Linear waves on a surface of vertical rivulet

The type of film flow whereby the fluid flows in the form of many streamlets is typically called a rivulet flow. Whereas an individual streamlet bounded by two contact lines is called a rivulet. Special attention has been paid to rivulet flows because of their practical value for a variety of devices in power engineering and chemical technology, such as absorbers, distillation columns, evaporators, and heat exchangers for the liquefaction of natural gas. In the present paper the waves in vertical rivulet are investigated analytically. The Kapitza-Shkadov model is used to describe the wavy rivulet flow since it was well proven in the study of nonlinear waves in falling liquid films over a wide range of Reynolds numbers. The equations of the wavy rivulet flow are derived on the basis of the weighed residual method. These equations turn out to be the projections of the Shkadov's model equations on system of basis functions, constructed in special way. Linearizing these equations results in the dispersion relations for plane waves. The stability criterion for rivulet flows is deduced, and the analysis of dispersion relations depending on dimensionless parameters is carried out.


Introduction
Compared to uniform liquid film flows, rivulet dynamics exhibit several interesting features, such as the presence of movable contact lines and wetting angle hysteresis. Most theoretical studies, beginning with the pioneering study of Towell and Rothfeld [1], are devoted to stationary and smooth (without waves) rivulets flowing down an inclined plane as well as curved surfaces [2,3]. Profiles of smooth rivulets and steady regimes of rivulet flows were calculated using both analytical methods in a lubrication approximation [2,3] and numerical methods based on Navier-Stokes equations [4,5]. In the simplest case of a thin rivulet flowing down a vertical flat surface, the rivulet profile was shown to represent an arc of a circle.
The stability of a moving rivulet has been studied in a limited number of works. A linear stability analysis was performed in [6][7][8][9][10] based on Navier-Stokes equations for the normal mode (plane waves). In these theoretical studies the stability calculations of rivulet are executed in lubrication approximation, i.e. at very small Reynolds numbers and for plane modes of perturbations only. Alekseenko, Markovich and Shtork [11] performed the first detailed experimental study of the wave motion of a rectilinear rivulet for the case of liquid flow along the lower outer wall of an inclined tube. With application of the imposed periodic pulsations of the flow rate the periodic non-linear waves have been studied, new wave modes are found, and comparison to known two-dimensional film flow is made. In the papers [12,13] method of PIV was applied in the first time to measure instantaneous velocity field in wavy rivulet. Alekseenko et al. [14] were the only researchers to present the results of experimental investigation on periodic, nonlinear waves on a rectilinear rivulet flowing down a vertical wall. Waves in rivulet flow were studied in a wide range of a Reynolds number and forcing frequency for various values of angles of wetting. In [15] the results of numerical simulations of threedimensional waves on the surface of a rivulet flowing down a vertical plate are presented. Various characteristics of linear and nonlinear regular waves in the rivulet are obtained through numerical calculations as a function of the forcing frequency at different Reynolds numbers and contact wetting angles. Calculations were executed only for those conditions which were implemented in experiments [14] and comparison with experimental data was made. The comparison shows that the applied model adequately describes the shape of the wave surface of a rivulet.
In the present paper the waves in vertical rivulet are investigated theoretically. On the basis of a weighed residual method the approach allowing simplify a problem is developed and dispersion relations for linear waves are deduced analytically at moderate Reynolds numbers.

Theoretical model
Let us consider a thin liquid layer flow over a vertical flat plate as a rivulet of constant width 2b. We introduce a Cartesian coordinate system Oxyz with the Ox-axis directed along gravity and the Oy-axis directed along the normal to the plate (figure 1b). We assume that the layer thickness h is substantially smaller than the wavelength  (long-wave approximation) and the rivulet semi-width b. All of these assumptions are in good agreement with experimental observations [14]. As a first approximation with respect to the small parameters , the waves in a liquid layer may be described by the integral boundary layer (IBL) model [15], which was developed for three-dimensional long waves in a falling liquid film. The corresponding momentum and mass balance equations in dimensionless variables are the liquid flow rates per unit width along the Ox and Oz axes, respectively, . We choose the unperturbed film thickness H 0 at the symmetry axis of the rivulet as the scale and introduce the scales of velocity is Kapitza number,  is the surface tension,  is the density, and  is the kinematic viscosity of the fluid. With regard to the rivulet flow, these equations must be supplemented by the follows boundary conditions at the contact line (fixed contact line): . It is follows from (1), that h and q are even functions of coordinate z, and m is odd function of coordinate z. In case of a undisturbed flow (without waves) the partial derivatives on x and t in the equations (1) and also the flow rate m are equal to zero, thus a cross-section of rivulet is a parabola   The function h , q , m should have the finite values at a contact line, thus boundary conditions will be automatically fulfilled, since 0 Using h , q , m , we rewrite the equations (1) as follows   . Here, As a whole, relations (3) do not satisfy (2) (substitution in the equations gives the residual). According to a weighed residual method, the projections of a residual to basis functions should be equated to zero, as a result we obtain the system of differential partial equations for H j , Q j , M j . Coefficients of these equations are calculated through integrals from product of basis polynomials on function from .
in the equations (2). We define a scalar product of polynomials . Note the important property of the proposed basis, namely, near to a contact line the profiles of a thickness and flow rate are similar geometrically to corresponding profiles for a undisturbed flow, that is h~ . Thus the contact angle varies according to a wave phase.

Plane waves
In the most simple case each basis consist of one element: . It corresponds to a case of plane waves for which profile of wavy rivulet remains a parabola in any section. Let's project the equations (2) on correspondent polynomials. For this purpose the first and third equations (2) multiply on 1 1   , and the second equation (2) Here, . Distinction between dispersion relations for waves in a rivulet and in a film consists not only in coefficients of (6) but also available additional parameter b. The system (6) the same as in case of a film flow, is easily reduced to one quadratic equation with respect to 2 16 Here, sign "plus" corresponds to unstable mode, and the sign "minus" corresponds to stable mode (in the same way, as for waves in a film flow). For unstable mode C > 0 and the perturbations are growing ( > 0) in some range of a wavenumber. For stable mode C < 0, and the perturbations are damping at any wavenumber ( < 0).

Stability criterion of rivulet flow
In an asymptotics 1 Re  m  (at k→ 0) from the second equation (6)    various values of Kapitza number. From figure it is clear that area of instability and the maximum of increment decrease with parameter Fi 1/3 growth.

Conclusions
The 3D wavy rivulet flow has been studied theoretically. The approach reducing a problem to 2D one has been proposed on the basis of weighted residual method. The equations of the 2D rivulet flow have been deduced using projections of initial equations on the system of orthogonal polynomials. The system of equations for plane waves in the vertical rivulet has been deduced, and dispersion relations have been obtained. The stability criterion of rivulet flow has been deduced, and the analysis of influence of dimensionless parameters on dispersion dependences has been carried out.