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.     

Nonlinear capillary-gravitational periodic waves on the charged surface of a finite-thickness viscous liquid

An analytical expression for the time evolution of the profile of a nonlinear periodic capillary-gravitational wave traveling over the charged surface of a viscous incompressible finite-thickness liquid is found. The calculation is carried out in the second order of smallness in wave amplitude. It is shown that the dependence of a nonlinear correction to a linear solution on the liquid viscosity and liquid layer thickness changes qualitatively in going from thick to thin liquid layers.     

Nonlinear periodic waves on the charged surface of a finite-thickness ideal liquid layer

In the fourth order of smallness in the amplitude of a periodic capillary-gravitational wave travelling over the uniformly charged free surface of an ideal incompressible conducting liquid of a finite depth, analytical expressions for the evolution of the nonlinear wave, velocity field potential of the liquid, electrostatic field potential above the liquid, and nonlinear frequency correction that is quadratic in a small parameter are derived. It is found that the dependence of the amplitude of the nonlinear correction to the frequency on the charge density on the free liquid surface and on the thickness of the liquid layer changes qualitatively when the layer gets thinner. In thin liquid layers, the resonant wavenumber depends on the surface charge density, while in thick layers, this dependence is absent.     

Modification of the boundary layer theory for analysis of wave motions in a cylindrical jet of a viscous liquid

The existent concepts of the boundary layer near the free surface of a viscous liquid, which is related to its periodic motion, are modified with the aim of analyzing the finite-amplitude wave motion on the surface of a thick charged jet of a viscous conducting liquid. To describe the flow in the boundary layer, a model problem is proposed that is simpler in statement compared with the complete problem and the solution of which uses the governing properties of the exact solution obtained in the low-viscosity asymptotics: the form of the dispersion relation, wave profile, and rate of velocity field viscous damping with time. An estimate is made of the boundary layer thickness at which the discrepancy between the exact solution and solution to the model problem (stated in terms of the theory proposed) falls into a given interval in the low-viscosity asymptotics. The domain of applicability of the modified theory is determined.     

Resonance effect of the bottom topography on the surface of an inclined layer of a viscous liquid

The reaction of the film interface to low-amplitude waviness of the wall was studied. A linearized version of the problem described by the Orr — Sommerfeld equation was considered; the solution was sought by asymptotic expansion in small parameter 1/Re, and usual spectral problem concerning stability to perturbations of exp[iα(x-ct)] type was solved. According to calculations, for some specially chosen wave numbers α the drift and dispersion effects balance each other, providing zero resulting velocity c _R = 0. If we assume that a rigid wall is corrugated with the same α, we can say that stationary waves caused by the wavy wall are in resonance with intrinsic perturbations of the second kind. The work was financially supported by the Russian Foundation for Basic Research (Grants Nos. 05-08-33585a and 06-08-96637-r-yug-a.)     

Free surface and flow problem for a viscous liquid

An exact closed system of equations is proposed for describing the shape of the free surface of a viscous steady-state liquid in the 2D case in terms of the surface itself. A method that lowers the dimensionality in the Navier-Stokes equation is demonstrated, and its application in problems of steady-state flow past solids is considered.     

Thickness of a boundary layer attributed to the wave motion on the charged free surface of a viscous liquid

It is shown that the analytical estimator for the boundary layer thickness that contains the wave frequency in the denominator and is proposed for approximate calculation of the wave motion on the free surface of a viscous liquid cannot be formally applied to the wave motion on the uniformly charged liquid surface. The fact is that, when the surface charge density attains a value critical in terms for the Tonks-Frenkel instability, the wave frequency tends to zero. From the analysis of liquid motions near the electric charge critical density, a technique is proposed for calculating the thickness of a boundary layer attributed to flows of various kinds. It is found that the thickness of the boundary layer due to aperiodic flows with amplitudes exponentially growing with time (such flows take place at the stage of instability against the surface charge) does not exceed a few tenths of the wavelength, whereas the thickness of the boundary layer due to exponentially decaying liquid flows is roughly equal to the wavelength.     

On the theory of a boundary layer associated with wave motion on the free surface of a liquid

A modified theory of a boundary layer associated with a periodic capillary-gravitational motion on the free surface of an infinitely deep viscous liquid is proposed. The flow in the boundary layer is described in terms of a simplified (compared with the complete statement) model problem a solution to which correctly reflects the main features of an exact asymptotic solution: the rapid decay of the flow eddy part with depth of the liquid and insignificance of some terms appearing in the complete statement. The boundary layer thickness at which the discrepancy between the exact asymptotic solution and model solution is within a given margin is estimated.     

Modification of the boundary layer theory for analysis of finite-amplitude oscillations of a charged bubble in a viscous liquid

The prevailing concepts concerning the boundary layer near the free surface of a viscous liquid associated with oscillatory motion are modified for calculating finite-amplitude linear oscillations of a charged bubble in this liquid. Equations of the boundary layer theory for the neighbourhood of the oscillating free spherical surface of a charged bubble in a dielectric liquid are derived, their analytic solution is obtained and compared with the exact solution, and the thickness of the boundary layer is assessed. The range of applicability of the modified theory is determined.     

Electrostatic stability of the liquid layer surface on a hard wettable cylindrical substrate

The wave motion in a cylindrical layer of an ideal conducting liquid on a hard rod kept at a constant electrical potential is calculated accurate to the first order of smallness in dimensional perturbation of the free surface. The instability of the free surface is also considered. A dispersion relation is derived. It is shown that the range of instability waves depends on only the electric field strength near the free surface and the instability increments of capillary waves decrease as the layer gets thinner. The influence of the hard rod becomes tangible only when its radius becomes comparable to the thickness of the liquid layer.     

Instability of a charged layer of a viscous liquid on the surface of a solid spherical core

The dispersion relation for the spectrum of capillary waves of a spherical layer of a viscous liquid coating a solid spherical core with a layer of finite thickness is introduced and analyzed. It is shown that the existence of two mechanisms for the viscous dissipation of the energy of the capillary-wave motions of the liquid, viz., damping in the bulk of the layer and on the solid core, leads to restriction of the spectrum of the realizable capillary waves of the liquid on both the high-and low-mode sides. At a fixed value of the system charge which is supercritical for the first several capillary modes, the maximum growth rates in the case of a small solid core are possessed by modes from the middle of the band of unstable modes, while in thin liquid layers the highest of the unstable modes have the largest growth rates. This points out differences in the realization of the instability of the charged surface of the spherical layer for small and large relative sizes of the solid core. Zh. Tekh. Fiz. 67, 8–13 (September 1997)     

Convective flows in a layer of a viscous liquid with the uniformly charged free surface

It is found theoretically that the critical conditions under which a charged liquid surface becomes unstable against the electric charge relax as a result of interaction between capillary-gravitational and convective flows in the liquid. As the surface charge density approaches a value critical in terms of development of Tonks-Frenkel instability, convection in the liquid arises at a temperature gradient however small, this effect depending on the liquid layer thickness.     

The modified absorption optic method for diagnostics of the wave liquid film on a rotating surface

The modification of absorption optic method for diagnostics of the wave liquid film on the surface of a rotating disk was proposed. Inaccuracy of the method was estimated analytically and experimentally. The field measurements of liquid film thickness on a rotating disk were carried out.     

On calculating the oscillations of a viscous liquid layer over a solid spherical core in terms of the boundary layer theory

The theory of a boundary layer near the periodically oscillating free surface of a spherical viscous liquid layer over a solid core (bottom) is modified. Two boundary layers are considered to adequately describe a liquid viscous flow in the system: one at the free surface of the liquid and the other at the solid bottom. The thicknesses of the boundary layers are estimated, which provide any given discrepancy between an exact solution to the model problem and a solution obtained in the small viscosity approximation. Taking into account the boundary layer near the solid bottom is shown to be significant only for lower oscillation modes. For higher modes, the flow near the core can be considered potential. In the case of lower modes and shallow liquid, the surface and bottom boundary layers overlap and an eddy flow occupies the entire volume of the liquid.     

Finite-amplitude waves on the surface of a viscous deep liquid

For the first time a rigorous solution to the problem on time evolution of the periodic wave shape on the surface of a viscous infinitely deep liquid is found in the quadratic approximation with respect to the wave amplitude. It is found, in particular, that the damping rate of the quadratic component with respect to the wave amplitude is twice as high as the damping rate of the linear term. It is shown that inclusion of viscosity leads to asymmetry of the wave profile.     

Boundary layer flow and heat transfer of a nanofluid over a permeable unsteady stretching sheet with viscous dissipation

A numerical study of the boundary layer flow past unsteady stretching surface in nanofluid under the effects of suction and viscous dissipation is investigated. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. A similarity solution is presented, which depends on the unsteadiness parameter A, Eckert number Ec, ζ suction or injection parameter, Prandtl number Pr, Lewis number Le, Brownian motion number Nb, and thermophoresis number Nt. The governing partial differential equations were converted to nonlinear ordinary differential equations by using a suitable similarity transformation, which are solved numerically using the Nactsheim-Swigert shooting technique together with Runge-Kutta six-order iteration scheme. The accuracy of the numerical method is tested by performing various comparisons with the previously published work, and the results are found to be in excellent agreement. Numerical results are presented both in tabular and graphical forms illustrating the effects of these parameters on thermal and nanoparticle volume fraction boundary layers. The thermal boundary layer thickens with a rise in the local temperature as the Brownianmotion, thermophoresis, and convective heating each intensify.     

Decay of capillary turbulence on the surface of a viscous liquid

Decay of the turbulence of capillary waves on the surface of a real liquid is studied in the presence of the viscous damping of the waves at all frequencies after stepwise removal of external pumping. The investigation is performed using two different models: the weak turbulence approximation and the local turbulence model in which the energy redistribution over frequencies is described by the polynomial expression in the wave-occupation number. It is shown that the decay of turbulence in the viscous liquid proceeds self-similarly and begins at high frequencies. In the decay process, the frequency distribution of the energy of waves is close to the stationary form E _ω ～ ω^?3/2 in a wide frequency range below the boundary frequency of the inertial range during a relatively long time after removal of the external force. The calculation results agree qualitatively with the results of the experiments on capillary turbulence on the charged surface of liquid hydrogen.     

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.     

Settling and fluidization of non Brownian hard spheres in a viscous liquid

A mean field approach is used to estimate the energy dissipation during the homogeneous sedimentation or the particulate fluidization of non Brownian hard spheres in a concentrated suspension of infinite extent. Depending on inertial screening and the range of the hydrodynamic interactions, the effective buoyancy force is determined either from the average suspension density in a Stokes flow or from the fluid density in the turbulent flow regime. An energy balance then yields a settling or fluidization law depending on the particle Reynolds number in reasonable agreement with the Richardson and Zaki correlation and recent experimental results for particle settling or fluidization. We further estimate the energy dissipation in the turbulent boundary layers around the particles to precise the Reynolds number dependence of the hindered settling function in the intermediate flow regime. Received 22 February 1999 and Received in final form 14 June 1999