Some solutions of the 3D Laplace equation in a layer with oscillating boundary describing an array of nanotubes with applications to cold field emission. II. Irregular arrays

As in the first part (J. Brüning, S.Yu. Dobrokhotov, D.S. Minenkov, 2011), we construct a family of special solutions of the Dirichlet problem for the Laplace equation in a domain with fast changing boundary. Using these solutions, we construct an analytic model of cold field electron emission from surfaces simulating arrays of vertically aligned nanotubes. Explicit analytic formulas lead to fast computations and also allow us to study the case of random arrays of tubes with stochastic distribution of parameters. We present these results and compare them with numerical approximations given in [1].     

Symmetry vector fields and similarity solutions of a nonlinear field equation describing the relaxation to a maxwell distribution

In the study of the formulation of Maxwellian tails the nonlinear partial differential equation ² u/x +u/x+u ²=0 arises. We determine the Lie point symmetry vector fields and calculate the similarity ansätze. Then we discuss the resulting ordinary differential equations. Finally, the existence of Lie Bäcklund vector fields is studied and a Painlevé analysis is performed.     

A solution of the sine-Gordon equation with boundary conditions appropriate to oscillating superfluorescence

A procedure is given for solving the sine-Gordon equation with boundary conditions which define the physical problem of superfluorescent emission from an inverted and spatially extended medium. The shape of the leading pulse is calculated. An important new result is a verification and extension of the form of the expression of MacGillivray and Feld and of Skribanowitz, Herman, MacGillivray and Feld for the delay time τ_D in oscillating superfluorescence.     

Extremely short electromagnetic pulses in an array of carbon nanotubes with a longitudinal field inhomogeneity

The propagation of bipolar electromagnetic pulses in an array of semiconductor carbon nanotubes has been investigated. The inhomogeneity of the pulse field along the axis of the nanotubes has been taken into account for the first time. The evolution of the electromagnetic field and charge density in the sample has been described by the set of Maxwell’s equations and the continuity equation. The possibility of stable propagation of bipolar electromagnetic pulses occurring in an array of nanotubes has been demonstrated by numerical simulation. It has been shown that the propagation of the electromagnetic pulses induces the redistribution of the electron density in the sample.     

Unsteady-state nonlinear processes accompanying the interaction of an electron beam with an electromagnetic field near the boundary of a transmission band. I. The high-frequency boundary

Saratov State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 31, No. 2, pp. 207–221, February, 1988.     

Interference heat emission in an absorbing layer in the presence of a two-wave field

The control of the interference heat emission upon the oblique incidence of two counterpropagating (with respect to the transverse component of the wavevector) identically linearly polarized coherent waves on the opposite sides of an absorbing layer is considered. The dependences of the interference heat emission on the angle of incidence and the layer thickness are established for various refractive indices and absorption coefficients in weakly and strongly absorbing media. The conditions for the maximum interference heat emission are determined.     

On the thickness of a boundary layer related to the oscillating free surface of a charged drop of a viscous liquid

The capillary oscillations of a charged drop of a viscous liquid are calculated in terms of the boundary layer theory in an approximation linear in oscillation amplitude. Calculation is accompanied with the estimation of a relative error that arises when the exact solution is replaced by an approximate one. It is shown that, for the calculation accuracy in the framework of the boundary layer theory to be about several percent, the thickness of the boundary layer near the free surface of the drop must be several times larger than that at which the intensity of the eddy flow caused by the oscillating surface decreases by e times. As the viscosity of the liquid grows, so does the thickness of the boundary layer.     

Trend to equilibrium of weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions

Recently R. Illner and the author proved that, under a physically realistic truncation on the collision kernel, the Boltzmann equation in the one-dimensional slab [0, 1] with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense. Here it is proved that when the Maxwellians associated with the boundary conditions atx=0 andx=1 are the same MaxwellianM _w , then the solution is uniformly bounded and tends toM _w fort.     

Response of an open curved thin shell to a random pressure field arising from a turbulent boundary layer

A method is presented to predict the root mean square displacement response of an open curved thin shell structure subjected to a turbulent boundary-layer-induced random pressure field. The basic formulation of the dynamic problem is an efficient approach combining classic thin shell theory and the finite element method, in which the finite elements are flat rectangular shell elements with five degrees of freedom per node. The displacement functions are derived from Sanders’ thin shell theory. A numerical approach is proposed to obtain the total root mean square displacements of an open curved thin structure in terms of the cross spectral density of random pressure fields. The cross spectral density of pressure fluctuations in the turbulent pressure field is described using the Corcos formulation. Exact integrations over surface and frequency lead to an expression for the total root mean square displacement response in terms of the characteristics of the structure and flow. An in-house program based on the presented method was developed. The total root mean square displacements of a curved thin blade subjected to turbulent boundary layers were calculated and illustrated as a function of free stream velocity and damping ratio. A numerical implementation for the vibration of a cylinder excited by fully developed turbulent boundary layer flow was presented. The results compared favorably with those obtained using software developed by Lakis and Païdoussis (J. Sound Vib. 25 (1972) 1–27) using cylindrical elements and a hybrid finite element method.     

Soliton solutions of the non linear Schrödinger equation with an external time-dependent field

A functional transformation between solutions of the one-dimensional nonlinear Schrödinger equation with time- and coordinate-dependent coefficients and solutions of the conventional nonlinear Schrödinger equation (NSE) is constructed. Exact solutions of the NSE with a homogeneous time-dependent external electric field and the NSE with oscillator potential are obtained.     

Two-dimensional solutions of the Dirac equation for the motion of an electron in a helical magnetic field

Quantum theory of a free electron in a helical wiggler magnetic field has been studied. In the previous studies the transverse momentum operators in the Dirac hamiltonian were ignored. In this note, the 2-D solutions of the Dirac equation, based on a hamiltonian which includes one of the transverse momentum operators (p_x ≠ 0, p_y = 0) and the wiggler field as an external potential is derived. It is shown that there are solutions in terms of the Mathieu functions of fractional order.     

Extremely short two-dimensional optical pulses in an array of carbon nanotubes in the presence of a constant electric field

The interaction between the electromagnetic fields of extremely short two-dimensional optical pulses propagating in an array of zigzag-type carbon nanotubes and an external constant electric field is investigated. The evolution of the investigated system’s electromagnetic field is described by Maxwell’s equations.     

On stability of stationary soliton solutions to the nonlinear field equation in a general spinor model

The problem of determining the stability domain (in Lyapunov sense) of three dimensional soliton solutions is considered. Some necessary conditions for stability are obtained and it is shown that the boundary of the stability domain is defined by the inequality ωⁱω^k(?Qⁱ/?ω_k < 0.     

Formation of the turbulent boundary layer at air blowing through a wall with an abrupt change in boundary conditions

Development of an incompressible turbulent boundary layer with air blowing through a finely perforated flat surface, consisting of a permeable region and impermeable region behind, was studied experimentally. The mass flow rate of injected air Q per an area unit was varied from 0 to 0.2 (kg/s)/m². Detailed data about the internal structure of the boundary layer in the flow region, characterized by an abrupt change in the flow conditions at the boundary of permeable and impermeable regions, were obtained. A consistent decrease in the local values of skin friction coefficient along a permeable sample and with an increase in the values of Q, reaching 90% at maximal Q, is shown. The role of the flow region behind the zone with an abrupt change in the boundary conditions, essential from the viewpoint of skin friction reduction, is revealed.