**1. Introduction**

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The quasi-stationary Stokes approximation (Frenkel, 1945; Happel & Brenner, 1965) is used to describe viscous flows with small Reynolds numbers. Two-dimensional Stokes flow with free boundary attracted the attention of many researches. In particular, an analogy is drawn (Ionesku, 1965) between the equations of the theory of elasticity (Muskeleshvili, 1966) and the equations of hydrodynamics in the Stokes approximation. This idea allowed (Antanovskii, 1988) to study the relaxation of a simply connected cylinder under the effect of capillary forces. Hopper (1984) proposed to describe the dynamics of the free boundary through a family of conformal mappings. This approach was later used in (Jeong & Moffatt, 1992; Tanveer & Vasconcelos, 1994) for analysis of free-surface cusps and bubble breakup.

We have developed a method of flow calculation, which is based on the expansion of pressure in a complete system of harmonic functions. The structure of this system depends on the topology of the region. Using the pressure distribution, we calculate the velocity on the boundary and investigate the motion of the boundary. In case of capillary forces the pressure is the projection of a generalized function with the carrier on the boundary on the subspace of harmonic functions (Chivilikhin, 1992).

We show that in the 2D case there exists a non-trivial variation of pressure and velocity which keeps the Reynolds stress tensor unchanged. The correspondent variations of pressure give us the basis for pressure presentation in form of a series. Using this fact and the variation formulation of the Stokes problem we obtain a system of equations for the coefficients of this series. The variations of velocity give us the basis for the vortical part of velocity presentation in the form of a serial expansion with the same coefficients as for the pressure series.

We obtain the potential part of velocity on the boundary directly from the boundary conditions - known external stress applied to the boundary. After calculating velocity on the boundary with given shape we calculate the boundary deformation during a small time step.

Based on this theory we have developed a method for calculation of the planar Stokes flows driven by arbitrary surface forces and potential volume forces. We can apply this method for investigating boundary deformation due to capillary forces, external pressure, centrifugal forces, etc.

Planar Stokes Flows with Free Boundary 79

The free boundary evolution is determined from the condition of equality of the normal velocity *Vn* of the boundary and the normal component of the velocity of the fluid at the

> *<sup>p</sup> <sup>F</sup> x*

> > <sup>0</sup> *<sup>p</sup> x*

Let's point out a specificity of the quasi-stationary Stokes approximation (1), (2). This system

*v v V ex*

approximation the total linear momentum and the total angular momentum are indefinite.

, *v v <sup>p</sup> v F t xx*

is the density of liquid, lead to the quasi-stationary Stokes equations (5) if the

  the shape of the boundary changes insignificantly, namely

 

 

convective and non-stationary terms in (9) can be neglected. The neglection of the

viscosity. The non-stationary term in the equation (9) can be omitted if during the velocity

*VT L* which again leads to the condition Re 1 . The change of the volume force *F*

 

**2.3 The conditions of the quasi-stationary Stokes approximation applicability** 

 *x*

 **x**

is the renormalized surface force.

**2.2 The transformational invariance of the Stokes equations** 

These values should be determined from the initial conditions.

convective term leads to the requirement of a small Reynolds number Re *VL*

during the time *T* should also be small:

is the characteristic velocity, *L* is the spatial scale of the region *G* , and

acting on G, the equation of motion takes the form

(4)

(7)

(8)

(9)

, where *V*

and

is the kinematic

(5)

(6)

is the unit antisymmetric tensor. Therefore, for this

one can renormalize the pressure *<sup>p</sup> p U* and

 **x**

boundary:

where *f*

where *V*

where

the surface force *f*

 and 

The Navier-Stokes equations

field relaxation time <sup>2</sup> *T L*

In case of a volume force *F*

present (3), (5) in the form

 *f Un* 

 

is invariant under the transformation

, *V vn <sup>n</sup>*

If the volume force is potential *<sup>U</sup> <sup>F</sup>*

, *pn f*

are constants, *e*

Taking into account the capillary forces and external pressure, the strict limitations for motion of the free boundary are obtained. In particular, the lifetime of the configurations with given number of bubbles was predicted.
