It means that M has singular points which forms the so-called break circle where the manifold loses smoothness. Also, we indicate that the Riemann tensor Rm = (Rijkl) contains only one essential component R1212 which do not vanish identically on the model manifold constructed.

Hence this manifold cannot be transformed into the Euclidean plane even locally. Finally, we investigate how terms of this family of quadratic forms influence the length scales of turbulent motion and demonstrate that the action of the two-parametric scaling group admitted by the von Krmn-Howarth equation in the limit of infinite Reynolds numbers and the one-parametric scaling group in the case of finite Reynolds numbers on the semi-reducible pseudo-Riemannian manifold constructed leads to the conformal invariance of the corresponding manifolds.

This work is the second paper of our series devoted to the geometric theory of isotropic turbulence [1].

This work was supported by IIP SB RAS (Grant No. 103).

References 1. Grebenev V. N., Oberlack M. A geometric interpretation of the second-order structure function arising in turbulence. Math. Phys. Anal. Geom. 2009, V. 12. N 1. P. 1–18.

Kolpakov A. G. ROLE OF NON DEGENERATED JOINTS IN TRANSPORT IN FRAMEWORKS A. G. Kolpakov Cassino University, Italy The problem of joint of structural elements traditionally attracted attention of engineers due to the important role of the joints in industry. In fact, they often determine the (mechanical, thermal, electrical, etc.) strength of structures and devices. In the last decades, the problem of joint attracted attention of mathematicians, see, e.g., [1, 2, 3].

The most of researches accepted ideal geometry and material inhomogeneity of the joints. These conditions are restrictive for practice (we remind that the basic types of joints: bolted, riveted and welded are inhomogeneous and have complex geometry). Now, there exists an interest to joint of molecular structures, e.g., nanotubes, which have significantly nonhomogeneous structure [4].

The analysis of joint of relatively complex shape was carried out for so called notched rods and beams. Paper [5] analyzes the general case of rods (beams) with local perturbation of diameter (but not with inhomogeneities, for example, holes). It predicts a nontrivial joint condition for degenerated joints (for example, if joint has thickness more less than the thickness of the rods, see also [6]).

Our paper concerns joints which have dimension compared with the thickness (diameter) of rods and material properties of the elements of the joint are different not strongly. We refer such joints as normal type joints to separate them from joints with deep notches [5, 6] and joints on the basis of soft glue layers [7] (which we referred as degenerated joints).

The results can be formulated in the form of two items:

1. The joints of normal type do not influence the property of the framework (in other words, singularities related to the joints do not manifest themselves on global level).

2. Since the joints do not manifest themselves on the global level, all the specific of the joints can be observed on the local level. Then, we pay attention to local field in and near joint. We show that the computation of this field is reduced to solution for an infinite rod subjected to uniform overall field. Since in the case under consideration strong localization of the perturbation of the local field [7] takes place, the infinity means, in accordance with our numerical computations, about 5 (maximum 10) diameters of the rod or periodicity cells. Thus, our method is easy realized with standard FEM software and it can be used in practice.

The research was supported through Marie Curie actions FP7, project PIIF2-GA-2008-219690.

References 1. G.P. Ciarlet. Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis.

Springer, Heidelberg, 1990.

2. H. Le Dret. Problmes Variationnels dans les Multi-Domaines. Modlisation des Jonctions et Applications. Masson, Paris, 1991.

3. A. Gaudiello, B. Gustafsson, C. Lefter and J. Mossino. Asymptotic Analysis of a Class of Minimization Problems in a Thin Multidomain. Calc. Var. 2002. vol. 15. N 2. p. 181–201.

4. M. Meyyappan (ed). Carbon Nanotubes: Science and Applications. CRC Press, Boca Raton, FL, 2004.

5. E. Cabib, L. Freddi, A. Morassi and D. Percivale. Thin notched beams. J. Elasticity. 2001.

Vol. 64. p. 157–178.

Kuibin P. A., Sharypov O. V. 6. J. Casado-Diaz, M. Luna-Laynez and F. Murat. The diffusion equation in notched beam. Calc.

Var. 2008. vol. 31. p. 297–323.

7. A.A. Kolpakov and A.G. Kolpakov. Capacity and Transport in Contrast Composite Structures:

Asymptotic Analysis and Applications. CRC Press, Boca Raton, FL, 2010.

DISSIPATION-INDUCED INSTABILITIES IN FINITE- AND INFINITE-DIMENSIONAL SYSTEMS R. Krechetnikov, J. Marsden University of California, USA In this talk a joint work with Jerrold Marsden on a coherent theory of the counter-intuitive phenomena of dynamical destabilization under the action of dissipation is presented. While the existence of one class of dissipation-induced instabilities in finite-dimensional mechanical systems was known to Sir Thomson (Lord Kelvin), until recently it has not been realized that there is another major type of these phenomena hinted by one of theorems due to Russian mechanician Merkin; in fact, these two cases exhaust all the generic possibilities in finite dimensions. We put the main theoretical achievements in a general context of geometric mechanics, thus unifying the current knowledge in this area and the multitude of relevant physical problems scattered over a vast literature.

Next we develop a rigorous notion of dissipation-induced instability in the infinite-dimensional case, which inherent differences from classical finite degree of freedom mechanical systems make uncovering this concept more intricate. In building this concept of dissipation-induced instability we found Arnold’s and Yudovich’s nonlinear stability methods, for conservative and dissipative systems respectively, along with some new existence theory for solutions to be the essential ingredients.

As a paradigm and the first infinite-dimensional example to be carefully analyzed, we use a twolayer quasi-geostrophic beta-plane model, which describes the fundamental baroclinic instability in atmospheric and ocean dynamics.

STRUCTURE OF FILM FLOW OVER PLATE WITH MOVING LOCAL HEAT SOURCE P. A. Kuibin, O. V. Sharypov Kutateladze Institute of Thermophysics, SB RAS, Novosibirsk Novosibirsk State University The work is devoted to theoretical study of the structure of gravity-driven liquid film flow in the presence of local heating by moving heat source. The film is supposed to be thin, and longwave approximation is used. The novelty of the problem statement is that both heat source motion and gravity-driven flow are taken into account. Unlike the previous works [1, 2], the temperature distribution at the free surface is unknown and the conjugate hydrodynamic and heat steady-state problem is solved under constant heat release and uniform temperature at the heater.

The hydrodynamic part of the problem was reduced to equation for film thickness in the accompanying frame [1, 2], which takes into account following factors: capillary, hydrostatic, thermocapillary, mass and inertia forces. This equation together with energy equation was solved numerically using finite-difference approximation and iteration method. As a result of numerical modeling it is shown that the changing of velocity profile (heat source velocity increase with slope Lavrenteva O. M., Rosenfeld L., Nir A. Fig. 1: The influence of heat source velocity on film deformation under constant flow rate, film thickness and heat release.

angle decrease under other equal conditions: fixed flow rate, film thickness and heat release) leads to dramatic amplification of thermocapillary deformation of the film, see Fig. 1.

The work was supported by RFBR (project No. 09-01-00765-а) and by Ministry of Education and Science of Russian Federation (Program “Development of high school scientific potential”, projects No. 2.2.1.1/1269, 2.1.2/1270, and Federal Target Program “Scientific and scientific-teaching personnel of innovation Russia”).

References 1. Sharypov O. V., Kuibin P. A. Thermal-wave-induced vorticity in a liquid film. Technical Physics Letters. 2008. Vol. 34. No. 10. P. 848–850.

2. Sharypov O. V., Kuibin P. A. Heat-wave induced vortex in a thin liquid layer. International Review of Chemical Engineering. 2009. Vol. 1. No. 2. P. 158–163.

MOTION AND DEFORMATION OF PARTIALLY ENGULFED COMPOUND DROPS O. M. Lavrenteva, L. Rosenfeld, A. Nir TECHNION Israel Institute of Technology Two-phase hybrid drops, which are comprised of immiscible phases, occur in various natural and technological processes and environments, e.g. the atmosphere, liquid membranes and liquid bilayers, direct contact heat transfer and phase separation processes. One phase of such an aggregate is completely or partially engulfed by the other one. A compound drop with partial engulfment has three interfaces between the components of the aggregate and facing the ambient fluid. At equilibrium, all three interfaces are segments of spheres. The angles at the three-phase contact line are determined solely by the ratios of the interfacial tensions, and the resulting configuration of the aggregate depends on the relative volumes of the drop’s components. Exact analytical solutions describing creeping motion of such a hybrid drop in an infinite viscous domain under the influence of Marangoni effect due to various temperature distributions are constructed in [1] and [2]. However, when the drop moves in a non-isothermal ambient medium, the interfaces are deformed due to viscous stresses and to the non-homogeneous surface tension. We assume that the surfaces deform as the drop moves through the ambient fluid, and consider the deformation Naumov I.V., Okulov V. L., Sorensen J. N. making use of perturbation method. It is assumed that the capillary numbers associated with all interfaces are relatively small, and thus, corrections of the solutions obtained in the undeformable case, can be constructed making use of a regular perturbation technique. The problem is reduced to a 6-th order system of ordinary differential equations with four boundary and two integral conditions, the latter reflecting conservation of mass for the two phases comprising the drop. Results concerning the axisymmetric deformation of drops undergoing Marangoni migration are presented for a variety of the physical parameters involved, such as viscosity ratios, initial configuration of the compound drop and temperature dependence of the surface tension of each interface. The cases of spontaneous thermocapillary migration and the motion in an externally imposed temperature gradient are considered. For the latter case the evolution of the interface of the drop propagating to hotter region is taken into account, while for spontaneous migration, the deformations are steady.

References 1. Rosenfeld L., Lavrenteva O. M., Nir A. Thermocapillary motion of hybrid drops. Phys Fluids 2008. V. 20, 072102.

2. Rosenfeld L., Lavrenteva O. M., Nir A. On the thermocapillary motion of partially engulfed compound drops. J. Fluid. Mech. 2009. V. 626, pp. 263–289.

MULTIHELIX VORTEX BREAKDOWN I.V. Naumov1, V. L. Okulov1,2, J. N.SorensenInstitute of Thermophysics, SB RAS, Novosibirsk, Russia Department of Mechanical Engineering, DTU, Lyngby, Denmark Vortex breakdown is a phenomenon inherent to many practical tasks of wing and rotor aerodynamics, because for example the tip vortex breakdown influences strongly on its lift and performance.

The breakdown of these vortices is associated with an abrupt deceleration of the axial velocity on the vortex axis which sometimes develops to a recirculation zone. The two predominant breakdown configurations over delta wing, the bubble and the spiral, were first identified by Lambourne & Bryer [1]. Now seven forms of vortex breakdown have been identified, and only one of them the double helix represents stable multiple form though the stability theory of the equilibrium array from helical vortices predicts an stable existence of doublet, triplet, and other multiplets from the helical vortices (with number N<7) [2, 3].

Visualization of double, triple and quadruple modes of vortex breakdown.

Here we report experimental observations of new multiple helix forms of vortex breakdown which have been discovered in the flow in cylindrical container with a rotating endwall. On the Okulov V. L., Sorensen J. N. basis of the recent solution of the Kelvin’s problem on stability of vortex polygons for helical vortices and idea to look the multiple vortex breakdowns for flows where the stable helical multiples may exist, we identified the well-known double helix mode and new stable forms: triple and quadruple helix modes of the vortex breakdown which was identified by two different type of visualization on two different setups and supported by PIV and LDA measurements. This work was supported in part by “Rosobrazovanie” (project no. 2.1.2/1270) and “Rosnauka” (contract no. 5099).

References 1. Lambourne NC, Bryer DW. The bursting of leading-edge vortices. Aeronautical Research Council, R and M 3282, 2. Okulov V. L. On the stability of multiple helical vortices // JFM, 2004. V. 521. p. 319–342.

3. Okulov V.L., Sorensen J.N. Stability of helical tip vortices in a rotor far wake // JFM. 2007.

V. 576, p.1–25.

OPTIMAL ROTORS by JOUKOWSKY and BETZ V. L. Okulov1,2, J. N. SorensenInstitute of Thermophysics, SB RAS, Novosibirsk, 630090, Russia Department of Mechanical Engineering, DTU, DK-2800 Lyngby, Denmark In the history of rotor aerodynamics two ’schools’ have dominated the conceptual interpretation of the optimum rotor. In Russia, Joukowsky [1] defined the optimum rotor as one having constant circulation along the blades, such that the vortex system for an N-bladed rotor consists of N helical tip vortices of strength and an axial hub vortex of strength -N. A simplified model of this vortex system can be obtained by representing it by rotating horseshoe vortices (Fig.1, left). The other school, which essentially was formed by Prandtl and Betz [2], assumed that optimum efficiency is obtained when the distribution of circulation along the blades generates a rigidly helicoidal wake that moves in the direction of its axis with a constant velocity. Betz used a vortex model of the rotating blades based on the lifting-line technique of Prandtl in which the vortex strength varies along the wingspan (Fig. 1, right).

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