Dynamic problem for a two-layer medium with a vibrational source at the boundary interface

The present study is concerned with an analysis of gravitational and acoustic waves which are excited by a vibrational source deeply placed in a liquid covered by ice. An analysis of the rigidity characteristics of ice modeled by an elastic layer or by a Kirchhoff plate is done by factorization of the solution to the integral equation equivalent to an initially combined boundary value problem. The uncombined boundary condition is used to solve problems for unrestricted ice fields in [1–3], whereas combined conditions with vibrational sources positioned at the boundary of the medium are used in [4].Translated from Zhurnal Prikladnoi Mekhaniki, No. 3, pp. 125–129, May–June, 1986.     

Swell wave propagation in an inhomogeneous ice sheet

The propagation of surface waves beneath a periodically inhomogeneous ice sheet is considered. Areas of broken ice and hummock ridges are considered as irregularities. It is shown that waves with frequencies corresponding to wind and swell waves are strongly scattered by the irregularities and are damped exponentially as they propagate beneath the ice.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 162–169, September–October, 1996.     

Mathematical model of the dissociation of gas hydrates coexisting with ice in natural reservoirs

The dissociation of gas hydrate coexisting with ice in a low-temperature natural reservoir is investigated. A mathematical model of the process consisting of a generalization of the Stefan problem and containing two unknown moving phase transition boundaries — the hydrate dissociation and ice melting fronts — is constructed. It is shown that in high-permeability reservoirs the velocity of the dissociation surface is higher than that of the ice melting surface. As the permeability decreases, the fronts change places. The problem is solved in the self-similar approximation.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 84–92, March–April, 1993.     

Surface wave diffraction at a crack in sheet ice

The time-periodic motions of a liquid layer of finite depth beneath an ice sheet with a straight infinite crack having a periodic dependence on the horizontal coordinate in the direction of the crack are considered. The ice sheet is simulated by a thin elastic plate. It is assumed that the thickness of the plate changes abruptly across the crack. The problems of plane-wave diffraction at a crack, plane-wave diffraction atN cracks in a uniform ice sheet, and plane-wave reflection from a rigid wall are solved. The effect of the pre-existing state of stress of the ice sheet on the properties of the reflected waves is investigated. The condition of nontransmission of fix-frequency waves beneath the edge of the ice is obtained.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 93–102, March–April, 1993.     

Some features of flexural-gravity wave propagation in the presence of a crack in sheet ice

The effect of a crack in an ice sheet on the propagation of surface flexural-gravity waves in a basin of constant depth is analyzed. The ice sheet is simulated by two floating semi-infinite fragments of a thin elastic isotropic plate. As the boundary-contact conditions on the line of contact between the parts of the plate the conditions of continuity of displacements for arbitrary slopes simulating one ice-floe overlying on another and free-edge conditions (crack) are considered. The dependence of the amplitude characteristics of the perturbations on the thickness of the ice, its degree of compression, the incident wave frequency, the depth of the basin, and the form of the boundary-contact conditions is investigated. Problems of wave diffraction on inhomogeneities of an elastic plate were solved in [1, 2], and on a crack in the ice sheet in [3, 4].Sevastopol. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 144–150, March–April, 1996.     

Unsteady gravity-elastic and gravity-capillary ship waves

The development of three-dimensional waves generated by a region of pressures moving uniformly and rectilinearly over the surface of a thin elastic isotropic plate covering an ideal fluid layer of finite depth is investigated. The pressures act starting at a certain instant. A qualitative similarity between the waves occurring and gravity-capillary waves is noted. The calculations are made for an ice cover. This model problem permits examining a number of properties of the oscillations of the ice cover occurring when hauling freight over ice roads, landing and takeoff of aircraft from ice fields, etc. [1]. The development of ship waves in a fluid of finite depth in the absence of a floating plate was investigated in [2, 3] and gravity-capillary waves were studied in [4–6]. Certain properties of steady three-dimensional waves occurring during movement of a load over the surface of a floating elastic plate were established in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 26–32, September–October, 1978.     

Impingement of surface waves on the edge of compressed ice

The impingement of small-amplitude surface waves on the edge of a solid compressed ice sheet in a basin of finite constant depth is considered. The influence of the cylindrical rigidity and the value of the compressing force on the dependence of the amplitude coefficients of reflection and transmission on the incident wave period is analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 121–126, May–June, 1995.     

Natural vibrations of a hummock ridge in an elastic ice sheet floating on the surface of an infinitely deep fluid

The properties of the natural vibrations of a hummock ridge in an elastic ice sheet are investigated. Typical shapes of the dispersion curves for symmetric and antisymmetric boundary waves which propagate along the hummock and damp exponentially with distance from the latter are obtained. It is shown that natural vibrations can initiate failure of the ice sheet at a certain distance from the hummock. Under compression this process leads to the formation of a parallel hummock ridge.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 99–105, November–December, 1995.     

Linear problem of water-ice phase transitions in unsaturated soils

The approach proposed by Menot is developed. The problem of the freezing of an unsaturated soil is formulated for a compressible gas and equal component pressures. An analytic solution is obtained in the linear approximation. The results indicate the strong dependence of the ice saturation on the permeability of the soil and the applied pressure gradient.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 68–73, May–June, 1990.     

Tire slip-angle force measurements on winter surfaces

Tire lateral force data on winter surfaces cannot be obtained with the traditional laboratory test technique of an instrumented tire on a moving belt surface. Furthermore, changing snow and ice conditions can drastically change the tire/surface interaction. In this study the Cold Regions Research and Engineering Laboratory’s (CRREL’s) Instrumented Vehicle (CIV) was used in a unique configuration to measure tire lateral force versus slip-angle data on ice and snow at various temperatures, moisture contents, depths, and densities. The vehicle is instrumented to record longitudinal, lateral, and vertical force at the tire contact patch of each wheel as well as vehicle speed, tire speed, and front tire slip angle. The tests were conducted at the Keweenaw Research Center (KRC) in northern Michigan in February 2005 and March 2006. Tests were conducted on ice, packed snow from 0.50 to 0.58 g/cc, remixed snow depths of 2.5–20.3 cm at 0.43 to 0.48 g/cc and freshly fallen snow with depths of 0.5–17 cm at 0.07 to 0.23 g/cc. Surface air temperatures during testing ranged from −14 to 1.6 °C. The data collected show that peak lateral force and the shape of the lateral force versus slip-angle curve are related to snow properties and depths.     

Shock Compression of a Plate on a Wedged–Shaped Target

Problems of compression of a plate on a wedge–shaped target by a strong shock wave and plate acceleration are studied using the equations of dissipationless hydrodynamics of compressible media. The state of an aluminum plate accelerated or compressed by an aluminum impactor with a velocity of 5—15 km/sec is studied numerically. For a compression regime in which a shaped–charge jet forms, critical values of the wedge angle are obtained beginning with which the shaped–charge jet is in the liquid or solid state and does not contain the boiling liquid. For the jetless regime of shock–wave compression, an approximate solution with an attached shock wave is constructed that takes into account the phase composition of the plate material in the rarefaction wave. The constructed solution is compared with the solution of the original problem. The temperature behind the front of the attached shock wave was found to be considerably (severalfold) higher than the temperature behind the front of the compression wave. The fundamental possibility of initiating a thermonuclear reaction is shown for jetless compression of a plate of deuterium ice by a strong shock wave.     

Behavior of a floating elastic plate during vibrations of a bottom segment

The Wiener-Hopf technique is used to obtain an analytical solution for the problem of vibrations of a floating semi-infinite elastic plate due to earthquake-induced vibrations of a bottom segment. An explicit solution is obtained ignoring the inertial term. The surface-wave amplitudes and ice-plate deflection are studied numerically as functions of the frequency and position of the vibrating bottom segment, ice thickness, and fluid depth.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 98–108, March–April, 2005.     

Energy-separating properties of two-phase flow

A new, substantially nondissipative process of energy separation in two-phase flows has been investigated. Mixtures of air with water, kerosene, and an aqueous solution of diethylene glycol were studied at initial pressures of 3–20 bar. It was found that ice was formed in an air-water mixture issuing from a supersonic nozzle, and for a mixture of air with a nonfreezing diethylene-glycol solution the liquid obtained after nonequilibrium separation had a negative temperature. The possibility of effective freezing out of moisture on an uncooled solid surface exposed to a current of moist air from a supersonic nozzle was demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 49–55, November–December, 1976.     

A direct relationship between bending waves and transition conditions of floating plates

Ocean waves travel deep into ice fields in the polar regions, both affecting the formation of sea-ice and causing its break-up. Recently, it has been shown that a relatively simple linear water and bending wave theory can predict the decay rate of the wave energy travelling through fractured ice sheets and floes at the geophysically important wave periods of 6–15 s. That work used simple free-edge conditions. A possible improvement to the current model is to better represent the effective connection due to partially frozen cracks that occur in practice. The Wiener–Hopf technique gives explicit formulae for the velocity potential and surface deflection, expressed as series expansions over the modes of the elastic plate floating on water of finite depth, with the coefficients in the expansion given as functions of four constants. These constants are determined by a system of four linear equations, represented by a 4-by-4 matrix and a four-element vector. The elements of the matrix are given as explicit functions of relationship between edge conditions. General connections between ice sheets may be interpreted as a vertical and a rotational spring providing transition conditions for the shear force and the bending moment. The reflection and the transmission of waves can then be simply calculated as direct functions of the connection conditions. Conversely, reflected and transmitted waves allow complete characterization of the effective connection conditions at a material discontinuity.     

Calculation of wind currents in the edge zone of a sea ice sheet

The steady wind currents in the ocean near the edge of an ice sheet simulated by a rigid cover are studied. The characteristic horizontal scales of the problem correspond to several tens of kilometers. In this case, depending on the depth and viscosity of the fluid, the effect of the Coriolis force on its motion may be comparable with the effect of the wind shear stresses applied to the ice-free surface. Taking the Coriolis force into account leads to the formation of a spiral flow structure in the vertical direction.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 132–141, November–December, 1992.     

Effect of cavity inclination on a temperature and concentration controlled double diffusive convection at ice plate melting

This paper is concerned with the double diffusive convection due to the melting of an ice plate into a calcium chloride aqueous solution inside a rectangular cavity. It is mainly considered the effect of the cavity inclination on the melting rate and the mean melting Nusselt- and Sherwood-numbers, experimentally as well as numerically. The ice plate melts spontaneously with decreasing temperature at the melting front even if initially there does not exist a temperature difference between the ice and the liquid. The concentration- and temperature-gradients near the melting front induce double diffusive convection in the liquid, which will affect the melting rate. Experiments reveal that the mean melting mass increases monotonically with increasing cavity inclination. The numerical analysis based on the laminar assumption predicts well the melting mass in the range of =0–90°, however, under-predicts the melting mass in the range of =90–180° as compared with the experimental results.     

One-dimensional problems of frozen soil thawing due to solution seepage

One-dimensional problems of the thawing of a porous frozen soil by intensive injection of brine are considered. Assuming local thermodynamic equilibrium and negligible diffusion, these problems prove to be self-similar and allow for complete analytical study.Four soil thawing regimes can be distinguished depending on the ice melting behavior at the temperature and concentration fronts. Criteria for their occurrence are derived in terms of the temperature and concentration of the injected brine. Possible simplified formulations statements are also discussed.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 149–160, September–October, 1995.     

Numerical investigation and self-similar solutions of the glacier-ocean interaction problem

The mathematical modeling of the heat and mass transfer processes in glaciers is an effective means of investigating and predicting their development. A full explanation of the problem of constructing appropriate mathematical models is given in [1–5]. By analyzing the equations involved [3, 6] it is possible to establish the principal factors and dimensionless numbers determining glacier dynamics and provide justification for neglecting the secondary terms. In particular, a simplified closed system of differential equations for the detailed calculation of all the hydrodynamic characteristics of the glacier can be obtained for K_hj « 1 up to O(K _h ² ), where K_h is the ratio of the vertical and horizontal scales of the ice mass investigated (K_h 10^–4–10^–6). In this case many of the qualitative characteristics of glacier dynamics are preserved even in one-dimensional models within the subisothermal approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–7, September–October, 1986.     

Self-similar solution of a problem of freezing of finely dispersed soils with allowance for moisture migration in thawed and frozen zones

Heat transfer in freezing and thawing soils is accompanied by various processes among which phase transition of moisture and mass transfer should be distinguished in both the thawed and the frozen zones. Their consequence is the formation of ice schlieren and the swelling associated with this. In developing the methods of calculation of moisture migration it was assumed (see, for example, [1, 2]) that the mass transfer occurs only in the thawed part of the soil and is realized predominantly in the liquid phase through the diffusion-film mechanism. It was assumed that the phase conversion of water into ice occurs wholly on the phase interface (the Stefan formulation) and at the same time supplementary conditions for the moisture function are specified on it. Not all these assumptions are valid. In particular, the marked redistribution of moisture in the frozen zone is an important factor in the freezing of moist rocks [3, 4]. This last is also observed in the thawing of dispersed rocks and in frozen samples which are under the influence of a temperature gradient. These phenomena were modeled in [5, 7] on the basis of a single mathematical model which describes the conductive heat transfer, the moisture transfer in thawed and frozen zones, the phase transition of moisture in the temperature range, and the kinetic relaxation effects of moisture crystallization and ice melting. Analysis of the solutions obtained by means of a finite difference method showed that the proposed method of calculation gives results near the experiment. The present paper is devoted to a further study of the model indicated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 113–120, March–April, 1986.     

Problem of a solid half-space melted by a hot rotating disk or ring

The sliding friction of solids at high speed and under heavy load may be accompanied by a transition to the plastic or fluid state in the friction contact zone [1]. The stage corresponding to a developed fluid layer is investigated without taking into account the plastic deformation of the rubbing bodies; it is assumed that all the heat released is expended exclusively on melting the solid. Previous attempts to investigate this stage theoretically have been based on the approximation of a fluid layer of constant thickness and the use of the heat balance equation [1, 2]. Here, the velocity and temperature profiles are approximated by relations quadratic in the transverse coordinate with coefficients that depend on the longitudinal coordinate. These are determined from the boundary conditions and the integral relations of boundary layer theory. The relations obtained are used to determine the rate at which a hot rotating ring melts through a block of ice.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 30–34, May–June, 1990.