Fingering Instabilities of a Miscible Fluid Annulus on a ...

International Journal of Dynamics of Fluids ISSN 0973-1784 Vol. 1, No.1 (2005), pp. 57-68 © Research India Publications http://www.ripublication.com/ijdf.htm

Fingering Instabilities of a Miscible Fluid Annulus on a Rotating Hele-Shaw Cell Ching-Yao Chen*1 and Yu-Chia Liu2 *1

Department of Mechanical Engineering, National Yunlin University of Science & Technology, Yunlin, Taiwan 640 Republic of China E-mail: [email protected] 2

Department of Mechanical and Automation Engineering, Da-Yeh University, Taiwan, Republic of China

Abstract Numerical simulations of interfacial stabilities for miscible interfaces with a confined annulus in a rotating Hele-Shaw cell are presented by means of highly accurate numerical schemes. The influences of Coriolis forces, rotating speed, viscosity contrast and Korteweg stresses are simulated. The fingering patterns are detailed demonstrated by images of concentration, vorticity fields and streamlines systematically to depict the influences of various parameters. Keywords: Fingerings Instability, Rotating Hele-Shaw cell, Coriolis forces, Korteweg stresses

1. Introduction The Hele-Shaw flows driven by a centrifugal force have been the subject of recent studies [1-5], due to their potential applications to the technology of spin-coating or organic solvents for cleaning or mixing purposes. However, on the common coating or cleaning processes, a stable interface is desirable. A possible way for interfacial control is the arrangement of composite layer of fluids, or a fluid annulus. If a more viscous and heavier droplet is placed on a rotating Hele-Shaw flow, the interfacial fingering instability is mainly dominated by two opposite mechanisms, the stable viscous damping effects and unstable centrifugal forces. Nevertheless, the situation is more complex if the droplet is confined by a fluid annulus of less viscosity and lighter density. While effects acting on the inner interface are exactly the same as the previous case of simple droplet, the outer interface behaves differently with stabilizing centrifugal forces and unstable viscous effects. An interesting question arises that if the fingers on the inner interface and confined within the fluid layer of

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Ching-Yao Chen and Yu-Chia Liu

lighter annulus would break through the stable constraint of annulus. Reversed fingerings [6,7] are observed on the similar composite fluid layer of rectilinear geometry but with viscous unstable effects. The present study tends to investigate the effects of centrifugal destabilized forces in a radial geometry. Other factors to be clarified are the influences of the Coriolis forces and unconventional stresses on miscible interface, or so-called Korteweg stresses postulated by Korteweg [8]. The Coriolis forces, which are not commonly considered in the previous analytical and numerical studies, are expected to have significant effects at high rotating speeds. As the influences of Korteweg stresses, numerous studies have been conducted [9-15], and confirm their existence and stabilized nature similar to surface tension of immiscible fluids. Here, the coupling effects of Korteweg stresses with centrifugal and Coriolis forces will be also simulated to show their influences to the miscible interface. Delicate graphics of flow fields generated by means of highly accurate numerical scheme will be displayed to assist the understandings of influences by various control parameters. The outline of this paper is as follows. After the formulation of the physical problem and a brief review of the computational technique in § II, Section III focuses on the computational results and their interpretations. Conclusions are provided in § IV.

2. Physical Problem and Governing Equations We study the interfacial dynamics of miscible fluids with composite layers. An DQQXOXVRIOHVVYLVFRXVYLVFRVLW\ DQGOLJKWHUGHQVLW\!) fluid is sandwiched by a KHDYLHU !h DQG PRUH YLVFRXV h) fluid in a rotating Hele-Shaw cell as shown in figure 1(a). The governing equations, in a reference frame rotating with the cell, of continuity, momentum of the augmented Hele-Shaw equations [1,4,15] and concentration conservation take the forms

G ∇⋅u = 0 ∇( p + Q) = −

(2.1)

G G G 12 G µu + ρωˆ 2 x + 2ρωˆeZ × u + ∇ ⋅ (δˆ(∇c) × (∇c)T ) 2 b

∂c G + ∇ ⋅ (u c ) = D ∇ 2c ∂t

(2.2) (2.3)

G

Here, u denotes the velocity vector, b the Hele-Shaw cell gap spacing, ωˆ the G G angular speed, x the position vector on x-y plane, eZ the unit vector in z-direction, c the concentration of heavier fluid, D the diffusion coefficient and p the pressure. The YLVFRVLW\LVLQGLFDWHGE\DQG δˆ is the Korteweg constant. Q is the additional pressure due to the Korteweg stresses expressed as [9] Q=

2δ 2∆ρ δˆ 2 ∇c − 2 ∇ 2c + µD ∇ 2 c , 3 3 3ρ h

where δ 2 is a property constant and

! !h –! denotes the density difference. It

Fingering Instabilities of a Miscible Fluid Annulus

59

should be noticed that the problem will be solved in the formulation of vorticitystreamfunction, as described later. As a result, the additional pressure Q is cancelled and not explicitly showing in the final equations. In addition to the conventional viscous term on the right-hand side of the Hele-Shaw equations (2.2), the extra terms are the centrifugal forces, the Coriolis forces and the Korteweg stresses respectively. In order to render the governing equations dimensionless, we take the outer diameter of the annulus Do and the density difference • ! as the characteristic scales. Since the simulation is carried in the reference frame rotating with the cell, centrifugal-LQGXFHG WLPH / b2•! ωˆ 2 is taken as a characteristic time scale. By IXUWKHU VFDOLQJ ZLWK YLVFRVLW\ h and pressure •! ωˆ 2 Do 2, a characteristic velocity scale b2 •! ωˆ 2 Do h is obtained. Associated with the definition of the viscosityconcentration dependence [16] and density-concentration forms

µ (c) = µhe R(1−c) ,

ρ (c) = cρ h + (1 − c)ρl .

Figure 1: Reference case: Pe=6x103, R=- 5H DQG/ ,PDJHV RI FRQFHQWUDWLRQ fields at (a) t=0, (b) t=2.5, (c) t=5 and (d) t=7.5.

The dimensionless momentum and concentration equations are rewritten in terms of vorticity (&) and streamfunction (%),

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∇2ψ = −ω ,

(2.4)

1  ∂c ∂c  2 Re ∂c ∂c  δ  ∂c  ∂3c ∂3c  ∂c  ∂3c ∂3c  ,  −  u + v  +   ω = −R∇ψ ⋅ ∇c +  y − x  + + + µ  ∂x ∂y  µ  ∂x ∂y  µ  ∂x  ∂x2∂y ∂y3  ∂y  ∂x∂y 2 ∂x3 

(2.5)

∂c 1 + u ⋅∇c = ∇ 2c , ∂t Pe

(2.6)

where u and v are the velocity components at x and y directions respectively. The dimensionless rotating Peclet number Pe, the Reynolds number Re that accounts the Coriolis forces, viscosity contrast R DQGWKH.RUWHZHJFRQVWDQW/take the forms as µ ∆ ρ ωˆ 2 b 2 D 0 ∆ ρ ωˆ b 2 δˆ , Re = , R = ln l , δ = 4 µh 12 µ h D 12 µ h ∆ ρ ωˆ 2 D 0 2

Pe =

The boundary conditions are prescribed as follows: x = ± 1 :ψ = 0 ,

∂c = 0 ∂x

y = ± 1 :ψ = 0 ,

∂c = 0 ∂y

(2.7)

The initial conditions assume a less viscous annulus with dimensionless thickness Ta=0.25, shown as figure 1(a). Both sides of the fluid annulus are bounded by steep concentration gradients in forms of error function along the radial direction. The numerical methods applied here are largely identical to the techniques presented Ruith and Meiburg [17]. To solve the stream function equation (2.4) by a pseudospectral method, a Galerkin-type discretization using cosine expansion is employed in the streamwise direction:

ψ ( x , y , t ) = Σ ψˆ k ( y , t ) cos[ 2πkx ], ω ( x , y , t ) = Σ ωˆ k ( y , t ) cos[ 2π kx ]. In the normal direction, discretization is accomplished by sixth order compact finite differences. Vorticity equation (2.5) is evaluated by sixth order compact finite difference schemes. The spatial datives in the concentration equation (2.6) are discretized by sixth and fourth order compact finite difference schemes for diffusion terms and convection terms respectively. A fully explicit third order Runge-Kutta procedure on time employed to solve equation (2.6) and advance in time as ∂c = F ( c ), ∂t so that k k −1 k −1 k −2 ci , j = ci , j + ∆ t[γ k F ( ci , j ) + η k F ( ci , j )],

γ 1 = 8 / 15,η1 = 0; γ 2 = 5 / 12,η2 = −17 / 60; γ 3 = 3 / 4,η3 = −5 / 12.


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Details on the implementation of these schemes are provided by reference [16,17]. It has been noticed that numerical simulations of fingering instabilities are sensitive to the grid distributions. In order to reduce this grid-dependence effect, a very fine grid 513x513 is employed. Even direct validations by comparing to the experimental or theoretical results, which are yet available, are not possible. The numerical code had been quantitatively validated by comparing the growth rates with the values obtained from the linear stability theory in both a plane and radial front [16,17]. In addition, excellent quantitative and qualitative agreements are reached by comparing fingering patterns and numbers of fingers to the correspondent experimental and theoretical results in similar flow fields of a rotating droplet [18]. The above validations have provided the credibility of the present numerical code.

3. Results and Discussions There are four dimensionless control parameters presented, such as the Peclet number Pe which can be interpreted as the dimensionless rotating speed, the viscosity parameter R representing the viscosity contrast, and the Reynolds number Re, which is the effect of the Coriolis forces and the KortewegFRQVWDQW/ These parameters will be analyzed systematically. 3.1 Reference Cases Representative calculation for R=-2, Pe=6x103/ LVILUVWO\SUHVHQWHG7KH5-value indicates that the droplet is about 7.4 times more viscous than the fluid annulus. In common spin-coating practices with a rotating frequency in the order of thousands rpm, the equivalent Reynolds numbers of the Coriolis forces are in the magnitude of Re < O(1), which is considered in the present simulation. Figure 1 displays the time sequences of the concentration fields. At t=2.5 shown in figure 1(b), even though the mixing front of droplet is viscously stable, very vigorous interfacial instability is triggered by the strong centrifugal force on the inner interface. To conserve the mass, opposite fingers of less viscous fluid penetrate toward the droplet origin. Nonlinear tip-splitting, which is commonly seen in the conventional viscous fingering pattern resulting from fluids' viscosity contrast, is observed as well. However, as the arrivals of fingers to the boundary of confined fluid annulus, the growths of fingers are limited, which are in line of previous findings [7]. Nevertheless, a weak but yet interesting secondary interfacial instability are induced at t=5, as shown in figure 1(c). At later stage t=7.5, the constrain of fingering growth by the annulus associated with the great dispersion around the central region eventually lead to significant mixing and destructs fingering structures eventually. Developments of fingering instabilities can also be understood by the generation of vorticities. Images of absolute values of vorticity scaled by the instant maximum, as displayed in figures 2(a,b), show the structures of dipole pairs with local maximum at the tips. The secondary interface is clearly shown in figure 2(b). While the vorticities diploes of main interface diminish significantly due to constrain of annulus, pairs of more active vorticities dipoles are induced and catch from behind. The fingerings induced by the centrifugal forces can also be visualized by contours of streamfunction, as shown in figures 2(c,d).

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Numerous eddy pairs that perturb fingers' growth can be identified on the mixing interfaces. The streamlines distributions are denser at the region of higher vorticities, which represent vigorous fingerings. For higher rotating speed often used in practical spin coating process, the effects of the Coriolis forces shall be taken into account. The effects of the Coriolis forces are presented by simulations of Re=0.1 at time t=2.5, 5, as concentration images shown in figure 3. Compared to the previous non-Coriolis forces case, the fingers show a clearly trend of rotation counter-clockwise. The roots of the fingers appear thicker and more irregular. In addition, no secondary interfacial instability is observed, that indicates a stabilized mechanism by the Coriolis forces. Above results can be understood by the tangential shear effects to the mixing interfaces. Rogerson and Meiburg [19] have found that a tangential shear stabilizes a plane mixing interface.

Figure 2: Reference case: Pe=6x103, R=-5H DQG/ ,PDJHVRIYRUWLFLW\ILHOGVDW (a) t=2.5, (b) t=5, and contours of streamlines at (c) t=2.5 and (d) t=5.


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Figure 3: Images for concentration fields: Pe=6x103, R=-5H DQG/ at (a) t=2.5 and (b) t=5. In the present situation, the Coriolis forces are oriented perpendicularly to both the rotating axis (z-direction) and the local velocity (radial-direction), as expressed in equation (2.2). Tangential shear stresses result from the Coriolis forces, and lea to a more stable mixing interface with rotating fingering patterns. Shown in figures 4 and 5 are the corresponding plots for vorticities and streamlines. They confirm both the more stable single fingering front with less numbers of vorticity dipole and streamline eddy pairs as well as the rotating orientation caused by the presence of Coriolis forces.

Figure 4: Images for vorticity fields: Pe=6x103, R=-5H DQG/ DW (a) t=2.5 and (b) t=5.

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Figure 5: Contours for streamlines: Pe=6x103, R=-5H DQG/ DW (a) t=2.5 and (b) t=5. The previous simulations are carried by an ideal circular initial condition. The fingerings are perturbed by purely physical mechanisms and grid distribution. Perfect symmetries are preserved. However natural random noises prevent the above ideal situations, rather an irregular pattern occurs. We also carry simulations that are more realistic to occur naturally. A small magnitude of random noises is applied to the positions of 0.5 concentration. The results are displayed in figure 6. Even though the fingers appear more irregularly due to the random perturbations both for Re=0, 0.1, the main feature, such as the numbers of fingers, remains quite similar to the previous ideal cases. As a result, in order to visualize the influences of individual parameters effectively, only ideal initial condition is applied on the following simulations.

Figure 6: Images for concentration fields: Pe=6x103, R=-DQG/ ZLWKLQLWLDO perturbation at t=2.5 for (a) Re=0 and (b) Re=0.1.


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Figure 7: Images for concentration fields: Pe=2x103, R=-DQG/ DWW for (a) Re=0 and (b) Re=0.1.

Figure 8: Images for concentration fields: Pe=6x103, R=-1 and / DWW for (a) Re=0 and (b) Re=0.1.

3.2 Influences of Control Parameters In order to identify the influences of various parameters, we vary an individual parameter at a time with the rest of the parameters kept identical to the reference cases. We first focus on the effects of Peclet number. The centrifugal force is proportional to the Peclet number Pe. For certain fluids combination, larger Pe implies higher cell rotating frequency. Therefore less vigorous fingerings, quantified by the numbers of perturbed fingers, are expected at lower value of Pe due to weaker centrifugal force. Shown in figures 7(a,b) are the situations for lower Pe=2x103. Compared to the reference case shown in figure 2, even though the basic fingering patterns remain similar to the reference cases, much milder fingering instabilities,

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such as insignificant splitting behavior, are resulted. Control parameters related to the fluid properties are viscosity contrast, expressed as viscosity parameter R. For a fixed Peclet number in which •h and the rest of properties are held constantly, a lower viscosity contrast parameter R=-1 indicates an increase of the viscosity of lighter fluid•l. As a result, a lower viscosity parameter R=-1, as displayed in figures 8(a,b), provides more stable viscous effects and leads to less vigorous fingering compared to the reference cases. These more stable effects can be identified by the slower growths of fingers as well as the less significant tip-splitting effects. As the influences of Korteweg stresses, it has been confirmed to mimic the surface tension. A stabilized effect, such as less number of fingers and insignificant secondary fingerings, is observed, as shown in figure 9, which in line of expectations. Similarly to immiscible situations, the effects of stabilization can also be confirmed by the growths of characteristic quantity, such as the mixing interfacial length L. However, unlike the immiscible case that a clear interface can be defined, the mixing region of miscible fluids is a layer. No accurate interfacial length can be measured. Nevertheless, in the region of significant concentration gradient, the mixing interfacial length can be well represented as 1

  ∂c  2  ∂c  2  2   dxdy L (t ) = ∫ ∫    +    ∂x   ∂ y   

(3.8)

Figure 9: Images for concentration fields: Pe=6x103, R=-DQG/ -10-6 at t=2.5 for (a) Re=0 and (b) Re=0.1. The interfacial length starts to increase once the fingering instability is triggered, therefore an earlier growth and higher growth rate of interfacial length reflect a more unstable interface. As shown in figure 10, more stable interfaces result from the presence of Coriolis forces of Re=0.1, lower rotational speed at Pe=2x103, smaller viscosity contrast of R=-1 and effects under interfacial stresses aW/ -10-6 as described above.


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Figure 10: Evolution of interfacial lengths for various parameters.

4. Conclusion Numerical simulations of interfacial stabilities for miscible interfaces in a rotating Hele-Shaw cell by means of highly accurate numerical schemes have been presented. The centrifugal force, if the central fluid is heavier and in general more viscous, provides an unstable outward driving force, while the viscous effect tends to stabilize the contacting front. The presence of confined annulus serves as a barrier that prevents the breakthrough of the fingering. Since the orientation of Coriolis forces apply tangentially to the droplet circumference, the effects of Coriolis forces are found to stabilize the interface. Also simulated are the influences of dimensionless rotating speed, viscosity contrast and Korteweg stresses. The fingering patterns are detailed demonstrated by images of concentration, vorticity fields and streamlines systematically to depict the influences of various parameters.

Acknowledgments Support by ROC NSC Research Grant 92-2212-E-224-012 is gratefully acknowledged.

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