Scattering of a plane monochromatic wave by a system of strips

Summary The diffraction of a plane wave by a general system of strips is treated by means of separation of variables. A new addition theorem for Mathieu functions is used to satisfy the boundary conditions on the strips. Numerical calculations are performed for the case of two strips lying in the same plane, the boundary conditions being grad _n =0, while the angle of incidence of the plane wave is arbitrary. The transmission coefficient for a system of two slits in a plane, perfectly conducting screen is calculated for a range of values of the parameters. An approximate expression relating the transmission coefficient for the system of two slits to the transmission coefficient for a system of a single slit is given. As the distance between the two slits is increased, the transmission coefficient for the system of two slitsvery rapidly becomes nearly identical with the transmission coefficient for a single slit.     

Three-dimensional problem of unsteady wave motions of a liquid in a region of variable depth

The three-dimensional problem of unsteady wave motions of a liquid above a plane inclined floor in the framework of a linear dispersion model was considered for the first time in [1] for the particular case =/4, where is the angle of inclination of the floor plane to the free surface of the liquid. The class of exact self-similar solutions of the problem for =/2(2m + 1), m=0, 1, 2,..., for the case of an initial perturbation of a free surface of a special type which is constant in the direction of the normal to the shoreline was found in paper [2]. The present paper is devoted to the investigation of wave motions of a liquid due to an initial perturbation of arbitrary form for the angles of inclination of the floor assumed in [2]. A complete system of eigenfunctions corresponding to the continuous and discrete spectra is found. The theorem of the expansion of an arbitrary absolutely integrable function with respect to the boundary values of the eigenfunctions is proved. An exact solution of the problem is obtained and its asymptotic analysis is carried out.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–112, March–April, 1986.     

Localized shear discontinuities near the tip of a mode I crack

In this paper we study the deformation and stress fields near the tip of a crack under plane strain mode I conditions. A fully nonlinear theory of finite deformations is used and the material, which is assumed to be homogeneous, isotropic, incompressible and elastic, is characterized by its stress-strain behavior in simple shear. For the class of materials considered the governing system of differential equations may lose ellipticity at sufficiently severe strains. The analysis is based on a direct asymptotic calculation. The results involve two curves, issuing from each crack-tip, across which the deformation gradient, the effective shear and the stresses are discontinuous.     

Three-dimensional fluorescent particle tracking at micron-scale using a single camera

This article reports a new approach to track (x, y, z, t) coordinates of multiple fluorescent particles (diameter range 1–10 m) simultaneously using a quantitative defocusing method. We find that the defocused image of a 1-m diameter fluorescent particle formed by the objective lens of a conventional microscope has a bright outer ring due to the spherical aberration of the lens system. The ring radius increases as the particle is moved away from its reference plane and closer to the lens. The reference plane refers to locations of the particle at which the projected image is in focus. The (x, y, z) coordinates of the particle are then inferred from the center location of the image ring as well as the ring radius. The described technique is implemented successfully for obtaining 3D trajectories of swimming Escherichia coli cells.     

Evolution of Domains of Attraction of a Forced Beam with Two-Mode Interaction

When a dissipative dynamical system has multiple attractors, it is a task to determine and recognize the global domain of attraction of each attractor. In this paper we study the global behavior of a forced hinged-clamped beam with two-mode interaction. The equation of motion of the beam is reduced to four first-order autonomous ordinary differential equations. The system has an internal resonance condition of 2 31, where 1 and 2 denote natural frequencies of the first and second modes, respectively. When the excitation frequency is near 1, the system can have three equilibrium solutions, among which two are asymptotically stable and one is unstable. We examine how the domains of attraction of two stable equilibrium solutions evolve as the forcing frequency is varied across jump points. By using a special plane which contains all equilibrium points, called the principal plane, the global domains of attraction can be discussed more effectively. Results show that knowledge of this evolution helps us better understand the jump phenomenon.     

Some exact solutions of the equations of a stationary monoenergetic beam of charged particles

We consider the class of invariant solutions which can describe only vortex flows (curl P 0, P is the generalized momentum) and show that they contain solutions corresponding to flows from a plane or cylindrical emitter with a linear voltage drop across it (direct heating) in the temperature-limited regime^*. The solution is obtained in analytic form for emission from a plane in a uniform magnetic field perpendicular to the flow plane. It also (for=0) defines a plane magnetron in the T-regime. The solution of the problem for a cylindrical emitter reduces to considering equations describing a cylindrical diode or magnetron in the T-regime, where the shape of the collector is given by the potential distribution curve for these cases. We can extend the results to a relativistic beam if restrictions are imposed on its relative dimensions which permit us to ignore the magnetic self-field. Brillouin type flows (including irrotational ones) are studied in which particles move without intersecting the equipotential surfaces along three-dimensional spirals on the surface of cones. An analytic solution is given for relativistic Brillouin flow in a conical diode when strict allowance is mede for the magnetic self-field.     

Investigation of a “hanging” shock wave near a generating point

The hodograph method is used to plot a hanging shock wave in the plane nonequilibrium supersonic flow of an ideal gas. This paper considers the general case of an analytical solution in the plane of the hodograph at the point of generation of the shock wave. A type of limiting line is established which makes it possible to plot a shock wave (it is found that the shock wave may not extend over the whole flow, with a convolution in the physical plane).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 30–37, November–December, 1971.     

Development of the region of a turbulized liquid in a stratified medium

The article discusses the plane unsteady-state problem of the development of a region of turbulent pulsations in an incompressible stratified liquid. At the initial moment of time, the energy of the turbulence is given inside a region of finite dimensions. A semiempirical system of equations describing this process is proposed. The article gives the data from numerical calculations, illustrating the original expansion of the region as a result of turbulent diffusion, its subsequent compression along a vertical (collapse) under the action of the forces of buoyancy, and the internal waves generated by the collapse.The work was reported at the International Symposium on Stratified Flows (Novosibirsk, August 29–31, 1972).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 45–52, May–June, 1974.     

螺桨转子颤振涡动的一种算法

介绍螺桨转子系统颤振涡动的一种算法。本算法考察桨叶的振动变形，桨轴－支承系统的振动特性，以及螺转子的不平衡力。利用本算法对一模型螺桨转子和一实际螺桨转子进行了分析计算，对螺桨转子轴头动刚度、浆叶弹性变形、桨盘不平衡度等因素对螺桨转子系统颤振涡动的影响进行了分析，表明螺桨转子轴头动刚度下降时，螺浆发生颤振涡动的极限飞行速度下降。模型螺桨转子的计算结果与风洞试验结果相当吻合，证明本算法是正确的。     

Singularity Curves of a Parallel Pointing System

This paper studies a parallel pointing system used in aerospace applications for orientating parabolic antennas. In the literature [6, 7], the position analysis of this device has been already solved in closed form, whereas simple and efficient tools to address singularities distribution are still lacking. In this paper, its velocity analysis is addressed, and a singularity locus analytic expression, containing the manipulator geometric parameters and the end-effector orientation parameters, is derived. Moreover, it is shown that the determined singularity locus can be represented by curves (singularity curves) on a Cartesian plane having the generalized coordinates of the mechanism on the coordinate axes. Finally, an example showing the use of the proposed relationships is given and discussed.     

Determination of current flow in a plasma by a simulation method

A method is proposed which, for specific assumptions, allows us to determine the density distribution of a constant current flowing between electrodes in a plasma for plane parallel or radially symmetric electric and magnetic fields, allowing for anisotropic conductivity.Notation e_r, e, e_z unit vectors in a cylindrical coordinate system - E, e_r, e_z electric field strength vector and its components - V electric field potential - H, H_r, H, H_z magnetic field strength and its components - j current density vector - e electron charge - m electron mass - c velocity of light - momentum transfer time - ₀ normal plasma conductivity - _e electron cyclotron frequency - h unit vector in the direction of the magnetic field     

Small parameter method for determining motion of a viscous incompressible fluid in a thrust bearing

We consider the plane stationary motion of a viscous incompressible fluid between two surfaces. The fixed surface is given by the equation y=h[1+f(x/h)], where the functionf(x/h=h) characterizes the deviation of the fixed surface from the plane y=h(h and , are constants). The moving surface is a plane which moves with constant velocity along the x axis and remains parallel to the plane y=h. The small parameter method is used to solve the problem. The problem formulation is presented in the first section, the solvability of the linear equations obtained using the small parameter method is investigated in the second section, and the third section studies the convergence of the method and finds the radius of convergence of the constructed series.     

Yield stress of a magnetic fluid in a magnetic field

The yield stress ₀ of a magnetic fluid in a plane channel and the shape of the chains restraining the motion of the fluid are determined. The equilibrium problem for a magnetic fluid in a plane channel in the presence of an external magnetic field perpendicular to the plane of the channel and a pressure difference between the channel ends is solved within the framework of the structured fluid model. It is shown that equilibrium is possible only when the pressure difference does not exceed a certain limit p _cr at which the shear stress on the channel wall has a maximum. In weak fields p _cr and the corresponding ₀ depend quadratically on the field and in strong fields tend to saturation. The phenomenological parameters of the model are estimated for the case in which the microstructure is a system of chains of magnetic particles. The results obtained are found to be in qualitative agreement with the experimentally observed dependence of p _cr and ₀ on the field and the magnetic phase concentration.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 4–10, May–June, 1992.The authors are grateful to V. V. Gogosov for useful discussions and his interest in the work.     

Spots and turbulent domains in a model of transitional plane Couette flow

A review of the globally subcritical transition to turbulence in shear flows is presented, with an emphasis on the cases of plane and circular Couette flows (pCf and cCf, respectively). A Swift–Hohenberg-like model is next proposed to interpret the behavior of plane Couette flow in the vicinity of its global stability threshold. We present results of numerical simulations supporting this proposal and helping us to raise good questions about the growth and decay of intermittent turbulent domains in this precise context, and more generally about the coexistence of laminar flow and turbulence in other spatio-temporally intermittent flows. PACS 47.27.-i, 47.54.-r, 05.45.-a     

Stability of plane oscillations of a satellite in a circular orbit

We study motions of a rigid body (a satellite) about the center of mass in a central Newtonian gravitational field in a circular orbit. There is a known particular motion of the satellite in which one of its principal central axes of inertia is perpendicular to the orbit plane and the satellite itself exhibits plane pendulum-like oscillations about this axis. Under the assumption that the satellite principal central moments of inertia A, B, and C satisfy the relation B = A + C corresponding to the case of a thin plate, we perform rigorous nonlinear analysis of the orbital stability of this motion.In the plane of the problem parameters, namely, the oscillation amplitude ε and the inertial parameter, there exist countably many domains of orbital stability of the satellite oscillations in the linear approximation. Nonlinear orbital stability analysis was carried out in thirteen of these domains. Isoenergetic reduction of the system of equations of the perturbed motion is performed at the energy level corresponding to the unperturbed periodic motion. Further, using the algorithm developed in [1], we construct the symplectic mapping generated by the equations of the reduced system, normalize it, and analyze the stability. We consider resonance and nonresonance cases. For small values of the oscillation amplitude, we perform analytic investigations; for arbitrary values of ε, numerical analysis is used.Earlier, numerical analysis of stability of plane pendulum-like motions of a satellite in a circular orbit was performed in several special cases in [1–4].     

Bifurcations of two-dimensional into three-dimensional wave regimes for a vertically flowing liquid film

The three-dimensional steady traveling wave regimes of a viscous liquid film flowing down a vertical wall which branch off from two-dimensional nonlinear waves are investigated. The numerical calculations are based on a model system of equations valid for intermediate Reynolds numbers. It is shown that there exist two fundamentally different types of three-dimensional steady traveling waves branching off from plane waves. One of these possesses checkerboard symmetry in the distribution of the maxima of the wave profile thickness and is the more interesting. An important difference in the breakdown of plane waves of the first and second families is also demonstrated. The wave characteristics of certain three-dimensional regimes are calculated as functions of the bifurcation parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 109–114, September–October, 1990.     

Experimental investigation of a three-dimensional viscous underexpanded jet

The three-dimensional interaction of jet issuing from two- and four-nozzle systems into ambient space or an outer flow has been investigated experimentally. The range of the important parameters include the following: pressure imbalance n=P_a=/P=10–1.5·10², Mach number at the nozzle exit M_a=3.15, Mach number of the outer flow M=0, 3.1, and 6, the flow is turbulent in the mixing layer (P_a and P are the static pressures at the nozzle exit and in the outer flow). It is shown that the interaction of the jets broadens a multinozzle jet considerably in the plane of interaction, which is a plane of symmetry and which passes through the axis of the system between neighboring nozzles. The cross-sectional shape of a four-nozzle jet is cross-like over the entire length of the initial segment of the jet. The width of the mixing layer in the plane of interaction is considerably larger than in the central plane, which passes through the axis of opposed nozzles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 21–26, September–October, 1974.     

Resonance Capture in a Three Degree-of-Freedom Mechanical System

We study the phenomena of resonance capture in a three degree-of-freedom dynamical system modeling the dynamics of an unbalanced rotor, subject to a small constant torque, supported by orthogonal, linearly elastic supports, which is constrained to move in the plane. In the physical system the resonance exists between translational motions of the frame and the angular velocity of the unbalanced rotor. These equations, valid in the neighborhood of the resonance, possess a small parameter which is related to the imbalance. In the limit 0, the unperturbed system possesses a homoclinic orbit which separates bounded periodic motion corresponding to resonant solutions from unbounded motion which corresponds to solutions passing through the resonance. Using a generalized Melnikov integral, we characterize the splitting distance between the invariant manifolds which govern capture and escape from resonance for 0. It is shown that as certain slowly varying parameters evolve, the separation distance alternates sign, indicating that both capture into, and escape from resonance occur. We find that although a measurable set of initial conditions enter into a sustained resonance, as the system further evolves the orientation of the manifolds reverses and many of these captured solutions will subsequently escape.     

Untersuchungen über den Einfluß einer Heckschraube auf die Druckverteilung und die Grenzschicht schiffsähnlicher Körper

Übersicht Es wird der Einfluß einer Heckschraube auf das Strömungsfeld am Schiffsrumpf untersucht. Als Ersatzkörper für das Schiff dienen verschiedene Rotationskörper und die längsangeströmte ebene Platte. An diesen Modellen werden die Druckverteilung, die Grenzschicht und die Ablösung unter Einfluß der Heckschraube berechnet und im Windkanal gemessen. Dabei ergibt sich für den Normalsog und den Reibungssog eine gute Übereinstimmung zwischen Theorie und Messung, obwohl das Strömungsfeld im einzelnen von der Theorie noch nicht vollständig erfaßt wird. Sowohl der Normalanteil als auch der Reibungsanteil des Soges nehmen mit wachsendem Propellerabstand und fallendem Propellerschub ab. Der Reibungssog ist im Vergleich zum Propellerschub immer sehr klein.

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Summary The influence of a stern mounted propeller on the flow field past a hull is investigated. The hull is replaced by different bodies of revolution and a flat plate respectively. For these models the pressure distribution, the boundary layer and the flow separation are measured in a wind tunnel and are calculated, when the propeller is in action. The agreement between the theoretical and the experimental values of normal and viscous thrust deduction is quite satisfactory, though the flow field in detail is not described completely by the theory. Both the normal and the viscous part of thrust deduction decrease with increasing distance between propeller and hull and with decreasing propeller thrust. The viscous part of thrust deduction is always very small in comparison to the propeller thrust.

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Gekürzte Fassung der von der Fakultät für Maschinenwesen der TU. Braunschweig genehmigten gleichnamigen Dissertation [23]. Berichter: Prof. Dr. phil. H. Schlichting und Prof. Dr.-Ing. K. Gersten. Die Untersuchungen wurden von der Deutschen Forschungsgemeinschaft im Rahmen des Schwerpunktprogramms Schiffbau unterstützt.