**5. Limitations for the motion of the boundary**

#### **5.1 The rate of change of region perimeter**

The strong limitation for the motion of the boundary is based on a general expression regarding the rate of change of perimeter *L* . To obtain this expression we use the fact (Dubrovin at al, 1984) that

$$\frac{d\left[\Gamma\right]}{dt} = \left[\upsilon\_{\alpha} n\_{\alpha} H d\Gamma\right],\tag{38}$$

where *n H x* is the mean curvature of the boundary. In the 2D case is the perimeter of the region, and in the 3D case is the area of the boundary. We introduce the operator of differentiation along the boundary *Dn n x x* . Then we can write (38) in the form

$$\frac{d\mathcal{L}}{dt} = \int \upsilon\_{\alpha} D\_{\alpha\beta} \mathfrak{n}\_{\beta} d\Gamma. \tag{39}$$

Using the identity

$$\int \mathcal{D}\_{a\beta} \Lambda d\Gamma = 0,\tag{40}$$

where is an arbitrary field which is continuous on the boundary, and also the equation of continuity (2) and the boundary conditions (3) we can write (39) in the final form

$$\frac{d\left|\Gamma\right|}{dt} = -\left[\frac{p + f\_{\alpha}n\_{\alpha}}{2\,\mu}d\Gamma\right].\tag{41}$$

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

> <sup>1</sup> , . <sup>2</sup> *v x G x*

 

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

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

<sup>1</sup> , . <sup>2</sup> *k k v e fd x*

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

The strong limitation for the motion of the boundary is based on a general expression regarding the rate of change of perimeter *L* . To obtain this expression we use the fact

, *<sup>d</sup> v n Hd*

 

. *dL v D nd*

 

 

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

continuity (2) and the boundary conditions (3) we can write (39) in the final form

*dt*

  *x x*

is an arbitrary field which is continuous on the boundary, and also the equation of

 

. <sup>2</sup>

 

   

(37)

(38)

. Then we can write (38) in the

(39)

(40)

(41)

is the area of the boundary. We introduce the operator

is the perimeter

 

*dt*

is the mean curvature of the boundary. In the 2D case

*dt*

> 

(36)

**5. Limitations for the motion of the boundary** 

of differentiation along the boundary *Dn n*

0, *D d*

**5.1 The rate of change of region perimeter** 

(Dubrovin at al, 1984) that

*n*

*x* 

of the region, and in the 3D case

where

form

where 

*H*

Using the identity

in the form

is the vortex part.

calculate the velocity on the boundary as

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

$$\frac{dL}{dt} = -\frac{1}{2\mu} \int (p + f\_{\alpha} n\_{\alpha}) d\Gamma. \tag{42}$$

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

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 boundary force has the form

$$f\_{\alpha} = -\sigma n\_{\alpha} \frac{\partial \mathfrak{n}\_{\beta}}{\partial \mathfrak{x}\_{\beta}} - p\_{k} n\_{\alpha'} \quad \mathbf{x} \in \Gamma\_{k'} \tag{43}$$

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

$$\frac{dL}{dt} = -\frac{1}{2\,\mu} \left[ \int p d\Gamma - \left( p\_0 L\_0 + p\_b L\_b \right) + 2\pi\sigma \left( m - 1 \right) \right],\tag{44}$$

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

Using (20) we obtain

$$
\int p dG = p\_0 S + \left(p\_0 - p\_b\right) S\_b + \frac{\sigma}{2} L\_\prime \tag{45}
$$

where *S* and *Sb* are the area of region and the total area of the bubbles. For *p*, , the expressions (19), (34), (37) give us

$$\begin{split} \int p^2 d\mathbf{G} &= \frac{\sigma}{2} \left( p\_0 L\_0 + p\_b L\_b + \int p d\boldsymbol{\gamma} \right) + p\_0^2 S + \left( p\_0^2 - p\_b^2 \right) S\_b - \\ &- \mu \left( p\_0 - p\_b \right) \frac{d S\_b}{d \mathbf{t}}. \end{split} \tag{46}$$

Using (44) - (46) and the inequality <sup>2</sup> <sup>2</sup> <sup>1</sup> *<sup>p</sup> dG pdG <sup>S</sup>* we obtain the differential inequality

$$\begin{split} &\mu \left[ \sigma \frac{d\mathcal{L}}{dt} + (p\_0 - p\_b) \frac{dS\_b}{dt} \right] \le \\ &\le -(p\_0 - p\_b) \left[ \sigma L\_b + (p\_0 - p\_b) S\_b \right] - \\ &- \frac{1}{S} \left[ (p\_0 - p\_b) S\_b + \frac{\sigma}{2} L \right]^2 - \pi \sigma^2 (m - 1). \end{split} \tag{47}$$

This expression gives us the possibility to obtain the strict limitations for the motion of the free boundary in some special cases.

Planar Stokes Flows with Free Boundary 87

The outer boundary of the region is a circle with a large radius *R* . The bubbles are localized

Because 0 *W* , this configuration exists without change of the number of bubbles during

0 2

, *n fn x x* 

, <sup>2</sup> *<sup>p</sup> dG d* 

, *G G p p*

1 1 , ,. <sup>2</sup> *<sup>G</sup> <sup>G</sup> f fdG f fd P SL S* 

<sup>2</sup> , *<sup>G</sup> <sup>G</sup> p pp*

 

> 

 

ln 1 0 0 . *<sup>b</sup>*

around the center of the circle. Using the expressions <sup>2</sup> , 2

2 2

*b b m m p p W W <sup>t</sup>*

*p p <sup>t</sup> L p pS*

0 0

*pp pp*

*dW*

*dt* 

*b bb p p* <sup>0</sup> *S* . Therefore, at 0 0 *<sup>b</sup> p p*

0

The expression (56) is valid for any harmonic function

**6.1 Calculation of pressure and velocity** 

In case of capillary forces action

and expression (19) takes the form

*b*

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

*p p m*

*arctg*

 

see that the inequality (47) in the limit *R* takes the form

**5.4 Bubbles in an infinite region** 

where *W L* 

the time

or

where

0 <sup>1</sup> <sup>1</sup> . <sup>1</sup> <sup>1</sup> *<sup>m</sup>*

*m LmL*

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

*p p W m*

 

(0) exp . *<sup>b</sup>*

<sup>0</sup>

*b b b*

(54)

(55)

(56)

(58)

*p* . Then we obtain

(57)

. Let's apply

(53)

 0

0

(52)

(50)

 *R S S L RL b b* , we can

(51)

*m LL*

#### **5.3 The influence of capillary forces only**

In this case the inequality (47) may be simplified:

$$\frac{d\mathcal{L}}{dt} \le -\frac{\sigma}{2\mu} \left[ \frac{\mathcal{L}^2}{2S} + 2\pi \left( m - 1 \right) \right].\tag{48}$$

where *m* is the number of bubbles. Let *L S* 2 be the asymptotic value of the perimeter and let 2 *t S* be the dimensionless time. Then, according with (48), , *L Lup* 

$$\begin{aligned} L\_{\text{up}} &= \frac{L\_0 + L\_{\infty} \text{th}\left(\tau\right)}{L\_{\infty} + L\_0 \text{th}\left(\tau\right)}, & m &= 0 \\ L\_{\text{up}} &= \frac{L\_0 L\_{\infty}}{L\_{\infty} + L\_0 \tau} \left(L\_{\infty} + L\_0 \tau\right), & m &= 1, \\ L\_{\text{up}} &= \frac{\sqrt{m - 1} \left(L\_0 - L\_{\infty} \sqrt{m - 1} \text{tg}\left(\sqrt{m - 1} \cdot \tau\right)\right)}{\sqrt{m - 1} + \lambda\_0 \text{tg}\left(\sqrt{m - 1} \cdot \tau\right)}, m &\ge 2 \end{aligned} \tag{49}$$

where *Lup* is the upper limitation for time dependence of the perimeter - see Fig.2. The perimeter of system *L* lies in the interval *L LL up* .

Fig. 2. The upper limitation for the time dependence of the perimeter for various number of bubbles *m* .

Therefore, if we have no bubbles in the region, the characteristic dimensionless time of relaxation of the boundary to the circle 0 1 . In case of one bubble *m* 1 , *L L up* at the time 1 0 1 *L L* . The system with this topology can exist in this time period only. The bubble must collapse or break into two bubbles in time \* 1 . In case of 2 *m* bubbles, such configuration will exist during the time

2

2 2 *dL L*

*dt S* 

> 

*L m*

0

0

0

*L L L LL m*

0

 

*m L L m tg m L m m tg m*

0

(48)

(49)

at

. In case of 2 *m* bubbles,

be the asymptotic value of the perimeter

, 2.

1 . In case of one bubble *m* 1 , *L L up*

 

.

2 1.

*m*

be the dimensionless time. Then, according with (48), , *L Lup*

, 0 ,

1 11

1 1

Fig. 2. The upper limitation for the time dependence of the perimeter for various number of

Therefore, if we have no bubbles in the region, the characteristic dimensionless time of

1 *L L* . The system with this topology can exist in this time period only.

The bubble must collapse or break into two bubbles in time \* 1

is the upper limitation for time dependence of the perimeter - see Fig.2.

, 1,

 

 

**5.3 The influence of capillary forces only**  In this case the inequality (47) may be simplified:

where *m* is the number of bubbles. Let *L S* 2

*up*

*up*

*up*

0

0

*L L*

The perimeter of system *L* lies in the interval *L LL up*

*L L th*

*L L th*

and let

2 *t*

where *Lup*

bubbles *m* .

relaxation of the boundary to the circle 0

such configuration will exist during the time

the time 1 0 

*S*

$$\tau \le \tau\_m = \frac{1}{\sqrt{m-1}} \operatorname{arctg} \left( \frac{\sqrt{m-1} \left( L\_0 - L\_\infty \right)}{L\_0 + (m-1)L\_\infty} \right). \tag{50}$$

### **5.4 Bubbles in an infinite region**

The outer boundary of the region is a circle with a large radius *R* . The bubbles are localized around the center of the circle. Using the expressions <sup>2</sup> , 2 *R S S L RL b b* , we can see that the inequality (47) in the limit *R* takes the form

$$
\mu \frac{d\mathcal{W}}{dt} \le -(p\_0 - p\_b)\mathcal{W} - \pi \sigma^2 m \,\,\,\,\,\tag{51}
$$

where *W L b bb p p* <sup>0</sup> *S* . Therefore, at 0 0 *<sup>b</sup> p p*

$$\mathcal{W} + \frac{\pi \sigma^2 m}{p\_0 - p\_b} \le \left( \mathcal{W}(0) + \frac{\pi \sigma^2 m}{p\_0 - p\_b} \right) \exp\left( -\frac{p\_0 - p\_b}{\mu} t \right). \tag{52}$$

Because 0 *W* , this configuration exists without change of the number of bubbles during the time

$$t \le \frac{\mu}{p\_0 - p\_b} \ln \left[ 1 + \frac{p\_0 - p\_b}{\pi \sigma^2 m} \left( \sigma \mathcal{L}\_b \left( 0 \right) + \left( p\_0 - p\_b \right) \mathcal{S}\_b \left( 0 \right) \right) \right]. \tag{53}$$
