Planar Stokes Flows with Free Boundary

The quasi-stationary Stokes approximation (Frenkel, 1945; Happel & Brenner, 1965) is used to describe viscous flows with small Reynolds numbers. Two-dimensional Stokes flow with free boundary attracted the attention of many researches. In particular, an analogy is drawn (Ionesku, 1965) between the equations of the theory of elasticity (Muskeleshvili, 1966) and the equations of hydrodynamics in the Stokes approximation. This idea allowed (Antanovskii, 1988) to study the relaxation of a simply connected cylinder under the effect of capillary forces. Hopper (1984) proposed to describe the dynamics of the free boundary through a family of conformal mappings. This approach was later used in (Jeong & Moffatt, 1992; Tanveer & Vasconcelos, 1994) for analysis of free-surface cusps and bubble breakup. We have developed a method of flow calculation, which is based on the expansion of pressure in a complete system of harmonic functions. The structure of this system depends on the topology of the region. Using the pressure distribution, we calculate the velocity on the boundary and investigate the motion of the boundary. In case of capillary forces the pressure is the projection of a generalized function with the carrier on the boundary on the subspace of harmonic functions (Chivilikhin, 1992). We show that in the 2D case there exists a non-trivial variation of pressure and velocity which keeps the Reynolds stress tensor unchanged. The correspondent variations of pressure give us the basis for pressure presentation in form of a series. Using this fact and the variation formulation of the Stokes problem we obtain a system of equations for the coefficients of this series. The variations of velocity give us the basis for the vortical part of velocity presentation in the form of a serial expansion with the same coefficients as for the pressure series. We obtain the potential part of velocity on the boundary directly from the boundary conditions known external stress applied to the boundary. After calculating velocity on the boundary with given shape we calculate the boundary deformation during a small time step. Based on this theory we have developed a method for calculation of the planar Stokes flows driven by arbitrary surface forces and potential volume forces. We can apply this method for investigating boundary deformation due to capillary forces, external pressure, centrifugal forces, etc.


Introduction
The quasi-stationary Stokes approximation (Frenkel, 1945;Happel & Brenner, 1965) is used to describe viscous flows with small Reynolds numbers.Two-dimensional Stokes flow with free boundary attracted the attention of many researches.In particular, an analogy is drawn (Ionesku, 1965) between the equations of the theory of elasticity (Muskeleshvili, 1966) and the equations of hydrodynamics in the Stokes approximation.This idea allowed (Antanovskii, 1988) to study the relaxation of a simply connected cylinder under the effect of capillary forces.Hopper (1984) proposed to describe the dynamics of the free boundary through a family of conformal mappings.This approach was later used in (Jeong & Moffatt, 1992;Tanveer & Vasconcelos, 1994) for analysis of free-surface cusps and bubble breakup.We have developed a method of flow calculation, which is based on the expansion of pressure in a complete system of harmonic functions.The structure of this system depends on the topology of the region.Using the pressure distribution, we calculate the velocity on the boundary and investigate the motion of the boundary.In case of capillary forces the pressure is the projection of a generalized function with the carrier on the boundary on the subspace of harmonic functions (Chivilikhin, 1992).We show that in the 2D case there exists a non-trivial variation of pressure and velocity which keeps the Reynolds stress tensor unchanged.The correspondent variations of pressure give us the basis for pressure presentation in form of a series.Using this fact and the variation formulation of the Stokes problem we obtain a system of equations for the coefficients of this series.The variations of velocity give us the basis for the vortical part of velocity presentation in the form of a serial expansion with the same coefficients as for the pressure series.We obtain the potential part of velocity on the boundary directly from the boundary conditions -known external stress applied to the boundary.After calculating velocity on the boundary with given shape we calculate the boundary deformation during a small time step.Based on this theory we have developed a method for calculation of the planar Stokes flows driven by arbitrary surface forces and potential volume forces.We can apply this method for investigating boundary deformation due to capillary forces, external pressure, centrifugal forces, etc.
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 with given number of bubbles was predicted.The free boundary evolution is determined from the condition of equality of the normal velocity n V of the boundary and the normal component of the velocity of the fluid at the boundary:

General equations
In case of a volume force F  acting on G, the equation of motion takes the form If the volume force is potential one can renormalize the pressure p pU  and present (3), ( 5) in the form where f fU n   is the renormalized surface force.

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 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.

The conditions of the quasi-stationary Stokes approximation applicability
The Navier-Stokes equations , p vv vF txx 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 V and the initial region scale 0 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 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 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 where S is the area of the region, is the average velocity of the system, and 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 : , where  is the angular velocity of the rigid-body rotation where 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 and the total moment of force is equal zero: 0 M  .In case of a small Reynolds number, the Coriolis force 2 ev      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.see (Landau & Lifshitz, 1986 ).In a general case, according with (18),  is an arbitrary harmonic function and

Pressure calculation
is the analytical function associated with  as 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).
Since ( 23) is valid for arbitrary variations of pressure p  and velocity v   we choose them such that p  is left unchanged: In this case (23) gives us We introduce the one-parameter family of variations . Then ( 24) and ( 25) take the form ( 18) and ( 19).Suppose

Velocity calculation
The stress tensor, expressed in terms of the Airy function where k 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 1 ,.
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 The expression (37) gives us the explicit presentation of the velocity on the boundary.

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 where 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 where  is the coefficient of surface tension.Using ( 42), ( 43) we get This expression gives us the possibility to obtain the strict limitations for the motion of the free boundary in some special cases.

The influence of capillary forces only
In this case the inequality (47) may be simplified: where m is the number of bubbles.Let  in agreement with (Levich, 1962).According with (64), a small boundary perturbation of characteristic with aR  and amplitude Ha  has a characteristic decay time ~a    .

The capillary relaxation of an ellipse
Let's test our theory on an example of a large amplitude perturbation.We calculate the capillary relaxation of boundary with initial shape 1 xx ab   in two ways -using the numerical calculation based on (6.4) and the finite-element software ANSYS POLYFLOW (see Fig. 3 and Fig. 4).These methods of calculation give us the same results with discrepancy about 1%.

The collapse of a cavity
Let's now consider a large amplitude perturbation in the shape of a cavity (Fig. 5).By symmetry, the pressure must be an even function with respect to 2 x , i.e.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  

Conclusion
We presented a method to calculate two-dimensional Stokes flow with free boundary, based on the expansion of pressure in a complete system of harmonic functions.The theory forms the basis for strict analytical results and numerical approximations.Using this approach we analyse the collapse of bubbles and relaxation of boundary perturbation.The results obtained by this method are correlating well with numerical calculations performed using commercial FEM software.

2. 1
Fig. 1.Region G with multiply connected boundary  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: 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 00 VT L  is ensured by the requirement of a small Reynolds number 0 Re constructed from the characteristic initial velocity 0


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 ,, is the coordinate of the center of mass in the initial system K , dx v dt    .In the new system the surface force is the same as in the initial systemf f    , but the volume force transforms to FF       and total force is equal to zero: 0     .So, we eliminated the total force   using a noninertial center-of-mass reference system K .
Let   and  be smooth fields in the region G related by 2of motion (1) by   , integrating over G , and using (2), (3),(18) 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 k  in the region G with the multiply connected boundary  consists of functions of the form points, each situated in one bubble.The complete set of harmonic functions k  can be obtained in the form of Re k  and Im k  .According with (1), (2) the pressure p is a harmonic function.We present it in the form .
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 .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: 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 , . . .,

Fig. 2 .
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   1 m  ,

Fig. 4 .
Fig. 4. Relaxation from ellipse to a circle in finite-element calculation.
 are the polar coordinates in the 12 , xx plane).Since the width 