**3. Pressure calculation**

Let and be smooth fields in the region *G* related by

$$
\frac{\partial \mathcal{X}\_{\alpha}}{\partial \mathbf{x}\_{\beta}} + \frac{\partial \mathcal{X}\_{\beta}}{\partial \mathbf{x}\_{\alpha}} = 2\mu \boldsymbol{\delta}\_{\alpha\beta}. \tag{18}
$$

Multiplying the equation of motion (1) by , integrating over *G* , and using (2), (3), (18), we obtain

$$\int p\varphi \, d\mathbf{G} = -\frac{1}{2} \int f\_a \chi\_a \, d\Gamma \tag{19}$$

In the special case when 1 the expression (18) gives us *x* and, according with (19),

$$\int p d\mathbf{G} = -\frac{1}{2} \int f\_a \mathbf{x}\_a d\Gamma \tag{20}$$

Planar Stokes Flows with Free Boundary 83

. *k k*

(27)

According with (1), (2) the pressure *p* is a harmonic function. We present it in the form

*k p p*

<sup>1</sup> , 0,1,... <sup>2</sup> *kn k <sup>n</sup>*

2 2 *<sup>p</sup>* , *xx xx*

 

 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

, . *k k e fd x*

 

2 . *p*

, *n n*

 

   

*d* 2 ,, *dv d x G*

1 2

 

*x x* 

, , *v v d i pi*

 

*n p*

 

 

 

2

 

 

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

*<sup>e</sup> f x* , , *x x*

(28)

(29)

(30)

(31)

2 1

(34)

(35)

*x x*

(32)

(33)

,

 

 

 *dG p f d n* 

Using the expression (19) we obtain the algebraic system for coefficients *<sup>k</sup> p* :

*k* 

The stress tensor, expressed in terms of the Airy function

 

*<sup>k</sup>* from a fixed point to an arbitrary one we obtain

*x*

*x*

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

1 2

 

is a harmonic function conjugate to *p* ,

**4. Velocity calculation** 

where

where

Therefore

boundary

 

see (Landau & Lifshitz, 1986 ). In a general case, according with (18), is an arbitrary harmonic function and 1 2 *i* is the analytical function associated with as

$$d\,\mathcal{X} = (\psi + i\alpha)dz\tag{21}$$

where is a harmonic function conjugate to .

The expressions (18) and (19) are basic in our theory. There is also an alternative way to derive them. The equations of motion (1), continuity (2) and the boundary conditions (3) can be obtained from the variation principle (Berdichevsky, 2009).

$$\delta \left[ \frac{1}{4\mu} \int (p\_{\alpha\beta} p\_{\alpha\beta} - 2p^2) d\mathcal{G} - \int f\_{\alpha} v\_{\alpha} d\Gamma \right] = 0 \tag{22}$$

or

$$\frac{1}{2\mu} \int (p\_{a\beta} \delta p\_{a\beta} - 2p \delta p) d\mathcal{G} - \int f\_a \delta v\_a d\Gamma = 0 \tag{23}$$

Since (23) is valid for arbitrary variations of pressure *p* and velocity *v* we choose them such that *p*is left unchanged:

$$
\delta p\_{\alpha\beta} = -\delta p \cdot \delta\_{\alpha\beta} + \mu \left( \frac{\partial \delta v\_{\alpha}}{\partial \mathbf{x}\_{\beta}} + \frac{\partial \delta v\_{\beta}}{\partial \mathbf{x}\_{\alpha}} \right) = 0. \tag{24}
$$

In this case (23) gives us

$$\frac{1}{\mu} \int p \delta p d\mathcal{G} + \int f\_a \delta v\_a d\Gamma = 0. \tag{25}$$

We introduce the one-parameter family of variations , <sup>2</sup> *v p* . Then (24) and

(25) take the form (18) and (19). Suppose *<sup>N</sup> x R* . Then it follows from (18) that

$$(N-2)\frac{\partial^2 \boldsymbol{\nu}}{\partial \mathbf{x}\_{\alpha} \partial \mathbf{x}\_{\beta}} = 0.\tag{26}$$

Therefore, in the three-dimensional case is a linear function. Only in the two-dimensional case can be an arbitrary harmonic function. Formulating in terms of (3.5), only in the twodimensional space there exists a non-trivial system of pressure and velocity variations providing zero stress tensor variation.

The complete set of analytical functions *<sup>k</sup>* in the region *G* with the multiply connected boundary consists of functions of the form , *<sup>k</sup> <sup>o</sup> z zz k m* , where *<sup>o</sup> zm* are fixed points, each situated in one bubble. The complete set of harmonic functions *<sup>k</sup>* can be obtained in the form of Re *<sup>k</sup>* and Im *<sup>k</sup>* .

*d i dz* 

 . The expressions (18) and (19) are basic in our theory. There is also an alternative way to derive them. The equations of motion (1), continuity (2) and the boundary conditions (3) can

> <sup>1</sup> <sup>2</sup> 2 0 <sup>4</sup> *p p p dG f v d*

 <sup>1</sup> 2 0 <sup>2</sup> *p p p p dG f v d*

 

consists of functions of the form , *<sup>k</sup> <sup>o</sup> z zz k m*

situated in one bubble. The complete set of harmonic functions

 

*i* is the analytical function associated with

 

 

*x x* 

<sup>1</sup> *p pdG f v d* 0. 

 

2 *N* 2 0. *x x* 

 can be an arbitrary harmonic function. Formulating in terms of (3.5), only in the twodimensional space there exists a non-trivial system of pressure and velocity variations

   

0. *<sup>v</sup> <sup>v</sup>*

 

  

(22)

(23)

*p* and velocity *v*

(25)

 

(26)

is a linear function. Only in the two-dimensional

in the region *G* with the multiply connected

, where *<sup>o</sup> zm* are fixed points, each

*<sup>k</sup>* can be obtained in the

 

we choose them

(24)

. Then (24) and

(21)

as

is an arbitrary

see (Landau & Lifshitz, 1986 ). In a general case, according with (18),

 

harmonic function and 1 2

where 

or

such that *p*

case 

boundary

form of Re *<sup>k</sup>*

In this case (23) gives us

(25) take the form (18) and (19).

 

is a harmonic function conjugate to

Since (23) is valid for arbitrary variations of pressure

is left unchanged:

Suppose *<sup>N</sup> x R* . Then it follows from (18) that

Therefore, in the three-dimensional case

providing zero stress tensor variation.

and Im *<sup>k</sup>*

The complete set of analytical functions *<sup>k</sup>*

.

be obtained from the variation principle (Berdichevsky, 2009).

 

*p p*

 

We introduce the one-parameter family of variations , <sup>2</sup> *v p*

 According with (1), (2) the pressure *p* is a harmonic function. We present it in the form

$$p = \sum\_{k} p\_{k} \nu\_{k}. \tag{27}$$

Using the expression (19) we obtain the algebraic system for coefficients *<sup>k</sup> p* :

$$\sum\_{k} \left( \int \boldsymbol{\nu}\_{k} \boldsymbol{\nu}\_{n} d\mathbf{G} \right) \boldsymbol{p}\_{k} = -\frac{1}{2} \int \boldsymbol{f}\_{\alpha} \boldsymbol{\chi}\_{\alpha n} d\boldsymbol{\Gamma} \,, \quad n = 0, 1, \ldots \tag{28}$$
