Mathematical Model of the Liquid Film Flow on the Flat Surface

A liquid film on surface of a body decreases frictional resistance and can be used as a boundary layer control element. This article contains a mathematical model of a film flow over a half-plane, directed at an angle to the horizon. Liquid flow depends on gravity and friction with the external air flow. A model of incompressible viscous liquid near the boundary layer is used as the flow model. Summands of motion equation are averaged over the film thickness by the Leibniz rule. The square low is assumed for distribution of longitudinal velocity in the cross-section of the film with regard to the friction at the film's surface. An approximate solution of the problem is received as power series in powers of small parameter. The results are presented in a form diagrams of the film thickness and the average longitudinal velocity over the length of the plate. The mathematical flow model can be used to define flat film flow performance.


Introduction
Liquid film flow problems are quite frequent for technical appendixes and works by many authors. For example, work [1] presents film research findings for a two-phase gas-liquid flow in short horizontal pipes. The obtained results are compared to the experimental data. Work [2] contains film flow dynamic properties under a counter current airflow. The design data meet the experiment.
Work [3] examines a three-dimensional dynamic model of film flow over a cylindrical surface. The film is exposed to gravity, and surface-tension, as well as a drag of a countercurrent gas flow. Work [4] presents solution for related problem for airflow over a flat plate with a thin liquid film at the surface. The film is formed by extruding the liquid through a porous wall. In work [5] the film is formed out of the environment (for example, rain drops). The impact of raindrops to the boundary layer is ignored. The mathematical flow model is developed for a zero approximation. Works [6] - [7] contain findings of theoretical and numerical research of liquid films, works [8] - [10] present experimental research.
Work [11] presents a model of a thin film flow over an inclined plane. Summands of the Navier-Stokes equation and the energy equation are averaged over the depth of the liquid.
Work [12] proposes a mathematical model for evaporation of a thin film. Work [13] contains experimental results and numerical simulation of a film flow over a cooled wall. The measurements were done at a flat plate. Work [14] analytically investigates a condensate film stability problem on a vertical surface. Work [15] presents an analytical model for a laminar flow of a condensate film over an external surface of an isothermal vertical pipe. Regardless of a big number of works on the flow of liquid films, the interest to this subject does not wane.
The purpose of this research is the development of a mathematical model of a liquid film flow over a flat surface, forced by gravitation and the external airflow for the further simulation of a variable thickness film wave flow.

Statement of Problem
Flying vehicles are operated in various climatic zones and under different weather conditions. It is not rare when a liquid film is formed on the surface of a flying vehicle. The occurrence of the film can be caused by atmospheric precipitation; a liquid on the surface of a vehicle can form as a result of condensation of humidity from the surrounding air or artificially, through squeezing-out from a porous wall.
During the flight of a flying vehicle the film influences the air boundary layer, thus the velocity field is restructured. In addition, already with comparatively small Reynolds numbers calculated according to the characteristics of the film, waves appear on the surface of the film. Thus, a smooth film reduces the friction resistance, whereas a wavy film increases the friction resistance. It should also be noted that the film may be used as means to control the boundary layer of the body. In any case the film changes the aerodynamic characteristics of the structural elements. That is why a liquid film should be taken into account in the flight of a flying vehicle.
Strictly speaking, the problem of body flow with film is a conjugate problem. Its solution poses considerable difficulties. It is necessary to calculate the characteristics of a boundary layer and a liquid film, which are influencing each other. As a rule, such problems are solved numerically. For an analytical solution of such problem an iterative process can be used. The iterative process consists from an exterior and an interior problem. The exterior problem contains a problem an air boundary layer with a movable wall (the thickness of a film is, as a rule, is neglected). The interior problem is a problem of a liquid film flow with friction on the interfacial area. The friction is determined by the solution of the exterior problem. The velocity of slippage on the wall, for the exterior problem, is determined as the longitudinal liquid velocity on the film surface. The solution process consists in successive solutions of the exterior and interior problems. The solution continues until as the friction and velocity on the film surface will not within the prescribed accuracy.
The proposed procedure assumes presence mathematical models for the exterior and interior problems. In this paper we will discuss the mathematical model of flow of a liquid film on a flat surface under the effect of gravity and interfacial friction. As a surface element we will take a flat plate that moves in immovable air with an angle to the horizon. By reversing motion, we will receive the homogeneous airflow around the plate parallel to the plane of the plate. The liquid enters the film evenly through the porous surface of the plate (Figure 1). The research is limited to a steady flow of the liquid in a variable film forced by gravity and a drag of a countercurrent airflow. Mathematical target setting includes motion equations for the near-the-boundary layer and continuity equation for an incompressible liquid Boundary conditions at the wall are as follows friction with the external airflow is set for the surface of the film where p − pressure, ρ − density, , ν µ − kinematic and dynamic viscosity, g -acceleration of gravity, τ − friction, δ − film thickness.

Method of Solution
The motion equations is completed with a kinematic condition for the film surface Let's combine the motion and the continuity equations Since the thickness of the film is small enough, viscosity forces will play a major part in the flow development. This case allows adopting the square law for distribution of longitudinal velocity over the film cross-section and using the velocity expression for a uniform thickness film, flowing down along a flat vertical wall with regard to the surface friction In this case the film thickness and the average longitudinal velocity are the functions of the longitudinal coordinate. Let's average the summands of the equation (6) over the film thickness. For this let's integrate the summands from 0 to δ in respect of y using the Leibniz rule (derivative of an integral with a variable upper limit)

∫ ∫
The motion equation (6) will change as follows (skew brackets of the average velocity are hereinafter omitted) Let's write down the mass conservation equation, for whatlet's integrate the continuity equation (2) from 0 to δ in respect of y Putting expression (9) into the equation (8), we receive Let's use the film surface kinematic condition (5) and, with regard to the expression (7), write down Combination of formulas (9) and (11) will give the mass conservation law Let's transform the motion equation (10) and the continuity equation (12) with the use of dimensionless variables Were performed some transformation Dimensionless motion equation will be as follows Putting dimensionless variables into the mass conservation equation (12) gives the second differential equation

Results of the Solution and Discussion
The   It is seen from the diagrams that increasing the plate angle increases the longitudinal speed and decreases the film thickness. The Reynolds number does not exceed the Re 10 = . And the film flow is not a wave flow.

Conclusion
The mathematical model of a thin liquid layer flow over a flat plate surface allows defining the film properties on the flat elements of an aircraft. The results can be used for calculation of frictional resistance, which is a component of an aircraft resistance. An approximate analytical solution was obtained as a series in terms of small parameter powers taking into account zero and first approximation. The results are presented as graphs of film thickness and average longitudinal velocity depending on the plate length for various angles of slope to the horizon. It is proposed to use the analytical solution of the problem in an iterative process for the conjugate problem, and also as part of the mathematical model of a flow of a wavy film with variable thickness.