Analytical study of nonlinear vibrations of a charged viscous liquid drop

The generatrix of a nonlinearly vibrating charged drop of a viscous incompressible conducting liquid is found by directly expanding the equilibrium spherical shape of the drop in the amplitude of initial multimode deformation up to second-order terms. A fact previously unknown in the theory of nonlinear interaction is discovered: the energy of an initially excited vibration mode of a low-viscosity liquid drop is gradually (within several vibrations periods) transferred to the mode excited by only nonlinear interaction. Irrespectively of the form of the initial deformation, an unstable viscous drop bearing a charge slightly exceeding the critical Rayleigh value takes the shape of a prolate spheroid because of viscous damping of all the modes (except for the fundamental one) for a characteristic time depending on the damping rates of the initially excited modes and the further evolution of the drop is governed by the fundamental mode. In a high-viscosity drop, the rate of rise of the unstable fundamental mode amplitude does not increase continuously with time, contrary to the predictions of nonlinear analysis in terms of the ideal liquid model: it first decreases to a value slightly differing from zero (which depends on the extent of supercriticality of the charge and viscosity of the liquid), remains small for a while (the unstable mode amplitude remains virtually time-independent), and then starts growing.     

Numerical analysis of decaying nonlinear oscillations of a viscous liquid drop

An adaptive grid numerical model is developed for simulating the dynamics of a viscous liquid drop whose initial shape is strongly disturbed by an external field. Simulated oscillations of a drop in microgravity and on a horizontal surface are compared with available numerical and experimental results.     

Stability of a charged drop of a viscous electrically conducting liquid in a viscous electrically conducting medium

A dispersion relation is derived for capillary oscillations of a charged electrically conducting viscous drop in an electrically conducting viscous medium. It is shown that aperiodic instability of the charged interface between the two media can arise in this system, with a growth rate that depends qualitatively differently on the ratio of their conductivities in different ranges of values of this ratio. In a certain range of conductivity ratios the drop undergoes oscillatory instability. Zh. Tekh. Fiz. 69, 34–42 (October 1999)     

On the oscillations of a charged drop of a finite-conductivity viscous liquid

A dispersion relation for the capillary oscillations of a charged spherical drop of a viscous incompressible finite-conductivity liquid is derived and analyzed. It is found that electric currents inside the charged drop equalize its potential and produce liquid flows interacting with both potential and eddy poloidal liquid flows inside the drop that are due to drop oscillations. Taking into account the finiteness of the rate of potential equalization over the drop surface leads to an additional damping of the capillary oscillations that arises because of the increased role of energy dissipation.     

On the thermophoretic motion of a heated spherical drop in a viscous liquid

Expressions for the force and velocity of the thermophoretic motion of a spherical drop in a viscous liquid are derived for arbitrary temperature differences between the surface of the drop and regions away from it. The temperature dependence of the viscosity is taken into account in the form of an exponential-power series.     

On nonlinear resonant four-mode interaction between capillary vibrations of a charged drop

Energy transfer from higher modes of capillary vibrations of an incompressible liquid charged drop to the lowest fundamental mode under four-mode resonance is studied. The resonance appears when the problem of nonlinear axisymmetric capillary vibration of a drop is solved in the third-order approximation in amplitude of the multimode initial deformation of the equilibrium shape of the drop. Although the resonant interaction mentioned above builds up the fundamental mode even in the first order of smallness, its amplitude turns out to be comparable to a quadratic (in small parameter) correction arising from nonresonant nonlinear interaction, since the associated numerical coefficients are small.     

On the motion of a uniformly heated drop in a viscous liquid under gravity

The motion of a uniformly heated spherical drop under gravity is theoretically studied within the Stokes approximation. The Stokes and Hadamard-Rybchinsky formulas are generalized so that the temperature dependence of the viscosity can be found in a wide temperature range. Also, the drag force and the velocity of gravity fall are calculated for an arbitrary temperature difference between the surface of the drop and distant points.     

Boundary layer theory modified for analyzing finite-amplitude oscillations of a viscous liquid charged drop

The existing concepts of the boundary layer arising near the free surface of a viscous liquid, which is related to its periodic motion, are revised with the aim to calculate finite-amplitude linear oscillations of a viscous liquid charged drop. Equations complementing the boundary layer theory are derived for the vicinity of the oscillating free spherical surface of the drop. An analytical solution to these equations is found, comparison with an exact solution is made, and an estimate of the boundary layer thickness is obtained. The domain of applicability of the modified theory is defined.     

On the possibility of corona-discharge ignition in the vicinity of a weakly charged drop executing nonlinear vibrations

An analytic expression for the electric-field strength in the vicinity of a charged drop of an electrically conducting liquid is obtained for the case where the initial shape of the drop executing nonlinear vibrations is specified by a virtual excitation of an arbitrary single mode of capillary vibrations. It turns out that, even at small charges (such that the Rayleigh parameter for the drop is equal to one-tenth of the critical value associated with stability against the intrinsic charge), the electric-field strength at the drop surface in the case of an initial excitation of one of high modes is sufficient for the ignition of a corona discharge.     

On nonlinear vibrations of a charged drop in the third-order approximation in the amplitude of initial single-mode excitation

Nonlinear axisymmetric motions of the free surface of a charged drop of an ideal liquid under the single-mode initial deformation of its equilibrium shape is investigated in the third-order approximation in the initial perturbation amplitude. An analytical expression for the drop shape generatrix is derived. Nonlinear corrections to the vibration frequencies for the initial perturbation of an arbitrary mode are found for the first time. The effect of vibration nonlinearity on the instability of the drop against its self-charge is studied.     

Stability analysis of a nonlinear rotating blade with torsional vibrations

This paper discusses the stability of a spinning blade having periodically time varying coefficients for both linear model and geometric nonlinear model. To obtain a reduced nonlinear model from nodal space, a standard modal reduction procedure based on matrix operation is developed with essential geometric stiffening nonlinearities retained in the equation of motion. For the linear model, the stability chart with various spinning parameters of the blade is studied via the Bolotin method, and an efficient boundary tracing algorithm is developed to trace the stability boundary of the linear model. For the geometric nonlinear model, the method of multiple time scale is employed to study the steady state solutions, and their stability and bifurcations for the periodically time-varying rotating blade. The backbone curves of steady-state motions are achieved, and the parameter map for stability and bifurcation is developed.     

Capillary oscillations and stability of a charged, viscous drop in a viscous dielectric medium

The scalarization method is used to obtain a dispersion relation for capillary oscillations of a charged, conducting drop in a viscous, dielectric medium. It is found that the instability growth rate of the charged interface depends substantially on the viscosity and density of the surrounding medium, dropping rapidly as they are increased. In the subcritical regime the influence of the viscosity and density of both media leads to a nonmonotonic dependence of the damping rate of the capillary motions of the liquid on the viscosity or density of the external medium for a fixed value of the viscosity or density of the internal medium. The falloff of the frequencies of the capillary motions with growth of the viscosity or density of the external medium is monotonic in this case. Zh. Tekh. Fiz. 68, 1–8 (September 1998)     

Regular and singular perturbation expansions in the analysis of extensional vibrations of plates

The Kane-Mindlin differential equations appropriate for the study of low and high frequency extensional vibrations in elastic plate strips are solved by means of perturbation techniques. These procedures permit the development of simple expressions that can be used for the rapid calculation of the natural frequencies of vibration as a function of material and geometric parameters. A regular perturbation expansion yields an approximate form of the frequency relationship that is useful in the high frequency range, whereas in the low frequency range a singular perturbation procedure is used since the expansion parameter appears in the coefficient of the highest order derivative in the fundamental differential equation. Approximate formulas and some numerical results are presented for three types of boundary conditions.     

Nonlinear vibrations of a charged drop in an electrostatic suspension

The problem of nonlinear vibrations of a charged drop of an ideal incompressible conducting fluid in an electrostatic suspension is analytically solved in an approximation quadratic in two small parameters: vibration amplitude and equilibrium deformation of the shape of the drop in an electrostatic field. To solve the problem analytically, the desired quantities are expanded in semiinteger powers of the small parameters. It is shown that the charge of the drop and the gravitational field influence the shape of the drop, nonlinear corrections to the vibration frequencies, and critical conditions for instability of the drop against the surface charge. At near-critical values of the charge, the shape of the nonlinearly vibrating drop falls far short of being a sphere or a spheroid, which should be taken into account in treating experimental data.     

Mass transfer due to nonlinear capillary-gravitational waves on the surface of a viscous liquid

An analytical expression of the second order of smallness in wave amplitude-to-wavelength ratio is derived for a horizontal flow arising in a finite-depth layer of a viscous liquid under the action of a periodic nonlinear capillary wave. It is found that the liquid flow is determined by the nonlinear component of the velocity field vortex part and the flow rate increases with increasing viscosity and decreasing wavelength irrespective of the layer thickness. In thin layers, the flow rate rapidly drops from its maximal value with increasing viscosity, wavelength, and surface charge density. If the liquid surface is charged, the horizontal liquid flow decreases rapidly as the surface charge density approaches the threshold of the Tonks-Frenkel instability.     

Experimental determination of damping of plate vibrations in a viscous fluid

A method of determining the aerodynamic-drag coefficient of flat vibrating plates from the vibrogram of free damping vibrations of cantilever-fixed duralumin samples has been developed. From the results of our experiments, simple approximating formulas determining the decrement of damping vibrations and the aerodynamic-drag coefficient through the dimensionless vibration amplitude and the Stokes parameter are proposed. The approach developed in this study for determining the aerodynamic-drag coefficient of a vibrating plate can be a useful alternative to purely hydrodynamic methods of finding the drag of vibrating solids.     

Stability analysis of nonlinear Roesser-type two-dimensional systems via a homogenous polynomial technique

<正>This paper is concerned with the problem of stability analysis of nonlinear Roesser-type two-dimensional(2D) systems.Firstly,the fuzzy modeling method for the usual one-dimensional(1D) systems is extended to the 2D case so that the underlying nonlinear 2D system can be represented by the 2D Takagi-Sugeno(TS) fuzzy model,which is convenient for implementing the stability analysis.Secondly,a new kind of fuzzy Lyapunov function,which is a homogeneous polynomially parameter dependent on fuzzy membership functions,is developed to conceive less conservative stability conditions for the TS Roesser-type 2D system.In the process of stability analysis,the obtained stability conditions approach exactness in the sense of convergence by applying some novel relaxed techniques.Moreover,the obtained result is formulated in the form of linear matrix inequalities,which can be easily solved via standard numerical software.Finally,a numerical example is also given to demonstrate the effectiveness of the proposed approach.     

On the time evolution of the surface shape of a charged viscous liquid drop deformed at zero time

The problem of capillary oscillations of the equilibrium spherical shape of a charged viscous incompressible liquid drop is solved in an approximation linear in amplitude of the initial deformation that is represented by a finite sum of axisymmetric modes. In this approximation, the shape of the drop as a function of time, as well as the velocity and pressure fields of the liquid in it, may be represented by infinite series in roots of the dispersion relation and by finite sums in numbers of the initially excited modes. In the cases of low, moderate, and high viscosity, the infinite series in roots of the dispersion relation can be asymptotically correctly replaced by a finite number of terms to find compact analytical expressions that are convenient for further analysis. These expressions can be used for finding higher order approximations in amplitude of the initial deformation.     

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.     

On internal nonlinear resonant interaction of capillary-gravitational waves on the flat surface of a viscous liquid

Mechanisms behind internal nonlinear resonant interaction of periodic capillary-gravitational waves on the uniformly charged flat surface of an infinitely deep viscous conducting liquid are considered. A mathematical procedure modifying the well-known method of many scales is proposed for constructing an asymptotically valid solution near the resonance. It is shown that the internal nonlinear resonant interaction results in effective energy transfer from long waves to shorter ones. An increase in the viscosity of the liquid diminishes the rate of energy transfer between resonantly interacting waves.