stokes flow inside a sphere in an inviscid extensional...

Z. Angew. Math. Phys. (2017) 68:81c© 2017 Springer International Publishing AG0044-2275/17/040001-13published online June 30, 2017DOI 10.1007/s00033-017-0828-2

Zeitschrift fur angewandteMathematik und Physik ZAMP

Stokes flow inside a sphere in an inviscid extensional flow

Joel D. Krehbiel and Jonathan B. Freund

Abstract. We derive the streamfunction solution for flow in and around a viscous sphere suspended in an inviscid extensionalflow with matched stress boundary conditions, which is a model for estimating the stresses on a tiny suspended organism bya nearby expanding and collapsing bubble. The boundary conditions are enforced in an easily resolvable form by expressingthe surface stresses as sums of Legendre and Gegenbauer functions. The flow inside the sphere reflects a balance of exteriorinertia with internal viscous forces, which together are shown to constitute the relevant flow Reynolds number. The solutionis evaluated to examine the flow field inside this sphere as a potential source of damage to the organism.

Mathematics Subject Classification. 35Q35, 76D07, 76D09.

Keywords. Stokes flow, Viscous sphere, Inviscid extensional flow.

1. Introduction

The flow from an expanding or collapsing microbubble near a cell shows promise as a drug deliverytechnique [4], to destroy cancerous cells [22], or to rupture algal cells [10]. In these cases, the typicallyhigh interior viscosity suggests viscous flow dynamics inside the cell while the exterior flow dynamics arerelatively inviscid. For example, the Reynolds number outside an algal cell near a collapsing bubble isestimated to be near 10, while the Reynolds number inside the cell is near 0.1 [9]. Provided the cell isfar from the bubble, the flow is extensional in the frame of reference of the cell [9,12]. Here we developa solution for and examine the fluid motion inside a viscous sphere suspended in an inviscid fluid withextensional flow representing an expanding or collapsing microbubble.

For a very viscous fluid, the flow is governed by the Stokes equation. Explicit solutions to this equationhave been found for uniform flow over a sphere [8], inside a viscous sphere [1], over a slightly deformedsphere [8,16,18], between eccentric rotating spheres [5], and both over and inside a porous shell [2,14,15,17]. Here we derive the streamfunction for flow inside a viscous sphere due to an exterior extensional flowwith matched stress boundary conditions at the sphere surface. The full series solution is obtained byexpressing the exterior and interior boundary conditions as sums of Gegenbauer and Legendre functions.

This flow model is based on several simplifying assumptions. The bubble and cell are assumed spherical,which is a good approximation for the initial bubbles and cells used in corresponding algae disruptionexperiments [10]. The line between their centers provides an axis of symmetry for this flow. The spacingbetween the cell and bubble is assumed large compared with the cell radius so the flow field in the cellframe of reference is well approximated by a purely extensional flow. Since the cell is far from the bubble,we assume the cell does not affect the bubble motion. The Weber number of the exterior flow and theCapillary number of the interior flow are both near 10 [9] suggesting that the tension in the membranewould need to be 10 times greater for surface tension effects to be O(1) in either the interior or exteriorfluids. Thus, we ignore curvature and surface tension effects in the current analysis.

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81 of 13 J. D. Krehbiel and J. B. Freund ZAMP

Fig. 1. A sphere with radius a and viscosity μ in an inviscid flow axisymmetric about the z-axis with strain rate γ anddensity ρ

2. Formulation

The specific configuration is shown in Fig. 1. The viscous sphere has radius a and is suspended in asteady irrotational extensional flow of fluid with density ρ, while the flow inside has ĎRe � 1 for interiorviscosity μ. Here we use the overbar to denote inside the sphere. Axisymmetric extensional flow with acharacteristic strain rate of γ acts on the sphere.

The interior flow is governed by Stokes equation:

E4ψ = 0 (2.1)

with

E2 =∂2

∂r2+

sinθ

r2∂

∂θ

(1

sinθ

∂

∂θ

), (2.2)

and ψ the streamfunction.The exterior flow is irrotational and so satisfies

E2ψ = 0. (2.3)

3. Boundary conditions

We match the stresses at the sphere surface so that

σrr(a, θ) = σrr(a, θ) (3.1)

σrθ(a, θ) = σrθ(a, θ). (3.2)

ZAMP Stokes flow inside a sphere in an inviscid extensional flow of 13 81

It is common to include curvature effects for an approximate fluid sphere in the radial stress balance at afluid–fluid interface [18]; however, for the problem of interest the Weber number, defined in this case as

We =ργ2a3

KA� 10, (3.3)

where KA is the membrane’s resistance to tension [9]. Thus, we can anticipate that the inertial forcesdominate the interface stresses in the exterior fluid. Likewise, the viscous effects of the interior fluid are10 times greater than surface tension effects based on a Capillary number of

Ca =μγa

KA� 10. (3.4)

Thus, curvature and surface tension effects do not need to be included in the radial stress balance.The exterior flow solution is obtained from the analytical solution for extensional flow around a rigid

sphere (i.e. ur = 0 for r = a). We are interested in the initial rate of the shape deformation of thecell; however, if we match the radial velocities at the boundary, the cell would stay spherical. Thus, weallow the interior radial velocity to be nonzero to calculate the initial shape deformation rate. While thiswould locally violate mass conservation were the deformations allowed to develop, it provides a helpfulunderstanding of the possible initial deformation of the sphere. The exterior solution assumes a thinboundary layer where stresses are equal at the top and bottom of the boundary layer. The difficulty offormulating the boundary conditions for this problem is further discussed in Sect. 6.

4. Method of solution

4.1. Interior flow

The general solution to (2.1) is known [20]. Here we follow the formulation of Happel and Brenner [8],which is restricted to realistically finite velocities (see appendix). This restricted general streamfunctionis

ψ(r, θ) =∞∑

n=2

(Anrn+2 + Cnrn

)In(ζ), (4.1)

where ζ = cos θ and In is the Gegenbauer function of degree −1/2 and order n, defined in terms of theLegendre functions Pn(ζ) as

In(ζ) =Pn−2(ζ) − Pn(ζ)

2n − 1. (4.2)

For n = 0 and 1:

I0(ζ) = 1, I1(ζ) = −ζ, (4.3)

and the orthogonality property is [8]∫ 1

−1

Im(ζ)In(ζ)1 − ζ2

dζ ={

0 for m �= n2

n(n−1)(2n−1) for m = n.(4.4)

Likewise, the Legendre functions have∫ 1

−1

Pm(ζ)Pn(ζ)dζ ={

0 for m �= n2

2n+1 for m = n.(4.5)


There are two coefficients in (4.1): An and Cn for any n. The stresses at the sphere surface are obtainedfrom the streamfunction and take the forms (Sects. 6, 6, 6)

σrr(a, θ) =∞∑

n=1

σnPn(ζ) (4.6)

σrθ(a, θ) =∞∑

n=2

τn

sin θIn(ζ), (4.7)

where

σn = μ

(2(−n2 + n + 3)

nAn+1a

n − 2(n − 1)Cn+1an−2

)(4.8)

and

τn = μ(2(n + 1)(n − 1)Anan−1 + 2n(n − 2)Cnan−3

). (4.9)

These follow directly from the general streamfunction solution for axisymmetric Stokes flow.

4.2. Exterior flow

The exterior flow is modeled as that around a rigid sphere (i.e. ur = 0 for r = a). In spherical coordinates,the streamfunction for irrotational axisymmetric extensional flow is [13]

ψ(r, θ) = −r3γ sin2 θ cos θ, (4.10)

where γ is a constant that describes the velocity scale of the flow divided by a characteristic length scale;thus, we call γ the characteristic strain rate of this flow. Using Butler’s sphere theorem [3], we obtain thestreamfunction for axisymmetric extensional irrotational flow around a rigid sphere with radius a:

ψ(r, θ) = −γ sin2 θ cos θ

(r3 − a5

r2

)(4.11)

with corresponding velocities

ur(r, θ) = − 1r2 sin θ

∂ψ

∂θ= γ(2 cos2 θ − sin2 θ)

(r − a5

r4

)(4.12)

and

uθ(r, θ) =1

r sin θ

∂ψ

∂r= −γ sin θ cos θ

(3r +

2a5

r4

). (4.13)

Since the exterior flow is irrotational, pressure follows from Bernoulli’s equation,

p(r, θ) = p0 − ρ

2u2(r, θ), (4.14)

where u2(r, θ) = u · u and p0 is the stagnation pressure. Taking p0 = 0, without loss of generality,

p(r, θ) = −ρ

2u2(r, θ), (4.15)

and for the velocity defined in (4.12) and (4.13), the pressure is

p(r, θ) = −ρ

2

[γ2(2 cos2 θ − sin2 θ)2

(r − a5

r4

)2

+ γ2 sin2 θ cos2 θ

(3r +

2a5

r4

)2]

. (4.16)


For an inviscid fluid, the only stresses are due to the pressure contribution, and

σrθ(r, θ) = 0, (4.17)

σrr(r, θ) =ργ2

2

[(2 cos2 θ − sin2 θ)2

(r − a5

r4

)2

+ sin2 θ cos2 θ

(3r +

2a5

r4

)2]

, (4.18)

which at r = a are

σrθ(a, θ) = 0, (4.19)

σrr(a, θ) =25ργ2a2 sin2 θ cos2 θ

2. (4.20)

For this axisymmetric exterior flow, we seek series sums of the forms

σrr(a, θ) =∞∑

n=0

σnPn(ζ) (4.21)

σrθ(a, θ) =∞∑

n=2

τn

sin θIn(ζ), (4.22)

so that they match the form of the interior representation of the boundary conditions (4.6) and (4.7). Thecoefficients σn and τn are found using the orthogonality of the Gegenbauer and Legendre polynomials.The orthogonality properties (4.4) and (4.5) can be used to show

σn =2n + 1

2

∫ π

0

σrr(a, θ)Pn(cos θ) sin θdθ, (4.23)

and

τn =n(n − 1)(2n − 1)

2

∫ π

0

σrθ(a, θ)In(cos θ)dθ = 0. (4.24)

The integrals (4.23) and (4.24) provide an explicit way to determine the coefficients of the stress condi-tions at the boundary. Once these coefficients are obtained, they may be matched with the correspondinginterior coefficients (4.8) and (4.9), and the streamfunction coefficients (An and Cn) may be solved.

4.3. Matched stress boundary conditions

By writing the exterior stresses in the forms given in (4.21) and (4.22), we can write the matched stressconditions as σn = σn and τn = τn.

We note that to match completely the radial stress at the boundary, it is necessary to add the constantstress term from the exterior flow so that

σrr(a, θ) = σ0 +∞∑

n=1

σnPn(ζ). (4.25)

Using (4.18) and (4.23), we obtain the radial stress coefficients σn:

σn =2n + 1

2

∫ π

0

σrr(a, θ)Pn(cos θ) sin θdθ =2n + 1

2

∫ π

0

25ργ2a2 sin2 θ cos2 θ

2Pn(cos θ) sin θdθ. (4.26)


This integral is evaluated to find

σ0 =53ργ2a2, (4.27)

σ2 =2521

ργ2a2, (4.28)

σ4 = −207

ργ2a2, (4.29)

and

σn/∈{0,2,4} = 0. (4.30)

4.4. Detailed coefficient matching conditions

With σn and τn, we can solve for the streamfunction coefficients in (4.8) and (4.9). Values for A3 and C3

follow from σ2 and τ3:

σ2 = μ(A3a

2 − 2C3

)= σ2 =

2521

ργ2a2 (4.31)

τ3 = μ(16A3a

2 + 6C3

)= τ3 = 0, (4.32)

which together yield

A3 =25ργ2

133μ(4.33)

and

C3 = −200ργ2a2

399μ. (4.34)

Likewise, values for A5 and C5, follow from σ4 and τ5

σ4 = μ

(−9

2A5a

4 − 6C5a2

)= σ4 = −20

7ργ2a2 (4.35)

τ5 = μ(48A5a

4 + 30C5a2)

= τ5 = 0, (4.36)

which yield

A5 = −200ργ2

357μa2(4.37)

and

C5 =320ργ2

357μ. (4.38)

Since σn = τn = 0 for all other n, An and Cn are zero for n �∈ 3, 5.The inner streamfunction simplifies to

ψ(r, θ) =ργa2

μγa

[(25133

r5

a3− 200

399r3

a

)I3(ζ) +

(−200

357r7

a5+

320357

r5

a3

)I5(ζ)

]. (4.39)

The coefficient is expressed this way to isolate the exterior flow velocity scale γa. The other collection offactors is a nondimensional group (ργa2/μ), which can be considered the ratio of inertial forces in theinviscid flow to the viscous forces in the sphere, and it is thus a Reynolds number.


x/a

Fig. 2. Streamlines for a viscous sphere in an inviscid extensional flow with ψ/(γa3) = ±10−3+0.4j and ψ/(ργ2a5/μ) =±10−6+0.5j for j = −10, 9, . . . 9, 10. The inset box shows vortices in the interior flow near θ = 0

5. Flow solution

The contours of the streamfunctions for a typical case are visualized in Fig. 2. The interior streamlinesterminate on the sphere, consistent with the expected radial deformation of the sphere. Small recirculatingflows seen near θ = 0 and θ = π are noted, especially since small flow features would potentially causerelatively sharp deformation, though here the velocities are so small this is unlikely to be important. Thepeak velocity in them is 20% of the peak within the sphere. These features are reminiscent of the internalcirculation in a viscous sphere in a uniform viscous flow [8].

The corresponding stresses for this case are visualized in Fig. 3. The maximum radial stress insidethe sphere is along the lines θ = ±π/4 and θ = ±3π/4. The maximum internal radial stress is at thesurface: σrr(a, π/4) = 25ργ2a2/12. The stress magnitude (σ2

rθ + σ2rr)

1/2 is a maximum on the surface ofthe sphere as shown in Fig. 4 and thus is also equivalent to 25ργ2a2/12 since there is zero angular stresson the boundary.

Even though σrθ = 0 outside the sphere, σrθ is nonzero inside the sphere with local maxima andminima close to θ = π/8 + nπ/4 for integer n. The locations of the extrema of angular stress areevaluated numerically, since the complicated nature of the analytical solution obfuscates the results. Theglobal extrema are at r/a = 0.613, ζ = ±0.414, providing σrθ = ±0.889ργ2a2. Local extrema occur atr/a = 0.757, ζ = ±0.929 giving σrθ = ±0.639ργ2a2.

For a membrane covering a cell, the areal strain might be particularly important. To visualize theexpected initial deformation, we artificially advect the surface based on interior surface velocities mul-tiplied by time t. The initial shape change and areal strain (ΔA/A) on the surface of the sphere areshown in Fig. 5. Here t is chosen so that the shape change is clearly visible. We confirm with this thatthe small recirculating zones do not have significant baring upon the overall deformations. The maximumtensile strain occurs when θ = 0.941, 2.20, 4.08, and 5.34 radians. Compressive strain is maximum atθ = ±π/2.


−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

0.5

1

1.5

2

2.5

3

z/a

x/a

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

z/a

x/a

(a) (b)

Fig. 3. a σrr and b σrθ scaled by ργ2a2 for a viscous sphere in an inviscid extensional flow

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

z/a

x/a

Fig. 4. Stress magnitude scaled by ργ2a2 for a viscous sphere in an inviscid extensional flow

6. Additional discussion

We have shown that the matching conditions for the purely extensional flow provide a solvable mathe-matical system, but this is not always the case. This is not the case for a general exterior flow, with thesame matching conditions on the sphere. The C2 term corresponds to uniform flow, and as such imposesno stress. Thus, the coefficient of the C2 term is zero in both (4.8) and (4.9), leaving two conditions forA2 to satisfy:

6μaA2 = τ2 (6.1)

and

6μaA2 = σ1, (6.2)


-1 -0.5 0 0.5 1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

z/a

x/a

0 1 2 3 4 5 6-0.015

-0.01

-0.005

0

0.005

0.01

0.015(a) (b)

Fig. 5. Representative a distortion and b local areal strain for ργa2/μ = 0.1 and tγ = 1.56

which is only possible if τ2 = σ1.Therefore, imposed boundary stress at the surface of a sphere is inconsistent if σ1 �= τ2. If σ1 = τ2, a

solution may exist; if σ1 �= τ2, a solution certainly does not exist. For our boundary conditions, motivatedby an expanding bubble somewhat removed from the viscous sphere, we do have σ1 = τ2 = 0.

The general case of inviscid flow over a viscous sphere provides significant challenges in matchingboundary conditions. It would seem that general solution would require a boundary layer on the sphere,arising due to the small but nonzero viscosity of the exterior fluid. However, another coupling arises inthis case: since the flow inside a viscous sphere only has two unknowns (An and Cn), only two boundaryconditions may be applied. It is not possible to match four conditions (two velocities and two stresses) ifthe exterior flow is completely specified.

Feng et al. [7] obtain a solution for a viscous sphere in a uniform exterior flow at small but finiteReynolds numbers through several assumptions. They assume a slip velocity at the sphere surface, con-tinuous shear stress, and no penetration. A singular perturbation approach allows a solution to the steadyNavier–Stokes equations at small Reynolds numbers. They also mention that their boundary conditionsseem to be only solvable numerically. Other results regarding the drag of viscous spheres at intermedi-ate Reynolds numbers suggest that the time-scales associated with interior and exterior flows are bothimportant. Thus, it is impossible to find an analytical solution for these general problems [6].

Further, the inviscid flow solution assumes a slip velocity above the boundary layer. It is necessaryto know the velocity profile in the boundary layer to estimate the stresses and velocities at the interfaceof the two fluids. Since the boundary layer develops in time, this requires a solution to the unsteadyboundary layer equations. A general solution does not exist, but solutions exist for an impulsively startedplate [19,21] or an impulsively stretching plate [11]. Further work is required to specify the form of theboundary layer near a viscous sphere in an inviscid flow.

A. Appendix: General solution of Stokes equation

The general solution for axisymmetric Stokes flow was first presented by Samson [20], and we use theformulation of Happel and Brenner [8], restricted to finite velocities along ζ = ±1:

ψ(r, θ) =∞∑

n=0

(Anrn+2 + Bnr−n+3 + Cnrn + Dnr−n+1

)In(ζ). (6.3)


Additionally, for the velocities to be bounded at the origin, Bn = Dn = 0. Thus, the streamfunctionis simplified to

ψ(r, θ) =∞∑

n=2

(Anrn+2 + Cnrn

)In(ζ). (6.4)

A.1. Velocities

The velocities are obtained by differentiating the streamfunction:

ur(r, θ) = − 1r2sinθ

∂ψ

∂θ, (6.5)

uθ(r, θ) =1

rsinθ

∂ψ

∂r. (6.6)

Using the following relationship

dIn(ζ)dζ

= −Pn−1(ζ), (6.7)

the velocities are written as

ur(r, θ) = −∞∑

n=1

(Anrn + Cnrn−2

)Pn−1(ζ) (6.8)

or

ur(r, θ) = −∞∑

n=1

(An+1r

n+1 + Cn+1rn−1

)Pn(ζ), (6.9)

and

uθ(r, θ) =∞∑

n=0

[(n + 2)Anrn + nCnrn−2

] In(ζ)sinθ

. (6.10)

The pressure is obtained by integrating the Stokes equation and gives

p(r, θ) = − μ

∞∑n=2

[2(2n + 1)

n − 1Anrn−1

]Pn−1(ζ). (6.11)

A.2. Radial stresses

The radial stresses for a Newtonian fluid are calculated from the general equation

σrr(r, θ) = −p(r, θ) + 2μ∂ur

∂r. (6.12)

Combining (6.8), (6.11), and (6.12) yields

σrr(r, θ) = μ

∞∑n=2

[2(2n + 1)

n − 1Anrn−1

]Pn−1(ζ) − 2μ

∞∑n=2

(nAnrn−1 + (n − 2)Cnrn−3

)Pn−1(ζ), (6.13)

σrr(r, θ) = μ∞∑

n=2

[2(−n2 + 3n + 1)

n − 1Anrn−1 − 2(n − 2)Cnrn−3

]Pn−1(ζ). (6.14)


This equation may be written simply as

σrr(r, θ) = μ

∞∑n=1

σnPn(ζ), (6.15)

with

σn = μ

(2(−n2 + n + 3)

nAn+1r

n − 2(n − 1)Cn+1rn−2

). (6.16)

A.3. Tangential stresses

The Newtonian fluid tangential stresses are

σrθ(r, θ) = μ

[− uθ

r+

∂uθ

∂r+

1r

∂ur

∂θ

]. (6.17)

The first term is

− uθ

r= −

∞∑n=2

[(n + 2)Anrn−1 + nCnrn−3

] In(ζ)sinθ

. (6.18)

The second term is

∂uθ

∂r=

∞∑n=2

[n(n + 2)Anrn−1 + n(n − 2)Cnrn−3

] In(ζ)sinθ

. (6.19)

The third term is

1r

∂ur

∂uθ= −

∞∑n=2

(Anrn−1 + Cnrn−3

) ∂

∂θPn−1(ζ). (6.20)

One can show [9] that the derivative

∂

∂θPn−1(ζ) = −n(n − 1)

In(ζ)sinθ

, (6.21)

so that

1r

∂ur

∂uθ=

∞∑n=2


)n(n − 1)

In(ζ)sinθ

. (6.22)

Finally, this result is combined with the previous two terms of the tangential stress condition and sim-plified.

σrθ(r, θ) =μ

(−

∞∑n=2

[(n + 2)Anrn−1 + nCnrn−3

] In(ζ)sinθ

+∞∑

n=2

[n(n + 2)Anrn−1 + n(n − 2)Cnrn−3

] In(ζ)sinθ

+∞∑

n=2


)n(n − 1)

In(ζ)sinθ

)(6.23)

σrθ(r, θ) =μ

( ∞∑n=2

[2(n + 1)(n − 1)Anrn−1 + 2n(n − 2)Cnrn−3

] In(ζ)sinθ

)(6.24)


The tangential stress is written as

σrθ(r, θ) =∞∑

n=2

τn

sinθIn(ζ) (6.25)

with

τn = μ(2(n + 1)(n − 1)Anrn−1 + 2n(n − 2)Cnrn−3

). (6.26)

Acknowledgements

The authors thank Dr. Lance Schideman for bringing this topic to their attention. This work was sup-ported in part by the National Institutes of Health (NIH) NIDDK Grant No. P01-DK043881.

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Joel D. KrehbielHesston College301 S. Main St.Hesston KS 67062USAe-mail: [email protected]

Jonathan B. FreundDepartment of Mechanical Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbana IL 61801USA

(Received: January 21, 2017; revised: June 14, 2017)

stokes flow inside a sphere in an inviscid extensional...

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