Effects of non-uniform interfacial tension in small Reynolds number ﬂow past a spherical liquid drop

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— journal of September 2011

physics pp. 493–507

Effects of non-uniform interfacial tension in small Reynolds number flow past a spherical liquid drop

D P MASON^1,^∗and G M MOREMEDI²

1Centre for Differential Equations, Continuum Mechanics and Applications,

School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa

2Department of Mathematical Sciences, University of South Africa, P.O. Box 392, Pretoria 0003, South Africa

∗Corresponding author. E-mail: David.Mason@wits.ac.za

Abstract. A singular perturbation solution is given for small Reynolds number flow past a spherical liquid drop. The interfacial tension required to maintain the drop in a spherical shape is calculated.

When the interfacial tension gradient exceeds a critical value, a region of reversed flow occurs on the interface at the rear and the interior flow splits into two parts with reversed circulation at the rear.

The magnitude of the interior fluid velocity is small, of order the Reynolds number. A thin transition layer attached to the drop at the rear occurs in the exterior flow. The effects could model the stagnant cap which forms as surfactant is added but the results apply however the variability in the interfacial tension might have been induced.

Keywords. Non-uniform interfacial tension; small Reynolds number flow; matched asymptotic expansions; transition layer.

PACS No. 47.15.Gf

1. Introduction

Taylor and Acrivos [1] considered the problem of small Reynolds number flow past a spherical liquid drop with uniform interfacial tension between the liquid drop and the surrounding fluid. The problem is a singular perturbation problem and Taylor and Acrivos calculated the solution to first order in the Reynolds number. They found that if the interfacial tension is uniform then the drop will deform and the deformation occurs at first order in the Reynolds number.

In this paper we shall consider small Reynolds number flow past a spherical liquid drop and calculate, to first order in the Reynolds number, the interfacial tension distribution which allows the liquid drop to remain spherical.

Small drops in low Reynolds number flow are found to be spherical [2]. Liquid drops will be approximately spherical if the interfacial tension and the viscous forces are more

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dominant than inertial forces. This will be satisfied if the Weber number, We, and the Reynolds number, Re, are less than unity. We shall consider a perturbation solution with Re<1 and we shall assume that Re and We have the same order of magnitude.

We find that, with variable interfacial tension, the solution of the fluid mechanics problem contains one undetermined parameter α which depends on the unknown terminal velocity of the drop and is related to the interfacial tension gradient. A physical mechanism which could produce the required non-uniform interfacial tension is surfactants which are impurities in a fluid that have an affinity for an interface.

Early experimental work on the effects of surfactants on liquid drops was done by Savic [3] and Griffith [4,5] who found that as the concentration of the surfactant is increased there is a decrease in the circulation velocity of the fluid inside the drop and that this decrease in velocity first occurs at the rear of the drop. Savic was the first to propose the stagnant cap model to describe this effect. Schechter and Farley [6], Wasserman and Slattery [7] and Levan and Newman [8] investigated theoretically the effects of surfactants on the terminal and interfacial velocities in creeping flow past a drop in which inertia effects are neglected. Sadhal and Johnson [9] derived an exact solution for creeping flow past a bubble or drop with a stagnant cap of surfactant film at the rear. They obtained a closed form expression for the drag force and found a shift in the centre of the internal vortex due to the stagnant cap. Oguz and Sadhal [10] extended the analysis for weakly inertial flows using a matched asymptotic expansion to first order in the Reynolds number. More recently, Li and Mao [11] have investigated numerically the effect of surfactant on the motion of a drop at intermediate Reynolds number. They found that the recirculating wake behind the drop becomes more closely attracted to the drop as the surfactant concentration is increased.

We shall make no assumption about the fluid velocity on the interface such as the exis- tence of a stagnant cap at the rear of the drop. Also we shall not specify a specific diffusion model for surfactants which could determine the parameterα. Instead, we shall give a general description of the effects of this variable interfacial tension on a spherical drop in small Reynolds number flow, correct to first order in the Reynolds number, independently of how the variability in the interfacial tension might have been induced.

2. Mathematical formulation

The fluid forming the spherical liquid drop and the exterior fluid, which is infinite in extent, are both viscous, incompressible and immiscible. Fluid variables inside the liquid drop will be denoted by a circumflex. A frame of reference fixed at the centre of the moving spherical drop, which has attained its terminal velocity U , will be used. We introduce spherical polar coordinates(r, θ, φ)with axisθ =0 in the direction of the terminal velocity U as shown in figure 1. Since the fluid flow is axisymmetric, there is no dependence on the angleφand Stokes stream functions,ψˆ andψ, inside and outside the liquid drop can be introduced.

The equations will be formulated in terms of(r, μ) where μ = cosθ and expressed in terms of dimensionless variables. All velocities are divided by the terminal velocity U , all stress byρU²whereρis the density of the surrounding fluid and all lengths by the radius of the spherical liquid drop, a. The interfacial tension,σ (μ), is the magnitude of the tensile

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er

g

U U U U

U U

ρ

ρ^ r

r θ θ φ

e_θ

e_θ O

ρ

Figure 1. Spherical polar coordinate system(r, θ, φ). The origin of the coordinate system is fixed at the centre O of the liquid drop. The velocity of the exterior fluid at infinity is the terminal velocity U of the drop. The axis of symmetry is defined by the gravity vector g through O. Ifρ > ρˆ , g points in the direction indicated. Ifρ < ρˆ , g points in the opposite direction.

force per unit length of a line on the interface. It acts normal to the test line and parallel to the tangent plane. It is made dimensionless by division byσ(1):

(μ)= σ(μ)

σ(1) , (1)=1. (1)

The dimensionless numbers are Re= ρaU

η , We= ρU²a

σ (1) , F=U²

ag , κ=ηˆ

η, γ = ρˆ

ρ, (2)

where Re is the Reynolds number of the exterior fluid, We is the Weber number, F is the Froude number,κ is the ratio of the shear viscosities,ηˆandη, of the interior and exterior fluids andγ is the density ratio. We consider the case in which Re, We and F are the same order of magnitude. The limit of a bubble rising through a fluid is obtained by letting κ→0 andγ →0 while the limit of a solid sphere is obtained by lettingκ→ ∞.

The formulation is similar to that of Taylor and Acrivos [1] except that the spherical liquid drop does not deform and the interfacial tension is not uniform. The Navier–

Stokes equations for the exterior and interior fluids, when expressed in terms of the stream functionsψ(r, μ)andψ(r, μ)ˆ are [1]

D⁴ψ = Re r²

∂ψ

∂r

∂

∂μ−∂ψ

∂μ

∂

∂r + 2μ (1−μ²)

∂ψ

∂r +2 r

∂ψ

∂μ

D²ψ, (3)

D⁴ψˆ = γRe κr²

∂ψˆ

∂r

∂

∂μ−∂ψˆ

∂μ

∂

∂r + 2μ (1−μ²)

∂ψˆ

∂r +2 r

∂ψˆ

∂μ

D²ψ,ˆ (4) where

D²= ∂²

∂r² +(1−μ²) r²

∂²

∂μ². (5)

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The problem is to solve (3) and (4) subject to the following boundary conditions:

(i) At the surface of the liquid drop r=1, τrr(1, μ) = ˆτrr(1, μ)+ 2

We (μ), (6)

τrθ(1, μ) = ˆτrθ(1, μ)+ 1

We(1−μ²)^1/2 d(μ)

dμ , (7)

vθ(1, μ) = ˆvθ(1, μ), (8)

vr(1, μ) = 0, vˆr(1, μ)=0. (9)

The boundary conditions (6) and (7) are obtained from the balance of normal and tangential stress at the interface [2]. The boundary conditions (8) and (9) describe continuity in the tangential component of the velocity at the interface and vanishing of the normal components of the velocity of each fluid at the spherical surface of the liquid drop.

(ii) At r = ∞,

r=→ ∞, ψ(r, μ)→ 1

2(1−μ²)r², (10)

p(∞,0)=0. (11)

The boundary condition (10) states that at large distances from the liquid drop, the fluid velocity tends to the terminal velocity of the drop. In condition (11) the exterior fluid pressure at r = ∞,μ=0 is arbitrarily set to zero [1].

(iii) At r =0,

rlim→0

1 r²

∂ψˆ

∂μ and lim

r→0

1 r

∂ψˆ

∂r are finite. (12)

The boundary condition (12) is the requirement thatvˆr andvˆ_θ must remain finite at the centre of the liquid drop.

The components of the Cauchy stress tensor are τrr(r, μ) = −p+ 2

Re

∂vr

∂r , (13)

τrθ(r, μ) = 1 Re

−1

r (1−μ²)¹^/²∂vr

∂μ +r ∂

∂r v_θ

r

, (14)

ˆ

τrr(r, μ) = − ˆp+ 2κ Re

∂vr

∂r , (15)

ˆ

τrθ(r, μ) = κ Re

−1

r (1−μ²)¹^/² ∂vˆr

∂μ +r ∂

∂r vˆ_θ

r (16)

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and the fluid velocity components are vr(r, μ)= −1

r²

∂ψ

∂μ, v_θ(r, μ)= − 1 r(1−μ²)¹^/²

∂ψ

∂r , (17)

ˆ

vr(r, μ)= −1 r²

∂ψˆ

∂μ, vˆ_θ(r, μ)= − 1 r(1−μ²)¹^/²

∂ψˆ

∂r . (18)

The fluid pressures p(r, μ)andpˆ(r, μ)which occur inτrr andτˆrrare determined from the Navier–Stokes equations inside and outside the liquid drop in the form

curlω = Re

−grad 1

2v²+p +v×ω+k

F(cosθer+sinθeθ)

, (19) curlωˆ = γ

κRe

−grad 1

2vˆ²+ ˆp + ˆv× ˆω+k

F(−cosθer+sinθe_θ)

, (20) where the vorticitiesωandωˆ are given by

ω= − 1

r(1−μ²)^1/2D²ψe_φ, ωˆ = − 1

r(1−μ²)^1/2D²ψeˆ _φ . (21) The constant k in the body force term in (19) and (20) takes the values±1; k = +1 when γ >1(ρ > ρ)ˆ and the liquid drop descends relative to the surrounding fluid while k= −1 when 0≤γ <1(ρ < ρ)ˆ and the liquid drop ascends relative to the surrounding fluid.

A special feature of the mathematical formulation is that the unknown quantity,(μ), occurs in the boundary conditions. The term d/dμ in (7) describes the non-uniform interfacial tension and vanishes when the interfacial tension is uniform [1].

We consider a perturbation solution and make straightforward perturbation expansions in powers of the Reynolds number. Let

ψ(r, μ) = ψ0(r, μ)+Reψ1(r, μ)+O(Re²), (22) ψ(r, μ)ˆ = ˆψ0(r, μ)+Reψˆ1(r, μ)+O(Re²), (23)

(μ)=0(μ)+Re1(μ)+O(Re²). (24)

We expand0(μ)and1(μ)in infinite series of Legendre polynomials as

0(μ)= ∞ n=0

AnPn(μ),

1(μ)= ∞ n=0

BnPn(μ). (25)

The lowest-order term in the expansion of the pressure is order Re⁻¹for the pressure gradient force to balance the viscous force in (19) and (20). The lowest order term in the perturbation expansion of the Cauchy stress tensor is also order Re⁻¹.

The solution is expressed in terms of the polynomials Qn(μ)=

_μ

−1

Pn(x)dx, (26)

where Pn(μ)is the Legendre polynomial of degree n. Since P0(μ)=1, P1(μ)=μ, P2(μ)= 1

2(3μ²−1), (27)

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it follows that

Q₀(μ)=1+μ, Q₁(μ)= −1

2 (1−μ²), Q₂(μ)= −1

2 μ(1−μ²). (28) We follow the well-established method of solution as devised by Proudman and Pear- son [12] and Taylor and Acrivos [1]. The calculations are lengthy. We shall therefore only present the results and verify that they agree with the established solutions in the appropriate limits.

3. Perturbation solution

The problem is a singular perturbation problem. The first-order term in the straightforward expansion (22) does not satisfy the boundary condition for large values of r . Inner and outer expansions exterior to the liquid drop are introduced. The outer variable is the contracted radial coordinateρ =r Re. The outer and inner expansions are matched using Van Dyke’s matching principle with n=m=2 [13].

The stream function inside the drop to first order in Re is ψ(ˆ r, μ) = 1

2

1−α

1+κ (r²−r⁴)Q₁(μ) +Re

3(2+α+3κ)

16(1+κ) (r²−r⁴)Q1(μ) +

−(2+α+3κ)(22−5α+17κ) 16(1+κ)²(4+κ) + (γ −1)

4(4+κ)

1−α 1+κ

2

×(r³−r⁵)Q₂(μ)

, (29)

whereαis defined in eq. (39) in terms of the interfacial tension gradient and is not determined in the fluid mechanics problem. The inner expansion of the stream function outside the drop to first order in Re is

ψ(r, μ) =

−r²+1 2

2+α+3κ 1+κ r−1

2

α+κ 1+κ

1 r

Q1(μ) +Re

1 8

2+α+3κ 1+κ

1

r −r² Q₁(μ) +

1 8

2+α+3κ

1+κ r²− 1 16

2+α+3κ 1+κ

2

r +(2+α+3κ)(κ²+20κ+2ακ+3α+18)

16(1+κ)²(4+κ)

−1 4

(γ −1) (4+κ)

1−α 1+κ

2

−(2+α+3κ)(α+κ) 16(1+κ)²

1 r

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+(2+α+3κ)(κ²−12κ+5α−18) 16(4+κ)(1+κ)²

1 r² +1

4

(γ −1) (4+κ)

1−α 1+κ

2 1 r²

Q2(μ)

. (30)

The outer expansion of the stream function outside the drop is (ρ, μ) = − 1

Re²ρ²Q1(μ)− 1 2Re

2+α+3κ

1+κ (1+μ)

×

1−exp

−1

2ρ(1−μ) +O(1). (31)

The coefficients Anand Bn in the expansions of0(μ)and1(μ)in (25) are A0 = 1

2 We

Re ˆ0= Uη

2σ(1) ˆ0, (32)

A1 = −|γ−1|(1+κ) We F +3

2 (2+3κ)We Re

= −|γ−1|(1+κ)ρga² σ (1) +3

2(2+3κ) Uη

σ(1), (33)

An =0, n≥2, (34)

B₀ = 1 2

ˆ1+ 1

16+(1+2γ ) 48

1−α 1+κ

2 We

Re, (35)

B1 = 3(2+α+3κ)(2+3κ) 16(1+κ)

We

Re, (36)

B2 = 5(1+κ) (4+κ)

−(2+α+3κ)(27κ²+58κ−8ακ+30−7α) 80(1+κ)³

+ (γ −1) 12

1−α 1+κ

2 We

Re, (37)

Bn =0, n≥3, (38)

whereˆ0+Reˆ1is the fluid pressure at the centre of the drop to first order in Re. The parameterαis defined as

α= −2 3

Re WeA₁= 2

3 |γ−1|(1+κ)ρga²

ηU −(2+3κ). (39)

It depends on the interfacial tension gradient through A₁and on the terminal velocity U . Solving (39) for U in terms ofαgives

U =2 3|γ−1|

1+κ 2+α+3κ

ρga²

η . (40)

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By considering an overall force balance on the drop it can be shown that the dimensionless drag on the liquid drop is

drag= 1 2

2+α+3κ 1+κ

1

Re+O(Re). (41)

The interfacial tension is

(μ) =(A0+Re B0)P0(μ)+(A1+Re B1)P1(μ)

+Re B2P2(μ)+O(Re²). (42) As(1)=1 and Pn(1)=1 for n≥0, it follows that

1=A₀+A₁+Re(B0+B₁+B₂)+O(Re²), (43) which determines the pressure at the centre of the drop,ˆ0+Reˆ1, in terms ofσ (1)and U . Subtracting (43) from (42) gives

(μ)=1+(A1+Re B₁)(P1(μ)−1)+O(Re²). (44) We shall now determine the range of values of α. Consider first the lower bound for αwhich corresponds to a spherical drop with uniform interfacial tension. From (44) the interfacial tension on a spherical drop will be uniform provided

A₁+Re B₁=0, B₂=0, (45)

that is, provided

α=(2+3κ)²

8(1+κ)Re+O(Re²), (46)

γ−1= 3(2+3κ)(27κ²+58κ+30)

20(1+κ) +O(Re). (47)

But when the interfacial tension is uniform the deformation,ζ(μ),of the drop is [1]

ζ(μ)= We

48(1+κ)²

γ−1−3(2+3κ)(27κ²+58κ+30) 20(1+κ)

P2(μ), (48) which vanishes and the drop will remain spherical when (47) is satisfied. If (46) and (47) are substituted into (29), (30) and (31) then the stream functions reduce to the corresponding stream functions derived by Taylor and Acrivos [1]. Further, if (46) is substituted into (41) then, correct to zero order in Re,

drag= 1 2

2+3κ 1+κ

1 Re+ 1

16

2+3κ 1+κ

2

, (49)

which agrees with the drag derived by Taylor and Acrivos [1] on a spherical drop with uniform interfacial tension. We therefore take (46) as the lower bound onα.

To obtain an upper bound onα, consider the limit of a solid sphere. For example, if a sufficient quantity of surfactant is added to the system then there is complete stagnation of the fluid within the drop [4,5]. If we let

α=1+9

8(1+κ)Re, (50)

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then the dimensionless drag (41) becomes, correct to zero order in Re, drag= 3

2Re+ 9

16, (51)

which is the drag for steady flow past a solid sphere derived by Proudman and Pearson [12]. Ifαin (50) is substituted into (29) and (30), the stream functions become correct to first order in Re,

ψ(r, μ)ˆ = ˆψsolid(r, μ)− 51

16(4+κ)r³(1−r²)Q2(μ)Re, (52) ψ(r, μ) = ψsolid(r, μ)+ 51

16(4+κ)

(r²−1)

r² Q₂(μ)Re (53)

and (31) becomes

(ρ, μ)=solid(ρ, μ)+O(1), (54)

whereψˆsolid(r, μ)=0 andψsolid(r, μ)andsolid(ρ, μ)are the inner and outer expansions of the stream functions for steady flow past a solid sphere [12]. The stream functions ψ(r, μ)ˆ andψ(r, μ) reduce to the corresponding stream functions for flow past a solid sphere to zero order in Re, but not exactly to first order in Re. However, the difference which is proportional to 51Re/16(4+κ)is small. Whenκ =2 and Re=0.1, 51Re/16(4+ κ)= 0.05. Whenαis given by (50), the flow inside the drop is almost stagnant and the flow outside the drop closely resembles flow past a solid sphere. We therefore take (50) as an upper bound onα.

The range of values ofαwhich we shall consider is therefore (2+3κ)²

8(1+κ) Re≤α≤1+9

8(1+κ)Re. (55)

4. Fluid velocity on the interface

From (18) and (29), the fluid velocity on the interface, r=1,is ˆ

vθ(1, θ)=− sinθ 2(1+κ)

1−α+3

8(2+α+3κ)

×Re

1+

−(22−5α+17κ) 3(1+κ)(4+κ) + 4(γ −1)(1−α)²

3(4+κ)(1+κ)(2+α+3κ)

cosθ .

(56) In general,vˆθ(1, θ) <0.However, if

α >1+3(1+κ)(3κ−5)

8(4+κ) Re, (57)

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thenvˆθ(1, θ) >0 for 0< θ < θ0where

cosθ0= 8(4+κ) 51(1+κ)Re

1+9(1+κ) 8 Re−α

. (58)

The right-hand side of (57) is less than the upper bound in (55). Whenαequals the upper bound in (55) then

ˆ

vθ(1, θ)= 51 sin 2θ

32(4+κ)Re. (59)

When plotting the fluid variables we shall choose Re = 0.1, We/Re = 1,κ = 2 and γ =2. Then from (55), 0.27≤ α≤ 1.34. In figure 2,vˆθ(1, θ)is plotted againstθfor a range of values ofα. Asαincreases the magnitude ofvˆθ(1, θ)decreases to order Re. When αsatisfies (57) a cap 0< θ < θ0 in whichvˆθ(1, θ) > 0 develops at the rear of the drop.

Whenαequals the upper bound in (57) we see from (58) and (59) that the cap extends to 0≤θ≤π/2 and the magnitude ofvˆθ(1, θ)is order Re.

In surfactant models, surfactant is convected to the rear of the drop, θ = 0, and as a result the interfacial tension gradient is greatest at the rear. The interfacial tension gradient counteracts the viscous stress on the interface which decreases the fluid velocity on the interface. The change invˆ_θ(1, θ)from negative to positive therefore first occurs at the rear.

Figure 2 agrees with experimental observations. As surfactant is added to the system, Griffith [4,5] observed that the fluid flow on the interface ceased and that this first occurred at the rear producing a stagnant cap. Although the interfacial velocity does not reduce to zero in figure 2, it becomes small of order Re.

Figure 2. Interfacial velocityvˆ_θ(1, θ)plotted againstθ for Re=0.1,κ = 2,γ = 2 andα=0.27,0.5,0.8,1,1.2 and 1.34.

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5. Fluid flow inside the liquid drop

The stream function (29) can be expressed in terms of r andθas ψ(r, θ)ˆ = −r²(1−r²)sin²θ

4(1+κ)

1−α+3

8(2+α+3κ)

×Re

1+

−(22−5α+17κ) 3(1+κ)(4+κ) + 4(γ −1)(1−α)²

3(4+κ)(1+κ)(2+α+3κ)

r cosθ . (60) In figure 3 the streamlines inside the liquid drop,ψ(ˆ r, θ)=constant, are plotted forα = 0.27,1, 1.2 and 1.34. The flow pattern forα=1 indicates a decrease in the fluid velocity at the rear of the drop because there is more space between neighbouring streamlines at the rear. When Re=0.1 andκ =2, condition (57) for reversed flow on the interface becomes α >1.02. Forα >1.02 an interior region, 0≤θ ≤θ0, with reversed circulation forms at the rear of the drop and it increases in size asαincreases. Forα=1.2,θ0=64^◦while for α=1.34,θ0=90^◦.

Figure 3. Streamlines inside the liquid drop (Re = 0.1, κ = 2, γ = 2) for α = 0.27,1,1.2 and 1.34. The direction of flow of the exterior fluid is from left to right.

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As well as decreasing the fluid velocity on the interface, the interfacial tension gradient reduces the forces within the drop which in turn reduces the fluid circulation. This first occurs at the rear of the drop, consistent with diffusion models for surfactants in which surfactants are convected to the rear of the drop. A stagnant fluid within the drop is produced when the interfacial tension gradient is sufficiently large that it balances the viscous stress.

Figure 3 is in good qualitative agreement with experimental observations. Griffith [4,5]

observed that as the surfactant concentration is increased the circulation velocity of the fluid inside the drop decreased and this first occurred at the rear of the drop. The addition of a sufficient quantity of surfactant resulted in complete stagnation of fluid within the drop.

6. Fluid flow outside the liquid drop

When reversed flow exists on the interface at the rear of the drop, the fluid flow will be altered outside the drop. We therefore investigate the external flow when (57) is satisfied.

The external flow close to the drop is described by the inner expansion of the stream function outside the drop, given by (30). Expressed in terms of r andθ, (30) is

ψ(r, θ)= (r−1)sin²θ 4r

2r²−

α+κ 1+κ r

−α+κ 1+κ +Re

4

2+α+3κ 1+κ

×

r²+r+1− 1

2r(2r³+Mr²+N r+S)cosθ , (61) where

M = − α+κ

1+κ , (62)

N = (16κ+ακ−α+18)

(1+κ)(4+κ) − 4(γ−1)(1−α)²

(1+κ)(4+κ)(2+α+3κ), (63) S = −(κ²−12κ+5α−18)

(1+κ)(4+κ) − 4(γ−1)(1−α)²

(1+κ)(4+κ)(2+α+3κ). (64) When eq. (57) is satisfied,α=1+O(Re)and therefore in terms of order Re in (61), αcan be replaced by unity. Nowψ =0 on the surface r =1 of the liquid drop, along the axis of symmetry,θ=0 andθ =π, and on the curve

cosθ=

2+3

4 Re r³− α+κ

1+κ −3

4 Re r²− α+κ

1+κ −3 4 Re r

3Re 8

2r³−r²+ 17

κ+4r+13−κ 4+κ

.

(65) Equation (65) defines the transition layer between the interior and exterior flows. The streamline (65) generates a surface of revolution about the lineθ=0.It is the boundary of

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the standing eddies downstream of the drop which are generated due to the reversed flow on the interface at the rear of the drop. When r = 1, θ = θ0 where cosθ0 is defined by eq. (58); cosθ0 < 1 since it is assumed that (57) is satisfied. The boundary of standing eddies is attached to the drop atθ =θ0.The downstream end of the boundary is atθ =0 and r=r₊. By putting cosθ= +1 in (65) it is found that r₊satisfies the cubic equation

F(r)=0, (66)

where

F(r)=2r³− α+κ

1+κ −9

8 Re r²− α+κ

1+κ −3 8

2κ−9 κ+4 Re r +3

8

κ−13

κ+4 Re. (67)

Now

F(0)= −3 8

13−κ 4+κ Re, F(1)= 2

(1+κ)

1+3(3κ−5)(1+κ)Re

8(κ+4) −α

. (68)

When eq. (57) is satisfied, F(1) <0.We assume that Re is sufficiently small that the sign of the coefficients in (67) is determined by the zero-order terms in Re.

Consider first 0 ≤ κ < 13. From Descartes rule of signs [14], F(r)=0 cannot have more that one positive root. Since F(∞)= ∞and F(1) <0 it follows that F(r)=0 has one positive root, r=r₊, which is greater than unity.

Consider nextκ > 13. From Descartes rule of signs, F(r)=0 cannot have more than two positive roots. Since F(∞)= ∞, F(1) <0 and F(0) >0, F(r)=0 has two positive roots, r₊>1 and 0<r₋<1.

Forκ=13, F(r)has one positive root which to first order in Re is, r₊=1+ 1

21

α−1−21

2 Re (69)

and r₊>1 by eq. (57).

Thus when eq. (57) is satisfied, F(r)=0 has one and only one positive root, r₊, which is greater than unity. It can be verified that to first order in Re,

r₊=1+ 2 3(1+κ)

α−1−3(1+κ)(3κ−5) 8(4+κ) Re

, (70)

and r₊>1 by eq. (57). For the upper bound in eq. (55) and to first order in Re, r₊=1+ 17

4(1+κ)Re. (71)

In figure 4 the streamlines inside and outside the liquid drop are plotted from the stream functions (60) and (61) forα=1.2.The fluid flow inside the liquid drop consists of two regions, a smaller region, 0≤ θ ≤ θ0 =64^◦, at the rear and a larger region at the front.

Because of the reversed flow on the interface at the rear there is a thin transition layer at

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Figure 4. Streamlines inside and outside the liquid drop forα=1.2 withκ=2,γ =2 and Re=0.1. The arrows indicate the direction of flow.

the rear outside the drop. The boundary of the transition layer is attached to the drop at θ=θ0=64^◦and the maximum thickness of the layer which occurs atθ=0 is

2 3(1+κ)

α−1−3(1+κ)(3κ−5) 8(4+κ) Re

=0.04. (72)

Since from eq. (57),α−1=O(Re), the thickness of the transition layer is of order Re. For the upper bound in eq. (55), the maximum thickness is 17Re/4(1+κ). There are standing eddies in the interior of the transition layer.

It is of interest to compare the results found here with the results for small Reynolds number flow past a solid sphere and past a liquid drop with uniform interfacial tension.

For small Reynolds number flow past a solid sphere the perturbation solution predicts that standing eddies exist only for Re > 8 and that the boundary of the standing eddies is attached to the surface of the sphere [12,13]. There is good qualitative agreement with numerical results and experiment even though Re is given values greater than allowed in the derivation of the perturbation solution.

For small Reynolds number flow past a liquid drop with uniform interfacial tension the perturbation solution predicts that standing eddies exist only for [15]

Re>8

1+ 8 3κ^1/2 +O

1

κ as κ → ∞, (73)

and that the boundary of the standing eddies is detached from the surface of the drop. There is only one region of flow inside the drop and a transition layer between the interior and exterior flow is not necessary. Again, there is good qualitative agreement with numerical and experimental results [16] even though the values of Re are greater than allowed for a perturbation solution.

In comparison, provided the interfacial tension gradient is sufficiently large that eq. (57) is satisfied, the transition layer found here exists for Re > 0. The perturbation solution should be valid for 0 < Re<∼1 and values of Re outside the range allowed by the perturbation solution do not need to be considered.

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7. Concluding remarks

The singular perturbation solution of Taylor and Acrivos [1] and the solution presented here represent two extreme cases. In the solution of Taylor and Acrivos the interfacial tension is uniform but the liquid drop deforms. In the solution given here the interfacial tension is non-uniform and is such that the liquid drop remains spherical.

We have investigated the general properties of small Reynolds number flow past a spherical liquid drop with non-uniform interfacial tension. The distribution of interfacial tension was determined by the condition that the drop remains spherical. We found that if the interfacial tension gradient is sufficiently large a region of reverse flow occurs on the interface at the rear and the flow inside the drop splits into two parts with reversed circulation at the rear. The magnitude of the interior fluid velocity is small, of order the Reynolds number. These results agree with experimental observations of the addition of surfactants to the fluid, which could therefore provide a mechanism to induce the distribution of interfacial tension which allows the drop to remain spherical in small Reynolds number flow.

Acknowledgements

This material is based on the work supported financially by the National Research Foundation, Pretoria, South Africa. The authors thank the referee for constructive comments.

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