Mathematical Model of the Liquid Film Flow on the Flat Surface

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1 Aerican Journal of Aerospace Enineerin 017; 4(1): doi: /j.ajae ISSN: (Print); ISSN: (Online) Matheatical Model of the Liquid Fil Flow on the Flat Surface N. I. Klyue Departent of Matheatical Modelin in Mechanics, Saara State Uniersity, Saara, Russia Eail address: To cite this article: N. I. Klyue. Matheatical Model of the Liquid Fil Flow on the Flat Surface. Aerican Journal of Aerospace Enineerin. Vol. 4, No. 1, 017, pp doi: /j.ajae Receied: Deceber 8, 016; Accepted: January 19, 017; Published: February 1, 017 Abstract: A liquid fil on surface of a body decreases frictional resistance and can be used as a boundary layer control eleent. This article contains a atheatical odel of a fil flow oer a half-plane, directed at an anle to the horizon. Liquid flow depends on raity and friction with the external air flow. A odel of incopressible iscous liquid near the boundary layer is used as the flow odel. Suands of otion equation are aeraed oer the fil thicness by the Leibniz rule. The square low is assued for distribution of lonitudinal elocity in the cross-section of the fil with reard to the friction at the fil's surface. An approxiate solution of the proble is receied as power series in powers of sall paraeter. The results are presented in a for diaras of the fil thicness and the aerae lonitudinal elocity oer the lenth of the plate. The atheatical flow odel can be used to define flat fil flow perforance. Keywords: Fil, Liquid, Boundary Layer, Flow, Friction, Sall Paraeter 1. Introduction Liquid fil flow probles are quite frequent for technical appendixes and wors by any authors. For exaple, wor [1] presents fil research findins for a two-phase as-liquid flow in short horizontal pipes. The obtained results are copared to the experiental data. Wor [] contains fil flow dynaic properties under a counter current airflow. The desin data eet the experient. Wor [3] exaines a three-diensional dynaic odel of fil flow oer a cylindrical surface. The fil is exposed to raity, and surface-tension, as well as a dra of a countercurrent as flow. Wor [4] presents solution for related proble for airflow oer a flat plate with a thin liquid fil at the surface. The fil is fored by extrudin the liquid throuh a porous wall. In wor [5] the fil is fored out of the enironent (for exaple, rain drops). The ipact of raindrops to the boundary layer is inored. The atheatical flow odel is deeloped for a zero approxiation. Wors [6] - [7] contain findins of theoretical and nuerical research of liquid fils, wors [8] [10] present experiental research. Wor [11] presents a odel of a thin fil flow oer an inclined plane. Suands of the Naier-Stoes equation and the enery equation are aeraed oer the depth of the liquid. Wor [1] proposes a atheatical odel for eaporation of a thin fil. Wor [13] contains experiental results and nuerical siulation of a fil flow oer a cooled wall. The easureents were done at a flat plate. Wor [14] analytically inestiates a condensate fil stability proble on a ertical surface. Wor [15] presents an analytical odel for a lainar flow of a condensate fil oer an external surface of an isotheral ertical pipe. Reardless of a bi nuber of wors on the flow of liquid fils, the interest to this subject does not wane. The purpose of this research is the deelopent of a atheatical odel of a liquid fil flow oer a flat surface, forced by raitation and the external airflow for the further siulation of a ariable thicness fil wae flow.. Stateent of Proble Flyin ehicles are operated in arious cliatic zones and under different weather conditions. It is not rare when a liquid fil is fored on the surface of a flyin ehicle. The occurrence of the fil can be caused by atospheric precipitation; a liquid on the surface of a ehicle can for as a result of condensation of huidity fro the surroundin air or artificially, throuh squeezin-out fro a porous wall.

2 N. I. Klyue: Matheatical Model of the Liquid Fil Flow on the Flat Surface Durin the fliht of a flyin ehicle the fil influences the air boundary layer, thus the elocity field is restructured. In addition, already with coparatiely sall Reynolds nubers calculated accordin to the characteristics of the fil, waes appear on the surface of the fil. Thus, a sooth fil reduces the friction resistance, whereas a way fil increases the friction resistance. It should also be noted that the fil ay be used as eans to control the boundary layer of the body. In any case the fil chanes the aerodynaic characteristics of the structural eleents. That is why a liquid fil should be taen into account in the fliht of a flyin ehicle. Strictly speain, the proble of body flow with fil is a conjuate proble. Its solution poses considerable difficulties. It is necessary to calculate the characteristics of a boundary layer and a liquid fil, which are influencin each other. As a rule, such probles are soled nuerically. For an analytical solution of such proble an iteratie process can be used. The iteratie process consists fro an exterior and an interior proble. The exterior proble contains a proble an air boundary layer with a oable wall (the thicness of a fil is, as a rule, is nelected). The interior proble is a proble of a liquid fil flow with friction on the interfacial area. The friction is deterined by the solution of the exterior proble. The elocity of slippae on the wall, for the exterior proble, is deterined as the lonitudinal liquid elocity on the fil surface. The solution process consists in successie solutions of the exterior and interior probles. The solution continues until as the friction and elocity on the fil surface will not within the prescribed accuracy. The proposed procedure assues presence atheatical odels for the exterior and interior probles. In this paper we will discuss the atheatical odel of flow of a liquid fil on a flat surface under the effect of raity and interfacial friction. As a surface eleent we will tae a flat plate that oes in ioable air with an anle to the horizon. By reersin otion, we will receie the hooeneous airflow around the plate parallel to the plane of the plate. The liquid enters the fil eenly throuh the porous surface of the plate (Fiure 1). ariable fil forced by raity and a dra of a countercurrent airflow. Matheatical taret settin includes otion equations for the near-the-boundary layer u u u u + = sinα + x y y p, 0 = (1) y and continuity equation for an incopressible liquid u + = 0. () x y Boundary conditions at the wall are as follows y = 0, u = 0, =, (3) friction with the external airflow is set for the surface of the fil u y =, µ = τ, (4) y where p pressure, ρ density,, µ ineatic and dynaic iscosity, acceleration of raity, τ friction, fil thicness. 3. Method of Solution The otion equations is copleted with a ineatic condition for the fil surface d y =, ( ) = u( ). (5) dx Let's cobine the otion and the continuity equations + = sinα + x y y u ( u) u. (6) Since the thicness of the fil is sall enouh, iscosity forces will play a ajor part in the flow deelopent. This case allows adoptin the square law for distribution of lonitudinal elocity oer the fil cross-section and usin the elocity expression for a unifor thicness fil, flowin down alon a flat ertical wall with reard to the surface friction y 1 y τ 3y u( x, y) = 3 < u > y +. (7) µ Fiure 1. Fil flow diara: u, the coponents of the elocity ector, elocity of injection ass, U elocity of the incoin flow, α anle to the horizontal plate, x, y - coordinate axis. In this case the fil thicness and the aerae lonitudinal elocity are the functions of the lonitudinal coordinate. Let's aerae the suands of the equation (6) oer the fil thicness. For this let's interate the suands fro 0 to in respect of y usin the Leibniz rule (deriatie of an interal with a ariable upper liit) The research is liited to a steady flow of the liquid in a

3 Aerican Journal of Aerospace Enineerin 017; 4(1): there fro we will find d u udy u( ) dy dt = + t t 0 0 u d dy = udy u ( ). dt dt t 0 0 The otion equation (6) will chane as follows (sew bracets of the aerae elocity are hereinafter oitted) du τ 1u 1u + 5 dx 4µ 0 13τu 3τ d µ 80µ dx τ τ 3τ 3u + u + + u( ) ( ) = sin α +. (8) 0µ x 3µ ρ Let's write down the ass conseration equation, for whatlet's interate the continuity equation () fro 0 to in respect of y du u d τ d ( ) = + +. (9) dx dx 4µ dx Puttin expression (9) into the equation (8), we receie du 9u 3τ 3u dx 10 10µ 10 τ 3τu d 3u µ 0µ dx τ dτ τ 3τ 3 u + + u + = sinα +. (10) 4µ 0µ dx 3µ ρ Let's use the fil surface ineatic condition (5) and, with reard to the expression (7), write down y =, d 3u τ d ( ) = u( ) = +. (11) dx 4µ dx Cobination of forulas (9) and (11) will ie the ass conseration law d du = +. (1) u dx dx Let's transfor the otion equation (10) and the continuity equation (1) with the use of diensionless ariables ξ = 3 6 x, u = U 3 ( ξ ), ( ξ) µ T =, τ =. (13) 3 then A 6 = 3, B =, C =. 3 µ D =, 3 d 3 3 d d ξ BA d, du du d ξ CA du dx dx 3ξ dx dx 3ξ = = = =. Diensionless otion equation will be as follows du 1 1 d 6 (3U T ) 6 ( U T + UT ) + 45ξ ( U dt + T sinα T + U + 3 ( U + T ) = 0. (14) 6 3 Puttin diensionless ariables into the ass conseration equation (1) ies the second differential equation du d + U ξ = (15) 3 0 and the initial conditions of the proble ξ = 0: U = = 0. (16) Solin this proble requires settin the fil surface friction. Let's oit the fil thicness and study the Blasius proble of strealine of a plate with forward air. When oin fro the ede of the plate the surface friction becoes τ = const. Therefore let's assue the friction alue τ = const, that is a sufficient approxiation if the ass is injected at a distance fro the ede of the plate. The alue of friction at the fil-air boundary is set as an aerae of friction fro the Blasius. Input data of the proble. Let a flat horizontal plate with the lenth of L = 1 is blown oer by airflow with elocity U = 14 / s at teperature 0 t = 0 C. The Reynolds 5 nuber for a forward air flow is Re = U L/ = 8,9 10, 1 1 where 1 the air ineatic iscosity. Let's tae the ass 5 injection elocity = 10 / s, then the diensionless 3 coordinate ξ = 1,8 10 << 1. Therefore, solution of (14) and (15) nonlinear differential equations can be presented as rows by the sall paraeter power U = αξ и = 1 = βξ, (17) = 1 where α and β ( = 0,1,,... ) coefficients are defined at a fixed anle of the plate and the ien friction alue T. Let's write down the solution for α = π / and T = 0 Were perfored soe transforation 3 U( ξ) = 0,63ξ 0,11 ξ +..., ( ξ) = 0,79ξ + 0,14 ξ +... (18) ξ = Ax 1/3, = B ( ξ), u = CU( ξ), τ = DT( ξ), where 4. Results of the Solution and Discussion The results of the solution are presented in diensional

The Reynolds nuber does not exceed the Re = 10. And the fil flow is not a wae flow. 5. Conclusion Fiure 3.

4 4 N. I. Klyue: Matheatical Model of the Liquid Fil Flow on the Flat Surface way at diaras on fiure and fiure 3 for α = π /and Re = 10, where Re = u / Fiure. Aerae lonitudinal elocity, 1 τ = 0; τ = 0,15. Fiure 5. Fil thicness. It is seen fro the diaras that increasin the plate anle increases the lonitudinal speed and decreases the fil thicness. The Reynolds nuber does not exceed the Re = 10. And the fil flow is not a wae flow. 5. Conclusion Fiure 3. Fil thicness, 1 τ = 0; τ = 0,15 As it was anticipated, the concurrent airflow increases the aerae lonitudinal elocity and decreases the fil thicness. Diaras (fiure 4 and fiure 5) show ariation of the aerae lonitudinal speed and the fil thicness for τ = 0 Re = 10 and for different plate anles 1 α = π /4, α = π /6 The atheatical odel of a thin liquid layer flow oer a flat plate surface allows definin the fil properties on the flat eleents of an aircraft. The results can be used for calculation of frictional resistance, which is a coponent of an aircraft resistance. An approxiate analytical solution was obtained as a series in ters of sall paraeter powers tain into account zero and first approxiation. The results are presented as raphs of fil thicness and aerae lonitudinal elocity dependin on the plate lenth for arious anles of slope to the horizon. It is proposed to use the analytical solution of the proble in an iteratie process for the conjuate proble, and also as part of the atheatical odel of a flow of a way fil with ariable thicness. References [1] López, J., Pineda, H., Bello, D., Ratoich, N. Study of liquidas two-phase flow in horizontal pipes usin hih speed filin and coputational fluid dynaics. Experiental Theral and Fluid Science. V.76, 016, pp [] Lu, H., Lu, L., Luo, Y., Qi, R. Inestiation on the dynaic characteristics of the counter-current flow for liquid desiccant dehuidification. Enery. V.101, 016, pp [3] Li, M., Lu, Y., Zhan, S., Xiao, Y. A nuerical study of effects of counter-current as flow rate on local hydrodynaic characteristics of fallin fils oer horizontal tubes. Desalination. V.383, 016, pp Fiure 4. Aerae lonitudinal elocity. [4] Klyue, N. I., Giadie, A. G., Kryuo, Y. A. Two-edia boundary layer on a flat plate. IJET. V.6, Issue 5, 014, pp

5 Aerican Journal of Aerospace Enineerin 017; 4(1): [5] Klyue, N. I., Kryuo, Y. A. Influence of fluid fil on friction of a flat plate. Russian Aeronautics (Iz. VUZ). V.57, Issue 4, 014, pp [6] Caassa, R., Orosy, H. R. On iscous fil flows coatin the interior of a tube: Thin-fil and lon-wae odels. Journal of Fluid Mechanics, V.77, 015, pp [7] Muraatsu, K., Youn, Y., Han, Y., Haseawa, Y., Shiazono, N. Nuerical study on the effect of initial flow elocity on liquid fil thicness of accelerated slu flow in a icro. International Journal of Heat and Fluid Flow. V.54, 015, pp [8] Han, Y., Kanno, H., Ahn, Y.-J., Shiazono, N. Measureent of liquid fil thicness in icro tube annular flow. International Journal of Multiphase Flow. V.73, 015, pp [9] Youn, Y. J., Muraatsu, K., Han, Y., Shiazono, N. The effect of initial flow elocity on the liquid fil thicness in icro tube accelerated slu flow. International Journal of Multiphase Flow. V.73, 015, pp [10] Ju, P., Broos, C. S., Ishii, M., Liu, Y., Hibii, T. Fil thicness of ertical upward co- current adiabatic flow in pipes. International Journal of Heat and Mass Transfer. V.89, 015, pp [11] Richard, G. L., Ruyer-Quil, C., Vila, J. P. A three-equation odel for thin fils down an inclined plane. Journal of Fluid Mechanics. V.804, 016, pp [1] Auş, Y., Dursunaya, Z. A new approach to thin fil eaporation odelin. International Journal of Heat and Mass Transfer. V.101, 016, pp [13] Liu, C. L., Liu, J. L., Zhu, H. R., Wu, A. S., He, Y. H., Zhou, Z. X. Fil coolin sensitiity of laidbac fanshape holes to ariations in exit confiuration and ainstrea turbulence intensity. International Journal of Heat and Mass Transfer. V.89, Issue 1145, 015, pp [14] Dean, G., Sanya, A., Aowanou, C. Lainar fil condensation alon a ertical plate ebedded in an anisotropic porous ediu with oblique principal axes. Heat and Mass Transfer. V.5, Issue10, 016, pp [15] Ki, S., Lee, Y. G., Jern, D. W. Lainar fil condensation of saturated apor on an isotheral ertical cylinder. International Journal of Heat and Mass Transfer. V.83, 015, pp

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