Interfacial instability of two superimposed immiscible viscous fluids in a vertical Hele-Shaw cell under horizontal periodic oscillations

(1)

Interfacial instability of two superimposed immiscible viscous fluids in a vertical Hele-Shaw cell under horizontal periodic oscillations

Jamila Bouchgl and Sa¨ıd Aniss^*

University Hassan II, Faculty of Sciences A¨ın-Chock, Laboratory of Mechanics, Casablanca, Morocco Mohamed Souhar

Lemta, UMR CNRS7563, Ensem,2avenue de la Forˆet de Haye, B.P.160, Vandoeuvre-L`es-Nancy54504, France (Received 22 February 2013; published 29 August 2013)

The effect of horizontal periodic oscillations on the interfacial instability of two immiscible, viscous fluids of different densities, confined in a vertical Hele-Shaw cell, is investigated. An inviscid linear stability analysis of the viscous basic flow leads to the periodic Mathieu oscillator describing the evolution of interfacial amplitude.

We examine mainly the effect of the periodic oscillations and the influence of the viscosity on the stability of the interface. The results show that a decrease in the viscosity contrast has a stabilizing effect on the Kelvin-Helmholtz instability, which is displaced towards the long-wave region. The effects of other parameters such as the frequency number and the Weber number are also examined.

DOI:10.1103/PhysRevE.88.023027 PACS number(s): 47.20.−k

I. INTRODUCTION

Several works have been carried out to study the interfacial instability of superposed layers subject to periodic oscillations both experimentally and theoretically [1–6]. For instance, the Kelvin-Helmholtz instability with oscillating flow was studied first by Kelly [2]. He focused his study on the stability of an interface between two superposed fluid layers in which the velocity fields are periodic. He reduced the linear stability problem to the periodic Mathieu equation and discovered cases where the oscillations stabilizes the unstable shear flow.

An application of Kelly’s problem [2] has been given by Lyubimov et al. [3], who studied the interfacial instability of two superposed immiscible fluids subject to horizontal oscillations. A model which takes into account dissipation processes due to the presence of viscous friction has also been constructed. In this study [3], authors have analyzed the behavior of the interface of fluid layers with different densities and they have found that the basic instability mode, associated with the development of the Kelvin-Helmholtz instability on the interface between counterstreaming flows, occurs and the possibility of a parametric resonance appears.

In the limit of zero viscosities, the first type of instability is characterized by a finite excitation amplitude threshold, whereas there is no threshold for the parametric instability.

However, for fairly high vibration frequencies, the parametric instability occurs over a narrow wave-number range and it is highly sensitive to viscous damping. Later, another study, similar to that in [2] and [3], was carried out by Khenner et al. [4], who treated the linear stability of the interface between two fluids filling a cavity which performs horizontal harmonic oscillations. Similarly to Kelly [3], they reduced the linear stability problem under the inviscid approximation to the Mathieu equation. Hereafter, they have examined the parametric resonant regions of instability associated with the intensification of the capillary waves at the interface and they have compared the results to those found in the viscous case

*s.aniss@etude.univcasa.ma

in a fully numerical investigation. Recently, the linear stability analysis model carried out by Khenner et al. [4] has been extended both theoretically and experimentally by Talibet al.

[5,6] to include the effect of the viscosities of the fluid layers of finite depth and confined between horizontal boundaries.

In these studies, the authors have solved numerically, using spectral methods, the linear stability problem for an exhaustive range of vibrational-to-viscous force ratios and viscosity contrasts. They have shown that the viscous model allows us to predict the onset of each mode of instability, particularly in the limit of high-viscosity contrast. Talibet al.[6] characterized the evolution of neutral curves from the multiple modes of the parametric resonant instability to the single-frozen-wave mode encountered in the limit of practical flows. They found that the interface is linearly unstable to a Kelvin-Helmholtz mode and to successive parametric-resonance modes. These two modes exhibit opposite dependencies on the viscosity contrast, which have been understood by examining the eigenmodes near the interface. More recently, Jalikopet al.[7]

investigated experimentally the dependence of the onset of the secondary instability on the viscosity ratio. They considered high viscosity ratios and they found that the primary wave loses stability to a secondary oscillatory transverse amplitude modulation, which is approximately subharmonic to the forcing frequency.

In this paper, we consider the case of two immiscible, incompressible viscous liquid layers confined in a vertical Hele-Shaw cell under horizontal sinusoidal vibration. Then we perform an inviscid linear stability analysis to describe the perturbation of the interface. This study aims to examine the effect of periodic oscillations and the influence of the viscosity, which acts on the vertical walls of the cell, on the threshold of the interfacial instability. The physical motivation of this work are varied and include the following.

(i) We extend the linear stability models studied in [3–5] to include the effect of the viscosity of the liquid layers confined in a vertical gap and of finite depth. In this configuration, the oscillation of the cell accelerated differentially the superposed layers of immiscible fluids because of their different densities and the resulting time periodic shear flow depends significantly

(2)

on the viscosity contrast between the fluids. We show that the instability of the interface depends significantly on this viscosity contrast. The results in the case of equal viscosities and in the limit of a large frequency number and low viscosities are compared to those found in the case of inviscid approximation performed in [3] and [4].

(ii) We discuss the conditions under which the analogy exists between a Hele-Shaw cell and a porous medium described by Darcy’s model. In this work, this analogy differs from the classical one in which the system is stationary and the depth-averaged velocity is proportional to the pressure gradient. Indeed, the incompressible liquids are contained in a Hele-Shaw cell with end walls, thus only the periodic oscillation, and not the pressure gradient, generates the time periodic flow, which consists of counterflowing layers. In addition, Darcy’s model and the Hele-Shaw equations include the inertial terms. We justify the Hele-Shaw analogy to simulate the interfacial instability in porous medium by determining, in Sec.II B2, the conditions under which the basic state, calculated from equations of motion in the Hele-Shaw cell, is identical to the one calculated from Darcy’s model. We show that this analogy exists for low-dimensional frequencies of oscillation and when the permeability of the porous medium is defined byK=e²/12, whereeis the thickness of the cell.

However, the analogy of the perturbed equations in the inviscid theory is obvious. In this relevant case of a low-dimensional frequency, the analogy can serve to simulate the influence of the periodic horizontal vibration on the threshold of interfacial instability in porous media.

II. FORMULATION A. Governing equations

Consider two immiscible, incompressible Newtonian fluid layers confined in a vertical Hele-Shaw cell (see Fig. 1).

The heavy fluid occupies the bottom region, of heighth₁, and the light one occupies the upper region of height,h₂. We denote the height of the cellh=h₁+h₂; the distance between the vertical walls,e; and the aspect ratio of the cell,=_h^e 1.

The valuesz= ±^e₂andy=0,hcorrespond to the boundaries of the cell. Each fluid layer is characterized by a densityρj

and a dynamic viscosity μ_j, where the subscripts j =1,2 denote the lower and the upper layer, respectively. Assume that the Hele-Shaw cell is submitted to horizontal oscillatory motion according to the law of displacementacos(ωt)x, where a andωdesignate, respectively, the displacement amplitude and the dimensional frequency of the oscillatory motion.

Therefore, the fluid layers are submitted to two volumic forces:

FIG. 1. Schematic of the arrangement of the Hele-Shaw cell.

the oscillatory force −ρ_jaω²cos(ωt)x and the gravitational force ρ_jg. The denser fluid is placed in the lower layer, so that the configuration is gravitationally stable. Under these assumptions, the physical problem is governed by the following Navier-Stokes equations:

∇·Vj =0 (j =1,2), (1)

dVj

dt = − 1 ρj

∇p_j+ν_jVj+aω²cos(ωt)x+g, (2) wherepj is the hydrodynamic pressure andVj =(uj,vj) the velocity in each fluid layer.

B. Base-flow solution

In the basic state, the fluid layers are separated by an interface which is initially planar, horizontal, and coincident with they =0 plane by choice of the coordinate system. To determine the basic flow we look for the one-component velocity field,V^b_j(z,t)=(U_j^b(z,t),0,0), which is periodic, parallel to thexaxis. In the traditional Hele-Shaw flow [8,9] where the aspect ratio of the cell is considered lower than unity,1, a first approximation is obtained from Eqs.(1)and(2)as follows:

ρj

∂U_j^b

∂t = −∂P_j^b

∂x +μj

∂²U_j^b

∂z² +ρjaω²cos(ωt), (3) 0= −∂P_j^b

∂y −ρjg. (4) The vertical end walls atx=0,Lof the Hele-Shaw cell gen- erate counter-flowing layers which arise when the horizontal volume flux is conserved. This is modeled by the integral condition of balance of the displacement volume of both fluids [3]:

₀

−h1

V^b₁·xdy= − h₂

0

V^b₂·xdy. (5) Using the no-slip boundary conditions on the vertical walls of the cell,U_j^b(z,t)=0 atz= ±₂^e, and condition(5), Eqs.(3) and(4)are integrated with respect to coordinatezto obtain the velocity and the pressure of the viscous basic state, written as U_j^b(z,t)=Fj(z) cos(ωt)+Gj(z) sin(ωt), (6) P_j^b = −ρjgy+Cj, (7) whereC_j are arbitrary constants and the functionsF_j(z) and G_j(z) are given by:

F₁(η)= −H F₂(η)

= −aω

H(ρ−1) ρH+1

sinh _η

e

sin 1−^η_e cosh( )+cos( ) +sinh

1−^η_e sin _η

e

cosh( )+cos( )

, G₁(η)= −H G₂(η)

= −aωH(ρ−1) ρH+1

−1+cosh _η

e

cos 1−^η_e cosh( )+cos( ) + cosh

1−^η_e cos _η

e

cosh( )+cos( )

,

(3)

with η=z+^e₂, H= ^h_h²₁, and = ^ωe₂² _μ^ρ¹₁^h_h²₂⁺₊^ρ_μ²₂^h_h¹₁ = σ₂ _μH^ρH⁺₊¹₁, where the parameter σ₂= ^ωe_2ν₂² is the frequency number,ρ= ^ρ_ρ¹₂is the density ratio, andμ= ^μ_μ¹₂is the viscosity ratio, which can be expressed as a function of the viscosity contrast, μ= ¹⁻₁₊^A_A, with A= ^μ_μ²₂⁻₊^μ_μ¹₁ and −1A1. Fur- thermore, the averaged components of the velocity field are

U¯_j^b=F_jcos(ωt)+G_jsin(ωt) (j =1,2), (8) with

F₁ = −aωH(ρ−1) ρH+1

1

sinh( )−sin( ) coth( )+cos( )

, (9) F₂ = −1

HF₁, (10)

G₁= aωH(ρ−1) ρH+1

1

− sinh( )+sin( ) cosh( )+cos( )

, (11) G₂= −1

HG₁. (12)

In the following we consider two limiting cases related to low and high frequencies.

1. Base-state solution in the high-frequency limit In the high-frequency limit,ω−→ ∞, corresponding to

−→ ∞, the viscous base-flow solution determined in this work, and given by Eqs.(8), tends to the solution corresponding to the inviscid approximation given by Khenneret al.[4] in the case of two fluid layers of infinite extent in the horizontal directions

U¯₁^b=aωh₂(ρ1−ρ₂)

h₁ρ₂+h₂ρ₁sin(ωt), (13) U¯₂^b = −h₁

h₂U¯₁^b. (14) 2. Hele-Shaw analogy in the low-frequency limit In the case of two superimposed immiscible viscous fluids saturating a porous medium with constant porosity and constant permeabilityK, the equations of conservation of mass and momentum in the presence of horizontal oscillations are written in Darcy’s model as

∇·Vpj =0 (15) ρj

ε ∂Vpj

∂t +Vpj

ε ∇·Vpj

= −∇pj−μj

KVpj +ρjaω²cos(ωt)x+ρjg, (16) where j =1,2 and Vpj designates the Darcy velocity [10].

The velocity of the basic state,V^b_pj(t)=(U_pj^b(t),0,0), is of the form

U_pj^b (t)=Fpjcos(ωt)+Gpjsin(ωt). (17)

Using Eqs.(5),(15), and(16),Fpj andGpj are written as

F_p1 = −H F_p2= aω²H(ρ−1) (ρH+1)

K_μ₁_h₂₊_μ₂_h₁

ρ1h2+ρ2h1

K_ω

ε

₂

+_μ₁_h₂₊_μ₂_h₁

ρ1h2+ρ2h1

₂,

G_p1 = −H G_p2= aω³H(ρ−1) (ρH+1)

K² εK_ω

ε

2

+ε_μ₁_h₂₊_μ₂_h₁

ρ₁h₂+ρ₂h₁

2.

In the low-frequency limit,ω→0, the velocity component of the base state in the porous medium has the expansion in terms ofωas follows:

F_p1 = −H F_p2 =aH(ρ−1) ρH+1

ρ₁h₂+ρ₂h₁

μ₁h₂+μ₂h₁Kω²+O(ω⁴), G_p1 = −H G_p2= aH(ρ−1)

ρH+1

ρ₁h₂+ρ₂h₁ μ₁h₂+μ₂h₁

₂ K²

ε ω³ +O(ω⁵).

Therefore, at orderω⁴ the velocity, (17), in each fluid layer becomes

U¯_p1^b =2aωH(ρ−1) ρH+1

2

pcos(ωt), (18) U¯_p2^b = −h₁

h₂U¯_p1^b , (19) where p = ^ωK₂ _μ^ρ¹₁^h_h²₂⁺₊^ρ_μ²₂^h_h¹₁.

Similarly, expressions(9)–(12)are expanded for smallωto obtain

F₁= −H F₂= aH(ρ−1) ρH+1

e²

12ω²+O(ω⁴), G₁= −H G₂ =aH(ρ−1)

ρH+1

2 e⁴ 120ω³ +O(ω⁵).

Then the velocity components at order ω⁴ of fluids in the Hele-Shaw cell are

U¯₁^b = aωH(ρ−1) ρH+1

2

6 cos(ωt), (20)

U¯₂^b= −h₁

h₂U¯₁^b. (21) Finally, in the low-frequency limit, the base flow in a porous medium, Eqs. (18) and (19), and that in a Hele-Shaw cell, Eqs.(20)and(21), are similar if the permeabilityK=e²/12.

C. Linear stability

We assume that the base state is disturbed so that the velocity and the pressure fields in the perturbed state are written as the sum of the base flow variables and small perturbations:

Vj =V¯^b_j +vj(u(x,y,t),v(x,y,t)), P_j =P_j^b+p_j(x,y,t).

(4)

Hence, the linearized system of the conservation equations in the inviscid approximation is

∂u_j

∂x +∂v_j

∂y =0, (22)

ρj

∂uj

∂t +U¯_j^b∂uj

∂x

= −∂pj

∂x, (23)

ρ_j ∂vj

∂t +U¯_j^b∂vj

∂x

= −∂pj

∂y . (24)

To investigate the dynamical evolution of the interface in a vertical Hele-Shaw cell, we describe the instantaneous interface asy=0+ξ(x,t), whereξ(x,t) is an infinitesimal perturbation of the horizontal interface. Note that the Hele- Shaw analogy with a porous medium is valid for inviscid perturbations, and by using the relationVj = ^V^pj.

Hereafter, we seek the solution of the system of Eqs.(22)–(24)in terms of normal modes as

[pj,u_j,v_j]=[ ˜p_j(t,y),u˜_j(t,y),v˜_j(t,y)]e^ikx, (25) ξ(x,t)=ξ˜(t)e^ikx, (26) wherei² = −1 andk is the wave number in thex direction.

Moreover, we consider the velocity potentialsφj, solutions of the continuity equation, Eq.(22), defined by

φj =φ˜j(y,t)e^ikx =

C_j¹(t)e^ky+C_j²(t)e⁻^ky

e^ikx. (27) The constants C_j¹ andC_j² are determined by using the slip boundary conditions at the horizontal walls, u_j(y,t)=0 at y= −h₁,y=h₂, and the kinematics conditions linearized at the interface,

dξ˜(t)

dt +ikU¯_j^bξ˜(t)= ∂φ˜j

∂y . (28)

To complete the set of equations we provide the dynamic boundary condition at the interface,

P₁^b+p₁

−

P₂^b+p₂

=γ∇·n, (29) where n is the unit vector normal to the interface, ∇·n=

−^∂_∂x²^ξ2(t) is the linearized form of the curvature interface, andγ is the surface tension. Hereafter, the total pressure is linearized aty =0 as follows:

P_j^b+pj =P_j^b(0)+∂P_j^b

∂y

y=0

ξ+pj(y). (30) With the above assumptions, Eq.(29)leads to a parametric differential equation for the amplitude ˜ξ(t) of the interface displacement from its equilibrium position,

d²ξ˜(t) dt² +2ik

R₁U¯₁^b+R₂U¯₂^b [R₁+R₂]

dξ˜ dt +

(ρ₁−ρ₂)

R₁+R₂ gk+ikR₁^d_dt^U^¯¹^b+R₂^d_dt^U^¯²^b R₁+R₂

−k²R₁ U¯₁^b₂

+R₂ U¯₂^b₂

R₁+R₂ + σ k³ R₁+R₂

ξ˜(t)=0, (31) where

R₁=ρ₁coth(kh1), R₂=ρ₂coth(kh2).

To eliminate the term which contains the first-order derivative in ˜ξ(t), we use the change of variable,

ξ˜(t)=ξ¯(t) exp

−ik

R₁U¯₁^b+R₂U¯₂^b R₁+R₂ dt

. (32) After that, the space is scaling by the capillary length,l_c= [_g(ρ^γ

1−ρ₂]¹², and the time by _ω¹. Hence, Eq.(31)is reduced to the nondimensional form,

d²ξ¯(t) dt² + 1

We

n(1+n²)(ρ−1) ρcoth(nH₁)+coth(nH₂) + 4n²B_v

ρU¯₁^bcoth(nH1)+U¯₂^bcoth(nH2)2

[ρcoth(nH1)+coth(nH2)]²

− ρU¯₁^b2

coth(nH₁)+U¯₂^b2

coth(nH₂) ρcoth(nH₁)+coth(nH₂)

ξ¯(t)=0, (33) where We=^ω_g²^l^c is the Weber number,H_j = ^h_l_c^j is the dimensionless layer height, andn=klc is the dimensionless wave number. The Bond number, B_v, characterizes the vibration intensity and it is defined by

Bv =a²ω² 4

ρ₁−ρ₂ gγ

¹₂ . Finally, Eq.(33)is then given by

d²ξ¯(t)

dt² +[δ+β₁cos²(t)+β₂sin²(t)+β₃sin(2t)] ¯ξ(t)=0, (34) which is a Mathieu equation, where the coefficients are

δ= n(1+n²)(ρ−1)

αWe , (35)

β₁ = 4n²Bv

αWe

ρcoth(nH1) ρ

αcoth(nH1)−1

F²₁ +coth(nH2)

coth(nH₂) α −1

F²₂ +2ρ

αcoth(nH₁) coth(nH₂)F₁F₂

, (36)

β₂ = 4n²Bv

αWe

ρcoth(nH₁) ρ

αcoth(nH₁)−1

G²₁ +coth(nH2)

coth(nH₂) α −1

G²₂ +2ρ

αcoth(nH1) coth(nH2)G1G₂

, (37)

β₃ = 4n²B_v αWe

ρcoth(nH1) ρ

αcoth(nH1)−1

F₁G₁ +coth(nH₂)

coth(nH2) α −1

F₂G₂ +ρ

αcoth(nH1) coth(nH2)(F1G₂+F₂G₁)

, (38) withα=ρcoth(nH1)+coth(nH2).

(5)

Equation (34) is solved numerically using the spectral method inMATLAB, this method provides a matrix formulation of the linear stability problem which corresponds to an eigenvalue problem. This problem can be also solved using a method based on the Floquet theory, which leads to the Hill determinant corresponding to the marginal stability curve [11]. This determinant can be written formally in the formf(We,n, Bv, ρ, σ₂, A, H₁, H₂)=0.

III. RESULTS AND DISCUSSION

The instability of the interface is studied by analyzing the influence of the physical parameters such as the viscosity of the upper fluid, the viscosity ratio, and the frequency of oscillations. These parameters are characterized by the frequency number, the contrast of viscosities, and the Weber number.

The variation of the Bond number, which characterizes the vibration intensity, as a function of the wave number represents the marginal stability curve, Bv(n), which is determined numerically for assigned values of the frequency number,σ₂, the viscosity contrast,A, the Weber number, We, the density ratio,ρ, and the dimensionless layer thicknesses.

In the following results, two types of instabilities occur at the interface: the Kelvin Helmholtz instability and the parametric resonance. The criterion of the Kelvin Helmholtz instability was determined by Chandrasekhar [12] in the case of constant velocities and by Kelly [2] in the presence of a time-dependent velocity. This criterion can be obtained approximately, as in [2], by vanishing the sum of the stationary coefficients of the second term in Eq. (34). Otherwise, the parametric resonance is similar to that which occurs when the suspension point of a simple pendulum is oscillating vertically.

In this situation, the motion of the pendulum is governed by Mathieu’s equation. One can show that the stable equilibrium state can be destabilized and the unstable one can be stabilized.

The bounded solutions of Mathieu’s equation are harmonic or subharmonic depending on the amplitude and the frequency of forcing. Note that the subharmonic solutions correspond to the situation in which the frequency of the response is half the frequency of oscillations. In this study and as in [3], we show numerically that the most dangerous modes correspond to harmonic solutions.

1. Fluids of equal dynamic viscosities:

Effect of the frequency number

We examine first the effect of viscosity via the frequency number, σ₂, for fluids of equal dynamic viscosities, A= 0(μ1 =μ₂), and when the lower fluid is heavier than the upper one,ρ= ^ρ_ρ¹₂ =2. We use the representative values of the Weber number and the dimensionless heights, We=10 andH₁=H₂=1 [3,4]. It is worth noting, from results in Figs. 2(a) and 2(b), that when the frequency number σ₂ increases from σ₂=0.1 to σ₂=50, the marginal stability curves converge, in excellent agreement, toward that obtained by Khenneret al.[4] in the limit case corresponding to the inviscid approximation and infinite extent of the fluid layers in the horizontal directions. This coincidence occurs from σ₂=50 in Fig.2(b). This result is expected since a decrease in the dynamic viscosity corresponds to an increase in the

0 1 2 3 4 5 6 7

0 0.5 1 1.5 2 2.5x 10⁶

n B_v

σ₂=0.1 σ₂=0.12

First region parametricof instability Kelvin−Helmholtz

Instability

(a)

0 1 2 3 4 5 6 7

0 5 10 15 20 25

n B_v

(b)

Kelvin−Helmholtz Instability

σ₂=2 σ₂=3

σ₂=50

FIG. 2. Variation of the Bond number, Bv, versus the wave number,n, for We=10,ρ=2,A=0,H1=H2=1, and different values of the frequency number,σ2.

frequency number, and therefore, there is convergence to the marginal stability zones in which the base flows are that of inviscid fluids [4].

The parametric instability occurs for arbitrarily low vibration amplitudes, i.e., for any small values of the Bond number. However, the development of the Kelvin-Helmholtz instability [12] is observed for a small wave number, i.e., in the long-wavelength mode excited at the onset. For the low-frequency number in Fig.2(a), it turns out that an increase in the kinematic viscosity of the upper fluid,ν₂, corresponds to decreasing the frequency number,σ₂, and acts to stabilize the Kelvin-Helmholtz instability. Indeed, whenσ₂ decreases, the Kelvin-Helmholtz instability threshold increases and tends to shift to higher values. This significant stabilizing effect is observed through the threshold values presented in Table I.

For example, in the case corresponding to fluids of equal viscosities,A=0,B_vo=2.77 for σ₂=3 andB_vo=2.7×10⁵ forσ₂=0.1. Furthermore, the periodic oscillations give rise

(6)

TABLE I. Kelvin Helmholtz instability threshold,Bvo, for different values of the frequency number,σ2, and viscosity contrast,A, with We=10,ρ=2, andH1=H2=1.

A σ2=0.1 σ2=3 σ2=50

−0.9 Bvo=2.70×10⁷ Bvo=35.82 Bvo=1.87

−0.5 Bvo=1.08×10⁶ Bvo=3.21 Bvo=1.76 0 Bvo=2.7×10⁵ Bvo=2.77 Bvo=1.744 0.5 Bvo=1.2×10⁵ Bvo=2.54 Bvo=1.73 0.9 Bvo=7.48×10⁴ Bvo=2.45 Bvo=1.72

to two first regions of parametric instability having the most unstable wave numbers at n=3 and n=4.9. We see that the wave numbers, corresponding to the resonance, are still independent of the frequency number.

0 1 2 3 4 5 6 7

0 0.5 1 1.5 2 2.5 3 3.5

4x 10⁷

n B_v

A=−0.9

A=−0.5

(a)

0 1 2 3 4 5 6 7

0 0.5 1 1.5 2 2.5x 10⁶

n B_v

A=0 A=0.5

A=0.9 Kelvin−Helmholtz

Instability

(b)

FIG. 3. Marginal stability curves: variation of the Bond number, Bv, versus the wave number, n, for σ2=0.1, We=10, ρ=2, H1=H2=1, and different values of the viscosity contrast,A.

2. Fluids of unequal viscosities: Influence of the viscosity contrast To examine the influence of the viscosity contrast, the neutral curves, Bv(n), are presented for different values of Aand for the following assigned values, We=10,ρ=2, and H₁=H₂=1 [Figs.3(a)and3(b)]. These curves are plotted in the pertinent case of a low frequency number, σ₂=0.1 (μ2 1). Thus, in this situation, A= −0.9 corresponds to μ₂ μ₁ and then the two fluids are very viscous, while A=0.9 corresponds toμ₂ μ₁and then the lower fluid layer is less viscous than the upper one. Similarly, the neutral curves in Figs. 4(a) and 4(b)are plotted for σ₂=3 (μ₂1); for A= −0.9, the upper fluid layer is less viscous than the lower one, and whenA=0.9 we are in the presence of two weakly viscous fluid layers. By inspecting these curves, we can see first how the Kelvin-Helmholtz threshold decreases dramatically as the viscosity contrast increases fromA= −0.9 toA=0.9.

The effect of the viscosity contrast is more pronounced in the case of a low frequency number, σ₂=0.1, than in the case

0 1 2 3 4 5 6 7

0 10 20 30 40 50 60 70 80 90 100

n Bv

A=−0.9

A=−0.5

(a)

0 1 2 3 4 5 6 7

0 2 4 6 8 10 12 14 16 18 20

n Bv

A=0.5

A=0.9 A=0 Kelvin−Helmholtz

Instability First region of

parametric instability

Stable Stable

Stable

(b)

FIG. 4. Marginal stability curves: variation of the Bond number, Bv, versus the wave number, n, for σ2=3, We=10, ρ=2, H1=H2=1, and different values of the viscosity contrast,A.

(7)

0 1 2 3 4 5 6 0

1 2 3 4 5 6 7

n Bv

First region of parametric instability

Stable Stable

Stable A=−0.9

A=0.9

A=0

FIG. 5. Marginal stability curves: variation of the Bond number, Bv, versus the wave number, n, for σ2=50, We=10, ρ=2, H1=H2=1, and different values of the viscosity contrast,A.

of the highest one,σ₂=3. Moreover, this result is confirmed forσ₂=50, where the effect of viscosity contrast disappears.

Indeed, Fig. 5 shows that for a high frequency number, σ₂=50, the marginal stability curves for different values of the viscosity contrast,A= −0.9 andA=0.9, coincide with stability boundaries of the inviscid case studied by Khenner et al.[4].

Finally, the results in Figs. 3(a) and 3(b) demonstrate that the decrease in viscosity contrast between the layers systematically increases significantly the stability threshold of the Kelvin-Helmholtz instability. This instability is displaced into a short-wave region in which the instability can be suppressed by viscosity. This stabilizing effect is due to the presence of vertical walls on which there is friction, and

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9 10x 10⁵

n Bv

We=10

We=50

We=100

FIG. 6. Marginal stability curves: variation of the Bond number, B_v, versus the wave number, n, for σ2=0.1, A=0.5, ρ=2, H1=H2=1, and different values of the Weber number,We.

this effect is more important for low values of the frequency number.

3. Dependence on the Weber number:

Effect of the oscillation frequency

We show in Fig.6 the dependence of the Bond number, Bv, on the wave number,n, forH₁=H₂=1,σ₂=0.1,ρ= 2, A=0.5, and different values of the Weber number, We.

We find that when the Weber number increases, i.e., with an increase in the vibration frequency, the resonance zones corresponding to the parametric instability shift to the right gradually. Indeed, Fig.6exhibits a significant increase in the wave numbers, which are displaced into the region of the short-wave perturbations. This displacement of the parametric instability regions is accompanied by a significant expansion of the Kelvin-Helmholtz instability region.

IV. CONCLUSION

In this study, we have performed a linear stability analysis of an interface between two viscous immiscible fluids of different densities confined in an oscillating vertical Hele-Shaw cell.

The linear problem has been reduced to a periodic Mathieu equation governing the evolution of the amplitude of the interface. We have focused our analysis on both the effect of the frequency number,σ₂, and that of the viscosity contrast,A, on the boundaries of the marginal stability. It was shown that a decrease in the frequency number has a strong stabilizing effect on the Kelvin-Helmholtz instability. However, an increase in this number leads, as expected, to previous results performed by Khenner et al. [3] in the situation of nonviscous fluid layers of infinite horizontal extent. Furthermore, the viscosity contrast has a stabilizing effect on the first mode of instability, i.e., the Kelvin-Hemholtz instability. This effect is more significant for a low frequency number. Indeed, in this pertinent case, the large friction at the vertical walls contributes to the decrease in the inertial effect and quite the same situation takes place in the case of a porous medium when the Darcy law is assumed for the resistance force. The effect of the Weber number has also been studied and we have shown that an increase in this number leads to an increase in the wave numbers where a short-wave parametric resonance occurs.

Finally, we note that experimental work taking into account the presence of the side walls, as in the Hele-Shaw situation, does not exist and the previous experiments in this area have focused on studying the interfacial instability in the presence of horizontal oscillations in the case of infinite lateral extent.

Therefore, an experimental study is considered as a perspective to this theoretical investigation.

ACKNOWLEDGMENTS

The authors would like to thank the unknown referees for their remarks and suggestions, which improved the work considerably. The authors gratefully acknowledge financial support from the Centre National de la Recherche Scientifique (CNRS), France, and the Centre National pour la Recherche Scientifique et Technique (CNRST), Morocco.

(8)

[1] G. H. Wolf,Z. Phys.227, 291 (1969).

[2] R. E. Kelly,J. Fluid Mech.22, 547 (1965).

[3] D. V. Lyubimov, M. V. Khenner, and M. M. Shotz,Fluid Dyn.

33, 318 (1998).

[4] M. V. Khenner, D. V. Lyubimov, T. S. Belozerova, and B. Roux, Eur. J. Mech. B: Fluids18, 1085 (1999).

[5] E. Talib and A. Juel,Phys. Fluids19, 092102 (2007).

[6] E. Talib, S. V. Jalikop, and A. Juel, J. Fluid Mech. 584, 45 (2007).

[7] S. V. Jalikop and A. Juel,Phys. Fluids24, 044104 (2012).

[8] S. Aniss, M. Souhar, and J. P. Brancher,Phys. Fluids 7, 926 (1995).

[9] A. Abidate, S. Aniss, O. Caballina, and M. Souhar,Phys. Rev.

E75, 046307 (2007).

[10] D. A. Nield and A. Bejan,Convection in Porous Media, 3rd ed.

(Springer-Verlag, New York, 2006).

[11] P. M. Morse and H. Feshbach,Methods of Theoretical Physics (McGraw–Hill, New York, 1953).

[12] S. Chandrasekhar,Hydrodynamic and Hydromagnetic Stability (Oxford University Press, London, 1961).