**2. General equations**

#### **2.1 The quasi-stationary Stokes approximation**

The equations of viscous fluid motion in the quasi-stationary Stokes approximation due to arbitrary surface force *f* and the continuity equation in the region <sup>2</sup> *G R* with boundary have the form

$$\frac{\partial p\_{\alpha\beta}}{\partial \mathbf{x}\_{\beta}} = \mathbf{0} \,, \tag{1}$$

$$\frac{\partial \mathbf{v}\_{\beta}}{\partial \mathbf{x}\_{\beta}} = \mathbf{0} \,, \tag{2}$$

where *<sup>v</sup> <sup>v</sup> p p x x* is the Newtonian stress tensor; *v*are the components

of the velocity; *p* is the pressure; is the coefficient of the dynamical viscosity, which is assumed to be constant. The indices , take the values 1, 2. Summation over repeated indices is expected. The boundary conditions have the form

$$
\mathfrak{p}\_{\alpha\beta}\mathfrak{n}\_{\beta} = f\_{\alpha\prime} \quad \mathfrak{x} \in \Gamma \tag{3}
$$

where *n* and *f* are the components of the vector of outer normal to the boundary and the surface force. Let <sup>0</sup> be the outer boundary of the region; ( 1,2,..., ) *<sup>k</sup> k m* - the inner *m*

boundaries (boundaries of bubbles); 0 *k k* - see Fig.1.

Fig. 1. Region *G* with multiply connected boundary 

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

The equations of viscous fluid motion in the quasi-stationary Stokes approximation due to

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

0

is the Newtonian stress tensor; *v*

<sup>0</sup> be the outer boundary of the region; ( 1,2,..., ) *<sup>k</sup>*


are the components of the vector of outer normal to the boundary and the

*v x* 

and the continuity equation in the region <sup>2</sup> *G R* with boundary

, (1)

, (2)

is the coefficient of the dynamical viscosity, which is

take the values 1, 2. Summation over repeated

(3)

*k m* - the inner

are the components

with given number of bubbles was predicted.

**2.1 The quasi-stationary Stokes approximation** 

*x x* 

indices is expected. The boundary conditions have the form

 

,

 **x**

0

*k* 

*k*

*m*

**2. General equations** 

arbitrary surface force *f*

where *<sup>v</sup> <sup>v</sup>*

 

of the velocity; *p* is the pressure;

assumed to be constant. The indices

boundaries (boundaries of bubbles);

 

, *pn f*

Fig. 1. Region *G* with multiply connected boundary

*p p*

where *n*

 and *f*

surface force. Let

have the form

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 boundary:

$$\mathbf{V}\_n = \mathbf{v}\_{\beta} \mathbf{v}\_{\beta}, \quad \mathbf{x} \in \Gamma \tag{4}$$

In case of a volume force *F*acting on G, the equation of motion takes the form

$$\frac{\partial p\_{\alpha\beta}}{\partial \mathbf{x}\_{\beta}} = -\mathbf{F}\_{\alpha} \tag{5}$$

If the volume force is potential *<sup>U</sup> <sup>F</sup> x* one can renormalize the pressure *<sup>p</sup> p U* and present (3), (5) in the form

$$\frac{\partial p\_{\alpha\beta}}{\partial x\_{\beta}} = 0 \tag{6}$$

$$
\eta\_{\alpha\beta}\eta\_{\beta} = f'\_{\alpha}, \quad \mathbf{x} \in \Gamma \tag{7}
$$

where *f f Un* is the renormalized surface force.

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

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

$$
v\_{\alpha} \to v\_{\alpha} + V\_{\alpha} + \mathfrak{e}\_{\alpha\beta} \mathfrak{x}\_{\beta} o\tag{8}$$

where *V* and are constants, *e* is the unit antisymmetric tensor. Therefore, for this approximation the total linear momentum and the total angular momentum are indefinite. These values should be determined from the initial conditions.

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

The Navier-Stokes equations

$$
\rho \left( \frac{\partial v\_{\alpha}}{\partial t} + v\_{\beta} \frac{\partial v\_{\alpha}}{\partial \mathbf{x}\_{\beta}} \right) = \frac{\partial p\_{\alpha\beta}}{\partial \mathbf{x}\_{\beta}} + F\_{\alpha\prime} \tag{9}
$$

where is the density of liquid, lead to the quasi-stationary Stokes equations (5) if the convective and non-stationary terms in (9) can be neglected. The neglection of the convective term leads to the requirement of a small Reynolds number Re *VL* , where *V* is the characteristic velocity, *L* is the spatial scale of the region *G* , and is the kinematic viscosity. The non-stationary term in the equation (9) can be omitted if during the velocity field relaxation time <sup>2</sup> *T L* the shape of the boundary changes insignificantly, namely *VT L* which again leads to the condition Re 1 . The change of the volume force *F* and the surface force *f*during the time *T* should also be small:

Planar Stokes Flows with Free Boundary 81

. In the new system the surface force is the same as in the initial system

> 

 

be smooth fields in the region *G* related by

*x x* 

, *<sup>d</sup> I M dt* 

<sup>2</sup> *F F ex ev x* 2 ,

and the total moment of force is equal zero: 0 *M* . In case of a small Reynolds number, the

is small compared with the viscous force.

eliminate them using the noninertial reference system with the rigid-body motion due to the

 

> 

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

1 the expression (18) gives us *x*

1 <sup>2</sup> *pdG f x d* 

 

 

is the moment of inertia of our system. In the new system the surface

 

2 .

 

The total moment of force in the new system stays unchanged: *M M* .To eliminate the total moment of force *M* we switch from the system *K* to the rotating reference system

 

is the coordinate of the center of mass in the initial system *<sup>K</sup>* ,

and total force is equal to zero:

(15)

(17)

and, according with (19),

using a noninertial center-of-mass reference

(16)

, but the volume force transforms to:

and total moment of force *M* not equal to zero we can

(18)

(19)

(20)

 

, integrating over *G* , and using (2), (3), (18), we

where <sup>1</sup> *x x dG S*

 

, but the volume force transforms to *F F*

, *v v ex*

is the angular velocity of the rigid-body rotation

. So, we eliminated the total force

force is the same as in the initial system *f f*

 

*d x*

0

system *K* .

*dt* 

 where *I x x dG* 

Coriolis force 2 *e v*

 

So in case of the total force

force and moment of force.

**3. Pressure calculation** 

In the special case when

Multiplying the equation of motion (1) by

*v*

*K* :

where 

Let and 

obtain

*f f* 

$$\frac{\delta F\_a}{\delta t} T << F\_{a\prime} \quad \frac{\delta f\_a}{\delta t} T << f\_{a\prime} \tag{10}$$

For the forces determined by the region shape (like capillary force or centrifugal force) the conditions (10) lead to Re 1 again.

The neglection of the non-stationary term is a singular perturbation of the motion equation in respect of the time variable. It leads to the formation of a time boundary layer of duration *T* , during which the initial velocity field relaxates to a quasi-steady state. The condition of a small deformation of the region during this time interval 0 0 *VT L* is ensured by the requirement of a small Reynolds number <sup>0</sup> Re constructed from the characteristic initial velocity <sup>0</sup> *V* and the initial region scale <sup>0</sup>*L* .

Let's integrate the motion equation (5) over the region *G* and use the boundary condition (3). As a result we obtain the condition

$$
\Delta \Phi\_{\alpha} = \int F\_{\alpha} \, d\mathcal{G} + \int f\_{\alpha} d\Gamma = 0. \tag{11}
$$

The equations of viscous fluid motion in the quasi-stationary Stokes approximation (5) have the form of local equilibrium conditions. Correspondingly, the total force which acts on the system should be zero. The same way, using (5) and (3) one can obtain the condition

$$M = \int e\_{a\beta} \mathbf{x}\_a \mathbf{F}\_{\beta} \, d\mathbf{G} + \int e\_{a\beta} \mathbf{x}\_a f\_{\beta} d\Gamma = 0. \tag{12}$$

where *e* is the unit antisymmetric tensor. Therefore, the total moment of force *M* acting on the system should be zero.

#### **2.4 The Stokes equations in the special noninertial system of reference**

Conditions (11) and (12) are the classical conditions of solubility of system (2), (5) with boundary conditions (3). Let's show that these conditions are too restrictive. For example, for a small drop of high viscous liquid falling in the gravitation field the total force is not zero, but equal to the weight of the drop. Therefore, we cannot use the quasi-stationary Stokes approximation to describe the evolution of the drop's shape due to capillary forces. But in a noninertial system of reference which falls together with the drop with the same acceleration, the total force is equal to zero.

In a general case, the total force and total moment of force *M* acting on the system are not equal to zero. The Newton's second law for translational motion has the form

$$
\rho S \frac{d \left< v\_{\alpha} \right>}{dt} = \Phi\_{\alpha'} \tag{13}
$$

where *<sup>S</sup>* is the area of the region, <sup>1</sup> *v v dG S* is the average velocity of the system, and

 is the total force. Let's choose the center-of-mass reference system *K* instead of the initial laboratory system *K* . The velocity and coordinate transformations have the form

$$
\sigma\_a' = \upsilon\_a - \langle \upsilon\_a \rangle, \qquad \mathfrak{x}\_a' = \mathfrak{x}\_a - \langle \mathfrak{x}\_a \rangle,\tag{14}
$$

 

*<sup>F</sup> <sup>f</sup> TF T <sup>f</sup> t t*

For the forces determined by the region shape (like capillary force or centrifugal force) the

The neglection of the non-stationary term is a singular perturbation of the motion equation in respect of the time variable. It leads to the formation of a time boundary layer of duration *T* , during which the initial velocity field relaxates to a quasi-steady state. The condition of a small deformation of the region during this time interval 0 0 *VT L* is ensured by the requirement of a small Reynolds number <sup>0</sup> Re constructed from the characteristic initial

Let's integrate the motion equation (5) over the region *G* and use the boundary condition

The equations of viscous fluid motion in the quasi-stationary Stokes approximation (5)

acts on the system should be zero. The same way, using (5) and (3) one can obtain the

*M e x F dG e x f d* 0.

Conditions (11) and (12) are the classical conditions of solubility of system (2), (5) with boundary conditions (3). Let's show that these conditions are too restrictive. For example, for a small drop of high viscous liquid falling in the gravitation field the total force is not zero, but equal to the weight of the drop. Therefore, we cannot use the quasi-stationary Stokes approximation to describe the evolution of the drop's shape due to capillary forces. But in a noninertial system of reference which falls together with the drop with the same

, *d v*

have the form of local equilibrium conditions. Correspondingly, the total force

*F dG f d* 0.

 

 

is the unit antisymmetric tensor. Therefore, the total moment of force *M* acting

 

 

**2.4 The Stokes equations in the special noninertial system of reference** 

not equal to zero. The Newton's second law for translational motion has the form

*S dt* 

*S*

laboratory system *K* . The velocity and coordinate transformations have the form

   

is the total force. Let's choose the center-of-mass reference system *K* instead of the initial

*vv v xx x* , ,

 

conditions (10) lead to Re 1 again.

velocity <sup>0</sup> *V* and the initial region scale <sup>0</sup>*L* .

(3). As a result we obtain the condition

condition

where *e*

 on the system should be zero.

acceleration, the total force is equal to zero.

where *<sup>S</sup>* is the area of the region, <sup>1</sup> *v v dG*

In a general case, the total force

, , *a a*

(10)

(11)

(12)

and total moment of force *M* acting on the system are

is the average velocity of the system, and

 

(14)

(13)

 which

 where <sup>1</sup> *x x dG S* is the coordinate of the center of mass in the initial system *<sup>K</sup>* ,

*d x v dt* . In the new system the surface force is the same as in the initial system *f f* , but the volume force transforms to *F F* and total force is equal to zero: 0 . So, we eliminated the total force using a noninertial center-of-mass reference system *K* .

The total moment of force in the new system stays unchanged: *M M* .To eliminate the total moment of force *M* we switch from the system *K* to the rotating reference system *K* :

$$
v\_{\alpha}"\to v\_{\alpha}' - e\_{\alpha\beta} \mathbf{x}\_{\beta}' \Omega,\tag{15}$$

where is the angular velocity of the rigid-body rotation

$$I\frac{d\varOmega}{dt} = \varOmega\_{\varomega} \tag{16}$$

 where *I x x dG* is the moment of inertia of our system. In the new system the surface force is the same as in the initial system *f f* , but the volume force transforms to:

$$F\_{\alpha}^{\prime} \rightarrow F\_{\alpha}^{\prime} + \rho \left( e\_{a\beta} \mathbf{x}\_{\beta}^{\prime} \stackrel{\bullet}{\Omega} + 2e\_{a\beta} \mathbf{v}\_{\beta}^{\prime} \Omega + \Omega^2 \mathbf{x}\_{\alpha}^{\prime} \right) \tag{17}$$

and the total moment of force is equal zero: 0 *M* . In case of a small Reynolds number, the Coriolis force 2 *e v* is small compared with the viscous force.

So in case of the total force and total moment of force *M* not equal to zero we can eliminate them using the noninertial reference system with the rigid-body motion due to the force and moment of force.
