**6. Motion of the boundary due to capillary forces**

#### **6.1 Calculation of pressure and velocity**

In case of capillary forces action

$$f\_{\alpha} = -\sigma n\_{\alpha} \frac{\partial n\_{\beta}}{\partial x\_{\beta}}, \quad x \in \Gamma \tag{54}$$

and expression (19) takes the form

$$
\int p\psi \, d\mathbf{G} = \frac{\sigma}{2} \int \psi \, d\Gamma \,\,,\tag{55}
$$

or

$$
\langle p\psi \rangle\_G = \langle p \rangle\_G \langle \psi \rangle\_{\Gamma'} \tag{56}
$$

where

$$
\left\langle f \right\rangle\_G = \frac{1}{S} \int f dG, \quad \left\langle f \right\rangle\_\Gamma = \frac{1}{L} \int f d\Gamma, \quad \left\langle P \right\rangle\_G = \frac{\sigma}{2S}.\tag{57}
$$

The expression (56) is valid for any harmonic function . Let's apply*p* . Then we obtain

$$
\left\langle p^2 \right\rangle\_G = \left\langle p \right\rangle\_G \left\langle p \right\rangle\_{\Gamma'} \tag{58}
$$

Planar Stokes Flows with Free Boundary 89

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

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

*n*

cos *<sup>n</sup> <sup>n</sup>*

is small <sup>2</sup>

 

functions in this space is

(*r* and

*<sup>n</sup> mn m n <sup>g</sup> R*

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

2( 1)

*n* 

*r n*

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

are the polar coordinates in the 1 2 *x x*, plane). Since the width

(65)

. Then the complete system of orthogonal harmonic

2( 1) cos . *<sup>n</sup>*

 2 2 1 21 Re , *<sup>H</sup> R H*

*<sup>p</sup> <sup>R</sup> <sup>R</sup> R z R R HR*

*r n n R*

(66)

It can be seen from (58) that

$$
\langle p \rangle\_{\varGamma} \ge \langle p \rangle\_{\varGamma}.\tag{59}
$$

Introducing the generalized function (simple layer)

$$\delta\_s(\mathbf{x}) = \int \delta(\mathbf{x} - \mathbf{y}) d\mathbf{l}\_{\mathbf{y}\prime} \tag{60}$$

we see that p is the projection of *<sup>s</sup>* onto the subspace of harmonic functions. Introducing in *G* a complete system of orthonormal harmonic functions *<sup>k</sup> <sup>k</sup>* <sup>0</sup> which obey the orthogonality condition *k n kn <sup>G</sup>* , we obtain from (56) the following expression for the pressure

$$\{p = \{p\}\_{\mathcal{G}} \sum\_{k=0}^{\infty} \Xi\_k \{\Xi\_k\} \, \, . \tag{61}$$

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

$$
v\_{\alpha} = \frac{1}{2\mu} (\sigma n\_{\alpha} - \Phi\_{\alpha}), \quad \mathbf{x} \in \Gamma. \tag{62}$$
