Instability in sugar–salt double-diffusive systems

In the beginning of the 1960s, it was shown in a series of experiments that a complex structure, termed as fingers, appears at the interface between the two layers of a double-diffusive system [1], such as when a horizontal layer of sugar solution is placed above a layer of denser salt solution [2].

A qualitative model of the phenomenon is as follows [3]: instability at the sugar–salt interface causes microscopic upwards and downwards protrusions. Since the molecular diffusivity of salt is about 3 times that of sugar, streams of sugar water gain much salt by horizontal diffusion but retain majority of sugar. They become denser and descend, while streams of salt water become less dense and ascend. This results in the formation of fingers over time, which appear as tiny, tube-like structures approximately 1 mm apart. The resulting outcome is the transport of sugar downwards and salt upwards.

The onset of double-diffusive convection has been extensively studied from an experimental and numerical point of view [4–7]. Sorkin et al [8] provides experimental verification of the predicted dependence of width h with length L of the fingers according to the relationship L ∼ h^1/4. Lambert et al [9] provides experimental verification of the 4/3 flux law. Yet, experimental verification of the instability condition is limited.

In section 2, the derivation of the instability condition is presented. In section 3, the experimental set-up and methodology are described. The qualitative model is investigated in section 4, and an explicit verification of the instability condition is presented. The work is concluded in section 5. These experiments can be reproduced in a high school or undergraduate laboratory, providing an introduction to the fields of fluid mechanics and double-diffusive convection for advanced high school or undergraduate students.

The derivation of the instability condition for double-diffusive systems is well-documented in literature [10–13], which we reproduce and summarise here for completeness. This section is intended primarily for educators and advanced students seeking a deeper understanding of the theoretical foundations underlying the instability condition verified in this work. Interested readers may consult standard references such as [14] for an introduction to mathematical tools such as partial derivatives and vector calculus used in the derivation. It is emphasised, however, that this section is not essential for following the qualitative model or experimental results presented later in the paper. Readers may choose to proceed directly to subsequent sections without loss of continuity.

We start with the stable stratification condition for a gravitationally stable system:

$$\begin{equation}{\left( {\beta \Delta S} \right)_0} < {\left( {\alpha \Delta T} \right)_0}\end{equation} \tag{ 1 }$$

where T is the salt concentration, S is the sugar concentration, ${\textrm{ }}\alpha = \frac{1}{{{\rho _0}}}\frac{{\partial \rho }}{{\partial T}}$ is the coefficient of salt cubical expansion, $\beta = \frac{1}{{{\rho _0}}}\frac{{\partial \rho }}{{\partial S}}$ is the coefficient of sugar cubical expansion, ${\left( {\alpha \Delta T} \right)_0}$ is the initial excess density due to salt, and ${\left( {\beta \Delta S} \right)_0}$ is the initial excess density due to sugar.

The fingers that appear at the interface between the two layers can be considered as instabilities growing exponentially in time. Using the continuity, Navier–Stokes and diffusion equations, the perturbations obey:

$$\begin{equation}0 = \nabla \cdot {\boldsymbol{u}}^{^{\prime}}\end{equation} \tag{ 2 }$$

$$\begin{equation}\frac{{\partial {\boldsymbol{u}}}}{{\partial t}} = - \frac{1}{{{\rho _0}}}\nabla \rho ^{\prime} - g\frac{{\rho ^{\prime}}}{{{\rho _0}}}\widehat {\boldsymbol{k}} + v{\nabla ^2}{\boldsymbol{u}^{\prime}}\end{equation} \tag{ 3 }$$

$$\begin{equation}\frac{{\partial T^{\prime}}}{{\partial t}} + w^{\prime}\overline {{T_z}} = {k_T}{\nabla ^2}T^{\prime}\end{equation} \tag{ 4 }$$

$$\begin{equation}\frac{{\partial S^{\prime}}}{{\partial t}} + w^{\prime}\overline {{S_z}} = {k_S}{\nabla ^2}S^{\prime}\end{equation} \tag{ 5 }$$

$$\begin{equation}\frac{{\rho ^{\prime}}}{{{\rho _0}}} = \beta S^{\prime} - \alpha T^{\prime}\end{equation} \tag{ 6 }$$

where $v$ is the kinematic viscosity, ${k_T}$ is the diffusivity of salt (1.54 × 10⁻⁵ cm² s⁻¹), and ${k_S}$ is the diffusivity of sugar (0.52 × 10⁻⁵ cm² s⁻¹).

We consider only linear perturbations proportional to ${{\textrm{e}}^{\lambda t}}\sin \left( {{k_1}x} \right)\sin \left( {{k_2}y} \right)$, since fingers typically exhibit much longer vertical scales than horizontal scales. We then obtain the equations relating $\lambda$ to ${\kappa^2} \equiv k_1^2 + k_2^2$:

$$\begin{equation}\lambda w^{\prime} = - g\left( {\beta S^{\prime} - \alpha T^{\prime}} \right) - v{\kappa^2}w^{\prime}\end{equation} \tag{ 7 }$$

$$\begin{equation}\lambda T^{\prime} + w^{\prime}\overline {{T_z}} = - {k_T}{\kappa^2}T^{\prime}\end{equation} \tag{ 8 }$$

$$\begin{equation}\lambda S{^{^{\prime}}} + w{^{^{\prime}}}\overline {{S_z}} = - {k_S}{\kappa^2}S^{\prime}.\end{equation} \tag{ 9 }$$

Equations (8) and (9) can be expressed in terms of $T^{\prime}$ and $S^{\prime}$, respectively, and after substituting into (7), we obtain a cubic equation in $\lambda$:

$$\begin{align}\lambda w{^{^{\prime}}} & = - g\left[ {\beta \left( {\frac{{ - w{^{^{\prime}}}\overline {{S_z}} }}{{\lambda + {k_S}{\kappa^2}}}} \right) - \alpha \left( {\frac{{ - w{^{^{\prime}}}\overline {{T_z}} }}{{\lambda + {k_T}{\kappa^2}}}} \right)} \right]\nonumber\\ &\quad - v{\kappa^2}w^{\prime}.\end{align} \tag{ 10 }$$

There are 3 solutions for $\lambda$: one real and two complex conjugates. The complex solutions represent slowly-decaying internal waves. The real solution represents double-diffusive instability if $\lambda$ > 0. To determine the instability criterion, we analyse the marginal stability condition, $\lambda$ = 0. We then find:

$$\begin{equation} g\left( {\frac{{\beta \overline {{S_z}} }}{{{k_S}}} - \frac{{\alpha \overline {{T_z}} }}{{{k_T}}}} \right) = v{\kappa^4}.\end{equation} \tag{ 11 }$$

This equation has solutions with $\kappa$ ε $\mathbb{R}$ provided that the LHS is positive. Therefore, we derive the instability condition for the formation of the fingers from infinitesimal perturbations:

$$\begin{equation}\frac{{{k_S}}}{{{k_T}}} < \frac{{{{\left( {\beta \Delta S} \right)}_0}}}{{{{\left( {\alpha \Delta T} \right)}_0}}} < 1.\end{equation} \tag{ 12 }$$

For the sugar–salt double-diffusive system, we obtain [2, 3, 10, 11]:

$$\begin{equation}\frac{1}{3} < \frac{{{{\left( {\beta \Delta S} \right)}_0}}}{{{{\left( {\alpha \Delta T} \right)}_0}}} < 1.\end{equation} \tag{ 13 }$$

Experiments were carried out at room temperature in a Plexiglas tank 20 cm long, 15 cm wide and 22.5 cm deep. The general procedure was to prepare separate solutions of sugar and salt, of the same volume, at the desired densities and mix each of them thoroughly using a magnetic stirrer. The tank was first filled with a layer of the sugar solution. Then, the salt solution was slowly introduced underneath the sugar solution by pouring through a funnel into a tube leading to the base of the tank. The tube is regulated with burette clips to control the flow rate, with the total filling time averaging 15–25 min. The tank was illuminated horizontally from the side using a collimated lamp, and the image of the fingers was projected onto a screen (figures 1 and 2). The illumination from the collimated lamp has a negligible change in diameter over 2 m distance, with intensity variation over the illuminated area not more than 5%. The tank was covered with a lid to minimise evaporation. The fingers were photographed at regular time intervals over a few hours. The fingers appear as polygons as viewed from the top, and columns from the side (figure 3).

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A video clearly illustrating the phenomenon and growth of the fingers over 2 h 30 min is shown (see online supplementary information for video). The increase in the height of the fingers with time are extracted in figure 4. The widths of the fingers are obtained by the following procedure (figure 5): adjust the image contrast to obtain a distinct line profile, then perform fast Fourier transform. The widths are typically found to be less than 1 mm.

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To verify the qualitative model predicting the transport of sugar downwards and transport of salt upwards, the property that sugar is optically active is used [3]. If a laser beam is shone through a tank with sugar solution, its polarization will be rotated by an angle proportional to the concentration of sugar. The tank can be gently lowered or raised using a lab jack, thus setting the laser beam at different depths [3]. Thus, horizontal averaged vertical profiles of sugar concentration can be obtained (figure 6).

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The polarimeter was first calibrated to ensure that the rotation of the plane of polarization of the laser beam was proportional to sugar concentration (figure 7). Then, the vertical profiles of sugar concentration at three different times for the top and bottom layers were plotted (figure 8). It is found that the difference in sugar concentration between the top and bottom layers, $\Delta S$, has decreased from 0.139 g cm⁻³ at 38 min, to 0.095 g cm⁻³ at 112 min, to 0.072 g cm⁻³ at 252 min. This confirms that sugar is indeed transported downwards via the finger interface over time, in accordance with the qualitative model [3].

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The dependence of the growth of the fingers on different sugar and salt concentrations are investigated in figures 9(a) and 9(b) respectively. It is found that an increase in sugar concentration leads to an increase in the growth rate of the fingers, while an increase in salt concentration leads to a decrease in the growth rate of the fingers.

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These trends can be explained with the qualitative model. Sugar columns grow downward due to gravitational forces while salt columns grow upward due to buoyancy forces. For fingers to grow, salt columns need to be less dense while sugar columns need to be denser. With larger salt concentration or smaller sugar concentration, the system is overall more gravitationally stable, causing the slower growth of the fingers.

Having investigated the qualitative model, we now provide an explicit experimental verification of the instability condition. A phase diagram with different values of ${\left( {\beta \Delta S} \right)_0}$/${\left( {\alpha \Delta T} \right)_0}$ is plotted in figure 10. The phase diagram shows broad agreement with equation (13), with the formation of fingers generally observed when the instability condition is satisfied. The experimental deviations from the theoretical prediction at the lower boundary are interesting, which reflect the limitations of the idealized linear analysis (valid for infinitesimally small perturbations), as well as the effects of imperfect initial layering in the experiment, where mixing and diffusion may have already altered the starting concentration gradient.

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This paper has revisited a classic problem in fluid dynamics, reintroducing it into the context of physics education through a relatively simple and cheap experiment, following [3]. By examining how the growth of sugar–salt double diffusive fingers depends on varying concentrations of sugar and salt, and confirming the associated downward flux of sugar, the study offers empirical support for the qualitative model. Then, an experimental verification of the instability condition for the sugar–salt double-diffusive system is presented.

This work showcases an example of a laboratory project that will expose advanced high school or undergraduate students to deeper concepts in physical chemistry and fluid mechanics, piquing their interest in this field. The apparatus required can be assembled at relatively low cost, allowing the vivid demonstration of the intriguing growth of sugar–salt fingers, which serves as further motivation for students to deepen their understanding of the theory. This experiment is hoped to be of interest to a range of students including budding physical chemists, physicists, and chemical engineers.

For future extension, experimental verification of the instability condition for the sugar-heat double-diffusive system (ratio of diffusivities ≈ 300), or in the presence of other solvents with different viscosities, can be conducted. It will also be interesting to investigate what happens when a layer of salt water is instead placed above a layer of sugar water, where a series of oscillatory motions are expected.

Finally, it should be noted that a related phenomenon manifests on a global scale where salt fingers lead to thermohaline circulation in the world’s oceans [10, 15, 16]. This illustrates how experimental investigations performed in the laboratory can relate to complex real-life problems, inspiring students in scientific research, in the spirit of the IYPT.

This work was conducted in Raffles Institution, Singapore. I thank Mr Sze Guan Kheng, Mrs Lim Siew Eng, and Mr Jee Kai Yen, for consultations and support.

All data that support the findings of this study are included within the article (and any supplementary files).