## 1. Introduction

Stokes steady-state approximation [1] is widely used for calculation of viscous flows with low Reynolds numbers. This approach was first applied to describe the flow with free boundaries in [2]. Many classical problems of high-viscous liquids hydrodynamics were solved in this approximation: the destruction of bubbles [3], the formation of the cusp on the liquid surface [4,5], the coalescence of liquid particles [6] and some problems of nanofluidics [7].

The idea of using inertia-less approximation to calculate wave relaxation on the surface of a fluid with high viscosity was suggested by [8]. The relaxation of harmonic perturbations in this approximation was considered by [9,10]. This approach has also been used in studying thin film flow [11], two-dimensional Stokes flow with free boundary [12-13].

In this work we study an aperiodic relaxation of a small localized perturbation of the planar surface of an incompressible fluid under the influence of gravity and surface tension. The description is given in the Stokes approximation. It is shown that this imposes limitations on the three-dimensional spectrum of the considered perturbations. An equation describing perturbation damping in Fourier space was obtained. It is shown that the volume of the perturbation becomes zero in the time that is small by comparison to the time of perturbation damping. In the short wave limit, the law of evolution of the perturbation permits a simple geometric interpretation.

The second part of the work is focused on theoretical investigation of instability of the surface shape under the influence of diffusion mass flux through the surface. A local eminence on the surface can lead to perturbation growth. Stability and instability of the perturbation will depend on which mechanism prevails. Diffusion causes distortion of surface shape, gravitational and capillary forces have the opposite effect.

## 2. Evolution of half-space free boundary perturbation

### 2.1. The general equations

We consider a half-space filled with a highly viscous incompressible Newton liquid of constant viscosity. At the initial moment of time the planar surface of half-space obtains a small local perturbation. We will investigate further the evolution of this perturbation due to gravitation, capillary forces and external mass flux through the free surface. We introduce the Cartesian coordinate system with axes

The indices

The following conditions in case of infinite half-space must also be met:

Here

Equation of free surface evolution and its initial condition have the form

where

Applying to (1)-(4) the two-dimensional Fourier transform with respect to the longitudinal coordinates

we obtain

Integrating (5) with boundary conditions (6), we obtain

It is apparent that the Fourier component

### 2.2. Relaxation of the initial perturbation of the liquid surface, exact solution

Consider the relaxation of the initial perturbation of the surface due to capillary and gravitation forces in the absence of mass flux at the boundary (

where

constitutes the "volume of perturbation". According to (10) this value becomes equal to zero due to the gravity force. Immediacy of this process is determined using a quasi-stationary Stokes approximation. Detailed discussion of this topic see in [14].

Switching to the case of arbitrary wave numbers, we introduce the dimensionless variables

Then (10) takes the form

Applying inverse Fourier transform to (11) we have

The expression (12) allows calculation of the surface perturbation in any time for an arbitrary initial perturbation.

### 2.3. Qualitative description of the perturbation relaxation

For a qualitative description of the relaxation of the initial perturbation is convenient to use the original expression (11). We have

From (13) it is clear that the configuration at the moment

By solving the equation

At

To estimate perturbation evolution we replace function

where

Returning to the expression (13) we see that the evolution of perturbations can be considered as a result the initial perturbation “cut-off” in Fourier space. The system under consideration behaves as filter which "cuts" in the spectrum of the initial perturbation wave numbers the annular region with characteristic dimensions

Spatial filtering of the Fourier transform of the initial perturbation in the region of small wave numbers is equivalent to subtracting its part localized in a circle of radius

Localization of the Fourier transform of the perturbation in a circle of radius

To evaluate the value of the perturbation at the origin of coordinates we use of the fact that

Then

The above statements are based on the approximate replacement of real

Let us consider the relaxation of perturbations with different spatial scale and shape. To illustrate the qualitative dependencies we will use graphics produced on the basis of the exact solution (12).

### 2.4. Relaxation long-wave perturbations by gravity

If the characteristic dimensionless scale of the region occupied by the perturbation

At the initial stage of relaxation process

Through this long-wavelength addition, the initial perturbation goes down (in case of positive volume) on

By the time

If the characteristic dimensions of the initial perturbation in two mutually perpendicular directions are very different, the relaxation process is more complicated. Note that for the perturbation, elongated along an axis

From the time moment

At the moment

### 2.5. Relaxation of short-wave perturbations due to capillary forces

The evolution of perturbation with the scale

Relaxation of short-wave perturbations is mainly determined by the action of surface tension forces, i.e. filtration its Fourier transform by the large wave numbers. The relaxation process begins at the moment

Finally let’s describe the relaxation of small-scale perturbations with significantly different scales along the axes

After the moment

The relaxation process ends substantially at the moment

## 3. Finite layer over solid bottom and over a half-space

### 3.1. Finite layer over solid bottom

In this section we study surface perturbations of a flat liquid layer with depth H resting on a solid bottom. We use system (1) and conditions on the free boundary (2) to calculate perturbation evolution due to gravity and surface tension. The conditions at infinity (3) are replaced by the no-slip condition along the solid surface

Again applying the Fourier transform in the longitudinal coordinates and solving the resulting system of ordinary differential equations, we obtain

where

The evolution equation of the free surface and the initial condition has the form

The solution of equation (19) is similar to (10)

where

At

where

### 3.2. Finite layer over a half-space

Consider a plane layer with thickness

where

### 3.3. Comparing the cases of finite layer and finite layer over a half-space

Fig.7 shows function

Fig. 8 and Fig. 9 show decrease of the amplitude for the perturbation with the same initial characteristics. The figures show that the perturbation on the surface layer over solid bottom decays faster than the perturbation on the surface layer over the half-space.

### 3.4. Comparing the relaxation of finite layer free surface perturbation in linear and non-linear approximation

Using finite-element software Comsol Multiphysics we developed a 2D axisymmetric model of the viscous flow to calculate transient relaxation of the perturbation on the free surface and compare it with analytical results reported in the previous sections. We used “Laminar Two-Phase Flow, Moving Mesh” module to enable deformation of the computational domain during the solution. Full Navier-Stokes equations are solved on the moving mesh to describe deformation of the domain due to capillary forces acting along the free surface.

Computational mesh was refined near the perturbation to resolve high curvature and velocity gradients (see Fig. 10).

Multiple cases were calculated corresponding to small and large non-dimensional perturbation amplitude and width. The evolution of the perturbation amplitude with time was compared for numerical and analytical models (Fig. 11). It can be seen that the analytics and FEM are in excellent agreement for small amplitude A=0.1 for both small and large width (a=0.1, a=1.0). Some discrepancy is observed in the case of large amplitude A=1.0 due to violation of the analytic model assumptions.

## 4. Evolution of small perturbations of half-space free boundary due to external mass flux, gravitation and capillary forces

Let’s describe the influence of external mass flow on stability of the free surface of half-space. A small perturbation on the free surface leads to a perturbation of the diffusion boundary layer above it. The corresponding mass flow perturbations on the free boundary defines the source term in the right hand side of (9)

where

## 5. Conclusion

The relaxation process of a small perturbation of the free planar surface of a viscous incompressible fluid under the action of capillary and gravitational forces was investigated. The case of half-space, layer over hard bottom and layer over the half-space was analyzed. We propose a method for qualitative analysis of the configuration of the perturbation based on spatial filtering the Fourier transform of the initial perturbation.

It is shown that under the gravity force there is an additional long-wave perturbation arising around the initial perturbation which provides a zero total volume of the perturbation. Capillary forces lead to an increase of the area affected by the perturbation. The influence of amplitude of perturbation on the rate of its relaxation was investigated numerically and compared with the analytics.

The stability of the free surface of the half-space with the mass flux at the boundary was studied.

## Acknowledgments

This work was financially supported by Government of Russian Federation, Grant 074-U01.