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Effect of Amplitude Modulation of Gravitational Vibration on Convective Instability of
Reaction Fronts in Porous Media
Karam Allali Department of Mathematics University Hassan II-Mohammedia Po. Box 146, FST-Mohammadia, Morocco
Mohamed Belhaq Department of Mechanics University Hassan II-Casablanca P.O. Box 5366, Maˆarif, Casablanca, Morocco
Abstract
This paper studies the influence of periodic and quasi-periodic ampli- tude modulations of gravitational vibration on the convective instability of reaction fronts in porous media. Specifically, two cases of amplitude modulation are investigated. In the first case, we consider that the frequency ν2 of the amplitude modulation is either double or half the frequency of the basic gravitational vibration ν1. In this case the mod- ulated gravitational vibration remains periodic. In the second case, we assume that the frequencyν2 is such thatν1andν2are incommensurate which forces the gravitational vibration to be quasi-periodic. The model considered in this study consists of the heat equation, the equation for the depth of conversion and the equations of motion under the Darcy law. The convective instability threshold is obtained. The linear stabil- ity analysis of the steady-state solution is performed and the obtained interface problem is solved numerically.
Mathematics Subject Classification: 35K57, 37M05, 76E15, 76S05 Keywords: convective instability, amplitude modulation, quasi-periodic modulation, gravitational vibration, linear stability analysis, porous media
1 Introduction
The study of convective instability of reaction fronts in porous media under the influence of an imposed periodic gravitational vibration (GV) has been reported in several works. For instance, the influence of a periodic vibration on convective instability of a reaction front in the case of frontal polymerization has been studied in [1] and it was shown that the front becomes more stable for small amplitudes of vibration, loses its stability for hight ones, and a slight increase of the frequency of vibration stabilizes the reaction front. A similar study was performed in the case of porous media focusing on the dependence of the convective threshold on the permeability parameter [2, 3].
While most of the studies were related to the influence of periodic vibrations on convective instability, only few works have been devoted to investigate the influence of quasi-periodic (QP) vibrations on convective instability. The first significant contribution that analyzed the effect of a QP gravitational modulation on the stability of a heated fluid layer was given in [4]. The threshold of convection corresponding to QP motions was determined in the case of heating from below or from above. Similar studies were performed to investigate the QP convective instability in Hele-Shaw cell [5] and in thermal instability in a horizontal Newtonian magnetic liquid layer with non-magnetic rigid boundaries and in the presence of a vertical magnetic field [6].
In all these later studies [4–6], the original problem is systematically re- duced to a standard QP Mathieu equation using Galerkin method. The marginal stability curves are then obtained by implementing the harmonic balance method coupled with Hill’s determinants [7, 8]. However, in the case of reaction fronts the method of reducing the original Navier-Stokes equa- tions to a certain QP Mathieu equation using Galerkin and harmonic balance methods cannot be used. This is principally due to the coupling of the concen- tration and the heat equations (i.e. reaction-diffusion problem coupled with the Darcy equation). Thus, to obtain the convective threshold in this case, the original reaction diffusion equations are first reduced to a singular perturbation problem using the matched asymptotic expansion, and then, a linear stability analysis is performed to solve the reduced interface problem using numerical simulations.
The aim of the present work is to examine the influence of the modulation amplitude Λ(t) of the periodic GV, Λ(t)sin(ν1t), on the convective instability of reaction fronts in porous media, where Λ(t) and ν1 are the amplitude and the frequency of the GV, respectively. Specifically, two cases are considered.
First, we examine briefly the case where the frequency ν2 of the modulation amplitude Λ(t) = λcos(ν2t) of the GV equals 2ν1 or ν21. This case imposes the GV to stay periodic. The second case considers that the frequency of the modulation amplitude ν2 is such that ν1 and ν2 are incommensurate, which
forces the gravitational vibration to be QP.
Notice that in a recent work [9], the QP vibration was introduced by impos- ing two additional vibrations with two incommensurate frequencies, while, in the present paper the QP vibration is implemented by modulating the ampli- tude of the GV. Our motivation here is that a modulation of the GV amplitude is easy to produce such as, for instance, shaking the system which is already under a GV. In contrast, introducing a QP vibration with two additional in- commensurate frequencies is more difficult to realize in practice. In addition, to the best of our knowledge, the effect of the amplitude modulation of a GV on convective instability in general, and on reaction fronts instability in particular, has not been considered in previous works.
The next Section introduces briefly the model, while Section 3 deals with the linear stability and approximates the infinite narrow reaction zone based on the formulation of the interface problem. Results and discussions are provided in Section 4 and we conclude in Section 5.
2 Governing equations
Consider that the reaction front in the porous medium propagates in the op- posite sense of gravity, and assume that the porous matrix is filled by an incompressible reacting fluid submitted to vibration with amplitude modula- tion. The problem is described by the following system of equations consisting in the reaction-diffusion equations coupled with the equations of motion
∂T
∂t +v.∇T =κΔT +qK(T)φ(α), (1)
∂α
∂t +v.∇α =dΔα+K(T)φ(α), (2)
v+K
μ∇p= gβK
μ ρ(T −T0)(1 +λcos(ν2t)sin(ν1t))γ, (3)
∇.v= 0. (4)
with the following boundary conditions:
T = Ti , α= 1 andv = 0 when y→+∞, (5) T = Tb , α= 0 andv = 0 when y→ −∞. (6)
Here T is the temperature, α the depth of conversion, v = (vx, vy) the fluid velocity, pthe pressure, κ the coefficient of thermal diffusivity, d the diffusion coefficient,qthe adiabatic heat release,gthe gravity acceleration,ρthe density, β the coefficient of thermal expansion,μthe viscosity andγ is the unit vector in the upward direction (in y direction). In addition, T0 is the mean value of temperature, Ti is an initial temperature while Tb is the temperature of the burned mixture given by Tb =Ti+q. The function K(T)φ(α) is the reaction rate where the temperature dependence is given by the Arrhenius law:
K(T) =k0exp − E
R0T
, (7)
where E is the activation energy, R0 the universal gas constant and k0 is the pre-exponential factor. For the asymptotic analysis of this problem we assume that the activation energy is large and we consider zero order reaction for which
φ(α) =
1 if α <1
0 if α= 1 . (8)
In order to obtain the dimensionless model, we introduce the spatial vari- ables x = xc1
κ , y = yc1
κ , time t = tc21
κd, velocity v
c1, pressure pκμ K with c1 = c/√
2 and frequencies σ1 = κ
c21ν1, σ2 = κ
c21ν2. Denoting θ = T −Tb
q and
keeping, for convenience, the same notation for the other variables, we obtain the system
∂θ
∂t +v∇θ= Δθ+WZ(θ)φ(α), (9)
∂α
∂t +v∇α=Δα+WZ(θ)φ(α), (10)
v+∇p=Rp(θ+θ0) 0
1
(1 +λcos(σ2t)sin(σ1t)), (11)
div(v) = 0 (12)
with the following conditions at infinity:
θ = −1 , α= 0 and v = 0 when y→+∞, (13) θ = 0 , α = 1 andv = 0 when y→ −∞, (14)
where = d/κ is the inverse of the Lewis number, Rp = Kc21P2R
μ2 , where R is the Rayleigh number and P is the Prandtl number defined, respectively, by R = gβqκ2
μc31 and P = μ
κ. In addition, we use the parameters δ = R0Tb E and θ0 = Tb −T0
q . The reaction rate is then given by:
WZ(θ) =Zexp
θ Z−1+δθ
, (15)
where Z = qE
R0Tb2 stands for Zeldovich number.
3 Linear stability analysis
3.1 Approximation of infinitely narrow reaction zone
We perform an analytical treatment by reducing the original problem to a singular perturbation one where the reaction zone is supposed to be infinitely narrow and the reaction term is neglected outside the reaction zone. A closed interface problem is obtained applying a formal asymptotic analysis assuming = 1
Z is a small parameter.
Let us denote byζ(t, x) the location of the reaction zone in the laboratory frame reference. The new independent variable in the direction of the front propagation is given by
y1 =y−ζ(t, x). (16)
We introduce new functions θ1, α1, v1, p1:
θ(t, x, y) =θ1(t, x, y1), α(t, x, y) =α1(t, x, y1),
v(t, x, y) =v1(t, x, y1), p(t, x, y) = p1(t, x, y1). (17) We re-write the equations in the form (the index 1 for the independent variables is omitted):
∂θ
∂t − ∂θ
∂y1
∂ζ
∂t +v.∇θ =Δθ +WZ(θ)φ(α), (18)
∂α
∂t − ∂α
∂y1
∂ζ
∂t +v.∇α =WZ(θ)φ(α), (19)
v+∇p=Rp(θ+θ0)(1 +λsin(σ1t)cos(σ2t))γ, (20)
∂vx
∂x − ∂vx
∂y1
∂ζ
∂x + ∂vy
∂y1 = 0, (21)
where we have set Δ = ∂2
∂x2 + ∂2
∂y21 −2∂ζ
∂x
∂2
∂x∂y1 + ∂ζ
∂x 2 ∂2
∂y12 − ∂2ζ
∂x2
∂
∂y1, (22)
∇ = ∂
∂x − ∂ζ
∂x
∂
∂y1, ∂
∂y1
. (23)
Next, we use the matched asymptotic expansions. The outer solution of the problem is sought in the form
θ =θ0+θ1 +..., α=α0+α1+...,
v=v0+v1+..., p=p0+p1+.... (24) where (θ0, α0,v0) is a dimensionless form of the basic solution.
In order to obtain the jump conditions in the reaction zone we consider the inner problem and we introduce the stretching coordinate η = y1/, with = 1/Z. Then, the inner solution is expanded in the form
θ=θ˜1+..., α= ˜α0+α˜1+...,
v= ˜v0+˜v1+..., p= ˜p0+˜p1+..., ζ = ˜ζ0+ζ˜1+.... (25) Substituting these expansions into (18)-(21), we obtain the first-order inner problem:
1 +
∂ζ˜0
∂x
2∂2θ˜1
∂η2 + exp θ˜1
1 +δθ˜1
φ( ˜α0) = 0, (26)
−∂α˜0
∂η
∂ζ˜0
∂η − ∂α˜0
∂η
˜ vx0∂ζ˜0
∂x −˜vy0
= exp θ˜1
1 +δθ˜1
φ( ˜α0), (27)
∂p˜0
∂η = 0, (28)
˜
vx0+∂p˜0
∂x − ∂ζ˜0
∂t
∂p˜1
∂η = 0, (29)
˜
vy0+ ∂p˜1
∂η =−Rpθ0(1 +λsin(σ1t)cos(σ2t)), (30)
−∂v˜x0
∂η
∂ζ˜0
∂x +∂v˜0y
∂η = 0. (31)
Then the matching conditions are η→+∞: θ˜1 ∼θ1|y1=0++η∂θ0
∂y1|y1=0+, α˜0 →0, v˜0 →v0|y1=0+, (32) η → −∞: θ˜1 →θ1|y1=0−, α˜0 →1, v˜0 →v0|y1=0−. (33) From (28) we obtain that ˜p0 does not depend on η, which implies that the pressure is continuous through the interface. Next, denoting bys the quantity
s= ˜vx0∂ζ˜0
∂x −v˜0y, (34)
we obtain from (31) thats does not depend on η. Finally, from (29), (30) and (34) we easily obtain that ˜v0x and ˜vy0 do not depend on η, which provides the continuity of the velocity through the interface.
We next derive the jump conditions for the temperature from (26) in the same way as it is usually done for combustion problems. From (27) it follows that ˜α0 is a monotone function and 0<α˜0 <1. Since we consider zero-order reaction, we have φ( ˜α0)≡1. We conclude from (26) that ˜θ1 is also a monotone function. Thus, multiplying (26) by ∂θ˜1
∂η and integrating, we obtain ∂θ˜1
∂η 2
η=+∞−∂θ˜1
∂η 2
η=−∞=−2 A
θ1
−∞
exp( τ
1 +δτ)dτ, (35) where we have set
A= 1 + ∂ζ˜0
∂x 2
. (36)
Subtracting (26) from (27) and integrating, we obtain
∂θ˜1
∂η
η=+∞−∂θ˜1
∂η
η=−∞=−1 A
∂ζ˜0
∂t +s
. (37)
Using the matching conditions and truncating the expansion:
θ0 ≈θ, θ1|y1=0−≈Zθ|y1=0 ζ0 ≈ζ, v≈v0, (38)
we obtain the jump conditions ∂θ
∂y1 2
y1=0+−∂θ
∂y1 2
y1=0−= 2Z
1 + ∂ζ
∂x
2−1 θ|y1=0
−∞
exp( τ
Z−1+δτ)dτ, (39)
∂θ
∂y1
y1=0+−∂θ
∂y1
y1=0−=− 1 +
∂ζ
∂x
2−1∂ζ
∂t + (vx∂ζ
∂x −vy)
y1=0
. (40)
3.2 Formulation of the interface problem
The interface problem will be written as a system of equations for the reactant and a system of equations for the product as well as the jump conditions.
We have for y > ζ (in the unburnt medium)
∂θ
∂t +v.∇θ= Δθ, (41)
α≡0, (42)
v+∇p=Rp(θ+θ0)(1 +λsin(σ1t)cos(σ2t))γ, (43)
∇.v= 0. (44)
In the burnt medium (y < ζ), we obtain the system
∂θ
∂t +v.∇θ= Δθ, (45)
α≡1, (46)
v+∇p=Rp(θ+θ0)(1 +λsin(σ1t)cos(σ2t))γ, (47)
∇.v= 0. (48)
We finally complete this system by the following jump conditions at the inter- face y=ζ
[θ] = 0, ∂θ
∂y
=
∂ζ∂t
1 + ∂ζ
∂x
2, (49)
∂θ
∂y 2
=− 2Z
1 + (∂x∂ζ)2
θ(ζ)
−∞
exp s
1/Z+δs
ds, (50)
[v] = 0. (51)
Here we denote by [ ] the quantity [f] = f|ζ−0−f|ζ+0.The above free boundary problem is completed with the conditions at infinity:
y→+∞, θ =−1 and v= 0, (52)
y→ −∞, θ = 0 andv= 0. (53)
3.3 Travelling wave solution
We perform the linear stability analysis of the steady-state solution for the interface problem. This interface problem has a travelling wave solution:
θ(t, x, y) =θs(y−ut), α(t, x, y) =αs(y−ut) andv= 0, (54) in which the steady-state solution θs is given by:
θs(t, y) =
0 if y <0
e−uy−1 if y >0, (55) and the steady-state solution αs is written as:
αs(t, y) =
1 if y <0
0 if y >0. (56)
where the number u stands for the stationary front velocity.
Introducing the coordinates in the moving frame defined by y1 = y−ut, the above travelling wave is now considered as a stationary solution of the following problem:
∂θ
∂t +u∂θ
∂y +v.∇θ = Δθ, (57)
v+∇p=Rp(θ+θ0)(1 +λsin(σ1t)cos(σ2t))γ, (58)
∇.v= 0, (59)
together with the jump conditions (49)-(51).
To perform the linear stability analysis, we introduce a small perturbation to the stationary solution. To this end, we consider a perturbation of the reaction front of the form
ζ(t, x) =ut+ξ(t, x), with ξ(t, x) =1(t)eikx. (60) The stability of the solution is carried out by assuming the solution of the problem in the perturbed form:
θ =θs+ ˜θ, v=vs+ ˜v, (61) where
θ(t, x, y) =˜ θj(y, t)eikx, forj = 1, 2,
˜
v(t, x, y) =vj(y, t)eikx, forj = 1, 2. (62) Here the index j = 1 corresponds to solutions forz <0 andj = 2 corresponds to those for z > 0. For simplicity, we eliminate the pressure p and the com- ponent vx of the velocity from the interface problem applying two times the operator curl. Thus, we obtain the following problem
For the burnt media (y <0 ):
v1−k2v1 =−Rpk2(1 +λsin(σ1t)cos(σ2t))θ1, (63)
∂θ1
∂t −θ1−uθ1+k2θ1 = 0. (64) For the unburnt media (y >0 ):
v2−k2v2 =−Rpk2(1 +λsin(σ1t)cos(σ2t))θ2, (65)
∂θ2
∂t −θ2−uθ2+k2θ2 =uexp(−uy)v2, (66) Taking into account that
θ|ξ=±0 =θs(±0) +ξθs(±0) + ˜θ(±0), (67) and
∂θ
∂y|ξ=±0 =θs(±0) +ξθs(±0) + ∂θ˜
∂y(±0), (68)
Substituting (60) into (49)-(51), we obtain the following jump conditions:
θ2(0, t)−θ1(0, t) =u1(t), (69) θ2(0, t)−θ1(0, t) =−1(t)u2−1(t) +v1(0, t), (70)
1(t)u2+θ2(0, t) =−Z
uθ1(0, t), (71)
v2(i)(0, t) =v(i)1 (0, t) i= 0,1. (72)
4 Numerical results and discussions
To construct the convective instability boundaries, we solve numerically the problem (63)-(66) with the jump conditions (69)-(72) for given values ofZ and k and for varying values of Rayleigh number Rp. The numerical accuracy is controlled by decreasing the time and space steps.
4.1 Periodic modulation case
We consider that the frequency ν2 of the modulation amplitude is equal to 2ν1 or toν1/2. Figure 1 shows the critical Rayleigh number as function of the amplitudeλfor the two different cases. It can be seen that in the absence of the modulation (σ1 = 0), we find the same result as the unmodulated case [2]. The curves indicate that for relatively small values of the modulation amplitude λ and for σ2 = σ1/2, a destabilizing effect is observed. In contrast, a slight stabilizing effect may be gained when σ2 = 2σ1. In other words, tuning the frequency of the modulation at half that of the vibration causes a destabilizing effect.
4.2 Quasi-periodic modulation case
We now assume that the amplitude Λ(t) of the gravitational modulation is modulated with the frequencyν2 such thatν1 andν2 are incommensurate (the frequencies ratio ν2/ν1 is irrational) such that the gravitational modulation is QP.
0 50 100 150 0
50 100 150 200 250
λ
RC
σ2 = 0 σ2 = 2σ1 σ2 = σ1/2
Figure 1: Critical Rayleigh versus λ for different values of frequencies ratio and for σ1 = 100.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 100 200 300 400 500 600
σ1
RC
σ2 = 21/2 σ1 σ2 = 31/2 σ1 σ2 = 51/2 σ1 σ2 = 111/2 σ1 σ2 = 371/2 σ1
Figure 2: Critical Rayleigh number versus σ1 for λ= 10.
Figure 2 illustrates for different frequencies ratio the critical Rayleigh num- ber as function of the frequency of the vibration, σ1, for the value of the ampli- tude λ= 10. This figure shows that for frequencies ratio σσ2
1 =√
2, stabilizing effect can be gained in a large interval ofσ1 (σ1 1400) and for relatively large values of Rayleigh number; see the pick in the figure for Rc 450. Beyond σ1 1400, the critical Rayleigh number drops to the value of the unmodulated case (Rc 26) causing an abrupt destabilization of the reaction front. The plots of Fig. 2 also show that increasing the value of the frequencies ratio,
0 500 1000 1500 2000 2500 0
50 100 150 200 250 300 350 400 450
σ1
Rc
σ2=51/2 σ1 σ2=111/2 σ1 σ2=371/2 σ1
Figure 3: Critical Rayleigh number versus σ1 in the case of two additional QP vibration (picked from [9]).
increases the maximum value of the pick and shifts the abrupt destabiliza- tion value left causing the domain of stability to decrease substantially in σ1 direction and to increases in Rc direction.
Note that this result obtained by modulating the amplitude of the GV (Fig.
2) is different from that given in the case of a QP vibration consisting of two additional excitations [9]. Indeed, in the later case, the shift phenomenon of the abrupt destabilization has not been observed (see Fig. 3 picked from [9]);
the shift occurs at the same value of the frequency σ1. It is also interesting to mention that for high values of σ1, the curves tend to the unmodulated case.
Figure 4 shows that decreasing the frequency ratio, a stabilizing effect is gained in a certain interval of the frequencyσ1 (σ1 ≈1300) and a destabilizing effect is observed beyond.
Finally, Fig. 5 depicts the influence of the amplitudeλof the QP vibration on the stability of the reaction front. This figure shows that increasing λ has a stabilizing effect on the reaction front.
4.3 Conclusion
In this work, the influence of the modulation amplitude of a GV on the convec- tive instability of reaction fronts in porous media was studied considering two cases. The first case assumes that the frequency of the modulation σ1 is either σ1 = 2σ2 or σ1 = σ22 (periodic modulation), while the second case considers that the amplitude of the GV is modulated with another frequency such that the resulting gravitational vibration is QP.
0 200 400 600 800 1000 1200 1400 1600 1800 2000 0
100 200 300 400 500 600
σ1
RC
σ2 = 21/2 σ1 σ2 = (1/2)1/2 σ1 σ2 = (1/11)1/2 σ1 σ2 = (1/37)1/2 σ1
Figure 4: Critical Rayleigh number versus σ1 for λ= 10.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 50 100 150 200 250 300 350 400 450 500 550
σ1
RC
λ = 10 λ = 30
Figure 5: Critical Rayleigh number versus σ1 for different values of the ampli- tude λ and for σ2/σ1 =√
2.
To approximate the convective instability threshold, the original reaction- diffusion problem was first reduced to a singular perturbation one using the matched asymptotic expansion. Then, the linear stability analysis of the steady-state solution for the interface problem was performed and the obtained reduced interface problem was numerically solved.
The results shown that in the case of a periodic modulation and for rel- atively small values of the modulation amplitude λ a destabilizing effect of the reaction front is observed when σ1 = σ22. In other words, the stability of
the reaction front can be obtained by tuning the frequency of the modulation amplitude to half that of the vibration.
In the case of a QP vibration, the results shown that for a certain value of the GV frequency σ1, an abrupt destabilization of the reaction front may occur. It was also shown that increasing the value of the frequencies ratio shifts this destabilization value toward smaller frequencies of the GV. This abrupt destabilizing of the reaction front occurring for different values of the frequencies ratio has not been observed in the case of a QP vibration with two additional excitations [9]. Also, it was noticed that increasing λ reduces drastically the stabilization region of the reaction front.
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