, is shaded. It is seen that the cellular structure is always observed only in the upper part of the maps, i.e. in the situation, when an acid solution is located in the upper layer. In the opposite configuration, namely when the upper layer is occupied by a base solution, the formation of the cells has never been found.

Figure 13.

Areas of parameter space for the CDD instability (shaded region) within the DC regime (white colour region) revealed experimentally for $\textrm {HNO}_3$–$\textrm {MeOH}$ and $\textrm {HCl}$–$\textrm {NaOH}$ systems.

To understand the reason for this asymmetry, it is necessary to analyse the density profile, which can be qualitatively reconstructed from the interferograms. An example of such a profile is shown schematically in figure 14 together with the corresponding interferogram. Note that the density distribution appears in the course of counter-diffusion of the salt and the reagent contained in the upper layer. If the salt contributes more to the density than the reagent, i.e. has the greater expansion coefficient, then the formation of reverse density distribution becomes possible. The smaller the ratio of solutal expansion coefficients of the reagent and salt, the deeper the potential density well that can be formed. In the case when the upper layer is formed by acid, this ratio is less than 0.5 (see table 1) for most pairs, which results in the formation of a relatively deep potential well (see the density profile in figure 14(a)) and, as consequence, the cellular convective structure. The formation of the potential well in a density-stratified medium is also strengthened by the concentration dependence of diffusion, which decreases with concentration for salts and increases for acids. As was shown in Bratsun et al. (Reference Bratsun, Kostarev, Mizev and Mosheva2015), the cellular structure is reproduced with numerical experiments only by taking into account the concentration dependence of diffusion. In the opposite case, when the base is dissolved in the upper layer, the ratio of expansion coefficients is larger 0.8 for the nitrogen acid salts. This turns out to be insufficient to form a deep enough potential well (see the density profile in figure 14(b)), whereby the liquid above the reaction front remains stable. In the case of hydrochloric acid salts, the ratio of the expansion coefficients of the base and salt is always greater than unity, which makes it impossible to form potential well in an initially stably stratified medium.

Figure 14.

Interference patterns and corresponding vertical density profiles for the potential well ($a$) and barrier ($b$).

The above reasoning also allows for the explanation of the absence of this regime in the experiments with lithium hydroxide. The $\textrm {HNO}_3$–$\textrm {LiOH}$ and $\textrm {HCl}$–$\textrm {LiOH}$ pairs are the only pairs where the cellular structure was not observed. It is seen from table 1 that the ratio of expansion coefficients is rather large for both salt–acid and salt–base pairs. Besides, the diffusion coefficient of the salt of both acids increases with concentration.

It is worth noting here that the cellular convective structure described above looks similar to that reported in Eckert, Acker & Shi (Reference Eckert, Acker and Shi2004). The authors observed formation of a horizontal line of convective cells below the interface for reactive immiscible systems with similar neutralization reactions. Indeed, at first glance, the structures obtained in Eckert et al. (Reference Eckert, Acker and Shi2004) and those presented in our paper look identical. However, there is big difference in the conditions under which they are formed. In Eckert et al. (Reference Eckert, Acker and Shi2004), the DLC that develops above the reaction front is bounded from above in its propagation by an interface that acts as an impenetrable boundary. Thus, the convective motion turns out to be restricted to a narrow layer, which predetermines the formation of a convective structure of the cellular type. In the miscible case, studied by us, the formation of a cellular convective pattern occurs in unbounded liquid due to the formation of a section on the density profile with unstable stratification, localized between two extrema. Thus, the main difference between the results obtained in Eckert et al. (Reference Eckert, Acker and Shi2004) and those presented by us consists in different density profiles formed as a result of the reaction. In the immiscible case, the density profile has one extremum above the reaction front, which is caused by the DLC mechanism. However, the restricted space available for convection to develop imposes the formation of a cellular-type motion. In our case, the DLC mechanism together with the CDD effect results in the formation of two extrema in the density profile above the reaction front which trigger a cellular structure to develop, when the local Rayleigh number exceeds a critical value (see figure 6 in Bratsun et al. (Reference Bratsun, Mizev and Mosheva2021)). We would like to note that the DLC mechanism alone, without taking the CDD effect into account, is not able to result in the formation of such a density profile (see figure 2 in Bratsun et al. (Reference Bratsun, Mizev and Mosheva2021)), and therefore cellular convective structure. This gives us the right to define this phenomenon as a separate type of instability, called by us the CDD instability (Bratsun et al. Reference Bratsun, Kostarev, Mizev and Mosheva2015).

4.5. Transition between the CC and DC modes

Here, we describe the possibility of a transition between the CC and DC regimes in the course of the experiment. In the CC regime, the high conversion rate of the reagents, together with the intensive convective motion in the upper layer, results in the reduction of the upper reagent concentration throughout the entire layer. At that, the concentration of the reagent dissolved in the lower layer remains unchanged. Thus, the ratio of the concentrations of the reagents near the reaction front changes all the time during the experiment performed in the CC regime. Due to this, the point describing the position of the system in the regime map is shifted along the axis towards the lower concentration values of the reagent dissolved in the upper layer. Let us consider for definiteness the experiment with the $\textrm {HNO}_3$–$\textrm {NaOH}$ pair in the situation when the aqueous nitric acid solution is located in the upper layer. We selected the initial state of the system within the parameter area corresponding to the CC mode very near the boundary of the zone, as shown in figure 15($a$).

Figure 15.

Schematic illustration of the dynamic transition from the CC mode (in the form of the SW) to the DC mode (first in the form of DLC instability, then in the form of the CDD cells) during one series of experiments.

Immediately after the solutions were brought into contact, the intensive convective motion developed in the upper layer in complete accordance with the CC regime scenario. Figure 15($b$) presents the temporal evolution of the reaction front coordinate and interferograms obtained experimentally at certain instants of time. It is seen that the development of the convective motion within the CC regime is accompanied by a fast enough front motion. As the acid is consumed during the reaction, the point marking the actual position of the system in the regime map is shifted to the left, which results in the transition of the system to the parameter area corresponding to the DC regime (see an open circle in figure 15($a$)). It is seen that this takes place approximately 300 seconds after the beginning of the experiment. Therefore, the reaction front velocity significantly decreases and a stable transition zone begins to form around the front. In the upper layer, the intensive convective motion is replaced by a weak enough DLC flow typical of the DC regime. Then, at approximately $t\approx 700$ s, a horizontal row of convective cells forms above the reaction front because the system is within the CDD area. This transition is accompanied by a slight increase in the reaction front velocity (figure 15$b$), which is caused by the appearance of an additional mass transfer mechanism above the front due to CDD convection.