Floquet Instability of Gravity-Modulated Salt Fingering in a Porous

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Floquet Instability of Gravity-Modulated Salt Fingering in a Porous Medium S. Saravanan* and S. Keerthana Department of Mathematics, Bharathiar University, Coimbatore 641 046, Tamil Nadu, India ABSTRACT: Fingering convection in a horizontal doubly stratified fluid-saturated porous medium is examined under the influence of modulating gravitational field. The Brinkman model of momentum transfer is considered, and the Boussinesq approximation is assumed for natural convection. Floquet theory is employed to determine the onset condition within the framework of linear theory. Continued fractions that can handle arbitrary modulational parameters has been used to predict the dynamic instability regions at the marginal state and critical instability boundaries. Certain combinations of solutal Rayleigh and Schmidt numbers give rise to the occurrence of doubly unstable regions. An increase in the modulation amplitude always encourages instability. The competition between synchronous and subharmonic instability modes is observed for a certain range of values of the parameters. Solutal convection is found to introduce closed disconnected instability loops at the marginal state. It is seen that gravity modulation effects disappear for higher modulational frequencies. The results could be of use in applications like enhanced oil recovery from geothermal reservoirs and solidification of binary alloys, among others.

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Nield5 was the first to examine the possibility of convection with double diffusion in porous media. Using linear perturbation analysis, he derived conditions for the onset of such a flow. Later, Tauton et al.6 extended his analysis and predicted that the wavenumber at the marginal state of diffusive convection depends on the concentration Rayleigh number alone whereas that of fingering convection also depends on the thermal Rayleigh number. A finite amplitude analysis of fingering convection in a fluid layer was examined by Straus,7 and it was found that small-scale motions alone are stable when the salinity gradient is larger than that required for the onset condition, and the wavelength of the most stable state agrees well with the wavelength that maximizes the salt flux. In an interesting work, Chen and Chen8 showed that the steady onset of fingering convection in a doubly diffusive layer is similar to Taylor−Couette flow in the small gap limit. Through an elaborate computational study, they predicted the occurrence of oscillations in nonlinear fingering convection in porous media as the solutal convection was increased gradually (Chen and Chen).9 They could delimit the regions of steady, periodic, and unsteady fingering convection based on thermal and solutal Rayleigh numbers. Several studies on salt fingers have been carried out in search of scales for this unique pattern (Wooding,10 Green,11 Shen,12,13 and Sorkin et al.).14 Unlike constant gravity field that was considered in the above studies, the presence of modulation in it can significantly affect

INTRODUCTION Double diffusive convection has been prominently studied because of its fascinating consequences and direct relevance to several processes of practical interest like interior flow in stellar bodies, mantle flow in Earth’s crust, extraction process from geothermal reservoirs, and melting and solidification of binary alloys, and so on. The original stimulation for the study of this type of convection (ref 1) came from the necessity to explain certain observed phenomena in oceanography. Moreover the applications for double diffusive convection include compositions of stars that generate gradients of hydrogen and helium across their core and two different solutes in metallic melts leading to inhomogeneities arising in the solidification process of chemical engineering. The difference in diffusivities of two constituent diffusing components which affect fluid density serves as the driving mechanism for this flow. Simultaneous variations of these two components modify the dynamics of the resulting convective transport. During the late 1960s it was demonstrated that this type of convection is also possible in porous media which motivated the researchers to explain filtration processes occurring in geothermal applications. Much interest lies when the distributions of the two components offer opposite effects to the density stratification: Convection of fingering type occurs when the component that diffuses more slowly (salt) is destabilizing; convection of diffusive type occurs when the component that diffuses faster (heat) is destabilizing. These types have been discussed elaborately by Turner2 and Huppert and Turner.3 Later studies on convection in porous media involving double diffusion with additional effects are summarized in the book by Nield and Bejan.4 © XXXX American Chemical Society

Received: October 7, 2016 Revised: February 16, 2017 Accepted: February 17, 2017

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DOI: 10.1021/acs.iecr.6b03866 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

presence of Soret effect using the method of averaging. Chen and Chen28 experimented with a laterally heated tank of fluid in the presence of a solute gradient. They demonstrated the appearance of a salt finger regime at reduced gravity levels owing to small mass diffusivity. Bardan et al.29 investigated double diffusive convection induced by the combined effect of thermal and solutal gradients in a two-dimensional rectangle-shaped enclosure subjected to high-frequency oscillations. The quasi-equilibrium state was linearly stable up to a critical value regardless of the nature of oscillations. In the case of vertical oscillations, increasing the oscillational amplitude decreased the subcritical solutions and even reversed them; the opposite trend occurred in the presence of horizontal oscillations. Bardan et al.30 continued the above study for vertical thermal and solutal gradients subjected to vertical oscillations. The oscillations were found to delay or advance the convection onset, depending on the governing parameters. The effect of oscillations on the flow behavior structure near the bifurcation point was predicted, showing different possible behaviors. Pillay and Govender31,32 investigated the influence of gravity modulation on convection arising in a mushy layer of solidifying binary alloy. It was found that up to a transition point increasing the frequency of modulation rapidly stabilizes solutal convection through synchronous mode. Beyond that point, further increases in the frequency tend to gradually destabilize the solutal convection through subharmonic mode. Yu and Chen33 carried out experimental cum numerical simulation studies to analyze instability features of a stratified fluid layer which was heated from below. They also analyzed the subsequent nonlinear evolution, in the presence of steady and modulated gravity. Strong34 investigated double diffusive convection in a tightly packed porous layer under the influence of vertical oscillations using continued fractions and an averaging method. Recently, Siddheshwar et al.35 have investigated the influence of gravity modulation on double diffusive convection arising in a fluid-saturated porous medium along with surface temperature modulation using weakly nonlinear analysis. In spite of several applications, fingering convection has been less studied compared to diffusive convection. Studies on convective instabilities with gravity modulation deal very little with fingering mechanism. A complete theoretical insight into salt fingering in the presence of gravity modulation is not yet clearly explored since the available works are restricted to either low-amplitude or high-frequency modulations due to the inadequacy of the method of solution adopted. These limitations do not apply to certain vibration technologies. For instance, the greater the amplitude of vibration, the greater the increment of oil production in the vibration production technology, discussed earlier, which is used in extraction process (see Wenfei and Xiaojiang).36 The purpose of the current investigation is to find stability characteristics corresponding to the onset of fingering convection under the influence of gravity modulation without restricting the modulational parameters to any particular region of validity.

the stability of a buoyancy-induced system by changing its sensitivity to convection. Its effect was first studied in a single diffusive horizontal fluid layer heated below and above by Gresho and Sani.15 Surprisingly, it was shown that a stable configuration of this system under constant gravity condition could be altered on introducing a time-dependent component in the gravity field. This developed great interest in understanding the control via gravity modulations. Random fluctuations of the gravitational field, in magnitude or direction, influence processes such as crystal growth, chromatographic segregation, bubble dynamics, polymer engineering, and ceramic processing. Its effect can also be observed on reaction driven convective instability in the case of frontal polymerization. Malashetty and Padmavathy16 studied the onset of convection in fluid and fluid-saturated porous layers under the influence of gravity modulations of small amplitude. They concluded that low-frequency modulations play an important role on the stability. Skarda17 analyzed gravitymodulated convection with surface tension variations. Zenkovskaya and Rogovenko18 used the averaging method to investigate filtration convection under high-frequency gravity modulations in an arbitrary direction. It was found that horizontal oscillation has a destabilizing effect in the cases of zero gravity and microgravity. Govender19,20 made stability analyses to investigate the influence of low-amplitude gravity modulation on convective heat transfer in a Darcy porous medium. He established that an increase in frequency of modulation leads to a stabilizing effect and then to a transition thereafter. Saravanan and Sivakumar21 studied the influence of gravity modulation on convective instability in a Brinkman porous medium that is heated either from below or from above. Applications wherein gravity-modulated double diffusive convection can occur is the enhanced hydrocarbon recovery process from geothermal reservoirs (see Beresnev et al.).22 Usually, less than 30% of the oil found deep inside Earth can be extracted through primary recovery mechanisms such as natural flow and artificial lift. The residual oil present in reservoirs, by capillary and viscous forces, is made mobile by injecting either heat or solvent or chemicals. In order to enhance oil recovery in cases of failure, vibrations are applied ultimately on the surface of the oil field or at the specific location through downhole vibration. This mechanism vibrates the underground media and changes the flow state of the remaining oil. Similarly, it is wellknown that solidification of binary alloys in a vertical Bridgman configuration either subjected to vertical ampule vibration or mounted in spacecraft maneuvers requires a thorough knowledge of doubly diffusive fingering convection with gravity modulations (Rudolph).23 Saunders et al.24 investigated the effect of vertical gravity modulation on the onset of thermosolutal convection in a fluid layer. A stably stratified layer corresponding to either a fingering or diffusive regime was considered. They discussed instability by plotting unstable regions as functions of modulation and inverse of frequency. It was found that in a fingering regime modulation inhibits instability that is relatively unaffected by steady acceleration. Terrones and Chen25 studied the effects of cross diffusion and gravity modulation in a double diffusive layer using the Galerkin method. They investigated the stability of doubly stratified fluid layers with stress-free and rigid boundaries when the stratification is either imposed or induced by Soret separation. Murray and Chen26 analyzed the influence of gravity modulations on thermosolutal convection during the process of directional solidification for large modulation frequencies. Gershuni et al.27 examined the impact of gravity modulation on a fluid layer in the

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FORMULATION OF THE PROBLEM A horizontally infinite fluid-saturated porous layer bounded by two surfaces z = 0 and z = h is considered. We use a Cartesian coordinate system with the z-axis pointing vertically upward. A binary, Newtonian, and incompressible fluid is taken. The solid phase of the porous medium is homogeneous, isotropic, nondeformable, and exhibits thermal equilibrium locally with the fluid. The bounding surfaces are kept uniformly at constant B


Article


Figure 1. Brinkman model Da = c = 10−2: Marginal curves with S (horizontal lines) and Sh (vertical lines) resonant loops for different values of −RaS with Sc = 10, η = 20, and ω = 10.

1 ∂v 1 1 ν + 2 v·∇v = − ∇p − λ v + ν∇2 v φ ∂t ρ K φ + (β T − β S ) g (t ) k ̂

temperatures T1 and T2 (T1 < T2) and constant solute concentrations S1 and S2 (S1 < S2), respectively. The variation in density is described by means of a linear relationship

s

ρ = ρ0 [1 − β(T − T0) + βs(S − S0)]

(1)

where ρ, T, S, β(>0), and βs(>0) represent the fluid density, temperature, solute concentration, temperature expansion coefficient, and solute expansion coefficient, respectively. The entire system is subjected to time-dependent gravity field modulating with time. The equations for the laminar flow through the porous medium with the Boussinesq approximation are

(2)

ϰ

∂T + v·∇T = χ ∇2 T ∂t

(3)

φ

∂S + v·∇S = Dm∇2 S ∂t

(4)

∇·v = 0

(5)

where v = (v1, v2, v3) is the filtration velocity, p is the pressure, φ is the porosity, K is the permeability, ν is the kinematic viscosity, C


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Figure 2. Brinkman model Da = c = 10−2: Marginal curve with S (horizontal lines) and Sh (vertical lines) resonant loops for different (−RaS, Sc) with η = 2 and ω = 10 in (a, b) and η = 20 and ω = 100 in (c, d).

ϰ = (ρCp)m/(ρCp)f is the heat capacity ratio (porous medium versus fluid), χ is the thermal diffusivity, Dm is the solutal diffusivity, and k̂ is the unit vector along the z-direction. One may observe that for λ = 0, φ = 1, and ϰ = 1, the above equations reduce to the case of a clear fluid. In the present study, we fix ⎡ ⎤ A λ = 1. The gravity is taken as g(t)k̂ = ⎣g0 + φ Ω2f ″ (τ )⎦k̂ , the sum of static and fluctuating components where A, Ω, and f(τ) the modulational amplitude, modulational frequency, and a 2π-periodic function having zero 2π-average, respectively, with τ = Ωt. The boundaries of the porous medium are taken to be tangential stress-free: v3 =

where u, q, Θ, and Φ are unsteady and small perturbations to the relevant quantities. We introduce scales for the variables: ⎞ ⎛ h2 ν ρν 2 (x , t , v , p , T , S ) → ⎜h , , , , ah , bh⎟ ⎠ ⎝ ν h K

where a = (T1 − T2)/h and b = (S1 − S2)/h. Then, the nondimensional equations governing the above perturbations are

∂v1 ∂v = 2 = 0 at z = 0 and h ∂z ∂z

We seek a quasi-equilibrium solution to the above set up in the form v = v0, T = T0(z), S = S0(z), and P = P0(z, t). This admits the following basic state given by 1 1 v = 0, T = T1 − (T1 − T2)z , S 0 = S1 − (S1 − S2)z , h h 0

∂u = −∇q − λ u + Da∇2 u + (Grθ − Grsϕ) ∂t (1 + η f ″ (τ ))k̂

(8)

ϰ

∂θ 1 2 − u3 = ∇θ ∂t Pr

(9)

φ

∂ϕ 1 − u3 = ∇2 ϕ ∂t Sc

(10)

c

0

⎡ ⎤ 1 p0 = ρ g (t )⎢(βT1 − βsS1)z − (β(T1 − T2) − βs(S1 − S2))z 2 ⎥ ⎣ ⎦ 2h (6)

∇·u = 0

(11)

where Gr = βah2g0K/ν2 is the thermal Grashof number, Grs = βsbh2g0K/ν2 is the solutal Grashof number, Da = K/h2 is the Darcy number, Pr = ν/χ is the Prandtl number, Sc = ν/Dm the Schmidt number, c = Da/φ the porosity−permeability parameter, η = AΩ2/φg0 is the modulational amplitude, and ω = Ωh2/ν is the modulational frequency. After operating curl curl on eq 8, the vertical component of the resulting

We now employ linear perturbation theory to assess the stability of the above state. For this purpose, consider v = v 0 + u, p = p0 + q , T = T 0 + Θ, and S = S 0 + Φ (7) D


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Industrial & Engineering Chemistry Research equation becomes ⎡∂ ⎤ c ⎢ + λ ⎥∇2 u3 = Da∇4 u3 + ∇12 [Grθ − Grsϕ] ⎣ ∂t ⎦ (1 + η f ″(τ ))k̂

where ∇12 =

(

2

∂ ∂x 2

2

+

∂ ∂y 2

) and ∇

2

(12)

= ∇12 +

2

( ) are the Lap∂ ∂z 2

lacian operators. The relevant nondimensional boundary conditions are u3 =

∂ 2u3 ∂z 2

= θ = ϕ = 0 at z = 0 and z = 1

(13)

We use normal modes to represent the vertical velocity component, temperature, and solute concentration as (u3 , θ , ϕ) = (u͠ 3(z , t ), θ (̃ z , t ), ϕ(̃ z , t )) ei(α1x + α2y)

(14)

where α1 and α2 represent wavenumbers in the x- and y-directions, respectively. Using this in eqs 9−12, we obtain ⎡ ∂ ⎤ 2 2 2 2 2 ⎢⎣c + λ ⎥⎦(D − α )u͠ 3 − Da(D − α ) u͠ 3 ∂t + (1 + η f ″(τ ))α 2[Grθ ̃ − Grsϕ]̃ = 0

(15)

ϰ

1 2 ∂θ ̃ (D − α 2)θ ̃ − u͠ 3 = 0 − Pr ∂t

(16)

φ

∂ϕ ̃ 1 − (D2 − α 2)ϕ ̃ − u͠ 3 = 0 Sc ∂t

(17)

where D ≡ ∂/∂z, and α = + is the overall horizontal wavenumber. We now separate the variables in eqs 15−17 using the following representation 2

α21

α22

̂ t ) sin(πz) [u͠ 3(z , t ), θ (̃ z , t ), ϕ(̃ z , t )] = [u3 ,̂ θ ̂, ϕ](

(18)

Substituting t = t ̂ Prc ϰ and ω = ω̂ / Prc ϰ , we arrive at a system of equations with periodic coefficients: c du 3 ̂ α2 = −(Da m2 + λ)u3 ̂ + 2 (1 + η f ″ (τ )) r dt ̂ m [Grθ ̂ − Gr ϕ]̂ s

(19)

ϰ dθ ̂ m2 ̂ = u3 ̂ − θ r dt ̂ Pr

(20)

φ dϕ ̂ m2 ̂ = u3 ̂ − ϕ r dt ̂ Sc

(21)

Figure 3. (a) Closed disconnected loops (S) for different values of Sc with RaS = −30, η = 20, and ω = 10. (b) Closed disconnected loops (S) for different values of ω with RaS = −10, Sc = 500, and η = 200. (c) Closed disconnected loops (S) for different values of RaS with Sc = 500, η = 200, and ω = 10.

an infinite system of equations with unknown coefficients wn, θn, and ϕn:

where m2 = α2 + π2 and r = Prc ϰ . We set f(τ) = cos ω t in eq 19 and omit tilde hereafter from t ̂ and ω̂ for notational convenience.

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⎞ 2m2 ⎛⎜ c (σ + inω) + λ + Da m2⎟wn= ⎠ α 2η ⎝ r

SOLUTION PROCEDURE Solutions to the above set of equations with time-periodic coefficients can be found via the Floquet theory. Accordingly, the solutions are expressed as +∞

̂ t ) = e σt (u3 ,̂ θ ̂, ϕ)(

∑ n =−∞

(wn , θn , ϕn) e

inωt

(22)

where σ is the Floquet exponent defining the behavior of the perturbations with time. By substituting this, eqs 19 and 20 yield E

⎞ ⎛2 ⎞ ⎛2 Gr ⎜ θn − θn − 1 − θn + 1⎟ − Grs⎜ ϕn − ϕn − 1 − ϕn + 1⎟ ⎠ ⎝η ⎠ ⎝η

(23)

⎛ϰ m2 ⎞ wn = ⎜ (σ + inω) + ⎟θn Pr ⎠ ⎝r

(24)

⎛φ m2 ⎞ wn = ⎜ (σ + inω) + ⎟ϕ Sc ⎠ n ⎝r

(25) DOI: 10.1021/acs.iecr.6b03866 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 4. (a) Brinkman model (Da = 10−2): −RaT,c against ω for different η with RaS = −10, Sc = 500, and c = 10−2. (b, c) αc against ω for different η with RaS = −10, Sc = 500, and c = 10−2.

By eliminating the variables wn and ϕn, we transform the

φ

where L = Le ϰ and P = Pr ϰ /c . Here RaT = Gr·Pr is the thermal Rayleigh number, RaS = Grs·Sc is the solutal Rayleigh number and Le = Sc/Pr is the Lewis number. We employ continued fractions method to solve the above linear algebraic system. The substitution ζn = θn−1/θn (θn ≠ 0) transforms eq 26 into

above system into an infinite tridiagonal system of equations (see ref 34) for θn: Mnθn + qn − 1θn − 1 + qn + 1θn + 1 = 0, n = ..., − 2, − 1, 0, 1, 2, ... (26)

⎛ q ⎞ Mn + ⎜qn − 1ζn + n + 1 ⎟ = 0, n = ..., −2, −1, 0, 1, 2, ... ζn + 1 ⎠ ⎝

⎞ ⎛ m2 Mn = (σ + inω)2 + ⎜ + (λ + Da m2)P ⎟(σ + inω) ⎠ ⎝ P ⎛ 2 ⎞ + ⎜m2(λ + Da m2) − qn⎟ η ⎠ ⎝

⎡ ⎛ m 2 ⎞⎤ (σ + inω) + P ⎟⎥ α 2η ⎢ ⎜ qn = Ra T − RaS⎜ m 2 ⎟⎥ 2m2 ⎢⎢ ⎝ L(σ + inω) + P ⎠⎥⎦ ⎣

(29)

From this, we derive the following two different recurrence relations for the unknown ζn (27)

ζn = − ζn = (28) F

q ⎞ 1 ⎛ ⎜M n + n + 1 ⎟ ζn + 1 ⎠ qn − 1 ⎝ −qn

Mn − 1 + ζn − 1qn − 2 DOI: 10.1021/acs.iecr.6b03866 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 5. (a) Brinkman model (Da = 10−2): −RaT,c against ω for different η with RaS = −10, Sc = 500, and c = 10−1. (b, c) −αc against ω for different η with RaS = −10, Sc = 500, and c = 10−1.

For σ = 0, the expressions for Mn and qn are simplified so that M−n = Mn and q−n = qn.

which in turn lead to two different continued fractions: ζn = −

qn + 1/qn − 1 Mn − q /q M qn − 1 − qn+1 − Mn+2 n+2 qn+3n / qn+1 −q

n

n+1

ζn =

−

−

Mn + 3 + ··· qn + 2

⎛ 2 ⎞ Mn = −n2ω 2 + ⎜m2(λ + Da m2) − qn⎟ η ⎠ ⎝

(30)

⎞ ⎛ m2 +⎜ + P(λ + Da m2)⎟inω ⎠ ⎝ P

qn −Mn − 1 −

qn − 1qn − 2 −M n − 2 −

qn − 2qn − 3 −Mn − 3 − ···

⎡ ⎛ m 2 ⎞⎤ inω + P ⎟⎥ α 2η ⎢ ⎜ qn = Ra T − RaS⎜ m 2 ⎟⎥ 2m2 ⎢⎢ ⎝ inωL + P ⎠⎥⎦ ⎣

(31)

Assigning n = 0 in eqs 30 and 31 leads to the following dispersion equation for σ in the explicit form M0 −

q0q1 M1 −

q1q2 M2 −

q2q3 M3 − ···

=

q0q−1 M −1 −

Thus, the dispersion equation (eq 32) transforms into

q−1q−2 M −2 −

q−2q−3 M−3 − ···

⎛ ⎞ ⎜ ⎟ q0q1 M0 Re⎜ ⎟= q1q2 2 ⎜ M1 − M − q2q3 ⎟ 2 ⎝ M3 − ··· ⎠

(32)

Now we can determine the values of σ. Equation 32 is reduced to the real form when σ = 0, corresponding to the synchronous (S) mode with period 2π/ω. G

(33) DOI: 10.1021/acs.iecr.6b03866 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 6. (a) Brinkman model (Da = 10°): −RaT,c against ω with RaS = −10, Sc = 500, c = 10°, c = 101, and η = 2. (b, c) αc against ω with RaS = −10, Sc = 500, c = 10°, c = 101, and η = 2.

When σ = iω/2 corresponding to the subharmonic (Sh) mode with period 4π/ω, the expressions for Mn and qn are simplified so that M−n = Mn−1 and q−n = qn−1

Transcendental eqs 33 and 34 are solved to obtain the marginal curves of thermal Rayleigh number RaT against wavenumber α for synchronous and subharmonic modes, respectively. Convergence of the continued fractions is verified numerically and truncated once the required accuracy (10−4) is obtained. The stability characteristics such as the critical Rayleigh number RaT,c, obtained by minimizing marginal RaT against α and the critical wavenumber αc, the α corresponding to RaT,c, are then calculated by fixing the values of other parameters.

⎛ ⎛ 1 ⎞2 2 ⎞ Mn = − ⎜n + ⎟ ω2 + ⎜m2(λ + Da m2) − qn⎟ ⎝ 2⎠ η ⎠ ⎝ ⎛ m2 ⎞ ⎛ 1⎞ +⎜ + P(λ + Da m2)⎟ i⎜n + ⎟ω 2⎠ ⎝ P ⎠ ⎝

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⎡ ⎛ 1 m 2 ⎞⎤ α 2η ⎢ ⎜ i n + 2 ω + P ⎟⎥ qn = ⎢Ra T − RaS⎜⎜ 1 m2 ⎟ ⎟⎥ 2m2 ⎢ ⎝ i n + 2 ωL + P ⎠⎥⎦ ⎣

( (

) )

RESULTS AND DISCUSSION This work is intended to examine the onset of fingering instability due to double diffusion in a binary fluid filled porous layer exposed to gravity modulation. This instability can arise when a slower diffusing component exhibits unstable stratification in the presence of a stable overall density distribution. A universal porosity permeability relationship that is applicable to all porous materials is not generally available. However, increasing porosity causes interconnected void spaces to expand, which in turn results in higher permeability. Hence, the porosities

Here the dispersion equation (eq 32) for this case becomes 2

M0 −

q0q1 M1 −

q1q2 M2 −

q2q3 M3 − ···

= q02 (34) H


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Figure 7. (a) Darcy model (Da = 10−4): −RaT,c against ω for different η with RaS = −10, Sc = 500, c = 10−2, and c = 10−3. (b, c) αc against ω for different η with RaS = −10, Sc = 500, c = 10−2, and c = 10−3.

are taken to be 0.01−0.1 for the Darcy model (Da = 10−4) and 0.1−1.0 for the Brinkman model (Da = 10−2), following Saravanan and Sivakumar.21 Taking σ = q = n = 0, eq 26 reduces to the unmodulated result Ra T,c =

(π 2 + α 2)2 (1 + Da(α 2 + π 2)) + RaS α2

Marginal curves are now constructed as functions of modulation amplitude and frequency for the Brinkman model Da = c = 10−2 for different RaS. These curves consist of an array of alternating loop shaped branches corresponding to S (horizontal lines) and Sh (vertical lines) modes of instability response. When plotted in the (−RaT, α) plane, they expose the regions of instability emerging due to presence of gravity modulation (Figure 1a). The global minimum in each marginal curve occurring at the bottom-most loop which may be either S or Sh type is referred to as the critical Rayleigh number RaT,c. The critical wavenumber αc is that α corresponding to RaT,c. It is seen that the loops occurring in regions of higher α get stretched along the direction of increasing α and are displaced upward leaving a thin Sh loop in the lower wavenumber region. In spite of the influence of Sc and RaS, the natures of the two bottom-most instability loops in Figure 1a are similar to Figure 5a of our earlier paper (Saravanan and Sivakumar).21 This indicates that the resonant behavior is qualitatively insensitive to lower values of Sc and RaS at this level of gravity modulation, η = 20 and ω = 10. As RaS increases, the loops of Sh type alone elongate significantly

(35)

In the absence of solute gradient (RaS = 0), one may find that RaT/Da → 27π4/4 and αc → π/√2 in the clear fluid limit Da → ∞ coinciding with the classical results of Rayleigh−Benard convection. Setting Da = 0, eq 35 reduces to the well-known result for the problem of double diffusive convection in the absence of modulation (see Nield and Bejan).4 In particular, eq 27 for Da = 0 corresponds to double diffusive porous convection using the Darcy model as studied by Strong,34 and for RaS = 0, it corresponds to single-component porous convection using the Brinkman model as studied by Saravanan and Sivakumar.21 The values of Pr and κ are fixed as 1.0 throughout the study. I


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Figure 8. (a) −RaT,c against −RaS with Sc = 5, 50, 500, 5000 for Da = 10−4: ■η = 20, ω = 10 and ▲η = 200, ω = 100; Da = 10−2: ●η = 20, ω = 10 and ⧫η = 200, ω = 100. (b) αc against −RaS with Sc = 5, 50, 500, 5000 for Da = 10−4: ■η = 20, ω = 10 and ▲η = 200, ω = 100; Da = 10−2: ●η = 20, ω = 10 and ⧫η = 200, ω = 100.

instability loop of the marginal curve mentioned earlier gets deformed by different parameters. Increasing Sc enlarges it until it reaches a saturation (Figure 3a). Figure 3b shows another interesting effect wherein the loop starts appearing for low ω, developing and then disappearing for intermediate values. During this change it gets shifted to a region of slightly higher α. Similarly, the effect of RaS shown in 3c is to enlarge the instability loop and elongate it toward higher −RaT region, as discussed earlier. The stability characteristics of the Brinkman model (Da = 10−2) are shown in Figures 4a and 5c for different values of c. Since solute is transported through the fluid alone whereas heat is transferred through both fluid and solid phases, c greatly affects the heat transfer within the layer. When RaS was sufficiently low, we found that −RaT,c → ∞ for ω → 0 and ω → ∞. It is not possible to destabilize a thermally stable porous medium at very low modulation frequencies; hence, the unboundedness of Rac can be expected near ω = 0. In contrast, as the modulation frequency reaches higher values, the porous medium starts acting as if it is not forced to vibrate; hence, the influence of gravity modulation disappears. It is obvious that the Sh mode always

and intersect with newly formed closed instability loop of S type representing the existence of doubly unstable regions, as observed earlier by Saunders.24 In particular, the bottom-most loop of the multilooped marginal curve in Figure 1b exhibits a Sh response but is partially overlapped by a new closed instability curve of S type. This originally appeared as an isle of instability well below the marginal curve when RaS = 24.5, referred to as closed disconnected loop (CDL) (see Saravanan and Sivakumar),37 before attaining the present position and shape. For a further increase in RaS, a gradual increase in the number of doubly unstable regions is observed as shown in Figure 1c−e. Figure 2a−d depicts the multilooped marginal curves for two different sets of the modulational parameters η and ω and the control parameters Sc and RaS. For lower η and ω,the onset of instability is through the S mode as shown in (Figure 2a,b). By increasing Sc and RaS, no significant effect is observed except a slight shrink in the S loop representing a slight reduction in the overall instability region. Moreover for lower η and ω, the loops are found to be very thin and narrow. However, for higher η and ω the loops become significantly wider and grow along the increasing α direction. Figure 3a−c displays how the closed J


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Figure 9. (a) −RaT,c against Sc with RaS = −10 for η = 20, ω = 10 and η = 20, ω = 100. (b) αc against Sc with RaS = −10 for η = 20, ω = 10 and η = 20, ω = 100.

Θ(z), and Φ(z) are given by F(z) = z2(1 − z)2, Θ(z) = z(1 − z) (1 + z − z2), and Φ(z) = z(1 − z) (1 + z − z2) (see Saravanan and Sivakumar).21 In choosing the approximations for Θ(z) and Φ(z), the additional conditions D2Θ = 0 and D2Φ = 0 at z = 0 and 1 are taken into account, provided by eqs 16 and 17. Following the steps already mentioned in the “Solution Procedure” section, we get another infinite tridiagonal system of equations for θn (see eq 26) with

remains the preferred mode of instability for ω > 102. From Figure 4a, one can immediately notice the jump in −RaT,c at ω = 78.9 for η = 200. The reason for this is the appearance of the CDL shown in Figure 3b. When η = 2, the two modes of instability alternate with each other in determining the criticality. These are associated with discontinuities in αc exhibiting jumps from higher to lower values against an increase in ω. These jumps show an increasing trend for an increase in ω. It is also visible that the instability set in via the S mode over a wide range of ω only for high η. The insets provided in Figure 4a,b show that the modulated results approach the marked unmodulated ones, determined by eq 35, in the limit of very large modulation frequency. Though the present study considers only stress-free boundaries, one may wish to compare the results with those corresponding to more physical rigid ones. These boundaries replace the second condition in eq 13 by ∂2 u3/∂ z2 = 0, and in this case, we adopt a single-term Galerkin method to solve eqs 15−17. Accordingly the amplitudes u͠ 3, θ̃ and Φ̃ are taken in the form u͠ 3(z, t) = a(t) F(z), θ̃(z, t) = b(t) Θ(z), and Φ̃(z, t) = c(t) Φ(z) respectively where the functions F(z),

⎛A ⎞ ⎜ + (λ + DaB)P ⎟(σ + inω) ⎝P ⎠ 2 + A(λ + DaB) − qn η (36)

Mn = (σ + inω)2 +

qn =

⎡ ⎛ (σ + inω) + A ⎞⎤ 121α 2η P ⎟⎥ ⎢Ra − Ra ⎜ T S⎜ A ⎟⎥ 248(12 + α 2) ⎢⎣ L in σ + ω + ( ) ⎝ P ⎠⎦ (37)

where A = (306 + 31α2)/31 and B = (Da(504 + 24α2 + α4)/ 12 + α2. The critical boundaries corresponding to the rigid boundaries are included in Figures 4a and 5a for comparison. K


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Figure 10. (a) −RaT,c against Pr for RaS = −10, Sc = 500, η = 20, and ω = 100. (b) αc against Pr for RaS = −10, Sc = 500, η = 20, and ω = 100.

tion frequencies compared to the other cases. Hence, the resulting convective pattern can be altered effectively by adjusting the frequency of gravity modulation suitably. The effect of RaS on the instability characteristics is depicted in Figure 8a,b for two different sets of modulational parameters. A closer watch of the results shows a sort of combined effect of those observed in previous works (Saravanan and Sivakumar;37 Saravanan and Jegajothi).38 −RaS delays the onset of convection initially and then suddenly starts advancing it after reaching a particular value, depending on Sc. An increase in −RaS beyond this limit causes the ensuing convection to oscillate in tune with the externally applied gravity modulation via the development of a CDL. The corresponding αc shows a decreasing trend in general that jumps down to a lower value at the emergence of the CDL for higher modulational parameters. The delay in the convective onset is more favored in the Darcy regime. Moreover, the destabilizing effect of −RaS is found to depend strongly on Sc. It can be observed that weaker mass diffusion induces conditions favorable for fingering convection and hence advances the onset of convection for higher values of Sc. As Pr is fixed, the diffusion of heat across the medium remains suppressed in case it is made

They exhibit almost a similar pattern but basically delay the onset of instability as anticipated. They suffer additional mode transitions for η = 2 and c = 10−2. Similar to Figures 4a and 5a, Figure 6a illustrates the stability behavior for Da = 10°. Unlike the previous case, the two modes of instability compete with each other in a wider range of ω (ω < 102) producing more number of cusps for low η. The parameter c has a significant role in delaying the onset of convection for high Da. The changes in wavenumber between S and Sh mode associated with the cusps in −RaT,c are displayed in Figure 6b,c. We note that as before αc shows an increasing trend against ω. It can be clearly observed that an increase in sparsity of the medium rises the competition for instability between the synchronous and subharmonic modes. Figure 7a demonstrates the stability characteristics for Da = 10−4 (Darcy model). As the porous medium becomes dense, tall circulation patterns are expected at the onset. In this case Sh mode is also the preferred mode of instability for higher values of ω. As ω nears zero, both the modes interact significantly for smaller η; hence, the critical RaT,c shows a nesting of the instability modes. The corresponding αc experiences abrupt changes at the transiL


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The Brinkman model always delays instability when compared to the Darcy model, with a decrease in the corresponding αc. The jumps in αc are increased as the parameter c takes small values. Higher values of c restrict the mode transitions to lower ω and destabilize the system for Da < 10−2, producing the opposite effect for Da > 10−2. The effect of RaS is found to strongly accomplish instability only for low vibrational parameters. As ω increases, it dominates the effect of solutal convection by stabilizing the system. The sparsity of the medium also contributes to the advance of instability along with RaS. The influence of Sc is felt through the Sh mode when it pairs with higher frequency for low Sc, and ω instability occurs via S mode also. In general RaT,c and αc for rigid boundaries are always greater than stress-free counterparts irrespective of all other parameter values. The presence of rigid boundaries makes a nest between the two modes of instability for a wider range of frequencies.

up of densely packed solid phase. This causes one to speculate the setting up of fingering convection in such a medium for much lower rates of mass diffusion compared to those in a sparsely packed one. This behavior can be easily observed in Figure 8a. It is also clear that the solutal mechanism alone is sufficient to induce the instability for sufficiently large values of Sc in the Brinkman model. Shown in Figure 9a is the effect of Sc as it is continuously varied. In general, it is seen that the stability limits exhibit changes for intermediate values of Sc and remain unaltered for its small and large values. Higher modulational frequency in the gravity field postpones convective instability that originates via denser fingers. For such higher frequencies, −Rac attains a maximum for intermediate values of Sc. This is accompanied by up and down going cells of tall dimension particularly visible in the Darcian regime. One may also notice that the Sh mode remains critical throughout the Sc range in this case. However, for lower modulational frequencies, the S mode is introduced at the criticality. Throughout the above discussion Pr was fixed at 1; hence, its influence at the onset of instability criteria is illustrated in Figure 10a,b. It can be clearly observed that for both high and low Pr the results approach the unmodulated ones as already reported in Saravanan and Sivakumar.21 For Pr < 0.1, it is interesting to note that RaT,c’s remain identical for both Darcy and Brinkman models when c = 10−2. Increase in c for both the models delay the onset of instability for Pr < 0.1 whereas it advances for Pr > 10. Corresponding results of αc are depicted in Figure 10b, where αc in general increases and reaches a maximum for intermediate Pr and then gradually decreases as Pr increases.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

S. Saravanan: 0000-0001-8004-2267 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to the University Grants Commission, India for its support through Special Assistance Programme (DRS) in Fluid Dynamics and financial support to S.K.

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CONCLUSION When a fluid under gravity modulation is subjected to thermal gradient, it becomes extremely sensitive near the critical point marking the convection threshold. Experiments have confirmed this trend. The reported results in this work show that in the presence porous media convective instability in such systems can be well-controlled by properly choosing the modulational amplitude and frequency. The onset of fingering convection in a double diffusive fluidsaturated porous medium was examined under periodically varying gravity field. Arbitrary modulational amplitude and frequency were considered. The results were analyzed by plotting marginal and critical stability boundaries and are summarized below. The occurrence of doubly unstable regions in the marginals is found when the rate of solutal convection is stronger than that of thermal convection. This result is very useful in suppressing unwanted buoyancy-induced motions during solidification process in chemical engineering applications. The solutal mechanism introduces CDLs at the marginal state which were not seen in an analogous problem containing a single diffusing component. Their existence show that the system is more prone to instability since they set in at very low RaT. The number of doubly unstable regions increase with increase in RaS. η always destabilizes the system irrespective of the value of ω. The transitions between the two modes of instability, viz., S and Sh modes, occur only at lower range of ω. Beyond that, Sh mode manages to emerge as the deciding one for higher values of ω when RaS takes smaller values. When it takes higher values, a transition to the S mode occurs at higher RaS, and the critical value jumps down and moves toward the unmodulated value in the large ω limit.

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REFERENCES

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Floquet Instability of Gravity-Modulated Salt Fingering in a Porous

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