Thermocapillary rivulets in the uniformly heated liquid film

The set of equations which describes a non-steady 3D flow of non-isothermal liquid film in the presence of thermocapillary effect is deduced in the long-wave approach. The used model is applicable for moderate Reynolds’s numbers Re∼10 and does not imply in advance set the profile of temperature in a film. The linear analysis of a stability of a film in relation to perturbations in spanwise direction is carried out and dispersion relations are gained. Nonlinear development of instability is investigated numerically in cases of 2D and 3D flow and appearance of quasistationary rivulet structure. Influence of dimensionless parameteres on characteristic scales of rivulet structures is revealed.


Introduction
Non-isothermal flows of liquid film are used in many technological processes. Dependence of the surface tension on temperature (thermocappilary effect) leads to occurrence of the shear stress which influences velocity of a fluid. Owing to thermocappilarity the hydrodynamics and heat transfer appear being interconnected. One of types of flow of liquid layers with a free surface, close to film flow, is rivulet flow which occurs in many technological apparatus. Theoretical studies of stability of a heated film and thermocapillary rivulets are based, as a rule, on long-wave approximation, but use two different approaches to the description of dynamics of film. In the first approach one evolutionary equation for a film thickness of type of equation Benney [1] can be deduced using lubrication approximation and taking into account a shear stress on a film surface owing to thermocapillarity. While deducing the evolutionary equation the temperature profile in a film is considered linear, that implies only conductive heat transfer and convection is neglected. Models [2−5] using the mentioned approach, are applicable only at small values of Reynolds number. In the second approach the Reynolds number is not supposed small, and dynamics of a heated film is described by system of the equations concerning a film thickness, flow rate and temperature of liquid. The theoretical models based on the second approach [6−8], have essentially wider applicability as they are deduced directly from the Navier-Stokes equations using some suppositions concerning a velocity profile and a temperature profile. The detailed review of studies on a problem of a stability of a heated films is available in [9]. The presented paper investigates nonlinear development of small spanwise-directed perturbation in the uniformly heated film which leads to 3D rivulets. Equations of IBL model [10] modified taking into account thermocapillarity together with the equation for temperature of liquid are applied to describe the dynamics of non-isothermal film. The used model is applicable in a wide range of hydrodynamic and thermal parametres and does not imply in advance set the profile of temperature.

Theoretical model
Let's consider a three-dimensional film of a viscous liquid flowing on uniformly heated infinite plate inclined under an angle  to horizon. The plate temperature is equal to T W , the film contacts to the motionless gas which has temperature T g , and heat exchange coefficient on interface is equal to .  The heated film flow is determined by the following dimensionless criteria: are determined only by the liquid properties and heating conditions. To unperturbed flow there corresponds solution

Stability and nonlinear development of perturbation in 2D statement
Let's consider the perturbed flow independent on coordinate x. In this case the flow remains quasithree-dimensional (the velocity component in the stream direction u also undergoes to perturbation), but a velocity field is identical in all sections of a film. Let's investigate stability of a base flow (2) concerning periodic on coordinate z perturbations with the set period L z . Let's put Linearizing the equations (1) relative to small perturbations Like it is made in [11], we present perturbations as follows Here   To investigate a nonlinear stage of instability development the equations (1) are solved numerically (for 2D flows at 0 /    x ) by finite difference method described in [11,12]. In an initial instant a perturbed film thickness      figure 3 it is visible, that two humps on interval edges grow in an initial stage of development, and in the middle of an interval (in an initial trough) additional hump appears with two new small humps from both sides (a curve 4 on fig. 3). In regions between humps new local minimums of film thickness appear (see fig. 4). Let's note, that the same development of instability was observed in 2D full-scale direct numerical simulation [13] fulfilled for a horizontal heated film.

Formation of 3D rivulet structure
The space development of periodic on z perturbations with wave length L z was simulated by means of the solution of the equations (1) by finite difference method. The numerical algorithm is analogous one applied in [11,12] for two-dimensional waves in the non-isothermal film. Initial conditions were set as small perturbation of a film thickness but unperturbed values of the flow rate and temperature Here H a is small amplitude of initial perturbation. The calculation area represented a rectangle . The size of calculation area in stream direction X end was large enough that it was possible to trace perturbation development downwards on a stream. On an inlet (i.e. at x = 0) the conditions (6) were supported, and on boundaries z = 0 and z = L z the periodicity conditions were set. Such statement of problem corresponds to real conditions of flow for a locally heated film in experiments [14][15][16] where developed stationary thermocapillary rivulets are observed. On an exit (at

Conclusions
In long-wave approach the system of equations which describes a non-stationary three-dimensional flow of heated liquid film at moderate Reynolds number Re~10 is deduced. On the basis of these equations the linear analysis of stability of 2D film flow relative to perturbations in the transverse direction is carried out. The simulation of a nonlinear development of small transverse perturbation having wavelength L z is fulfilled. In 2D calculations it is shown, that except the basic humps corresponding to wavelength L z the additional humps separated by most thin segments develop in the film. Further an asymptotic stage occurs, on which humps grow insignificantly, and the minimal film thickness h min monotonously decreases and asymptotically tends to be zero. Space development of perturbation on an inlet downwards on a stream was investigated in 3D calculations. It is shown, that downwards on a stream from region of linear development, where the amplitude of initial perturbation grows, the rivulets with high amplitude are formed. Development of a film cross-section in 3D simulations qualitatively matches to a film evolution in a time obtained by 2D simulations. Developed 3D rivulets have quasisteady character which corresponds to an asymptotic stage in 2D simulations. The amplitude of rivulets on an asymptotic region grows slightly with distance, and the minimum film thickness h min between humps gradually decreases and asymptotically tends to be zero. The effect of unlimited decreasing of h min is interpreted as a thermocapillary rupture of a film.

Acknowledgments
The work was supported by the Russian Science Foundation (grant No. 14-29-00093).