Spontaneous Interaction of Drops, Bubbles and Particles in Viscous

Ind. Eng. Chem. Res. 2002, 41, 357-366

357

Spontaneous Interaction of Drops, Bubbles and Particles in Viscous Fluid Driven by Capillary Inhomogeneities O. M. Lavrenteva, A. M. Leshansky, V. Berejnov,† and A. Nir* Department of Chemical Engineering, Technion, Haifa 32000, Israel

Spontaneous interaction of drops, bubbles, or particles in a suspension occurs when the continuous and dispersed phases are not in thermodynamic equilibrium. Interfacial mass or heat transfer between a bulk fluid and a drop in the presence of an adjacent particle leads to a spatially inhomogeneous concentration or temperature distribution. This nonuniformity causes surface tension gradients along the interface that induce Marangoni type flows in the vicinity of the interface and an eventual migration of droplets or bubbles relative to each other. We review the various cases of interaction of small inclusions at small and large Peclet numbers and a range of nonlinear unsteady effects implied by convective transport and deformation of surfaces. The interaction of solid spheres with nonisothermal free interfaces and the effect of gravity are also presented. 1. Introduction If a drop or a bubble is embedded into another immiscible fluid that is not isothermal, the nonuniformity of the interfacial tension results in a jump of the tangential viscous stresses across the interface, which affects the flow in the vicinity of the surface and the motion of the particle. This phenomenon, addressed as a thermocapillary or Marangoni effect, may have a considerable influence on the efficiency of many important technological processes that involve multiphase flows, such as liquid-liquid and liquid-gas separation, composite material production, and the processes with phase change. The motion of drops and bubbles is affected not only by the temperature gradients but also by the concentration of surfactants on the interfaces. This kind of Marangoni effect is more complicated because in addition to the surface tension the interface mobility can depend strongly on the surfactant concentration that, in turn, is determined by the different processes on the interface, such as adsorption-desorption, surface diffusion, and others. A discussion of the influence of various interfacial phenomena on the Marangoni flow is given in ref 1. Numerous recent theoretical studies of this effect were conducted for the limiting case of insoluble surfactant, with the concentration governed by the process on the interface itself. In the opposite limiting case when the surfactant concentration on the interface is predominantly determined by the mass transfer in the bulk and the surfactant does not affect any material properties except for the interfacial tension coefficient (weak surfactant), the governing equations are the same as in the case of thermocapillary motion. Most of the theoretical studies devoted to the thermocapillary migration of drops and bubbles assume a uniform temperature (concentration) gradient far from the bodies. A comprehensive review of theoretical and experimental developments of the subject is given by * To whom correspondence should be addressed. E-mail: [email protected]. Fax: 972-4-8230476. † Current address: Physics Department, Brandeis University, P.O. Box 549110, MS057, Waltham, Ma 02454-9110.

Subramanian.2 However, local inhomogeneities of concentration or temperature can appear also in suspensions in the absence of an externally imposed field if mass or heat transfer occurs between the phases. These induce spontaneous thermocapillary flow that may exert a considerable effect on the motion of droplets in the suspension. In particular, this effect may explain a strong influence of the direction of the interfacial mass transfer on the rate of droplets coalescence in emulsions as was demonstrated experimentally by Groothuiz and Zuiderweg,3 Jeffreys and Lawson,4 Gourdon and Casamatta,5 and Skelland and Kanel.6 It was shown that the coalescence rate substantially increases if the surfactant is transferred from the dispersed phase into the continuous one, and it decreases in the opposite case. In this paper, we present a review of the recent studies of the spontaneous interaction of drops, bubbles, and particles induced by interfacial mass or heat transfer. The general formulation is followed by the results for quasistationary droplet interaction, unsteady weak and strong convective effects, and new results of the influence of droplet deformability. The combined effect of gravity and spontaneous thermocapillarity is discussed as well and contains new consideration of the effect of thermal wake shedding from one drop onto another. 2. General Problem Formulation Consider a collection of drops of radii ai submerged in an unbounded Newtonian fluid, which is quiescent and has a uniform solute concentration C0 at infinity. The initial concentration inside the drops, C1, is constant but different from the value that would be in equilibrium with the outer concentration C0. Hence, mass transfer occurs between the phases (see Figure 1). The solute substance is either a weak surfactant (solute molecules that consist of short hydrophobic tails and weakly polar hydrophilic head, e.g., acetic acid, ethanol, acetone, etc.) or a very dilute strong surfactant so that the concentration has a negligible effect on any physical properties of liquids in the bulk and at the interfaces except for the interfacial tension which depends linearly on the surface concentration, γ ) γ0 +

10.1021/ie010099y CCC: $22.00 © 2002 American Chemical Society Published on Web 06/23/2001

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Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002

The governing parameters are the capillary number and the Pećlet number defined as

Ca )

where D0 refers to thermal or molecular diffusivity. The problem is completed by initial conditions for the concentration and the positions of the drops and by the equations governing interfacial mass transfer. The latter consists of mass balances involving equilibrium or nonequilibrium adsorption-desorption kinetics on the interfaces. In a general form, these equations are available (see, e.g., ref 1). In this paper, we use simplified asymptotic models of the interfacial mass transfer that would be specified in each section. Note that if the solute is not adsorbed at the surface the concentration and the mass flux is continuous across the interface. In this special case, the mathematical model specified above describes also heat transport and the so-called thermocapillary induced motion.

Figure 1. Interfacial transport in a suspension of droplets.

(∂γ/∂Γ) (Γ - Γ0(C0)). Here ∂γ/Γ is a negative constant and Γ0 is a surface concentration that would prevail in equilibrium with the bulk concentration C0. Suppose further that the surface concentration Γ is in equilibrium with the local concentration in the continuous phase fluid, Γ ) k0C and that mass transfer between the interface and the continuous phase liquid is governed by the diffusion in the bulk phase. We introduce nondimensional variables choosing the following scaling: the radius of a reference drop a for length

v)

|(∂γ/∂Γ)| |C1/ke - C0| η0k0

for velocity, a/v for time, and vη0/a for pressure. Here ke is the phase distribution coefficient (the ratio of equilibrium concentration inside and outside the drops), η0 is the viscosity at the continuous phase fluid and F0 is its density. Subscripts 0 and 1 denote the continuous and the dispersed phases, respectively. Assume that the Reynolds number Re ) vF0a/η0 is sufficiently small so that inertia effects are negligible. The velocity and pressure fields, v(t,x) and p(t,x), satisfy the Stokes equations. The velocity is continuous across the interfaces and vanishes far from the drops. The balance of stresses on each drop interface reads

Π0‚n - Π1‚n )

(Ca1 - c)Hn - ∂τ∂c τ,

x ∈ ∂Ωi (1)

Here x is a radius vector to a point in the coordinate system with its origin located at the center of inertia of the reference drop, ∂Ωi denotes the interface between the continuous phase and the ith drop, Π ) -pI + λ[∇v + (∇v)†] with λ ) η1/η0 in the drop and λ ) 1 in the continuous phase fluid is the viscous stress tensor, n and τ are unit vectors normal and tangential to the corresponding interface, and H ) ∇‚n is its mean curvature. c(t,x) ) (C - C0)|C1/ke - C0| is the dimensionless concentration in the continuous phase fluid and it satisfies the following balance:

Pe

(∂c∂t + v‚∇c) ) ∆c

vη0 va , Pei ) γ0 D0

(2)

3. Quasistationary Transport Suppose that the adsorption-desorption mass flux on the interfaces is balanced by fluxes from the two phases, whereas the diffusion and convection along the interface are negligible at the leading order. The surface concentration is in equilibrium with the bulk concentration of the continuous phase, whereas within the dispersed phase, a considerable resistance to mass transfer is concentrated in the vicinity of the interface, so that, at the leading order, the concentration inside the drop maintains its initial value. The mass balance results in a Robin type boundary condition for the concentration in a continuous phase fluid (see ref 7 for the details).

∂c ) Sh(c ( 1) ∂n

x ∈ ∂Ωi

(3)

with Sh denoting the Sherwood number. The “-” sign in the right-hand side of (3) corresponds to the case when the concentration inside the drop exceeds the equilibrium value and the mass transfer occurs from the drop to the continuous fluid; the “+” sign corresponds to the opposite case. If the Pećlet and the capillary numbers are sufficiently small, convective transport and the deformability of the drops can be neglected a priori. The drops maintain their initial spherical shape. The concentration field in the bulk can be found as a harmonic function that satisfies boundary conditions (3) independently of the velocity field. As soon as the interfacial concentration is known, the velocity field and the velocities of the drops Vi can be obtained solving the hydrodynamic part of the problem for each configuration of the droplets. Note that the concentration and velocity fields depend only on the instantaneous system geometry and not on the evolution of the process. This type of process is termed quasistationary. The evolution of the positions of the centers of the nondeformable drops Zi can be determined from

dZi ) Vi dt

(4)

The axisymmetric case of two drops in an infinite liquid was treated in great detail by Golovin et al.7 The

Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 359

Figure 2. Spontaneous thermocapillary interaction of two drops.

Figure 4. Aggregate migration velocity of two drops in contact. The maximum is at R ) 0.468.

Figure 3. Velocities of two unequal spherical drops during spontaneous interaction, (a) larger drop R2 ) 2 and (b) smaller drop R1 ) 1.

geometry of the system is shown in Figure 2. The analytical solution was facilitated by the use of bispherical coordinates. The dependence of the velocities on the parameters of the problem, Sherwood number, viscosity ratio, radii ratio R, and the separation distance h was examined. In the case when the drops have equal sizes and equal concentration of a surfactant, they approach each other if mass transfer occurs from the dispersed phase and separate if the transfer is directed oppositely. The velocity achieves a maximum value at a certain separation distance that is of O(1). Interaction of the drops in close proximity was analyzed making use of a combination of a lubrication analysis of the motion in the gap region and tangent sphere coordinates for the outer flow. It shows that, similar to the cases of other driving forces (gravity and externally imposed concentration gradient), the relative velocity decays as xh and, hence, the drops collide in finite time. An unusual interaction pattern is observed in a nonsymmetric case, when the drops are either of different sizes or have a different concentration of a surfactant. Drops of different radii, that are initially well separated, with mass transfer into the continuous phase begin to move in opposite direction toward each other. The larger drop moves with smaller velocity. When the drops come closer, the direction of the motion of the larger drop reverses. The two drops move now in the same direction. The individual velocities of the drops are depicted in Figure 3 versus separation distance. The separation distance proceeds to decrease, and the drops collide in finite time forming an aggregate that migrates in the direction of the larger drop. The dependence of

the aggregate migration velocity on the radii ratio for Sh ) 1 is depicted in Figure 4. This velocity vanishes in the limiting cases of R ) 1 (equal size drops) and R f 0 (vanishing radius of one of the drops), and a maximum velocity is achieved for R = 0.468. The other type of nonsymmetric interaction is described in ref 7. It occurs when the concentration of a surfactant inside one of the drops is less than the equilibrium value, whereas inside the other one it exceeds this value. In this case, drops of equal size chase each other in the same direction at any separation. The drift velocity decreases with the separation distance and has a maximum when the drops are in contact. 4. Convective Transport Effects When the Pećlet number is nonzero, a convective term in eq 2 provides an additional coupling of the hydrodynamics and transport processes. The effect of convective transport on the spontaneous thermocapillary motion was studied in ref 8 in the limiting case of low Pe, making use of perturbation methods. The deformability of the drops and inertia effects were neglected. The perturbations were found to be singular, and a method of matched asymptotic expansions was employed. The quasistationary solution described in the previous section served as a zero-order expansion term. The relative approach velocity in the case of two drops (as other field variables) was expanded as

V ) V0 + f(Pe)V1 + o(f(Pe))

(5)

with V0 denoting the velocity obtained for the case Pe ) 0. It was found that the process becomes timedependent because of the temporal changes in mutual positions of the drops. The unsteadiness appears in the form of a Basset type history term of O(Pe1/2) in the concentration field near a drop. Because in the presence of a Marangoni effect the force exerted on the drop depends on the concentration distribution on its surface, a history term appears also in the force balance and, hence, in the expansion of the drops migration velocity. For moderate initial separation, this effect prevails, and thus, f(Pe) ) Pe1/2, whereas the correction to the relative velocity between the drops is

V1 ) b(t)V0 )

V0 xπ

dτ ∫0t da dτ x

t-τ

(6)

360


with a(t) ) lim|x|f∞c0(t,x)/|x|. This term decays with the growth of separation distance. When h g O(1/Pe), the leading order correction term diminishes and is of O(Pe). 5. Deformable Drops 5.1. Asymptotic Solution for Small Capillary Number. If the capillary number is small, Ca , 1, correction to the solution obtained in the undeformable case, Ca ) 0, can be constructed making use of a regular perturbation technique. It is convenient to describe the interfaces using separate polar coordinates, ri, θi, φi, with origins at the centers of the undeformed drops. In the axisymmetric case depicted in Figure 2, the shape of the interface of each drop is described by ri ) Ri(θi) ) Ri + Cafi(cos(θi)) +.... All of the other variables are assumed to have similar representations. The O(1) terms of the normal stress balance (1) are of the form

[

]

dfi d (1 - µ2i ) + 2fi ) Qi(µi) + bi dµi dµi

(7)

where Qi(µi) ) 2Ric0 - Ri2[n‚Π0‚n], µi ) cos θi, and [n‚Π0‚n] denotes the jump of the normal stresses across the interfaces evaluated at the unperturbed boundaries. Thus, as soon as the undeformable solution is known, the perturbation of the spherical shape can be found independently from perturbations of the other variables. bi is the constant increase of the pressure inside the drop and is found from the requirement that the drop is incompressible. The homogeneous solutions of (7) are s1(µ) ) µ and s2(µ) ) F(-1/2, 1, 1/2, µ2), with F being the hypergeometric function. Note that s2 has logarithmic singularities at µ ) (1. The nonsingular solution of (7) can be written as

fi ) µi

∫0µ Qi(µ)s2(µ) dµ + s2(µi)∫-11µQi(µ) dµ + bi/2 i

(8)

For widely separated drops, where Z ) | Z1 - Z2| . 1, the functions fi can be approximated as

fi )

( )

4ShR12 R22 4 + λ 1 1 P (µ ) + O 4 1 + RjSh 1 + λ Z3 2 i Z

i ) 1, 2; i * j

The surfaces have the form of prolate spheroids. Note that this shape coincides with the shape of a drop in the presence of a point heat source produced by the other drop. The deformations decay monotonically with the growth of the viscosity ratio. When the distance between the drops becomes smaller, the deformation can be calculated according to formula 8, where the zero-order approximation of the interface stress jump was computed using bispherical coordinates technique. The function f(θ) in the case of equal size drops is shown in Figure 5 for various separation distances and viscosity ratios. For drops of different radii, the functions fi(θi) are shown in Figure 6. One can see that when the drops are well separated their form is close to prolate spheroids with the elongation in the gap region being a little larger than in the opposite regions. When the drops come closer, at separation distance of O(10-1), the interface in the gap region begins to flatten. The maximum distortion from the spherical shape is now in the region opposite to the gap. At closer proximity (separation distance of O(10-2)), the

Figure 5. Small capillary number deformation pattern of interacting equal drops for various separation distance and viscosity ratios: h ) 1 (dotted-dashed); h ) 0.5 (dotted); h ) 0.1 (solid); h ) 0.05 (dotted); λ ) 0.1 (a); λ ) 1 (b); λ ) 10 (c).

distortion from the spherical shape in the gap region becomes negative and a dimple is formed near the axis. With the growth of the viscosity ratio, the deformations become less pronounced and the above-mentioned qualitative change of the interface form in the gap region occurs only for smaller separations. The above analysis should be modified if the drops are at near contact, i.e., the separation between them is of the same order of magnitude as the deformation of the interface. The problem can be solved by matched asymptotic expansions for small Ca following the analysis of Yiantsios and Davis9 performed for the gravity-


Figure 6. Small capillary number deformation pattern of interacting unequal drops for R ) 0.4, λ ) 1, and various separation distances: h ) 1 (dotted-dashed); h ) 0.1 (solid); h ) 0.04 (dashed); h ) 0.02 (dotted).

driven motion. The outer solution, for two spherical droplets moving in apparent contact, determines a constant contact force, Fc. This force is used as a driving force for an inner solution that is governed by a set of integrodifferential equations coupling the flow in the gap region to the flow inside the drops. The analysis of ref 9 is directly applicable in our case after replacing the gravity induced contact force by the thermocapillaryinduced one. Thus, we can conclude that, for long times, a dimple is formed in the gap region. The thickness of the inter-drop film at the axis of symmetry decays as t-1/3, and the minimum thickness behaves as t-2/3. 5.2. Moderate Capillary Number. For the case of a moderate capillary number, a boundary integral approach was employed to study the evolution of droplets’ shapes and positions. The concentration and the velocity on the interfaces are governed by the following system of integro differential equations

(

2πc(x) + I∂Ω

)

Sh ξ‚n(y) + c(y) dSy ) ξ ξ3 1 ShI∂Ω dSy, x ∈∂Ω ξ

1-λ v(x) + 2 I K(ξ):v(y)n(y) dSy ) 1 + λ ∂Ω ∂c 1 1 I J‚ - c(y) Hn - τ dSy, Ca ∂τ 4π(1 + λ) ∂Ω x ∈ ∂Ω

{[

]

}

where ξ ) x - y, ξ ) |ξ|, ∂Ω denotes the union of all ∂Ωi, and the kernels for the single and double layer potentials for Stokes flow are, respectively

J(ξ) )

I ξξ 3 ξξξ + , K(ξ) ) ξ ξ3 4π ξ5

When the viscosities of the two fluids are equal, λ ) 1, the coefficients of the double layer integral vanish and the velocity is expressed explicitly in terms of the single layer potential with known density. For the sake of simplicity, our calculations are given for this case. The evolution of two identical initially spherical drops that start their motion at a separation distance of one radius is given in Figure 7 for Ca ) 1, the maximum possible capillary number, for which the surface tension remains positive. We observe that when the drops come closer they flatten in the gap region, and after that, the relative motion is drastically retarded. The appearance of dimples with negative curvature in gap regions is apparent. The shapes and the mutual position of the

Figure 7. Evolution of shape and mutual positions of two equal drops for Ca ) 1. (a) t ) 0; (b) t ) 64.5; (c) t ) 125; (d) t ) 187.5; (e) t ) 250.

drops at the final stage of the process are shown in Figure 8 for various values of Ca. For small Ca, the drops come to near contact preserving an almost spherical form, whereas for larger Ca, a liquid film with evidently flat interfaces or negative curvature develops between the drops. The radial size of the film and its width grow with the capillary number. When the drops have different sizes, the interaction pattern is more complex. Figure 9 shows the result of a dynamic simulation with R ) 0.6 and initial distance that equals the radius of the larger drop, h0 ) R1 ) 1. The drops begin to move in opposite directions with the smaller one having a substantially larger velocity. When they come closer, the gap regions flatten and a small dimple is formed on the interface of the larger drop. As in the case of nondeformable drops (see section 3), the direction of the motion of the larger drop reverses, the two drops move in the same direction with almost equal velocities, and the deformations drastically slow. The evolution of the individual velocities of the centers of mass of the drops for R ) 0.6 and various capillary number is shown in Figure 10. 5.3. On the Stability of Interdroplet Film. Leshansky10 has studied the linear stability of the interdroplet film in the presence of interfacial mass transfer under some simplifying assumptions. Specifically, the

362


Figure 8. Shape and mutual positions of two equal drops at t ) 250 for various capillary numbers. (a) Ca ) 1; (b) Ca ) 0.8; (c) Ca ) 0.6; (d) Ca ) 0.4; (e) Ca ) 0.2.

unperturbed state of the narrow liquid film that develops between the inviscid drops was regarded as a plane sheet that does not change its thickness with time. The results of the numerical simulations described above confirm the applicability of these assumptions especially in the case of equal-size drops for moderate Ca. It was shown in ref 10 that if the mass transfer occurs into the sheet the system is unconditionally unstable to the long wave perturbations. The development of this instability may lead to the breakup of the film and eventually to the coalescence of the drops. 6. Unsteady Heat Transfer Consider now adjacent drops submerged into initially an isothermal immiscible fluid. The initial temperature of the drops is supposed to be uniform but different from that of the continuous liquid. As it was mentioned above, the temperature Θ and velocity field in this case can be found as solutions of the boundary value problem outlined in section 2, satisfying the conditions of continuous temperature and heat flux across the interfaces. Because the initial temperature in the phases is not in thermodynamic equilibrium, then, over an initial period of time, “shock” boundary layers are formed near the interfaces of the droplets, whereas in the core region,

Figure 9. Evolution of shape and mutual positions of two unequal drops for R ) 0.6, Ca ) 0.7. (a) t ) 0; (b) t ) 64.5; (c) t ) 125; (d) t ) 187.5; (e) t ) 250.

the initial values of the temperature are reserved. We consider only the time scale, in which the boundary layer of the temperature field is maintained inside the drops. As we are interested mostly in the change in the mutual positions of the drops, we consider cases where the typical time of the existence of the internal thermal boundary layer exceeds the time required for the thermocapillary forces to move the drop to a distance comparable with its radius. With the natural choice of the time scale as the ratio between the length scale given by the radius of one of the drops and the Marangoni velocity generated by the initial temperature difference, it follows that the characteristic Pećlet number of the dispersed phase is large. If the Pećlet numbers of the continuous and the dispersed phases, Pe0 and Pe1, are high and of the same order of magnitude, thermal boundary layers on the


Figure 10. Evolution of the velocities of centers of mass of two deformable drops at R ) 0.6. (a) Ca ) 0.3; (b) Ca ) 0.5; (c) Ca ) 0.7.

inner and outer sides of the surfaces of the drops exist simultaneously. On the other hand, if the resistance to heat transfer is concentrated mostly in the dispersed phase, i.e., the Pećlet number of the continuous phase is small, there is a time interval in which thermal boundary layers inside the drops are present, while the temperature field outside the drops is fully established. The two asymptotic cases were studied in ref 11 in the nondeformable limit, Ca ) 0, and demonstrate qualitatively different patterns of heat transfer and of the dynamics of the drops. 6.1. The Case Pe0 , 1. If the Pećlet number of the ambient medium is negligible, the temperature in the continuous phase fluid is quasistationary at the leading order and the resistance to the heat transfer is concentrated mostly in the dispersed phase. At t e O(Pe) with Pe ) Pe1 . 1, the heat transfer inside the droplets is governed, to the leading order, by eq 2 with 1/Pe ) 0, and hence, the temperature remains equal to its initial value, Θ0i ) 1. In the continuous phase, the leading order approximation of the temperature Θ00 is a harmonic function that vanishes at infinity and satisfies the boundary conditions

Θ00

)1

x ∈ ∂Ωi

(9)

Note that Θ00 depends on time parametrically via the evolution of the geometry of the system. The zero-order solution does not satisfy the energy flux conditions on the interfaces

∂Θi 1 ∂Θ0 ) ∂ri R ∂ri

x ∈ ∂Ωi

φ (τ,µi)

∫0t xi

t-τ

dτ + o()

with 0

φi(t,µ2) )

dh ) - d(λ, R)xth dt with d(λ,R) being a constant coefficient derived in ref 11, and hence, the drops collide in finite time. If the initial separation h0 is small enough, then, in the case of attraction, the evolution of the separation distance is given by the solution of the asymptotic equation above which satisfies the initial condition h(0) ) h0

h(t) ) h0[1 - (t/t0)3/2] It is shown in ref 11 that a dimensionless time, in which the drops approach and collide, is t0 ) (9h0Pe/d2)1/3. The mean relative velocity is V0 ) h0/t0 ) -(h0d/3)2/3(Pe)1/3. The instantaneous relative velocity

V(t) ) 3V0

1 ∂Θ0 , ri ) Ri, i ) 1, 2, ) Pe-1/2 R ∂ri

()[ ()] t t0

1/2

1-

t t0

3/2

vanishes at t ) 0 and at the moment of collision, t ) t0. The absolute value of V achieves a maximum

Vmax )

and hence, an internal boundary layer must be considered along the surface of each drop. The solution of the boundary layer problem that was constructed in ref 11 for the axisymmetric case of two drops implies that on the interfaces the temperature can be expanded as

1 Θi(t,µ2) ) 1 + xπ

Note that the boundary layer arises as a result of a discontinuity in the normal derivatives of the zero-order approximation rather than of the functions themselves. In a sense it is “softer” than classical boundary layers. The resulting normal component of the temperature gradient is of O(1). Yet it is still much higher than its tangential component, being of O(), and the boundary layer approximation is applicable. As soon as the temperature distributions along the interfaces are known, the thermocapillary-induced flow and the drift velocities of the drops, which are linear operators on the tangential derivatives of these functions, can be evaluated. Because the temperature difference across the boundary layer and along the interface are of O(), the simultaneous droplets’ velocities are of the same order of magnitude and depend strongly on time. It was shown in ref 11 that the time and the separation variables can be scaled as t ) -2/3s and h(t) ) δ(s). Thus, the total approach time was found to be of O(-2/3) ) O(Pe1/3). The motion of drops in close proximity was studied combining the tangent sphere coordinates approach and a lubrication analysis in the gap region. Because at small h the lubrication resistance Fl ) O(h-1/2), whereas the driving thermocapillary force is regular, the evolution of the separation distance in this case is governed by the equation

9 |V0| = 1.4174|V0| 28/3 t)

at t0

2

4/3

= 0.3969t0, h )

9 h 16 0

It follows that at close proximity all of the relative approach velocities can be normalized to a single curve. For the case of larger initial separation, the velocities were computed making use of bispherical coordinates for h(t) > h* and the lubrication approximation for smaller separations. Here h* is an auxiliary number that was found from the condition that the velocity computed using bispherical approach becomes close enough to that obtained by the lubrication analysis. Figure 11 depicts the evolution of the relative velocities as a function of time and of the initial surface-to-surface

364


h(t) ) h0[1 - (t/t0)]2 where t0 ) 2xh0/d1(λ,R) is the time in which the drops approach each other up to collision after the beginning of the thermal boundary layers interaction in the gap. d1(λ,R) is a proportionality coefficient given in ref 11. In this case, the mean relative velocity is V0 ) h0/t0 ) d1xh0/2, and the instantaneous relative velocity, V(t) ) - 2V0(1 - t/t0) decays linearly with time and vanishes at the moment of collision, t ) t0. 7. Spontaneous Thermocapillary Migration of Solid Particles

Figure 11. Evolution of the relative velocity of two equal drops at Pe1 . 1 and Pe0 ) 0 for various initial separations. (a) Repulsion; (b) attraction.

distance h0. Part a of Figure 11 corresponds to the case of repulsion, whereas part b corresponds to the case of attraction. The dashed curves are asymptotic evaluations for drops in close proximity. The induced thermocapillary motion has small velocities, but for moderate separation distance, it results in the attraction of the drops and their collision after time of O(Pe1/3) that is much shorter than the time of the boundary layer approximation applicability. 6.2. The Case Pe0 . 1. If the Peclet numbers of the two phases are comparable and large, boundary layers exist simultaneously inside and outside the drops. A substantial thermocapillary motion occurs only after temperature disturbances that are produced by one drop reach the interface of the other drop. For a moderate separation distance, this time is very large (of O(Pe)). For small separation distances, h e O(Pe-2/3), the boundary layers interact and induce high-temperature gradients on the interface in the small gap region near the other drop that result in a thermocapillary force of O(1) and lead to collision of the drops. The evolution of separation distance is given by11

A hot solid particle in a cooler ambient liquid generates a nonuniform temperature distribution. If the particle is isolated or if only solid boundaries are present in the system, no thermocapillary motion occurs. However, if it is placed near a free liquid-liquid or liquidgas interface, say a drop or a bubble, a thermocapillary flow is generated by tangential stress on these interfaces that, in turn, may lead to the motion of the solid particle. This kind of spontaneous thermocapillary interaction was first studied by Golovin12 neglecting convective transport and deformability of the bubble. The motion of a solid particle radius R1 ) 1 and an attached bubble with radius R2 ) R and the motion of separated particle and bubble in close proximity was considered by Leshansky et al.13 It was shown that well-separated bodies migrate in the opposite directions toward each other. At large a distance, the velocity of a bubble decays as h-2, whereas the velocity of a solid decays much faster, as h-5. When the bodies come closer, their velocities decrease and there is a critical distance at which the motion of the solid particle changes its direction. The bubble pushes it in front of itself. The velocity of the bubble remains larger than that of the particle, and the bodies continue to approach each other. A lubrication analysis of the flow in the gap region performed in ref 13 showed that the relative velocity decays as h at h f 0. Hence, in contrast to the case of two drops that collide in finite time, it will take the bubble an infinite time to catch up with the solid particle. The individual velocities tend to the same finite value as the separation distance diminishes. This velocity has a maximum value V ) V1 ) V2 = 0.053 at R = 2.68, and it vanishes in the limiting cases: R f 0 (a vanishing bubble) and R f ∞ (a flat interface). The dependence V(R) is shown in Figure 12 (solid line) together with the individual velocities of the bubble (dashed curve b) and of the solid particle (dashed curve s) at separation h ) 0.01. The effect of convective transport on this kind of thermocapillary-induced motion was studied in ref 8. It was shown that, similar to the case of drop’s interaction, a Basset-type term appears in the force and the correction to the migration velocities of the bodies are given by formula 6. 8. Combined Effect of Gravity and Spontaneous Thermocapillarity In all the cases described above, the flow was driven solely by the Marangoni effect. In real applications, however, the presence of another forcing, e.g., gravity, is typical. In this case, the governing parameters


Figure 12. Migration velocities of a bubble (dashed curve b) and a solid particle (dashed curve s) when the separation distance h ) 0.01 as a function of the radii ratio. The cluster velocity of a bubble and a particle in a point contact is given by the solid curve.

introduced in section 2 are not enough to describe the drop’s interaction and two more are needed. When the drops are falling or rising in the gravity field, their characteristic velocity is determined by the buoyancy driven motion. The velocity of the leading drop was used as a velocity scale in our considerations. The two additional governing parameters are the Marangoni number

Ma )

|∂γ/∂Θ| ∆Θ ∆Fga2

that in this case characterizes the relative influence of thermocapillarity and gravity, and W, the ratio of velocities of two drops when thermocapillarity and hydrodynamic interaction are absent. Note that both parameters can be negative. Negative Ma corresponds to the case of a cold leading drop, whereas negative W means that in the absence of interaction the two drops would move in the opposite directions. For the quasistationary processes, described in sections 3 and 7, the linearity of the problem implies that the velocity field is a superposition of that driven by the Marangoni effect and a well-known buoyancy induced motion. For two drops, the combined effect was studied in ref 7 and for a bubble and a solid particle in ref 13. It was demonstrated that the self-induced Marangoni effect may be rather strong, e.g., it can drive the motion of one of the two bodies against gravity. When the convective transport is taken into account, the interaction between the buoyancy and the thermocapillary forcing is not linear. However, in the case of small Pe the correction to the migration velocity can be constructed in the form (6).8 In this case, V0 denotes not a quasistationary velocity but the velocity that the drops would have in the absence of gravity. In contrast to this, if the Pećlet number defined by the gravity induced velocity is large, the pattern of the heat transfer is qualitatively different from that in the case without gravity. Thermal wakes are formed inside and outside the drops,14 where the convective transport is dominant. The temperature in the wake does not change along the streamlines at the length scale less than O(Pe). Convective heat transfer from the leading drop to the trailing one through the thermal wake yields a temperature

Figure 13. Temperature profiles on the surface of a drop trailing behind a thermal wake of a leading drop under the action of gravity and thermocapillarity for various values of Marangoni number.

gradient along the interface of the trailing drop that results in a Marangoni force of O(1) acting on this drop. We considered in details the limiting case when the drops are far apart and their hydrodynamic interaction becomes negligible while the thermocapillary interaction is still of O(1). For simplicity, we considered the case where the heat capacity of the ambient liquid is much smaller than that of the leading drop and much larger than that of the trailing one. In this case, at the leading order, the interface of the leading drop is isothermal and the interface of the trailing one is insulated, and thus, none of them is subjected to a thermocapillary force in the absence of another drop. In contrast to this, if the drops are not isolated and they are falling in a gravity field, one behind another, the heat transfer from the leading drop to the trailing one through the thermal wake results in a temperature variation along its interface, and in a considerable enchancement or retardation of the motion. Analysis of the temperature distribution in the boundary layers and in the wake regions similar to the one given in ref 15 reveals that along the interface of the trailing drop

Ts(θ) )

2 arctan xπ

x

2 3(1 + λ)τ(θ)

(10)

where

τ)

Ψ(r,η)

sin η dη ∫0θlim rfR R - r

(11)

and (r,θ) are spherical coordinates in a reference frame moving with the second drop and Ψ is a stream function of the flow in this reference frame. It follows from the representations given above that the interface of the trailing drop is not isothermal and that the temperature variation along it is of O(1). Hence, a thermocapillary flow with velocity of the same order of magnitude is induced near the interface. Note that the function τ(θ) is not known a priori, but it is related to the stream function in the vicinity of this drop that should be determined simultaneously with the temperature field.

366


Acknowledgment This research was supported by the Israel Science Foundation founded by the Academy of Science and Humanities. O.M.L. acknowledges the support of the Israel Ministry for Immigrant Absorption. A.N. owes much of his interest in problems involving hydrodynamics, transport, and surface deformations to many stimulating discussions with Professor John A. Quinn ever since 1974 when he spent a year at the University of Pennsylvania. Literature Cited

Figure 14. Migration velocity of a drop trailing behind a thermal wake of a leading drop under the action of gravity and thermocapillarity for various values of Marangoni number. The abscissa correspond to the velocity in the absence of thermocapillary effect.

The problem is thus reduced to a system of nonlinear functional equations on Ts and τ that was solved numerically. The results are partially presented in Figures 13 and 14. Temperature profiles on the interface of the trailing drop are shown in Figure 13 for various positive values of the Marangoni number, i.e., in the case when the leading drop is hotter than the ambient liquid. In Figure 14, the migration velocity of the trailing drop is plotted versus the parameter W (the velocity that this drop would have when isolated) for various values of Marangoni number. Recalling that the positive values of the velocity correspond to the drift in the direction of the leading drop, we conclude that at large enough Ma the trailing drop moves against the buoyancy force. For the buoyancy-driven motion of highly deformable drops, numerous recent studies revealed a rich variety of interaction patterns depending on the Bond number and the initial configuration of the system (see, e.g., refs 16 and 17 and the literature cited). Numerical simulations of an axisymmetric buoyancy-driven interaction of a leading drop with a smaller trailing drop were recently reported by Davis.18 It was demonstrated that the initially spherical trailing drop elongates considerably because of the hydrodynamic influence of the leading one. Afterward, a critical behavior is exhibited: depending on the governing parameters, the drops may either separate and return to a spherical shape or the trailing drop may be captured by the leading one. In some cases the drops may break up. No study of the combined effect of the gravity and spontaneous thermocapillarity has been reported so far. We anticipate a pronounced influence of interfacial heat or mass transfer on the motion of highly deformable drops, in particular in the regions of parameters where the behavior is critical.

(1) Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transport Processes and Rheology; Butterworth-Heinemann: Woburn, MA, 1991. (2) Subramanian, R. S. The motion of bubbles and drops in reduced gravity. Transport Processes in Bubbles, Drops, and Particles; Hemisphere: New York, 1992; p 1-42. (3) Groothuis, H.; Zuiderweg, F. G. Influence of mass transfer on coalescence of drops. Chem. Eng. Sci. 1960, 12, 288-289. (4) Jeffreys, G. V.; Lawson, G. B. Effect of mass transfer on the rate of coalescence of single drops at a plane interface. Trans. Instn. Chem. Engrs. 1965, 43, 294-298. (5) Gourdon, C.; Casamatta, G. Influence of mass transfer direction on the operation of a pulsed sieve-plate pilot column. Chem. Eng. Sci. 1991, 46, 2799-2808. (6) Skelland, A. H. P.; Kanel, J. S. Transient drop size in agitated liquid-liquid systems, as influenced by the direction of mass transfer and surfactant concentration. Ind. Eng. Chem. Res. 1992, 31, 2556-2563. (7) Golovin, A. A.; Nir, A.; Pismen, L. P. Spontaneous motion of two droplets caused by mass transfer. Ind. Eng. Chem. Res. 1995, 34, 3278-3288. (8) Lavrenteva, O. M.; Leshansky, A. M.; Nir, A. Spontaneous thermocapillary interaction of drops, bubbles and particles: Unsteady convective effects at low Pećlet numbers. Phys. Fluids 1999, 11, 1768-1780. (9) Yiantsios, S. G.; Davis, R. H. Close approach and deformation of two viscous drop due to gravity and van der Waals forces, J. Colloid Interface Sci. 1991, 44, 412-433. (10) Leshansky, A. M. On the influence of mass transfer on coalescence of bubbles. Int. J. Multiphase Flow 2001, 27, 189196. (11) Lavrenteva, O. M.; Nir, A. Spontaneous thermocapillary interaction of drops: Unsteady convective effects at high Peclet numbers. Phys. Fluids 2001, 13, 368-381. (12) Golovin, A. A. Thermocapillary interaction between a solid particle and a gas bubble. Int J. Multiphase Flow 1995, 21, 715719. (13) Leshansky, A. M.; Golovin, A. A.; Nir, A. Thermocapillary interaction between a solid particle and a liquid-gas interface. Phys. Fluids 1997, 9, 2818-2827. (14) Polyanin, A. D. Unsteady-state extraction from a falling droplet with nonlinear dependence of distribution coefficient on concentration. Int. J. Heat Mass Transfer 1984, 27, 1261 1276. (15) Polyanin, A. D. Method for solution of some nonlinear boundary value problems of a nonstationary diffusion-controlled (thermal) boundary layer. Int. J. Heat Mass Transfer 1982, 25, 471-482. (16) Manga, M.; Stone, H. A. Buoyancy-driven interaction between two deformable viscous drops. J. Fluid Mech. 1993, 256, 647 683. (17) Zinchenko, A. Z.; Rother, M. A.; Davis, R. H. Cusping, capture and breakup of interacting drops by a curvatureless boundary integral algorithm. J. Fluid Mech. 1999, 391, 249-293. (18) Davis, R. H. Buoyancy-driven viscous interaction of a rising drop with a smaller trailing drop. Phys. Fluids 1999, 11, 1016 1028.

Received for review February 1, 2001 Revised manuscript received April 30, 2001 IE010099Y

Spontaneous Interaction of Drops, Bubbles and Particles in Viscous

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