**6.2 Relaxation of a small perturbation of a circular cylinder**

Consider a small perturbation of the circular cylinder boundary, given by *rRh t* , , *h R* . Then we have from (62)

$$\frac{\partial \mathbf{l}}{\partial t} = -\frac{\sigma}{2\mu R} \sum\_{k=-\phi}^{\phi} |k| \exp(ik\rho \sigma) h\_{k'} \tag{63}$$

$$h\_k(t) \equiv \int\_0^{2\pi} \exp(-ik\rho) h(\rho, t) \frac{d\rho}{2\pi} = h\_k(0) \exp\left(-\frac{\sigma |k| t}{2\mu}\right) \tag{64}$$

in agreement with (Levich, 1962). According with (64), a small boundary perturbation of characteristic with *a R* and amplitude *H a* has a characteristic decay time ~ *a* .

#### **6.3 The capillary relaxation of an ellipse**

Let's test our theory on an example of a large amplitude perturbation. We calculate the capillary relaxation of boundary with initial shape 2 2 1 2 2 2 <sup>1</sup> *x x a b* in two ways - using the numerical calculation based on (6.4) and the finite-element software ANSYS POLYFLOW (see Fig. 3 and Fig.4). These methods of calculation give us the same results with discrepancy about 1%.

#### **6.4 The collapse of a cavity**

Let's now consider a large amplitude perturbation in the shape of a cavity (Fig. 5). By symmetry, the pressure must be an even function with respect to 2 *x* , i.e. 1 2 12 *p x x* , , *p x x* .

*p p* . 

( ) , *<sup>s</sup>*

 

0 . *<sup>G</sup> k k*

 

<sup>1</sup> , . <sup>2</sup>

exp , <sup>2</sup> *<sup>k</sup>*

*k*

*vn x*

 

Consider a small perturbation of the circular cylinder boundary, given by *rRh t*

( ) exp , (0)exp , 2 2 *k k <sup>d</sup> k t h t ik h t h*

in agreement with (Levich, 1962). According with (64), a small boundary perturbation of

Let's test our theory on an example of a large amplitude perturbation. We calculate the capillary

calculation based on (6.4) and the finite-element software ANSYS POLYFLOW (see Fig. 3 and Fig.4). These methods of calculation give us the same results with discrepancy about 1%.

Let's now consider a large amplitude perturbation in the shape of a cavity (Fig. 5). By symmetry, the pressure must be an even function with respect to 2 *x* , i.e.

2 2 1 2 2 2 <sup>1</sup> *x x a b*

*k <sup>h</sup> k ik h*

*t R* 

characteristic with *a R* and amplitude *H a* has a characteristic decay time ~

   

Introducing in *G* a complete system of orthonormal harmonic functions *<sup>k</sup> <sup>k</sup>* <sup>0</sup>

 

*p p*

onto the subspace of harmonic functions.

 

(59)

, we obtain from (56) the following

which

, ,

*a*

.

*dl* **<sup>y</sup> x xy** (60)

(61)

(62)

(63)

(64)

 

in two ways - using the numerical

It can be seen from (58) that

we see that p is the projection of *<sup>s</sup>*

*h R* . Then we have from (62)

expression for the pressure

Introducing the generalized function (simple layer)

obey the orthogonality condition *k n kn <sup>G</sup>*

In case of capillary forces the expression (37) takes the form

**6.2 Relaxation of a small perturbation of a circular cylinder** 

2

0

**6.3 The capillary relaxation of an ellipse** 

relaxation of boundary with initial shape

**6.4 The collapse of a cavity** 

1 2 12 *p x x* , , *p x x* .

Fig. 3. Computational domain used in finite-element calculation of ellipse relaxation.

Fig. 4. Relaxation from ellipse to a circle in finite-element calculation.

We introduce a space of two-variable harmonic functions which are even with respect to the second argument, and choose in it the complete system of functions in the form cos *<sup>n</sup> <sup>n</sup> r n* (*r* and are the polar coordinates in the 1 2 *x x*, plane). Since the width is small <sup>2</sup> 2( 1) *<sup>n</sup> mn m n <sup>g</sup> R n* . Then the complete system of orthogonal harmonic functions in this space is

$$
\Delta \Xi\_n = \sqrt{2(n+1)} \left(\frac{r}{R}\right)^n \cos(n\phi). \tag{65}
$$

Inserting (65) in (61) and summing the series yields

$$p = \sigma \left| \frac{1}{R} - \frac{H}{\pi R^2} - \frac{2}{\pi} \text{Re} \left( \frac{1}{R - z} - \frac{R - H}{R^2 - (R - H)R} \right) \right| \tag{66}$$

Planar Stokes Flows with Free Boundary 91

We presented a method to calculate two-dimensional Stokes flow with free boundary, based on the expansion of pressure in a complete system of harmonic functions. The theory forms the basis for strict analytical results and numerical approximations. Using this approach we analyse the collapse of bubbles and relaxation of boundary perturbation. The results obtained by this method are correlating well with numerical calculations performed using

The authors would like to express their sincere gratitude to Prof. V. Pukhnachov and Prof.

Antanovskii, L.K. (1988).Interface boundary dynamics under the action of capillary forces.

Berdichevsky, V. (2009). *Variational principles of continuum mechanics*, Vol. 1, Springer-Verlag,

Chivilikhin, S.A. (1992). Plane capillary flow of a viscous fluid with multiply connected boundary in the Stokes approximation, *Fluid Dynamics*, Vol. 27, No. 1, pp. 88 - 92. Frenkel, J. (1945). Viscous flow of crystalline bodies under the action of surface tension,

Dubrovin, B.A., Fomenko, A.T. & Novikov, S.P. (1984). *Modern Geometry. Methods and* 

Happel, S.J. & Brenner, H. (1965). *Low Reynolds Number Hydrodynamics*, Prentice-Hall,

Hopper, R.W. (1984). Coalescence of two equal cylinders: exact results for creeping

Ionesku, D.G. (1965). Theory of analytic functions and hydrodynamics, *in: Applications of the* 

Jeong, J.-T. & Moffatt, H.K*.* (1992). Free-surface cusps associated with flow at low Reynolds

Landau, L.D. & Lifshitz E.M. (1986). Theory of Elasticity. Course of Theoretical Physics.

Levich, V.G. (1962). *Physicochemical hydrodynamics*, Prentice-Hall, Englewood Cliffs, New

Muskeleshvili, N.I. (1966). *Some Fundamental Problems of the Mathematical Theory of Elasticity*

Pozrikidis, C. (1997). Numerical studies of singularity formation at free surfaces and fluid interfaces in two-dimensional Stokes flow, *J. Fluid Mech*. 331, pp. 145-167. Richardson, S. (2000). Plane Stokes flow with time-dependent free boundaries in which the fluid occupies a doubly-connected region, *Eur. J. Appl. Math.*, 11, pp. 249-269.

viscous plane flow driven by capillarity, *J. Am. Ceram. Soc.*, Vol. 67, No. 12, pp.

*Theory of Functions in the Mechanics of Continuous Media*, Vol. 2, Mechanics of

Quasisteady-state plane-parallel motion., *J. Appl. Mech. and Techn. Phys.* Vol. 29, No

**7. Conclusion** 

commercial FEM software.

3, pp. 396-399.

Berlin-Heidelberg

Englewood Cliffs

262 - 264.

Jersey

C. Pozrikidis for their attention to this research.

*Journal of. Physics*, Vol. 9, No. 5, pp. 385-391

Liquids and Gases [in Russian], Nauka, Moscow.

number, *J. Fluid Mech.*, Vol. 241, pp. 1-22.

Vol.7. Butterworth-Heinemann.

[in Russian], Nauka, Moscow

*Applications.* Part 1. Springer–Verlag

**8. Acknowledgment** 

**9. References** 

Fig. 5. Cavity perturbation.

whence, using (35), we have

$$\boldsymbol{\Phi} = \sigma \left[ \left( 1 - \frac{H}{\pi R} \right) \frac{z}{R} + \frac{2}{\pi} \ln \left( \frac{R^2 - (R - H)z}{(R - H)R} \right) \right]. \tag{67}$$

In spite of the logarithm, (67) is a single-valued analytical function in *G* , because the boundary perturbation constitutes a branch cut. If we insert (67) in (62), we find that the normal velocity of the cut edges 2 *V* (in the zero approximation with respect to the small parameter *<sup>H</sup>* ). The edges close up after a time . Although capillary forces generally tend to flatten the boundary perturbation, in this case they produce the opposite effect. Acting to reduce the length of the cut, the capillary forces generate a flow of scale *H* in the region. The velocities along 1 *<sup>x</sup>* and 2 *<sup>x</sup>* have the scales *H* and , respectively. If we equate the work of surface-tension force with the rate of energy dissipation by viscous forces, we find that 2 *<sup>H</sup>* <sup>2</sup> *H H H* or *<sup>H</sup>* ; this conforms to the rigorous result we obtained before.
