Two-media boundary layer on a flat plate

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1 Two-media bondary layer on a flat plate Nikolay Ilyich Klyev, Asgat Gatyatovich Gimadiev, Yriy Alekseevich Krykov Samara State University, Samara,, Rssia Samara State Aerospace University named after academician S.P. Korolev (National Research University), Samara, 86, Rssia Abstract: The present paper provides a soltion to the problem of a flow over a flat semi-infinite plate set at an angle to the horizon, and having a thin liqid film on its srface by external airflow. The film is formed by extrsion of liqid from the poros wall. The paper proposes a mathematical model of a two-media bondary layer flow. The main characteristics of the flow to a zero and a first approximation are determined. A drop of frictional stress is obtained. Keywords: plate, bondary layer, film, friction. INTRODUCTION Phenomena accompanied by a flow of flid films are broad and varied. The main experimental and theoretical stdies of sch films were carried ot in the XX centry. Among the pblications regarding films condensation (Koh et al., 96), (Tets Fjii and Haro Uehara, 97), (Mayhew and Aggarwal, 97), (Stephan, 6) shold be noted. Qestions concerning evaporation and boiling are qite well smmarized in recent books of (Baehr and Stephan, ) and (Ahsan, ). Resistance redction sing long-moleclar polymers or sspensions is also a well characterized phenomenon. The book (Hgh, 98) deals with varios aspects of resistance redction. Resistance redction by addition of polymers to a flow is among the phenomena occrring in the bondary region. Concentrated soltions of additives are continosly introdced into the bondary area where a thin layer of a non-newtonian liqid is formed in order to increase the effectiveness of the polymers addition. In this paper we consider a steady-state flow of a non-newtonian liqid film with a variable thickness on a flat plate nder the inflence of an incoming air flow, velocity vector of which coincides with the plate plane (Figre ). Let s sppose that the plate is at [alpha] angle to the horizon. Then a liqid flows de to gravity and friction on the oter srface of the film. Let s define the inflence of the film on the friction vale in the bondary layer. In general the problem is conjgate, inclding a problem of a film flow (internal problem) and a problem of the bondary layer of incoming air (external problem). To solve the dal problem a method of sccessive approximations is sed. The method consists in the fact that the external and internal problems can be solved separately, bt logically or iteratively. At each new approximation the inner problem is solved with regard to the friction reslting from the external problem, the soltion of which, in trn, takes into accont speed at bondary srface, obtained in the previos approximation from the interior problem. Ths, the iterative process contines ntil the speed and, conseqently, the friction at the phase interface change little from iteration to iteration. Fig.. Flow diagram. - film, - plate, - bondary layer, U incident flow velocity, g free fall acceleration, [alpha] - angle of inclination of the plate to the horizontal, x and y - Cartesian axis, L - length of the plate, vk mass addition speed Let the film thickness [delta] - m, and imposition of the film changes geometry of the plate slightly. Therefore, to solve the external problem film srface crvatre can be neglected. ISSN : 97- Vol 6 No Oct-Nov 68

2 . METHOD The task of the bondary layer. A mathematical formlation of the exterior problem incldes motion and continity eqations in the approximation of the bondary layer (index "" corresponds to the bondary layer, index "" - to the film) + v = ν () x v + = () x at bondary conditions: y =, = (x), v = v (x); y =, = U. () Bondary layer eqations are not applicable in the intermediate vicinity of the plate edge. Therefore, in the vicinity of a point x= initial (marked by index «s») velocity profiles and bondary layer thickness are defined. x =, =, v = v, = () s s s Let s introdce non-dimentional variables and save their former notations x y v x =, y =, =, =,v = () L L U U Sbstitting (.) into the eqation of motion (.), we obtain y v + =, (6) x Re UL where Re = - Raynolds nmber,, v - velocity projections on x and y axes correspondingly, - ν thickness, ν - kinematic viscosity, a stroke means derivative with respect to x. Continity eqation () will be as follows y v + =, (7) x bondary conditions () and () will be written over as follows y =, = / U, v = v U; y =, =, (8) x =, = / U, v = v / U, = / L (9) s s s The eqation of motion () and continity () correspond to the classical formlation (Blasis problem), and the bondary condition at infinity is replaced by an approximate one, namely, instead of y, U () is sed. Eqations (6) and (7) contain three nknown fnctions, v,. To close the system one more eqation is added, and for this prpose an eqation of motion over the bondary layer thickness is integrated. dy dy+ x () v + dy+ =. Re = y Dring the integration some members are dropped de to the se of the condition of smooth closing of the longitdinal velocity with the incident flow velocity / y = at y =, which is not contained explicitly in the classical formlation, bt was introdced in addition to the bondary condition (). Ths, the third linearly independent eqation is obtained (). ISSN : 97- Vol 6 No Oct-Nov 69

3 The problem for a flowing liqid film. In the approximation of the bondary layer theory let s write the eqation of motion and continity for incompressible flid + v = g sin α + ν, x v + =. x Let s establish a condition of adhesion and mass addition at the plate y =, =, v = vk, on the film srface at y = - friction obtained by solving the bondary layer problem τ = μ. Let s spplement the eqation of motion with a kinematic condition at the interface bondary v = () Let s introdce non-dimensional variables for the film (their notation are saved) x y x =, y =, =, =, L v =, τ = τ. vk μ v Taking into accont () the motion eqation () will be as follows ν g sin α ε + v = + x where and - thickness and longitdinal velocity scales correspondingly, ε = / L - small parameter. To determine the scale of thickness of and the longitdinal velocity of the film let s se the balance of gravity forces and viscos stress for a free draining film on a vertical wall (Baehr and Stephan, ). Then from the eqation (), scales connection has the following correlation ν g = () v L/ From the continity eqation () we obtain the scales connection =, and considering () we will have v Lν v L g k k =, =. g ν Zero-order approximation. Let s represent the nknown fnctions in expanded forms in powers of a small parameter, neglecting items of second and higher orders of smallness (hereinafter in the section sbscripts refer to the order of approximation and variables withot a sbscript correspond to the film) = + ε, v v v = + ε, ε velocity has the form = +, then in a zero approximation the bondary vale problem for the longitdinal = = = τ sin α,, y= y y= Let s write the integral (6) y y = sinα + τ y Let s define the transverse velocity profile, integrating the continity eqation, taking into accont (.7) and the bondary condition y =, v = k () () () (6) (7) ISSN : 97- Vol 6 No Oct-Nov 7

4 y v = ( τ + sinα ) (8) Sbstitting the non-dimensional variables () and then the expression (7) and (8) into (), after simple transformations, we obtain the eqation for ( ) sinα + τ x =, =. First approximation. Let s write the eqations of motion and continity v = Re + v, + =, x x which are spplemented with bondary conditions at the plate y = + ε = where, Re / ν = - Raynolds nmber. + p ( p ) y Re y =, =, v =, and on the film srface Let s expand the fnction in a series in the neighborhood of [delta] with the first order of accracy ( + ε) = ( ) + ε ( ) = () = ( ) + ε( ) + ε ( ) + ε ( ) Let s rewrite (9) sbstitting the expansion () and making the coefficients eqal at [epsilon] y =, = sinα. Let s sbstitte the nknown fnctions expanded in a series with the first order of accracy in the kinematic condition) v v y =, ( v + εv) + ε + ε = () = ( + ε) + ε + ε ( + ε) Eqating the coefficients at [epsilon] in (), and making some simple transformations, we obtain the final form of the kinematic condition y =, v = τ + Omitting the procedre of nknown fnctions finding in the first approximation, which coincides with the zeroorder approximation, let s write the soltion, y y y = Re p + ( p ) sinα (9) ISSN : 97- Vol 6 No Oct-Nov 7

5 p y ( p ) y v = Re sinα y + p ( p ) Re Re = ( p ) + p, q Where the notation p = τ + sin α, q = p+ + τ is introdced.. RESULTS Fig. and show graphs of variation of the friction coefficient cf τ ( ρu ) = and the longitdinal velocity U at the interface along the plate length at different Re and constant [alpha] and Re. Figre also shows the friction distribtion along the plate length calclated by the Blasis formla (Schlichting, 969). Fig. and show the variation of the interfacial friction coefficient and longitdinal velocity as a fnction of [alpha]. As expected, with increase of Re and inclination of the plate, the velocity at the interface increases, and the friction decreases. In fig. - all vales are provided taking into accont the first approximation. As for npblished reslts we shold note the inflence of Re on the process characteristics: with an increase of this parameter from its minimm vales to the critical ones ( Re ) the impact of the film on the frictional resistance decreases. Ths, when the frictional resistance Re = 6.7 in the presence of the film frictional resistance may many times vary depending on the angle of inclination and Re, whereas at Re =.7 the maximm difference will be no greater than %. /Блазиус Blasis/ Fig. Variation of friction coefficient at the phase contact area along the plate length at Re. Re. =, α = 9 and different ISSN : 97- Vol 6 No Oct-Nov 7

6 Fig. Variation of longitdinal velocity at the phase contact area along the plate length at different Re. Re. =, α = 9 and Fig. Variation of frictional coefficient at the phase contact area along the plate length at different α. =, Re = 99 and Re 6.7 Fig. Variation of longitdinal velocity at the phase contact area along the plate length at different α. =, Re = 99 and Re 6.7. DISCUSSION The system of integro-differential eqations (6), (7) an () of the bondary layer is solved sing a nmerically implicit scheme with the help of a method of delayed coefficients. In order to make sre that the selected method is correct the said system with bondary conditions of adhesion and impermeability was tested sing analytical soltion of Blasis (Schlichting, 969). The maximm difference between the estimated vale of frictional resistance and the exact soltion is not more than % in the immediate vicinity of the plate beginning and monotonically decreases to a vale less than.% at a length L. The difference between the velocity profiles does not exceed % over the entire length of the plate. Ths, the system of eqations (6), (7) and () ISSN : 97- Vol 6 No Oct-Nov 7

7 can be sed to calclate the characteristics of the bondary layer. In solving the dal problem analytical profiles of Blasis can be sed as the initial approximations s,vs, however linear profiles deliver good final reslts as well. It is known that a liqid film on the srface of a body blown over by an airflow changes characteristics of the bondary layer. The works of K.K. Fedyaevskiy (Fedyaevskiy, 9), (Fedyaevskiy, 9) and L.G. Loitsianskiy (Loitsyanskiy, 9) present two-media bondary layers, when a flid of one type is present or added in a wall-adjacent region and the whole system is placed in an incoming flow of a liqid of another type. In the work of (Loitsyanskiy, 9) a qestion regarding assessment of a possible gain in resistance de to introdction of a liqid to the bondary layer (mixing with the srronding liqid de to diffsion and convection), and conting of the reqired consmption of the liqid is considered. In the papers of (Shakhov et al., ) and (Shakhov et al., ) two-media bondary layer when the media do not mix are considered. It is assmed that applied throgh a streamlined srface medim is enclosed in a thin region where of the longitdinal velocity distribtion can be considered linear. In (Shakhov et al., ) a method of integral relations for the case of flow over a flat plate is sed, whereas in (Shakhov et al., ) a method of finite differences for an arbitrary body is developed.. CONCLUSION In general, the soltion to the dal problem depends on three parameters - Re,Re and angle α. The 6 soltion to the problem of a laminar bondary layer on a flat plate is limited by Re < and ( L) < > L laminar flow of a liqid film does not exceed Re = <, ( L) where < > L = ( Lyd, ) y ( L) - - average longitdinal velocity of the liqid film on the length L (Ktateladze and Styrikovich, 976) (Schlichting, 969). Therefore, the proposed mathematical model can be sed with the above restrictions. In fig. Re <. The analysis of the reslts allows to conclde that the liqid film on the srface of a body redces the frictional resistance. For blff bodies friction contribtion to the overall resistance of the body is small. On the contrary, for streamlined bodies, friction plays a defining role. Conseqently, when calclating the aerodynamic characteristics of a streamlined body the presence of a film on the body srface shold be taken into accont. LITERATURE [] A. Ahsan,. Evaporation, condensation and heat transfer, ISBN , pp: 8 [] JCY Koh, EM Sparrow and JP Hartnett. The two phase bondary layer in laminar film condensation, Int. J. Heat Mass transfer. Vol., pp Pergamon Press 96. [] HD Baehr, K. Stephan, Heat and Mass Transfer. Springer-Verlag Berlin Heidelberg, ISBN , pp: 77. [] K. Stephan. Interface temperatre and heat transfer in forced convection laminar film condensation of binary mixtres, International Jornal of Heat and Mass Transfer 9 (6) [] Ktateladze S.S., Styrikovich M.A. Hydrodynamics of gas-liqid systems. M., Energy, c. [6] Loytsyanskiy L.G. Abot resistance variations of bodies by way of filling of the bondary layer with flids with other physical constants // PMM. 9 T. VI. Isse, pp 9-. [7] Shakhov V.G., Wang Binbin, Ji Simei. Calclation of laminar bondary layer by integral method for two-flids pon flat plate / Aircraft indstry of Rssia. Problems and prospects. Proceedings of symposim with Intern. participation. Samara: Samara State Aerospace University,.pp. 8-. [8] Shakhov V.G., Xie Wei, Ji Simei. Two-flids bondary layers / Traffic management and aircraft navigation // Collection of scientific papers. XVI of All-Rssian scientific and technological seminar. P.. Samara: Samara State Aerospace University, pp.8 -. [9] Tets Fjii and Haro Uehara. Laminar filmwise condensation on a vertical srface, Int. J. Heat Mass transfer. Vol., pp. 7-. Pergamon Press 97. [] Fedyaevskiy K.K. Compressible gas trblent bondary layer skin friction // Works of Central Institte of Aerohydrodynamics. 9. Isse. 6. [] Fedyaevskiy K.K. Redction of frictional resistance by changing physical constants of the liqid near the wall // Izvestiya AN SSSR, Ser. OTN, 9, 9-. [] Schlichting G. Bondary layer theory. M.: Naka, p. [] Y.R. Mayhew and J.K. Aggarwal Laminar film condensation with vapor drag on a flat srface Int. J. Heat Mass transfer. Vol.6, pp Pergamon Press 97 ν ISSN : 97- Vol 6 No Oct-Nov 7