**4. Velocity calculation**

The stress tensor, expressed in terms of the Airy function,

$$p\_{\alpha\beta} = \frac{\hat{\alpha}^2 \varphi}{\hat{\alpha} \mathbf{x}\_{\alpha} \hat{\alpha} \mathbf{x}\_{\beta}} - \frac{\hat{\alpha}^2 \varphi}{\hat{\alpha} \mathbf{x}\_{\gamma} \hat{\alpha} \mathbf{x}\_{\gamma}} \delta\_{\alpha\beta\prime} \tag{29}$$

satisfies the equation of motion (1) identically. The boundary conditions (3) take the form

$$e\_{\alpha\beta}\tau\_{\gamma}\frac{\partial^2\varphi}{\partial x\_{\beta}\partial x\_{\gamma}} = -f\_{\alpha\prime} \quad \text{x} \in \Gamma\_{\prime} \tag{30}$$

where are the components of the unit tangential vector to the boundary, its direction being matched to the direction of circulation. Integrating (30) along the component boundary *<sup>k</sup>* from a fixed point to an arbitrary one we obtain

$$\frac{\partial \mathcal{Q}}{\partial \mathbf{x}\_a} = e\_{a\theta} \int f\_a d\Gamma\_{k\prime} \quad \mathbf{x} \in \Gamma\_k. \tag{31}$$

Using (1), (29) and the explicit form of the stress tensor, we get

$$d\left(\frac{\partial\varphi}{\partial\mathbf{x}\_{\alpha}}\right) = 2\,\mu dv\_{\alpha} + d\Phi\_{\alpha\prime} \quad \text{x} \in \mathcal{G}\_{\prime} \tag{32}$$

where

$$d\left(\varPhi\_1 + i\varPhi\_2\right) = \left(p + i\varOmega\right), \quad \varOmega = \mu \left(\frac{\partial v\_1}{\partial \mathbf{x}\_2} - \frac{\partial v\_2}{\partial \mathbf{x}\_1}\right), \tag{33}$$

is a harmonic function conjugate to *p* ,

$$\frac{\partial \Phi\_{\alpha}}{\partial \mathbf{x}\_{\beta}} + \frac{\partial \Phi\_{\beta}}{\partial \mathbf{x}\_{\alpha}} = 2p \delta\_{\alpha \beta}. \tag{34}$$

Therefore

$$\Phi\_{\alpha} = \sum\_{n} p\_{n} \chi\_{\alpha n'} \tag{35}$$

( 

where 

For *p*, 

 

free boundary in some special cases.

boundary force has the form

bubbles correspondingly. Using (20) we obtain

Planar Stokes Flows with Free Boundary 85

This expression is valid for any flow of incompressible Newtonian liquid (without Stokes approximation), generally speaking, with variable viscosity. We will use it for a 2D flow

*<sup>p</sup> fn d dt*

Let's take into account the capillary forces on the boundary, the external pressure 0 *p* and the pressure inside of the bubbles , 1,2,..., *k b p pk m* , equal in every bubble. Then the

*n f n pn x x* 

*dL pd p L p L m dt* 

where *L*0 and *Lb* are the perimeter of external boundary and the total perimeter of the

0 0 , <sup>2</sup> *b b pdG p S p p S L*

*b b b b*

2 2 2 2 0 0 0 0

*<sup>p</sup> dG pdG <sup>S</sup>*

0 0

*b b*

0

*dL dS p p dt dt*

*b*

*p p L p pS*

*p pS L m <sup>S</sup>*

<sup>1</sup> 1 . 2

This expression gives us the possibility to obtain the strict limitations for the motion of the

*b b bb*

2

*b*

2

*p dG p L p L pd p S p p S*

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

 

, , *k k*

 

(45)

we obtain the differential inequality

(46)

(47)

(44)

(43)

 

0 0 <sup>1</sup> 2 1, <sup>2</sup> *b b*

(42)

=L is the perimeter of region), in case of constant viscosity:

*dL*

**5.2 The dynamics of bubbles due to capillarity and air pressure** 

where *S* and *Sb* are the area of region and the total area of the bubbles.

.

*b*

0

*dS p p dt*

2

*b*

0

Using (44) - (46) and the inequality <sup>2</sup> <sup>2</sup> <sup>1</sup>

 

, the expressions (19), (34), (37) give us

is the coefficient of surface tension. Using (42), (43) we get

where *<sup>k</sup> p* are the coefficients of the pressure expansion (27). These coefficients are the solution of the system (28). According with (32) the velocity in the region *G* can be presented in the form

$$
v\_{\alpha} = \frac{1}{2\,\mu} \left( \frac{\partial \rho}{\partial \mathbf{x}\_{\alpha}} - \Phi\_{\alpha} \right) \quad \text{x} \in \mathcal{G}.\tag{36}
$$

The first term in the right-hand part of (36) is the potential part of velocity; the second term is the vortex part.

The gradient of the Airy function on the boundary was calculated in (31). Then we can calculate the velocity on the boundary as

$$
\sigma v\_{\alpha} = \frac{1}{2\mu} \left( e\_{a\beta} \int f\_a d\Gamma\_k - \Phi\_a \right), \quad \mathbf{x} \in \Gamma\_k. \tag{37}
$$

The expression (37) gives us the explicit presentation of the velocity on the boundary.
