ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2016, Vol. 57, No. 7, pp. 1226–1238. © Pleiades Publishing, Ltd., 2016. Original Russian Text © E.V. Aitova, D.A. Bratsun, K.G. Kostarev, A.I. Mizev, E.A. Mosheva, 2015, published in Vychislitel’naya Mekhanika Sploshnykh Sred, 2015, Vol. 8, No. 4, pp. 345–358.
Convective Instability in a Two-Layer System of Reacting Fluids with Concentration-Dependent Diffusion E. V. Aitovaa*, D. A. Bratsuna,c**, K. G. Kostarevb***, A. I. Mizevb****, and E. A. Moshevab***** a
Perm State Humanitarian–Pedagogical University, Perm, 614990 Russia Institute of Continuous Media Mechanics, Ural Branch, Russian Academy of Sciences, ul. Akademika Koroleva 1, Perm, 614000 Russia c Perm National Research Polytechnic University, Komsomol’skii pr. 29, Perm, 614990 Russia e-mail: *
[email protected], **
[email protected], ***
[email protected], ****
[email protected], *****
[email protected] b
Received September 4, 2015; in final form, December 30, 2015
Abstract—The development of convective instability in a two-layer system of miscible fluids placed in a narrow vertical gap has been studied theoretically and experimentally. The upper and lower layers are formed with aqueous solutions of acid and base, respectively. When the layers are brought into contact, the frontal neutralization reaction begins. We have found experimentally a new type of convective instability, which is characterized by the spatial localization and the periodicity of the structure observed for the first time in the miscible systems. We have tested a number of different acid–base systems and have found a similar patterning there. In our opinion, it may indicate that the discovered effect is of a general nature and should be taken into account in reaction–diffusion–convection problems as another tool with which the reaction can govern the movement of the reacting fluids. We have shown that, at least in one case (aqueous solutions of nitric acid and sodium hydroxide), a new type of instability called as the concentration-dependent diffusion convection is responsible for the onset of the fluid flow. It arises when the diffusion coefficients of species are different and depend on their concentrations. This type of instability can be attributed to a variety of double-diffusion convection. A mathematical model of the new phenomenon has been developed using the system of reaction–diffusion–convection equations written in the Hele–Shaw approximation. It is shown that the instability can be reproduced in the numerical experiment if only one takes into account the concentration dependence of the diffusion coefficients of the reagents. The dynamics of the base state, its linear stability and nonlinear development of the instability are presented. It is also shown that by varying the concentration of acid in the upper layer one can achieve the occurrence of chemo-convective solitary cell in the bulk of an almost immobile fluid. Good agreement between the experimental data and the results of numerical simulations is observed. Keywords: convective instability, neutralization reaction, nonlinear diffusion, miscible fluids DOI: 10.1134/S0021894416070026
1. INTRODUCTION The chemohydrodynamic structure formation in a system of two reacting fluids was experimentally observed for the first time likely in 1888 by Quincke [1]. He observed spontaneous emulsification at the interface between the oil solution of lauric acid and the aqueous solution of sodium hydroxide in the process of neutralization reaction. However, the really consistent detailed study of the processes of diffusion and hydrodynamic phenomena began only after a century because of appearance of important technological applications such as petroleum processing [2], combustion processes [3], and separation of uranium ore [4]. The mutual effect of chemical reactions and heat mass transfer in fluids is of special interest because reactions can significantly change the density, viscosity, heat transfer, and surface tension of fluids. This promotes the formation of dissipative structures of a new type transforming the initial distribution of reagents up to a change in the character of the reaction. The hydrodynamics of the exothermic neutralization reaction has been actively studied in recent years because of a comparatively simple but nonlinear kinetics of the reaction and numerous promising appli1226
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cations. It appeared that the development of the reaction in miscible systems is significantly different from that in immiscible systems. Two-layer systems of immiscible reacting solutions were considered in [5–11]. The chemisorption of carbon dioxide by an alkali is characterized by the same kinetics of the neutralization reaction [12, 13], which occurs at the “gas–fluid” interface. In particular, the experimental works [5, 6, 8] describe new effects associated with the interaction of the exothermic neutralization reaction and the “fluid–fluid” interface of the two-layer system placed in a Hele–Shaw cell. One layer of this system is an acid solved in an organic solvent and the second layer is an aqueous solution of the alkali. We revealed that different structures could be obtained by varying types of reagents and their initial concentrations. In particular, the authors of [5] observed the disordered propagation of finger convective structures in the lower layer in the reaction of carboxylic acid (alcoholic solution) with an aqueous solution of sodium hydroxyl, which is typical of such systems. Since acid coming from the upper layer is heavier than the base, the behavior of the system was easily explained by the Rayleigh–Taylor instability [6, 7]. Such a structure formation also occurs in the lower (liquid) phase at the chemisorption of CO2 by an alkali [12], which can also be attributed to the development of the mentioned instability [13]. However, the authors of [8] surprisingly obtained an ideally periodic system of chemoconvective cells slowly growing from an interface between the solution of propionic acid and aqueous solution of a more complex base—tetramethylammonium hydroxide (TMAH). It was theoretically demonstrated in [9] that the extraordinary regularity of the system is due to the balance between by the Rayleigh–Taylor and Rayleigh–Bénard instabilities. Surface instability mechanisms associated with the deformation of the interface [10], change in the surface tension owing to either heat release [11] or difference between concentrations of the reagents [11] or different rates of the adsorption–desorption processes [14] can also play an important role in the formation of a convective flow in immiscible solutions. The listed effects are involved in the fine adjustment of the configuration of chemoconvective cells equalizing them along the interface upward and over the ends of salt fingers downward. The situation with the study of miscible systems of reacting fluids was recently different [15–19]. In this case, where the interface is absent, the main causes of the development of chemoconvection are gravitational mechanisms of motion owing to the formation of regions with the instable stratification of the density because of different rates of diffusion of reagents [15, 16]. In binary systems, depending on the relation between the diffusion coefficients of the components, double-diffusion instability or DD convection, diffusive layer convection or DLC, and delayed double-diffusion instability or DDD convection can occur. It is also necessary to take into account the appearance of the third component—reaction product—which is formed at the reaction front and diffuses into the bulk of the initial reagents, which can also initiate the appearance of double-diffusion instability. A number of experimental and theoretical studies [16–19] of these phenomena revealed chemoconvective structures with a high degree of irregularity of salt fingers, which propagated on both sides of the interface between the layers. It is noteworthy that all known studies of chemoconvection imply the constant diffusion coefficients of reagents, whereas the actual diffusion coefficients always depend on the concentration of the solution. However, this dependence in hydrodynamics is usually neglected as being weak. Concentration-dependent diffusion coefficients were rarely considered only in pure reaction–diffusion problems without convection (see, e.g., [20, 21]), but only quantitative rather than qualitative changes were considered in those works. This work continues study [22] and concerns miscible reacting systems, where the structure formation is revealed for the first time in the form of an ideally periodic system of chemoconvective cells, which are formed parallel to the reaction front and perpendicular to the direction of the gravitational force. It is shown that the macroscopic motion of the fluid appears through a specific instability mechanism, which is due to the dependence of the diffusion coefficients of the reagents on their concentrations. Since this type of instability is described for the first time, we call it concentration-dependent diffusion instability or CDD convection. Some properties of this new type of instability are theoretically and experimentally studied and we present reasons why CDD convection has been yet unknown. The existence of convective cells surrounded by a quasi-immobile fluid is discussed for the first time. 2. THEORETICAL DESCRIPTION OF CDD CONVECTION 2.1. Mathematical Model Since the phenomenon significantly depends on laws of diffusion of the reagents, the theoretical description will be given for a particular system. Let two miscible fluids at the initial time fill the upper and lower layers in the Hele–Shaw cell. The upper layer is filled with the aqueous solution of nitric acid HNO3 with the concentration A , whereas the lower layer is filled with the aqueous solution of sodium JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS
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hydroxide NaOH with the concentration B . Chemical kinetics in this system is as follows. Acid in the aqueous solution dissociates into the hydroxonium cation and the acid radical anion: NO2OH + H2O → NO2O– + H3O+, and the base decays as follows:
Na OH → Na + + OH − . Then, H 3O + cations find anions OH − and form water, whereas base cations meeting acid radical anions form a salt (sodium nitrate with the concentration S ): NO2O– + Na+ → NaNO3.
(1)
Thus, the neutralization reaction occurs at the interface between the solutions, generally with heat release. However, we consider only an isothermal problem in this work, and the heat release effect will be disregarded in the equations. This assumption is based on experimental data and, furthermore, the walls of a cuvette can always be made quite heat conductive for heat to dissipate and to hardly affect the processes. The resulting simplified kinetics of reaction (1) can be represented in the form
A + B → S.
(2)
Since both reagents are dissolved in water, their mixing begins at the contact of initially separated layers. It is noteworthy that the problem is nonautonomous, because reagents are not added in the reaction and concentration profiles change irreversibly. We list the main assumptions underlying the theoretical model: —the gap h between wide walls is narrow enough to assume that the flow of a fluid in the Hele–Shaw assumption is almost two-dimensional; —the properties of the fluids are independent of the concentrations of the reagents except for the case mentioned below; —the problem is isothermal; —the rate of the neutralization reaction K does not change; —the initial concentrations of acid A0 and base B0 are the same. Let x and z be coordinates along the wide faces of the Hele–Shaw cell and the z = 0 plane is the initial interface between the layers. The boundaries of the cell are specified as 0 ≤ x ≤ H and −L ≤ z ≤ L . We represent the length in h, time in h2/Da0, velocity in D a0 h , pressure in ρ 0ν Da0 h 2 , and concentration in A 0 . Here, Da0 is the tabulated diffusion coefficient of nitric acid in water at a temperature of 25°C and ρ 0 and ν are the density and kinematic viscosity of water, respectively. Under the above assumptions, the convection–reaction–diffusion equations in the Hele–Shaw approximation have the dimensionless form
1 ⎛ ∂Φ + 6 ∂(Ψ, Φ) ⎞ = ∇ 2Φ − 12Φ − R ∂ A − R ∂ B − R ∂ S , a b s ⎜ ⎟ Sc ⎝ ∂ t 5 ∂(z, x) ⎠ ∂x ∂x ∂x
(3)
∂ A + ∂(Ψ, A) = ∇ D ( A)∇ A − α AB , a ∂t ∂(z, x)
(4)
∂ B + ∂(Ψ, B ) = ∇ D (B )∇ B − α AB , b ∂t ∂(z, x)
(5)
∂ S + ∂(Ψ, S ) = ∇ D (S )∇ S + α AB . s ∂ t ∂(z, x)
(6)
Here, the two-field representation of the equation of motion was used, Ψ is the stream function, Φ = −ΔΨ is vorticity, and the reaction terms are based on kinetic equation (2). Equation (3) differs from the standard Navier–Stokes equation in an additional term proportional to the vorticity, which describes the hydrodynamic drag of wide faces of the Hele–Shaw cell. The action of the faces is similar to the Darcy force in a porous medium. The diffusion terms in Eqs. (4)–(6) are represented in the most general form [23], allowing the dependence of the coefficients on the concentrations of the corresponding substances. JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS
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D × 105, cm2/s 5.0 Yeh, Wills (1971), HNO3 Nisancioglu, Newman (1973), HNO3 Chapman (1967), HNO3 Fary (1966), NaOH Noulty, Leaist (1984), NaOH Yeh, Wills (1970), HNO3 Harned, Shropshire (1958), NaNO3
4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0
0.5
1.0
1.5
2.0
2.5
3.0 C, mol/L
Fig. 1. Diffusion coefficients of (△, ◇, ♦) nitric acid, (■, □) sodium hydroxide, and (○, d) sodium nitrate versus their concentrations up to 3 mol/L at a temperature of 25°C according to the experimental data reported in [24–30]. Solid lines are the least squares approximations of the corresponding experimental data.
Equations (3)–(6) are supplemented by the boundary conditions
z = ±L :
x = 0, H :
Ψ = 0,
∂Ψ = 0, ∂z
∂A = 0, ∂z
∂Ψ = 0, ∂x
∂A = 0, ∂x
z Rb > Ra ), but it is simultaneously the least mobile because it has the smallest diffusion coefficient, see Eqs. (12). Salt in the process of reaction is concentrated near the reaction front, weakly diffusing from this region. However, only these obvious reasons are insufficient to explain an unusual density profile shown in Fig. 2a. Figure 2b shows the total density profiles calculated (line 1) with the constant diffusion coefficients given in Eqs. (12) and (line 2) with the data obtained using functional dependences (11). It is clearly seen that the difference of the density profile in the former case from the known profile for classical DLC instability is small rather than qualitative. Only allowance for the dependence of the diffusion coefficients on the concentrations makes it possible to obtain a local maximum of the density at the reaction front (Fig. 2b; line 2), which qualitatively changes the structure formation in the system. The analysis of Fig. 1 provides understanding the process of reaction. First, salt is released at the front and its amount increases with the time. With an increase in the concentration, the diffusion coefficient of salt decreases noticeably, complicating its removal from the region of the reaction front. However, the collection of salt does not terminate the reaction, because rapidly diffusing acid penetrates to the region below the reaction front and the diffusion coefficient of nitric acid increases with its concentration. Thus, the second minimum of the density appears only because of the concentration dependence of diffusion coefficients. For the corresponding convective instability, we propose the term “concentration-dependent diffusion instability or CDD convection.” The density minimum located above the front and the corresponding rising concentration plumes have a traditional origin: they are the manifestations of the DLC mechanism. We now analyze the stability of the base state specified by Eqs. (14)–(16) with respect to small monotonic perturbations:
⎡Φ(t, x, z )⎤ ⎡ 0 ⎤ ⎡ ϕ(t, x, z) ⎤ ⎢Ψ(t, x, z )⎥ ⎢ 0 ⎥ ⎢ψ(t, x, z)⎥ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ + ⎢ a(t, x, z) ⎥ exp(Ikx) , ( , ) A t z (17) ( , , ) A t x z = ⎢ ⎥ ⎢ B(t, x, z ) ⎥ ⎢B 0(t, z)⎥ ⎢ b(t, x, z) ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ 0 ⎢ S ( t , x , z ) ⎣ ⎦ ⎣S (t, z)⎥⎦ ⎣ s(t, x, z) ⎦ where ϕ, ψ , a, b , and s are the corresponding amplitudes of perturbations, k is the wavenumber, and I is the imaginary unit. JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS
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Substituting expansions (17) into Eqs. (3)–(10) and linearizing these equations near the base state specified by Eqs. (14)–(16), we obtain the following system of time-dependent amplitude equations for the determination of critical perturbations:
1 ∂ϕ = Δϕ − 12ϕ − k 2(R a + R b + R s), a b s Sc ∂ t
(18)
0 0 2 0 0 ∂ a = D ( A 0 )Δ a + dDa ( A ) ⎛ 2 ∂ A (z, t ) ∂ a + a ∂ A (z, t ) ⎞ − α A 0(z, t )b − α B 0(z, t )a + ψ ∂ A (z, t ), a ⎜ ⎟ 0 2 ∂t ∂z ∂z ∂z dA ⎝ ∂z ⎠
(19)
0 0 2 0 0 ∂ b = D (B 0 )Δ b + dD b (B ) ⎛ 2 ∂ B (z, t ) ∂ b + b ∂ B (z, t ) ⎞ − α A 0(z, t )b − α B 0(z, t )a + ψ ∂ B (z, t ) , b ⎜ ⎟ 0 2 ∂t ∂z ∂z ∂z dB ∂z ⎝ ⎠
(20)
0 0 2 0 0 ∂ s = D (S 0 )Δ s + dDs (S ) ⎛ 2 ∂ S (z, t ) ∂ s + s ∂ S (z, t ) ⎞ + α A 0(z, t )b + α B 0(z, t )a + ψ ∂ S (z, t ) s ⎜ ⎟ 0 2 ∂t ∂z ∂z ∂z dS ⎝ ∂z ⎠ with the boundary conditions
(21)
z = ±L :
ϕ = 0,
ψ=
∂ψ = 0, ∂x
a = 0,
b = 0,
s = 0,
(22)
x = 0, H :
ϕ = 0,
ψ=
∂ψ = 0, ∂x
a = 0,
b = 0,
s = 0.
(23)
Here, Δ ≡ ∂ 2 ∂ z 2 − k 2. Thus, the total spectral amplitude problem includes equations (18)–(23) for the determination of perturbations, the system of equations (14)–(16) for the determination of the base state, and laws of diffusion (11). A nonstandard feature of the formulated problem is the time dependence of not only the amplitudes of perturbations but also the base state given by Eqs. (14)–(16). To solve the problem, we used the initial value problem method [11]. It is known that this method provides adequate results always except for a short initial time interval, when the base state varies rapidly. In this work, the problem of the initial time interval is solved automatically because critically growing perturbations appear only in a certain time after the beginning of evolution that is certainly larger than the relaxation time in the initial value problem method. The general scheme of the numerical analysis included the joint time integration of Eqs. (14)–(16) for the base state and Eqs. (18)–(23) for small normal perturbations for a fixed wavenumber by the finite difference technique. The following quantity λ(t ), which has the meaning of the Lyapunov exponential, was calculated in the process of evolution: N
λ(t ) = 1 N
∑ Δ1t ln j =1
a j (t + Δt ) , a j (t )
where Δt is the integration step, N is the number of independent realizations (usually 10–15), and a(t ) are the perturbations of the concentration field of acid. Each independent integration of the system began with random initial conditions for all perturbations with amplitudes of no more than 10–4. The occurrence of instability (or exit from it) was identified by a change in the sign of the increment λ(t ) averaged over realizations. Figure 3a shows neutral curves for (dotted line) DLC instability and (solid line) CDD instability. Since the system is nonautonomous, the time is a parameter of the problem. It is seen in the figure that both modes are stable at the very beginning of evolution. At the time t ≈ 0.15, perturbation with the wavenumber k ≈ 4.6 , which corresponds to concentration-dependent diffusion, first loses stability. DLC instability appears approximately at t ≈ 0.25 and is characterized by the critical wavenumber k ≈ 0.75 because the slope of the density profile for DLC instability is smaller (Fig. 2a). A significant difference between initial perturbations is caused by different widths of instability regions (Fig. 2a). With the time, both instabilities are shifted toward longer wavelengths, which is due to the gradual diffusion-induced expansion of the convective region. The rate of growth of perturbations in the dependence on the wavenumber can be estimated from Fig. 3b, where the evolution of the maximum perturbation of the stream function for CDD instability is JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS
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ψ 10−5 (a)
k = 0.5 k = 1.0 k = 2.0 k = 3.0 k = 4.0 k = 5.0
(b) 10−6
1 10−7 0.1 0.01
CDD instability
10−8
DLC instability
0.1
1
10 k
0.1
1 t
Fig. 3. (a) Neutral curves for two modes of instability: (dotted line) diffusion layer convection and (solid line) CDD instability. (b) Evolution dependence of the maximum perturbation of the stream function ψ for various wavenumbers in the case of CDD instability.
shown. It is clearly seen that perturbations begin to grow exponentially at the entry into the bag of instability. 2.3. Numerical Results for the Total Nonlinear Problem We now discuss the results of direct numerical simulation of the problem specified by Eqs. (3)–(11). In this work, to obtain a numerical solution, we used the finite difference technique, which is described in detail in [9]. In our case, the general calculation scheme is even simpler because the fluids are mixed and the free surface is absent. We used the explicit scheme, where the stability of the method was ensured by the choice of the time step in the form:
Δt =
Δz 2 . 2(2 + max( Ψ , Φ ))
The accuracy of the integration of the Poisson equation was 10 −4 . Calculations were performed on a uniform grid, where a unit square region included 6 × 6 sites (Δ x = Δ z = 0.2). Such a resolution was chosen on the basis of the preceding experience of the numerical simulation of chemoconvective structures in the calculation domain of a similar geometry [6, 7, 9]. A random distribution of the stream function with an amplitude of no more than 10 −3 was specified as the initial condition. Figure 4 shows the vorticity contours and the total density field of the medium at the time t = 3 , when both types of perturbations have been already formed and become finite. An important feature of the system is independence of instabilities: both types of perturbations appear in different parts of the calculation domain and hardly interact with each other at the beginning of evolution. It was observed that the convective motion below the initial interface (line with the coordinate z = 0 in Fig. 4) is first exited. The wavelength of perturbation at the initial time is in good agreement with the linear analysis (see Fig. 3). The density field in Fig. 4b illustrates the nonlinear development of perturbations: a layer of the comparatively heavy fluid at the very beginning of evolution overhangs a layer of the light fluid, thus creating conditions for the appearance of instability, whereas a system of salt drops is formed below the z = 0 line already at t = 3. Since the maximum density near the reaction front is local, the structure is equalized owing to the denser medium located below. Thus, local chemoconvective motion is observed in the region between two massive immobile fluids (this is clearly seen in Fig. 4a). As was mentioned above, the instability band is slowly expanded because of the diffusion of the reagents (see Fig. 2), and the downward motion of heavysalt-enriched finger structures follows this expansion. The traditional convective motion in the form of rising concentration plumes occurs in the upper instability band. Their upward propagation becomes disordered very soon and is limited only the size of the calculation domain. The convection of the upper diffusion layer (DLC instability) has a larger wavelength and slightly affects the form of concentration fields at the beginning of evolution (Fig. 4). Both instabilities are separated by a thin layer of an almost immobile fluid, which can be deformed but holds integrity under convection conditions. This layer exist because of JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS
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z 10
(а)
z 10
5
5
0
0
−5
−5
−10
0
10
20
30
40
50 x
−10
(b)
0
1200
10 1300
20 1400
1500
30 1600
50 x
40 1700
1800
1900
Fig. 4. (a) Vorticity field Φ( x, z) and (b) total dimensionless density ρ( x, z) at the time t = 3 obtained by the numerical solution of the total nonlinear problem given by Eqs. (3)–(11) for the fontal neutralization reaction.
z 10
(а)
z 10
5
5
0
0
−5
−5
−10
0
10
20
30
40
50 x
−10 0 1200
(b)
10 1300
20 1400
1500
30 1600
50 x
40 1700
1800
1900
Fig. 5. (a) Vorticity field Φ( x, z) and (b) total dimensionless density ρ( x, z) at the time t = 3 obtained by the numerical solution of the total nonlinear problem given by Eqs. (3)–(11) for the neutralization reaction occurring only in the segment 30 < x < 40 of the contact line.
the presence of a narrow region above the reaction front with the stable configuration of the density profile (Fig. 2). The numerical study of the frontal neutralization reaction (Fig. 4) remarkably showed that chemoconvection can occur and exist for a quite long time in a local region of an immobile continuous medium. This is due to the formation of the density “pocket” inside which convection is developed. The formation of a localized convective structure is very surprising for classical thermal convection, which is rapidly expanded to the entire working space of the cuvette. To simulate the chemoconvective cell limited by density barriers from all sides, we calculated the reaction occurring only in a certain segment of the interface. This formulation reproduces a situation with two counterpropagating jets of reagents whose densities coincide with the densities of the corresponding layers in two mixing nonreacting fluids. Reagents flow to each other (acid from above and base from below) in the band of 30 < x < 40 . Thus, the reaction at the initial time occurs only at the intersection of this band and z = 0 interface. Further, the diffusion of reagents results in the bulk reaction. Figure 5 shows the vorticity contours and total density field of the medium at the time t = 3. It is clearly seen that edge effects affect the shape of the reaction front below which CDD instability appears locally again and above which two rising plumes of DLC convection are developed (Fig. 5b). The strongest convective vortices appear in the upper layer, where vertically oriented density gradients occur (Fig. 5a). 3. EXPERIMENTAL RESULTS The experiment was performed in the vertical Hele–Shaw cell formed by two plane–parallel glasses separated by a spacer specifying the inner sizes of the cavity 9.0 cm × 2.4 cm × 0.12 cm. Shallow horizontal grooves for a diaphragm separating the initial reagents were made in walls of the cuvette along its middle JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS
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(а)
(b)
(c)
(d)
(e)
(f)
1
(g)
5
(h)
10
(i)
Fig. 6. (Left panels) Visualized flow structure, (middle panels) concentration distribution, and (right panels) pH level for three times from the beginning of the experiment (a–c) 300 , (d–f) 1100, and (g–i) 2500 s.
plane. The reacting fluids were aqueous solutions of nitric acid (upper layer) and of sodium hydroxide (lower layer) in the equimolar concentration. For the start of the reaction, the diaphragm was removed from the cavity of the cell and fluids contacted each other forming the horizontal reaction front. For the complex study of processes accompanying the reaction, we used several experimental methods simultaneously. In particular, an autocollimation Fizeau interferometer visualized the distribution of the JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS
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λ, mm z, mm 9 (a) 0 8 7 −5 6 5 −10 4 3 −15 2 Experimental wavelength 1 Theoretical wavelength 0 −20 1000 1500 2000 2500 3000 3500 4000 0 t, s
(b)
Experimental upper boundary Theoretical upper boundary Experimental lower boundary Theoretical lower boundary
500
1000
1500
2000 t, s
Fig. 7. Evolution of the (a) wavelength and (b) boundaries of the chemical structure according to the (symbols) experiment and (lines) theory.
refractive index caused both by the variation of the concentrations of substances involved in the reaction and by the heat release in the exothermic reaction. The maximum change in the temperature measured by a thermocouple probe near the reaction front was no more than 1 K; consequently, the resulting interference patterns provide the refractive index field generally representing the distribution of the general density of the mixture in an almost isothermal situation. The addition of light-scattering particles to the solutions of reagents and the use of a light knife made it possible to determine the structure of the appearing convective motion. The spatial distribution of the reagents and the reaction product was controlled by means of pH indicators. Figure 6 illustrates the evolution of the neutralization reaction in the chosen system of fluids. In particular, immediately after contact between them, a transient region including the reaction front appears (Figs. 6a–6c, t = 300 s). The transient region itself remains immobile, whereas the transfer of the reagents and the reaction product in it is ensured by diffusion. With the time, a thin layer with a low concentration of acid, which is burned in the reaction, appears in the region immediately above the transient region. As a result, an unstable stratification of the density is created in the lower part of the upper layer and a weak convective motion begins in the entire upper layer. In several minutes after the beginning of the experiment, the convective motion but in the form of a horizontal array of convective cells is also developed in the transient region (Figs. 6d–6f, t = 1100 s). This convective structure exists between two immobile parts of the transient region, which indicates the formation of a localized density pocket with an unstable density distribution. The distribution of the pH level inside the layer is uniform, whereas this distribution in the region of cells is more complex: rising flows are enriched in the reaction product formed at the reaction front. On the contrary, descending flows are enriched in acid captured by the convective motion from the upper part of the transient region. The reaction product is collected primarily in the region of cells. The detected structure exists for several hours. With the time, the cells remain inside the transient region, their sizes increase, but their number decreases (Figs. 6g–6i, t = 2250 s). The vertical size of the transient region also increases with the time owing to diffusion. We emphasize good agreement between the experimental data and theoretical calculation both for the wavelength of concentration-dependent diffusion instability (Fig. 7a) and for the dynamics of expansion of the local convective region (Fig. 7b). 4. DISCUSSION AND CONCLUSIONS At least two important new results have been obtained in this work. The appearance of chemoconvection has been demonstrated for the first time inside an immobile fluid. A similar structure was observed experimentally in [8], where a system of two immiscible reacting fluids with the pronounced interface between them was studied. As was shown in a number of theoretical works [7, 9, 11], the interface plays an important role in the arrangement of the cells. In this work, the cells are equalized because of the formation of a local density pocket, where convection is developed. Beyond this region, all convective perturbations damp. In the classical convection problems, when the inhomogeneity of the density of the medium was specified by varying the boundary conditions or external actions (inhomogeneous heating, JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS
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flow of a substance, external force field, etc.), the local existence of convection is fundamentally impossible because inhomogeneity is formed in the entire space filled with the fluid. Convective motion can be localized inside the density pocket only if the change in the density responsible for this motion is also local. A chemical reaction is ideally appropriate for this aim because the concentrations of substances and the density of the medium vary sharply near the always-local reaction front. The simple mixing of substances without a reaction between them can hardly form the local density pocket because diffusion initially creates the nonlocal mass transfer of substances to completely equalizing the concentrations. This is possibly why the classical types of double-diffusion convection (DD, DDD, and DLC [15, 16]) in the presence of gravitation also always lead to the nonlocal motion of the fluid. The second achievement of this work is the discovery of a new type of instability—concentrationdependent diffusion (CDD) instability—in the double-diffusion convection family. It has been shown that the motion of the fluid in the initial two-component system appears because of the dependence of diffusion coefficients of the reagents and the reaction product on their concentrations. The mechanism of instability can be briefly described as follows. Since the diffusion coefficient of salt formed in the reaction decreases significantly with an increase in its concentration, the reaction product is concentrated near the front. The reaction in the bulk does not stop in this case, but is slowed slightly because the diffusion coefficient of acid decreases with a decrease in its concentration. Both these facts are responsible for the formation of conditions for the local development of convective motion near the reaction front. This physical mechanism is discussed for the first time in hydrodynamics. The reason is that the gradients of the concentrations of substances are weakly manifested at convection in systems of concentrated solutions because diffusion rapidly equalizes them in space. This effect can be completely manifested only in the presence of a chemical reaction, where the gradients of the concentrations of reagents are large near the front. Since chemoconvection is a comparatively new field in hydrodynamics, the described phenomena not immediately attracted attention of researchers. Although both results are connected to each other, they are not identical. The numerical experiments have demonstrated that the density pocket can be formed in the process of reaction even without variations of the diffusion coefficients of reagents. To summarize, both effects require further independent studies. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (project nos. 13-01-00508-a and 14-01-96021-r_ural_a) and the Ministry of Education and Science of the Permskii krai (assignment no. S-26/004.4). REFERENCES 1. Quincke, G., Über periodische Ausbreitung an Flüssigkeitsoberflächen und dadurch hervorgerufene Bewegungserscheinungen, Ann. Phys., 1888, vol. 271, no. 12, pp. 580–642. 2. Dupeyrat, M. and Nakache, E., Direct conversion of chemical energy into mechanical energy at an oil water interface, Bioelectroch. Bioener., 1978, vol. 5, no. 1, pp. 134–141. 3. Kolesnikov, A.K., Thermal explosion in a layer with boundaries at different temperatures in the case of transverse reagent motion, Fiz. Goreniya Vzryva, 1984, vol. 20, no. 3, pp. 64–65. 4. Thomson, P.J., Batey, W., and Watson, R.J., Interfacial activity in the two phase systems UO2(NO3)2/Pu(NO3)4/HNO3-H2O-TBP/OK, in Proceedings of the Extraction’84, Symposium on Liquid–Liquid Extraction Science, Scotland, Dounreay, November 27–29, 1984, vol. 88, pp. 231–244. 5. Eckert, K. and Grahn, A., Plume and finger regimes driven by an exothermic interfacial reaction, Phys. Rev. Lett., 1999, vol. 82, no. 22, pp. 4436–4439. 6. Bratsun, D.A. and de Wit, A., Control of chemoconvective structures in a slab reactor, Tech. Phys., 2008, vol. 53, no. 2, pp. 146–153. 7. Bratsun, D.A. and de Wit, A., Buoyancy-driven pattern formation in reactive immiscible two-layer systems, Chem. Eng. Sci., 2011, vol. 66, no. 22, pp. 5723–5734. 8. Eckert, K., Acker, M., and Shi, Y., Chemical pattern formation driven by a neutralization reaction. Mechanism and basic features, Phys. Fluids, 2004, vol. 16, no. 2, pp. 385–399. 9. Bratsun, D.A., On Rayleigh–Bénard mechanism of alignment of salt fingers in reactive immiscible two-layer systems, Micrograv. Sci. Tech., 2014, vol. 26, no. 5, 293–303. 10. Shi, Y. and Eckert, K., Orientation-dependent hydrodynamic instabilities from chemo-Marangoni cells to large scale interfacial deformations, Chin. J. Chem. Eng., 2007, vol. 15, no. 5, pp. 748–753. JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS
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Translated by R. Tyapaev
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