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December 22, 2016
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July 31, 2009
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April 18, 1958
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USAF review completed. Ey the work of rrandtl and his school the theory of boundary layers has become one of the universal methods in the mechanics of the continua during th,u last decades. This theory, however, to-day is a.,plied for solving several questions which are not directly connected with the problem of motion of viscous fluids. In the SSSF: the boundary layer theory has bu:en applied during the last time for investigations of the following kind : V (U,-. Cernij). 2) Calculation of instationary filter processes (r.Ja. Kocin, 3) Calculation of instationary heat transfer for a heat-transfer coefficient depending on the temperature (Ju.IT. Dem'janov). The aim of this report is to call the researcher's attention still to another field of application of the boundary layer theory. "ere I mainly think of the application of the boundary layer theory to the so-called "double-velocity hydro- and aerodynamics" according to Landau's terminology. In order to clear the problem I :!rant to give a survey on some concrete tasks of double-velocity hydra- and aerodynamics which were formulated corresponding to the boundary layer problems : 1) The flow of a homogeneous fluid over a porous surface. In this case the task can be formulated in such a way that a part of the ideal fluid with the mean density y1 and the speed V~ flows through the surface, while an2other part with the mean density 2 and the speed V2 flows along. If it is put = p1 + K2 as real density, then the boundary layer conditions are evidently (1.1) k VIr = p (1.2) V2n = 0 furthermore for (1 .3) V1 V 2 -a Vo x + y2 + z' oo it holds : Although such a double-velocity state is everywhere theoretically possible, it really occurs in that range only which lies directly on the porous wall. Fur ideal or viscous fluids the boundary layer near the wall must be investigated. This is, however, not the classi- cal boundary layer problem for which the flow is given in the layer, but a case in which the boundary layer and the external flow mutually influence each other - similar to the case of high supersonic veloci- ties. )'hereby the flow around a porous plate in supersonic must be solved as the first task. 2) The problem of porous cooling. In its character this problem does not differ from the first one; only the condition (1.1) must be replaced by a corresponding other one. 3) The problem of the boundary-layer control. After the things mentioned above it is not necessary to explain this in detail. 4) Furthermore the double-velocity hydrodynamics is of importance for the motion of such liquids and gases which are saturated with macroscopic dust particles. Here the question is essentially a generalization of the filter theory. The last case shall be discussed somehat more detailed. The deduction of the differential equations for the boundary layer in a liquid with double velocities : I think that we can consider with good reason liquids and gases saturated with macroscopic dust particles as a mixture of several continuous media. (Footnote Prof. Voronec, '3elgrad called my attention to the possibility to consider the mentioned media as continuous. This is most suitable also in the sense of a verdict of Zukovskij : "Mechanics is the art to set up integrable differentil equations".) It is evident that the tension and deformation relations are valid for each of the media taking part in the motion. (Footnote: see Pahma_tulin : "Foundations of gas dynamics of mutually penetrat- ing motions of compressible media", Fri'-:lad.IMIat.L1ech.20, 1956. A su:.i:-ary of the results of this paper was published in the reports of the Congress for Theoretical and Applied Llechanics in Brussels. - u i div Vi correspondingly iy oui dv. i=/`i( ) xy d y x 171 correspondingly r iyz and ?ixz under the penetration of two media the index i runs from 1 to 2. We introduce the expressions : f1 f 41w 2 ~2w where ~1 and ~2 are the mean densities and 41w and ~ 2w the real ones of the two media. The following equations of :-potion are obtained with the aid of these expressions : dui fi - + a f1 ,, . 1 (2 at c)x d x - 3 div Vi) 3y (correspoudinzcly also an equation with vi) u (Qiui) G(9 y ) + = U ax N y Yfi = 0 if the densities q and e2 are assumed to be constant in the equations of motion so tnat the condition f1 + f2 = 1 is satisfied, then six equations with five unknowns remain over. Therefore the mean densities Q1 and ~)2 are not allowed to be assumed as constant in a double-velocity fluid, each component of which is incompressible. Nevertheless this assumption for the motions in the boundary layer does not lead to a redundancy in determination of the ;)roblem, since in this case two equations drop out. Namely, if it is put Q1 = const and const and, if it is estimated for the motions in the G b.,urdary layer, then it remains F2 du2 = - f2= + i Y2 2 + I{(u,i - u2) dt 1' x --y 2 du 1~ dt1 f1 L'" + f1 /41 3 2r + IC(u2 - u1) u, V 0 Pnus one obtains four equations for four unknowns. If the pressure gradient vanishes, then the integral relations of v.:Kurman can be applied and one obtains according to Fohlhausen r1 [. S1 - (32 )3 i for 0 < y 17 r3 _ 1,y )3 J ~~ t2 3' '2 J 0 E y < 2 u2 = U for 0 2 \< y ,< d -_ 1 _ 2 + 22?2(5-6A + Ili) with Il =2 ?1 Approved For Release 2009/07/31 : CIA-RDP80T00246AO03500230002-6 It ccn be shown that for 2< 'i it always holds 2/-2+ ' 2c2(5 - 6 1+3 ) p1(r1 - 2) d; with 2 = 2 1 dc' 1 = f(c1) with iure 1 shows a picture of I the isoclines. From this it ~zl 2 = 2(.1) lies below a straight line. A detailed cal- culation shows that 2 very quickly tends to C1 and both tend to a value a which corresponds to the boundary- layer thickness of a liquid with the density + If a pressure gradient exists, then u1 and u2 and consequently also - 1 and 2 are different from each other in every arbitrary coint. Therefore a double-velocity theory in this case surely gives results which differ from the corresponding results in a one- comnonent medium. Approved For Release 2009/07/31 : CIA-RDP80T00246AO03500230002-6 In his former paper on "The origin of vortices in ideal fluids" Ludwig Prandtl directed to the possibility and suitability of investigating the origin and further development of spiral sur- faces of discontinuity on shelves, particularly the similar flows of this kind. Investigations of this form of motion which I shall denote as "second form of motion", have become particularly necessary during the last time, since the forces acting on thin wings of small aspect ratio for an afflux in fluids or gases under nonvanishing angle of incidence are mainly determined by the second form of motion. Furthermore the flow around obstacles by impact waves even depends on the second form of motion. If the body with borders or edges which is assumed to be two- dimensional rests in the moment t = 0 and moves for t) 0 according to the law v = vo(t), then the second form of motion develops on the edges for t t